Effective Field Theory of Surface-mediated Forces in Soft Matter
Cem Yolcu 2012
Department of Physics Carnegie Mellon University Pittsburgh, PA
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy
advised by Prof. Markus Deserno
beni pamuklara sarıp kocaman olayım diye okullara yollayan canım arkada¸sıma
Abstract We propose a field theoretic formalism for describing soft surfaces modified by the presence of inclusions. Examples include particles trapped at a fluid-fluid interface, proteins attached to (or embedded in) a biological membrane, etc. We derive the energy functional for nearflat surfaces by an effective field theory approach. The two disparate length scales, particle sizes and inter-particle separations, afford the expansion parameters for controlling the accuracy of the effective theory, which is arbitrary in principle. We consider the following two surface types: (i) one where tension determines the behavior, such as a fluid-fluid interface (referred to as a film), and (ii) one where bending-elasticity dominates (referred to as a membrane). We also restrict to rigid inclusions with a circular footprint, and discuss generalizations briefly. As a result of the localized constraints imposed on the surface by the inclusions, the free energy of the system depends on their spatial arrangement, i.e. forces arise between them. Such surface-mediated interactions are believed to play an important role in the aggregation behavior of colloidal particles at interfaces and proteins on membranes. The interaction free energy consists of two parts: (i) the ground-state of the surface determined by possible deformations imposed by the particles, and (ii) the fluctuation correction. The former is analogous to classical electrostatics with the height profile of the surface playing the role of the electrostatic potential, while the latter is analogous to the Casimir effect and originates from the mere presence of constraints. We compute both interactions in truncated expansions. The efficiency of the formalism allows us to predict, with remarkable ease, quite a few orders of subleading corrections to existing results which are only valid when the inclusions are infinitely far apart. We also found that the few previous studies on finite distance corrections were incomplete. In addition to pairwise additive interactions, we compute the leading behavior of several many-body interactions, as well as subleading corrections where the leading contribution was previously calculated.
Acknowledgments Let me tell you, these weren’t the best five years of my life. Not by a long shot. I snapped. More than once. Among the strongest forces that kept me from walking out was my advisor Markus Deserno’s unwavering support. His sincere concern and understanding is invaluable to me. I thank him from the bottom of my heart for accommodating all the quirks and twists I and my life came with. Personal matters aside, there have been times I thought he was more interested in my growth as a physicist than I was. I am among a lucky few who can say that. Shifting gears to more practical issues, I would finally like to acknowledge that Figure 1.12 (and Figure 2.9 in part) was made by him. As I am sure Markus would also agree, at a moment when we were both deeply concerned about the pace and direction of my research, Ira Rothstein’s involvement turned things around. Despite our uncanny inability to be on the same page when we attempted to communicate, the collaboration was fruitful, incredibly beneficial for me, and culminated in this thesis. In retrospect, the combined supervision of Markus and Ira was no doubt a blessing. The remaining examiners of my dissertation, Robert Sekerka, Robert Tilton and Michael Widom, did their best to monitor my research progress and to help me see it from a more professional perspective, for two years. I am pleased that they were a part of this. In the early days when Markus and I were both new to fluctuation-induced forces and grappling desperately with it, Martin Oettel shared his knowledge and insight with us generously and patiently. Many thanks for that, as well as providing his numerical data for one of the plots in Figure 1.12. Also, it would be unfair to overlook the assistance of a fellow graduate student, Robert Haussmann, in the later phases of my research. I was usually able to get a better hang of things after talking to him or reading his notes. I was given the marvelous gift of having a home away from home in Pittsburgh. Levent Yılmaz and Yonca Karakılı¸c (and their cats Marul, Sandy and Bulut) were my family here. Words cannot describe how precious that is to me. They have always been there for me, on the darkest of days as well as the brightest. I am humbled by their grace, generosity and kindness. I am also indebted to Levent for guiding me through the overwhelming (at least for me) confusion of being a stranger in a strange land. They were, fortunately, not the only people to make my Pittsburgh episode bearable— and sometimes pleasant, even. I thank (in a somewhat chronological order) Korhan Turan, ˙ glu, Can Akta¸s, Markus Deserno, Tristan Bereau, Mingyang Hu, G¨ uzide Atasoy, Nilay Ino˘ Karpur Shukla, Xiaofei Li, Patrick Diggins, Venky Krishnamani and Agnieszka Kalinowski for wonderful memories of friendship. As colleagues, Tristan, Mingyang and Venky also “nourished” me with their knowledge of physics, biology and computing. When I lost my drive and motivation, and succumbed to the beckonings of procrastination, Agnieszka’s classic brand of elder-sisterly authority put me back in my place.
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It seems that in the world we live, the worth of a man is measured in terms of how much money he commands. I am grateful that I have/had a mom and dad who did not fall prey to this hideous and fateful misconception of our time. Had the circumstances been otherwise, walking the path I walked would have been much, much more painful—perhaps impossible for an ill-willed weak-spirited being like myself. Even though I did not get this degree for anyone, I have to admit I am a bit disappointed to have one less proud parent than I anticipated in the beginning. Looking even further back, I still remember how much influence my sister had—then only seventeen—on which high school I ended up going to. It might come as an unnecessary dose of nostalgia but that is where you’ll find the beginnings of my intimacy with physics. Finally, let me seize the opportunity to express my appreciation towards those other people who in many ways influenced my career in academia up to this point and the person I am. Heartfelt thanks to Ersin Erol and Mehmet Gen¸c for their enthusiastic encouragement very early on (of which they may not even be aware), to Burak Han Alver, Kaan Atak, Ahmet Baykal, Erol Ertan and Selin Manukyan along with Yorgo Istefanopulos, Mutlu Koca, Muhittin Mungan, Alpar Sevgen and Teoman Turgut for their much-needed support and guidance when it was time to leave home for graduate school, to Erol Ertan, Bahar Esen, C ¸ i˘ gdem Evrandır and Muhittin Mungan for helping me through rough patches even transcending the thousands of miles between us, and to my dear friend Selin Manukyan for wrapping me up snug in soft cosmic cotton so it doesn’t hurt when I fall.
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List of Publications The original content of this thesis has appeared or will appear in the following publications:
Chapters 1 and 2: C. Yolcu, I. Z. Rothstein and M. Deserno, Effective field theory approach to Casimir interactions on soft matter surfaces, Europhysics Letters 96 (2011), 20003.
Chapter 1: C. Yolcu, I. Z. Rothstein and M. Deserno, Effective field theory approach to fluctuationinduced forces between colloids at an interface, Physical Review E 85 (2012), 011140.
Chapter 2: C. Yolcu and M. Deserno, Membrane-mediated interactions between rigid inclusions: an effective field theory, submitted to Physical Review E.
Chapter 3: C. Yolcu and M. Deserno, Surface-mediated interactions between flexible inclusions on a membrane under tension, in progress.
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Contents Introduction
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1 Interactions of rigid particles on a film 1.1 Overview of the system . . . . . . . . . . . . . . . 1.1.1 The film . . . . . . . . . . . . . . . . . . . . 1.1.2 Boundary conditions . . . . . . . . . . . . . 1.1.3 Surface-mediated interactions . . . . . . . . 1.2 Effective field theory . . . . . . . . . . . . . . . . . 1.2.1 Permanent sources . . . . . . . . . . . . . . 1.2.2 Induced sources . . . . . . . . . . . . . . . . 1.3 The interaction free energy: fluctuation-induced . . 1.3.1 Pair interactions . . . . . . . . . . . . . . . 1.3.2 Multibody interactions . . . . . . . . . . . . 1.4 The interaction free energy: elastic . . . . . . . . . 1.4.1 Interaction between permanent quadrupoles
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2 Interactions of rigid particles on a membrane 2.1 System and effective theory . . . . . . . . . 2.1.1 Surface energetics of the membrane . 2.1.2 Boundary conditions . . . . . . . . . 2.1.3 Effective theory . . . . . . . . . . . . 2.2 Entropic interactions . . . . . . . . . . . . . 2.2.1 Diagrams and rules . . . . . . . . . . 2.2.2 Pair interactions . . . . . . . . . . . 2.2.3 Multibody interactions . . . . . . . . 2.3 Elastic interactions . . . . . . . . . . . . . . 2.3.1 Pair interactions . . . . . . . . . . . 2.3.2 Triplet interactions . . . . . . . . . .
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3 Outlook 3.1 Anisotropic particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mixed Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Soft inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Summary and Conclusion
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Contents A Gaussian Integrals A.1 Gaussian averages and Wick’s theorem . . . . . . . . . . . . . . . . . . . . .
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B Index contractions in complex coordinates B.1 Harmonic kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C Renormalization of polarizabilities and counterterms
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D Binomial cycle simplification of the entropic pair free energy on a film
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E Induced monopoles on a film
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List of Figures
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List of Technical Notes
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Bibliography
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Introduction Particles adsorbed on soft surfaces are observed everywhere. Surely, we have all seen dust particles, an eyelash, a small insect (perhaps, and hopefully, not the last one) etc. on the interface between air and the glass of water we are about to drink. More appetizing examples would be pieces of breakfast cereal which happened not to get too wet and sink, but instead kept afloat, or even particles that are not so different from the surface itself: for example, the froth on our cup of coffee [VM05]. In fact, such situations where particles are trapped at or stuck to surfaces are even dearer to us. Our cells and most organelles within them are enveloped by membranes where many proteins are adsorbed or embedded, which perform vital functions, quite literally, such as fighting osmosis to maintain ion levels in the cytoplasm (ion pumps), communicating with the exterior environment (receptors) [AJL+ 83] or inducing necessary deformations [MG05, ZK06, VPS+ 06, SVR06] (e.g. tubulation, endo/exocytosis). Of course, unpleasant objects such as viruses can stick to the membranes as well [GSF05]. One does not always just run into particles bound to surfaces, but can also intentionally place them there. Particle-stabilized emulsions (or Pickering emulsions) [Pic07, Ram03, Bin02] are the quintessential example, where droplets of a fluid are dispersed in another fluid immiscible with the first one, by the help of solid particles that prefer to adsorb at the interface between the two fluids. Normally, such a mixture is unstable, and it will separate to minimize the unfavorable interface area. Emulsions are, as such, typically stabilized by surfactants that reduce the tension (free energy per area) of the interface. However, with solid particles that adsorb at the interface irreversibly, it is possible to achieve very stable emulsions and also droplets of one species dispersed in droplets of another species dispersed in yet another species [MMY+ 12]. Microsphere assemblies on thin films can act as nuclei for colloidal crystals [DVK+ 92, DVK+ 93] which, in turn, are used as templates for materials of controlled porosity [GYX99, HBS98], finding applications in catalysis and separation technologies. Such assemblies have also been utilized as lenses with tunable focal length [BKT+ 09]. In addition to practical applications, these colloidal systems make very good model systems for fundamental condensed matter systems where the “atoms” could be of micron size [ZM00, BBC+ 03, HACB03]. On the biological front, Nature is believed to make use of membrane-curving proteins [ZK06, VPS+ 06, SVR06, MG05] (such as endophilin, ampiphysin, epsin and clathrinadaptor-protein complexes) in membrane remodelling events such as endocytosis and exocytosis [TSHDC99, RIH+ 07, Koz07, AG09], or in the biogenesis of highly-curved organelles such as the endoplasmic reticulum and the Golgi apparatus [FRT+ 01, IER+ 05, VPS+ 06]. Interactions between the adsorbed particles affect the way they arrange on the surface.
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Introduction The first interactions to come to mind are of electrical origin: Coulombic forces between charged particles, van der Waals attraction without the need for excess charge, and specific chemical interactions, for example, between proteins on a membrane. However, this thesis, as its title suggests, concerns interactions not mediated by electric fields but those mediated by deformations (permanent or fluctuating) on the surface itself. The presence of particles on the surface can modify its shape and also its fluctuation modes. The first is perhaps easier to imagine. The line of contact between the particle and the surface may be deformed. This could happen due to the shape of the particle or, in the case of a particle at a fluid-fluid interface, irregular wetting of the particle surface by the two sides of the interface. Alternatively (or on top of that), the slope of the surface normal to this contact curve may be constrained at a given three-phase contact slope by Young’s relation [Saf94]. Such deformations are reminiscent of the way charge distributions or constrained boundaries deform electrostatic fields, or the way mass distributions deform space-time. The latter is perhaps a better analogy since the surface is the space where the particles live. However, under the assumptions we will make in this thesis, the deformations will be described as a field defined over a flat space—much like the electrostatic potential or the gravitational potential in Newtonian (linear) gravity— making analogies to electrostatics very useful. Indeed, it is immediately obvious that the equilibrium shape and overall (free) energy of the system will be determined by these sources of deformation and constraints just as in a typical electrostatics problem, and forces will emerge. The behavior of a surface, however, may be affected by adsorbed particles in another manner: by modifying its fluctuation modes, as mentioned above. The system is in contact with a heat reservoir and thereby subject to thermal fluctuations. But a surface forced to satisfy constraints (wetting, contact angle, etc.) around the particles fluctuates differently from one that is free of perturbations. In other (and appropriately theoretical) words, the perturbed surface explores the points in its phase space with different probabilities than its free counterpart. The free energy of the fluctuating system is therefore modified in a way dependent on the constraints—and in particular their spatial arrangement—hence giving rise to effective forces. This is an analog of the Casimir effect [Cas48, BMM01, Mil01] in quantum electrodynamics where uncharged conductors attract each other as a result of their equipotential constraint on quantum fluctuations of the electric field. Due to the way they arise, such interactions are also dubbed fluctuation-induced. So, where do we encounter surface-mediated forces? We see bubble “particles” clustering on the surface of our soda or coffee due to such attractions. One can also carry out one’s own experiments by spreading thumbtacks, needles, paper clips or cut pieces of soda straw [VM05, CFHQ02] on the surface of some still water.1 Observations that require more technical resources involve the assembly of submillimeter and microscopic colloidal particles at a fluid interface [BSR00, DKYN95, DPK01, LAZY05, SJ05]. How the curvature of the interface might be used to align anisotropic particles has also been 1 We would like to avoid any misunderstanding here. None of these examples fall within the special cases treated in this thesis. Although, it will be clear that it generalizes.
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Introduction investigated [LCJB+ 10, CJBL+ 11]. While most of these studies have an emphasis on the control of microparticle assemblies at interfaces, Yunker et al. recently suggested that the surface-mediated attraction between anisotropic particles on a droplet may overcome their tendency towards the three phase contact, and thereby suppress the formation of a “coffeering” as the drop evaporates [YSLY11]. The only observation of membrane-mediated interaction between proteins we are aware of was made by Casuso et al. [CSRS10], whereas interactions and aggregation of micron-sized colloidal beads on lipid vesicles were observed using light microscopy earlier by Koltover et al. [KRS99]. The fluctuation-induced component of surface-mediated interactions has so far been illusive, but research is underway [MZ]. The problem of surface-mediated interactions is interesting in purely academic terms as well. Finding the equilibrium shape of a surface with adsorbed particles is a boundary value problem with several compact boundaries (those between the particles and the surface). While the statement of the problem is usually trivial, the solution is certainly not. For example, the electrostatics problem with a point charge outside a conducting sphere is a standard textbook affair, whereas a configuration of three charged conducting spheres is far from it. Typically, one makes simplifying assumptions to gain some traction on the problem. Assuming point particles that encapsulate the presence of the particles only as seen from infinitely far away is one [SDJ00, Net97, DF99a, DF99b, BDF10, MM02]. It is sometimes also possible to assume that the boundary conditions are satisfied on each boundary separately, i.e. a superposition approximation [Nic49, GS71, CHJR81, vNSH05, VM05]. However, such assumptions only work when the particles are infinitely far away from each other, and consequently, interactions have been worked out only in this limit in the literature (at least in a consistent way). It may be possible to satisfy two boundary conditions simultaneously using bipolar coordinates [KPIN92, PKDN93] for two axisymmetric particles but one can see how difficult it would be to do the same for more than two particles or for particles of different shape. The interactions induced by fluctuations, on the other hand, require the evaluation of a partition function where the fluctuation degrees of freedom of the surface are constrained locally by the particles [GBP93a, GBP93b, GGK96b, GGK96a, PL96, LOD06, LO07, NO09, DF99a, DF99b, BDF10]. This constrained sum also complicates things to such an extent that it was not possible to obtain more than the large distance behavior (at least until recently [EGJK08] for the electromagnetic Casimir effect). Even though it is possible to apply the latter to surface-mediated forces, the method relies on matrix operations for the evaluation of forces and hence obscures insight into the problem a little. It also relies on the linearity of the underlying field theory, and therefore it is unlikely to be extended to surface-mediated forces in its full glory—which we mentioned to be more akin to gravity than electrostatics.2 The main theme of this thesis is that, since one is already describing the surface shape using a continuous field, the most straightforward way to attack the problem is through 2
Let us remind that in this thesis, we will make simplifying assumptions that will restrict us to the same “linear” regime. The nonlinear aspects are currently being studied and will likely be the topic of another Ph.D. thesis.
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Introduction the idea that lies at the heart of all local (Wilsonian [WK74]) field theories: separation of scales. By treating the boundary conditions around the particles as physics on a disparate (and shorter) length scale, one can deal with the fluctuations of the intervening medium at the scale where the boundaries have shrunk to points. But this is done in an informationconserving way; the points are not approximate descriptions of the particles. Although the problem will not be considered in full generality here, this approach can be adopted independent of the shape of the boundaries, how many there are, what the “mechanics” of the intervening medium is, whether it is linear, the nature of the boundary conditions, etc. This approach allows the analytic computation of physical observables in a very efficient and transparent way. We are able to calculate forces up to accuracy levels much higher than before and properly address many-body interactions, thanks to these two aspects of the formalism. Of course, there is nothing special about two dimensions. The same effective field theory (EFT) formalism can be used for classical electrodynamics problems, interactions in a correlated fluid or gravity in four dimensions. In fact, the flavor of EFT we will use in this thesis was developed to study the gravity wave profile for in-spiralling black holes [GR06], and has subsequently been utilized to derive further new results in gravitational wave physics [PRR11] as well as to calculate the leading order finite size correction to the Abraham-Dirac-Lorentz radiation reaction force law in classical electrodynamics [GLR10]. A close relative of the EFT formalism was also used to study the so-called critical Casimir forces [Kre94, KG99, Gam09] on colloidal particles suspended in a critical mixture [BE95, ER95, HSED98]. We begin the discussion with rigid disks trapped at a fluid interface. Instead of having a separate chapter for technical background, necessary concepts will be introduced and illustrated in context, mostly within this first chapter. Some digression into essential and yet somewhat auxiliary technical issues is distributed into footnotes and “Technical Note”s within the chapters, as well as appendices at the end of the text. The next chapter on interactions of particles on a membrane does not share the same pedagogical character of the first one, and was written in a more concise manner. We then conclude by wrapping up the merits and results of the work done here.
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1 Interactions of rigid particles on a film We begin with interactions mediated by the simplest imaginable surface; one whose behavior is determined by surface tension. We will refer to such a surface as a film, as the typical example is a soap film. Since this choice of surface comes with the tidiest mathematics, the introduction of most of the relevant physical concepts for the thesis—elementary or otherwise—will take place in this chapter, and hence is perhaps the most instructive. We restrict our attention to rigid particles here, and defer flexible particles to future work. The particles may be flat or they may enforce a deformed contact line on the surface. We will address both situations in the discussion regarding boundary conditions. However, note that in this thesis, whether the particles are flat or deformed, they are assumed to have a circular projection.1
1.1 Overview of the system Let us get a clear and more quantitative picture of the physical situation of interest before moving on to the main objective of the thesis. We will first discuss the surface, and then the particles in terms of how they affect the surface.
1.1.1 The film In this chapter, the medium where the particles are situated is a surface which we refer to as a film. The defining characteristic of a film is that its (stable) equilibrium shape is determined by the requirement of minimal area, under given boundary conditions, reminiscent of a soap film. This is equivalent to the statement that this surface owes its energy to surface tension, or mathematically that the energy functional (or Hamiltonian) of the surface is given as Z Hfilm = σ
dA ,
(1.1)
where dA is the proper area element on the surface. This expression is covariant, i.e. independent of how the surface is parameterized. For the computations in this thesis, the surface will be parameterized in the so-called Monge gauge (see, for example the book by Safran [Saf94]) which describes the surface as a height z(x, y) =: h(r) above the flat real plane (see Figure 1.1). Clearly, not every surface shape can be represented like this; we restrict to those that can. 1 It may help at this point to imagine a corn flake that has a saddle-like shape but looks like a disk when viewed from the top.
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1 Interactions of rigid particles on a film
h(x, y)
∂S2 ∂S1
x ∂S3
y
Figure 1.1: The surface is parameterized in the Monge gauge as a height function z = h(x, y) over the real plane. The circular projections of the inclusions are also depicted.
By simple geometry, it is seen that in this parameterization, the surface patch q above the
rectangular area element of area dxdy at the point (x, y) has an area of dxdy 1 + (∇h)2 . Therefore the surface Hamiltonian as a functional of the height field h(r) becomes Z q 2 (1.2) Hfilm [h] = σ d r 1 + (∇h)2 . S
Here, S denotes the projection of the surface onto the real plane. Soon, we will be computing partition functions with Boltzmann factors derived from this Hamiltonian. Unfortunately, the dependence of Eq. (1.2) on the field is not suitable for this purpose in practice, and therefore we will have to make one further assumption on the shape of the surface to make the problem tractable; that the surface is only weaklydeformed, such that |∇h| ≪ 1. This allows us to expand the square root in Eq. (1.2) and obtain Z 1 Hfilm [h] = σAS + σ d2 r (∇h)2 , (1.3) 2 S up to quadratic order in the field, where AS is the area of projection of the film onto the real plane. Minimization of Hfilm [h] requires it to be stationary with respect to variations around the minimizing profile h(r). With fixed boundaries, the change in the energy (1.3) resulting from a variation h → h + δh in the field is Z (1.4) δHfilm [h] = Hfilm [h + δh] − Hfilm [h] = σ d2 r ∇h · ∇δh , S
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1 Interactions of rigid particles on a film to lowest order, which integrates by parts to Z Z 2 2 δHfilm [h] = σ d r δh −∇ h + σ S
∂S
ˆ · ∇h , dℓ δh n
(1.5)
ˆ respectively denote the boundary of the domain S, a line element where ∂S, dℓ and n on the curve ∂S and the outward-pointing normal to the boundary at the position of dℓ. The stationarity condition, δHfilm [h] = 0, along with the arbitrariness of δh(r) requires the integrands in Eq. (1.5) to vanish point-by-point. The second integrand automatically vanishes in the case of Dirichlet boundary conditions, since the value of the field is prescribed on the boundary and hence δh(r ∈ ∂S) = 0. Other boundary condition types require a bit more attention [CH53]. The bulk term, on the other hand, yields ∇2 h = 0
(1.6)
irrespective of the boundary conditions. This differential equation which determines the equilibrium field configuration under given boundary conditions is usually referred to as the field equation, or sometimes the Euler-Lagrange equation of the problem. We have discussed how the surface behaves under imposed boundary conditions. These boundary conditions stem from the contact between the surface and the particles. Let us now explore those.
1.1.2 Boundary conditions If the particles are labeled with indices a, the surface has the boundaries ∂Sa (which we assume to be a circle in the present thesis) at the circumference of each particle, and an outer boundary. The requirement at the rim ∂Sa of each particle is that the surface simply attaches to the particle. Along with conditions at the outer boundary (to be specified later), this is sufficient to solve the second order differential equation ∇2 h = 0. The particles, and therefore the contact lines, are assumed to be rigid in this chapter. However, we will consider both flat (planar) and deformed contact lines. Notice that the shape of the contact line is what matters, not so much the particle. A flat disk, a sphere, an oblate sphere, etc. are all equivalent for us so long as a circular line of contact is enforced, e.g. due to two halves of the object having wetting properties favoring either side of a fluid interface (a so-called Janus colloid). Similarly, when we talk about a saddle-shaped particle, it may be a flattened object like a corn flake or an extended object whose contact with the surface is wavy. Let us be more quantitative. The contact line shape (not the shape of the footprint but the out-of-plane deformation) can be expressed as the following expansion in terms of the coordinates: (1) (2) hcl (r) = η (0) + ηi ri + ηij ri rj + . . . ∀r ∈ ∂Sa , (1.7) where repeated indices are summed over.2 Since the only possible motions of a rigid object are translation and rotation, the only parameters that may fluctuate freely are η (0) and 2
Also note that this is just a Taylor expansion and does not assume a specific shape for the domain Sa .
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1 Interactions of rigid particles on a film
free η (1)
free η (0)
Figure 1.2: The bobbing (η (0) ) and tilting (η (1) ) degrees of freedom of a rigid disk.
η (1) , encoding a height and a tilt, respectively (see Fig. 1.2). Whether the height and tilt are free or fixed depends on the extrinsic conditions imposed on the motion of the particle. On the other hand, the intrinsic rigidity requires that all the tensors η (>2) are fixed. If they are fixed to zero, then the contact line is confined into a plane determined by η (0) and η (1) , since Eq. (1.7) becomes the equation of a plane with z = h(x, y). This would be the flat disk case, which we will take up first. If there is a non-flat contact line at hand, the tensors η (>2) are still fixed but at nontrivial values. As for the outer boundary, we assume the surface is boundless, or that the outer boundary is at infinity, to unencumber the math. When the domain is infinite in size, chances are, some solutions to the field equation will not be square-integrable in the sense that Eq. (1.3) blows up. It will be clear in the next section that such a square-integrability condition renders an explicit constraint on the field at infinity, such as ∇h(r → ∞) → 0, unnecessary. The result is, a boundary value problem with prescribed values at the compact boundaries ∂Sa is adequately specified—without the need for a second piece of information such as the normal gradient at these boundaries—if the solution is to be square-integrable (i.e. physical, since otherwise it costs infinite energy). Finally, note that the boundary conditions at ∂Sa are of the Dirichlet type, as they specify the value of the function h(r) at the boundary. A Neumann type boundary condition, i.e. one that specifies the normal gradient of the field, could also be the case, for example, with smooth cylinders on which the position of the contact line is not constrained but the contact angle is. For smooth spherical particles trapped at a fluid-fluid interface, the position and normal gradient of the contact line are related (in a linear fashion for small deformations), which is yet another type of boundary condition. We do not consider these in this thesis. The response of a single disk In order to make the current and subsequent discussions more concrete, we will now consider the following problem as an example: a single disk of radius R experiencing a background deformation. One can imagine this background being the culmination of other particles’ effects, or due to thermal fluctuations.
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1 Interactions of rigid particles on a film To this end, we rewrite the field outside the particle as hout = hbg +hresp . We are looking for a square-integrable response hresp (r) that obeys ∇2 hresp = 0 with the condition that hout (r) must meet the contact line at r = R (clearly, we assume that the origin is at the center of our particle), i.e. hbg (R, ϕ) + hresp (R, ϕ) = hcl (R, ϕ) .
(1.8)
In polar coordinates, the most general harmonic function has the form h(r) = a ˜0 log kr +
∞ h X ℓ=1
i aℓ r−ℓ cos(ℓϕ − ℓϕℓ ) + a ˜ℓ rℓ cos(ℓϕ − ℓϕ′ℓ ) .
(1.9)
However, for those eigenfunctions whose prefactors are marked by a tilde, square-integraR bility is not satisfied, i.e. S d2 r (∇h)2 → ∞, due to their behavior as r → ∞. Thus, the most general acceptable form for the response hresp (r) becomes resp
h
(r, ϕ) =
∞ X ℓ=1
aℓ r−ℓ cos(ℓϕ − ℓϕℓ ) .
(1.10)
The solution of the boundary value problem is then to simply determine the coefficients aℓ and phase angles ϕℓ that satisfy the boundary condition (1.8). Next, to set up the problem, the contact line shape and background must be specified. We have already written the former in Eq. (1.7) but not so much in a useful form. After some explicit calculation (with x = r cos ϕ and y = r sin ϕ), it is not hard to convince yourself that from each coefficient η (ℓ) in Eq. (1.7), a series of terms rℓ cos ℓϕ, rℓ cos(ℓϕ − 2ϕ), rℓ cos(ℓϕ − 4ϕ) and so on (up to phase angles) arises. But at the end, when we set r = R for the boundary, it just becomes a Fourier series of the form cl
h (R, ϕ) =
∞ X ℓ=0
ηℓ Rℓ cos(ℓϕ − ℓαℓ ) .
(1.11)
It is not necessary to bother with how the scalars ηℓ are related to the tensors η (ℓ) in Eq. (1.7). For the sake of completeness, suffice it to say that the coefficient ηℓ is determined by the tensors η (>ℓ) . The background we will use to “excite” the particle will similarly have the form bg
h (r, ϕ) =
∞ X ℓ=0
bℓ rℓ cos(ℓϕ − ℓβℓ ) .
(1.12)
Note that this background consists entirely of harmonic functions, that is, those that obey the field equation. This did not have to be the case, since what matters is its value at r = R and nothing else for the boundary value problem. However, when we speak of backgrounds and responses further in the chapter, these will be more convenient.
9
1 Interactions of rigid particles on a film With the boundary value problem specified completely, we turn to its solution. Since the response (1.10) does not involve a monopole (ℓ = 0) term (recall square-integrability), this order is trivial, but also illustrates an important aspect of boundary motions. The ℓ = 0 term of the boundary condition (1.8) merely implies η0 = b0 . What this condition therefore means is that the parameter η0 —the average height of the contact line—must be free, so that whenever the disk experiences a monopole background in its vicinity, it can move up or down appropriately to match it. The rest of the multipoles yield the conditions ηℓ
aℓ cos ℓβℓ sin ℓβℓ cos ℓαℓ sin ℓαℓ − bℓ = 2ℓ = ηℓ − bℓ cos ℓϕℓ cos ℓϕℓ sin ℓϕℓ sin ℓϕℓ R
(1.13)
for aℓ and ϕℓ , after which the response is finally determined to be hresp (r) =
∞ h i X ηℓ R2ℓ r−ℓ cos(ℓϕ − ℓαℓ ) − bℓ R2ℓ r−ℓ cos(ℓϕ − ℓβℓ ) .
(1.14)
ℓ=1
Note how the response decouples into two parts. The induced deformation hresp ind (r) = −
∞ X ℓ=1
bℓ R2ℓ r−ℓ cos(ℓϕ − ℓβℓ )
(1.15)
stems from the existence of a rigidity constraint. That is, it arises from the fact that ηℓ>0 are fixed (perhaps with the exception of η1 ; see the last paragraph) but not the specific values they are fixed at. The permanent response hresp perm (r) =
∞ X ℓ=1
ηℓ R2ℓ r−ℓ cos(ℓϕ − ℓαℓ )
(1.16)
is created by (vertical) deviations of the contact line from a flat circle. That is, it requires the amplitudes ηℓ>0 of the contact line irregularities to be nonzero. We will actually refer to these results in the next section. Also note that with an appropriate choice of background (bℓ and βℓ ), the induced response can be made to cancel the permanent response. Contact line motions. We mentioned that for the system to be physically sensible, the vertical motions of the disk must be unconstrained. The same is not true for the tilt motion, i.e. η1 does not have to be free. But it can. It should be instructive to observe that if this is the case, in the response (1.14), the effect of a possible background tilt (b1 6= 0) can be cancelled out by η1 assuming an appropriate value.
1.1.3 Surface-mediated interactions Having discussed how the surface behaves and how the presence of particles affect it, we now move on to the interactions between the particles mediated by the surface.
10
1 Interactions of rigid particles on a film
Figure 1.3: A permanent quadrupole deformation arising from a saddle-shaped contact line (left) and an induced dipole deformation arising from a planar background impinging on a horizontally-fixed contact line (right).
Ground state interactions The source of the inter-particle forces is simply the presence of compact regions (occupied by the particles) with boundary conditions around them. If at least one of the particles impart a permanent deformation on the surface, then, in its ground state, the shape of the surface—and thereby its energy via Eq. (1.3)—is determined by the Euler-Lagrange equation under the constraints prescribed on the boundaries of these compact regions. It is then not at all unreasonable to think that for each different choice of distance (and orientation, if applicable) between these regions, the ground state shape and its corresponding energy will have a different value. An analogy with electrostatics will be of greatRhelp here: The problem of finding the electrostatic potential φ(r) and the total energy 12 ε d3 r(∇φ)2 of a configuration of conducting spheres, where at least one of them is charged, is exactly of the same form. While the problem of one conducting sphere at a certain distance from a point charge is a standard textbook exercise—which is typically used to illustrate the method of images—one can imagine it becomes way more intractable when the point charge is exchanged with a charged conducting sphere, not to mention when there is more conducting spheres lying around. Fluctuation-induced interactions There is another possibility of forces arising, that does not require any particle to impart a deformation on the surface. These are the so-called fluctuation-induced interactions, and they are reminiscent of the Casimir effect in electrodynamics. Dutch theoretical physicist, Hendrik B. G. Casimir, proposed in 1948 that even if they are charge-neutral, there should be an attraction between two conducting plates in vacuum [Cas48, BMM01, Mil01]. The reason is the equipotential constraint the conductors place on quantum fluctuations of the electric field and the attractive energy per area was predicted to be ∼ ~c/d3 , where d is the distance between plates. In our case, the surface and the particles are in contact with a thermal reservoir and therefore undergo fluctuations around the ground state configuration. However, these fluctuations are still subject to the constraints prescribed on the contours of the particles, akin to the Casimir effect. The result is that no phase space point (shape
11
1 Interactions of rigid particles on a film of surface) will be visited by the system if it violates the boundary conditions. In other words, Boltzmann factors e−βHfilm [h] pertaining to field configurations h(r) that possess unallowable features near the particle locations are excluded from the partition sum. Then, again, it is highly plausible that the spatial distribution of the particles will influence the partition function, and hence the free energy of the system. Clearly, as opposed to the ground state forces mentioned in the previous paragraph, these fluctuation-induced forces are only present under finite temperatures. The dimensionless free energy βF of the system is given by βF = − log Z, where Z is the canonical partition function. Since each possible profile h(r) is a microstate R of the system, the sum over all microstates becomes a functional integral, denoted with Dh. Hence, the free energy is written as Z βF = − log Dh e−βHfilm [h] . (1.17) Now, since the film Hamiltonian (1.3) is quadratic in the field h(r), we know how to take the functional integral, in principle; it is much like a multivariate Gaussian integral (see Appendix A) and the free energy would be βF ∼ log (det K) = Tr (log K), with K being the kernel of the quadratic functional in the exponent. If we perform an integration by parts on our energy functional (1.3) to bring it to a matrix product form, we have Z Z 1 1 2 2 ˆ · ∇h . dℓ h n (1.18) Hfilm [h] = σ d r h(−∇ )h + σ 2 S 2 ∂S Even though we Rcannot write down the kernel in Eq. (1.18) explicitly—but perhaps implicitly as (1.18) = d2 r d2 r′ h(r ′ )K(r ′ , r)h(r)—it is clearly a differential operator that acts on functions defined over the subset S of R2 and subject to boundary conditions described in Section 1.1.2. These features of the kernel have two consequences: (i) the resulting functional integral and free energy will depend on the boundary conditions—and most importantly for us, where they are imposed—and (ii) it is nowhere as straightforward a functional integral to perform as with a translationally invariant kernel, such as −∇2 acting on all functions defined over R2 with a square-integrable gradient.
1.2 Effective field theory Above, we motivated, in a somewhat quantitative fashion, why the forces of interest to this research arise and pointed out how the difficulties stem from the local boundary conditions. The main theme of this thesis is a reformulation of the problem which facilitates dealing with the issue. We achieve this using a worldline effective field theory, introduced by Goldberger and Rothstein for the gravity of extended objects [GR06]. In a nutshell, this approach frees the field from the constrained regions, and instead encodes the boundary conditions right into the energy functional in extra terms localized at discrete points where the particles
12
1 Interactions of rigid particles on a film are located—these would be trajectories or “worldlines” of the particles had we been interested in time dependences. This reformulation of the problem makes it much more straightforward to handle. The bulk theory (1.3) along with its boundary conditions is exchanged for an effective theory Heff [h] = H[h] + ∆H[h]
(1.19)
where there are no explicit boundary conditions and H[h] is the unrestricted bulk Hamiltonian Z 1 d2 r (∇h)2 . (1.20) H[h] = σ 2 R2 Then the free energy can be computed using the machinery of perturbation theory as a cumulant expansion (refer to Technical Note 1.1 for more information): βF = −
∞ X 1 h(−β∆H)q ic . q!
(1.21)
q=1
Furthermore, since the theory is local, i.e. no long range interactions in the Hamiltonian, P each particle/boundary contributes separately to the effective theory as ∆H = a ∆Ha , where a is a particle label. So how do we find these extra terms making up ∆H so that the effective theory is statistically similar to the original theory? We follow a renormalization argument, backwards: Analogous to block spin renormalization, if we were to coarse-grain and renormalize the original theory until all the boundaries shrunk to points, we would finally arrive at a renormalized Hamiltonian, statistically similar to the original theory, but generally with a multitude of new terms allowed by symmetries, with their magnitudes (“coupling constants”) resulting from the renormalization group flow. Armed with this knowledge, one can write down the most general effective Hamiltonian by augmenting the unrestricted bulk Hamiltonian by all localized terms that the symmetries in the problem allow, up to unspecified prefactors. These prefactors, also known as Wilson coefficients, are then fixed by matching the value of some suitably chosen observable (ideally something easily calculable) across the full and effective theories. This matching step ensures that out of a family of possible effective theories, the correct one is indeed chosen. As such, the effective theory is not a model, or an approximation of the original system; it is a reformulation of it. The above arguments imply that each ∆Ha is a derivative expansion comprising scalars obtained from derivatives of the field h(r) at the point r a . In general, the derivative expansion contains all polynomials consistent with the symmetries of the problem. But, as our bulk Hamiltonian was truncated after quadratic order in the field, we will do the same with the effective Hamiltonian.
13
1 Interactions of rigid particles on a film Technical Note 1.1: Cumulant expansion The Helmholtz free energy βF = − log Z of the system is Z Z βF = − log Dh e−βHeff [h] = − log Dh e−βH[h] e−β∆H[h] R by definition, where Dh symbolizes “sum” over all microstates h(r) of the surface. Note that if the R partition function for the unperturbed surface is Z0 = Dh e−βH ,
βF = − log Z0 − log e−β∆H ,
and the average is taken in the free ensemble (∆H = 0). Since the free energy − log Z0 of the unperturbed surface bears no dependence on the presence of particles, we absorb it in the definition of βF = 0. Then the (interaction) free energy reads, after expanding the exponential in a power series, " # ∞ X 1 q βF = − log 1 + h(−β∆H) i . q! q=1
Discussing the relation between the statistical objects moments, hX q i, and cumulants, hX q ic , is beyond the scope of the present thesis. One can refer to many textbooks on probability theory. The upshot of it is that the multitude of combinations of moments that the series expansion of the logarithm generates can be grouped with respect to the total power of ∆H, forming cumulants—and this can be taken as the definition of cumulants—which results in Eq. (1.21) [BDFN95]. When we begin using Feynman diagrams to track the many terms in the cumulant expansions, this will be equivalent to removing disconnected diagrams.
1.2.1 Permanent sources Linear scalars can be constructed from localized derivatives of the field such as Q(0) h(r a ), (1) (2) Qi ∂i h(r a ), Qij ∂i ∂j h(r a ), etc.3 The (tensor) Wilson coefficients are named appropriately as Q since they are analogous to electrostatic charges; finding a stationary point of the effective theory in the presence of a linear term is equivalent to finding the field deformation generated by a source (see Technical Note 1.2). For example, the “quadrupole” term Z (2) (2) −Qij ∂i ∂j h(r a ) = − d2 r δ(r − r a ) Qij ∂i ∂j h(r) Z (2) = − d2 r [∂i ∂j δ(r − r a )] Qij h(r) , (1.22) where an integration by parts was performed in the last step, augments the unperturbed Hamiltonian by a source (2)
ρ(r) = Qij ∂i ∂j δ(r − r a ) 3
Just in case it is not obvious, ∂j = ∂/∂rj and repeated indices are summed over.
14
(1.23)
1 Interactions of rigid particles on a film Technical Note 1.2: Euler-Lagrange equation in the presence of sources Consider the theory with a general source distribution ρ(r): Z Z 1 2 2 H[h] = σ d r (∇h) − d2 r ρ(r)h(r) , 2 where the integrals encompass the entire R2 . Then, the first variation is found as Z δH[h] = d2 r δh(−σ∇2 h − ρ) upon discarding the boundary term at infinity. Stationarity implies the inhomogeneous differential equation −∇2 h(r) =
1 ρ(r) . σ
The Green function of the problem, G(r, r ′ ) is the inverse of the operator −∇2 in position space, therefore defined as −∇2 G(r, r ′ ) = δ(r − r ′ ). Thus, convolving both sides of the equation with the Green function yields Z 1 d2 r′ ρ(r ′ )G(r ′ , r) . h(r) = σ The reason we insist on the minus sign in front of the operator ∇2 is that −∇2 is the positive definite kernel when we deal with Gaussian integrals.
as in Technical Note 1.2, and generates the deformation Z 1 1 (2) h(r) = d2 r′ ρ(r ′ )G(r ′ , r) = Qij ∂i ∂j G(r a , r) . σ σ
(1.24)
If the last step is inspected carefully, it is seen that the final derivative on the Green function is on its first argument, evaluated at r ′ = r a .4 Matching Earlier, we calculated the deformation that emanates from contact line irregularities of a finite-sized particle, i.e. in the full theory. The deformation (1.24) is the counterpart of the same physical observable in the effective theory; the field created by the “mock” point (2) charge we replaced the finite-sized object with. We can therefore fix the coefficient Qij by equating the two observables. First things first, the Green function for the harmonic operator in two dimensions is G(r ′ , r) = G(r, r ′ ) = −
1 log(r − r ′ )2 . 4π
(1.25)
4 Over the course of this research, the author’s calculations were interrupted countless times by a minus sign hunt, because of overlooking this.
15
1 Interactions of rigid particles on a film Then it is easily computed that (ri − ri′ )(rj − rj′ ) 1 δij − 2 . ∂i ∂j G(r , r) = − 2π(r − r ′ )2 (r − r ′ )2 ′
(1.26)
Let us place the particle at the origin h as iwe did earlier so that r a = 0 and write the (2) quadrupole tensor explicitly as Qij = kq −kq .5 Then Eq. (1.24) becomes k q h(r) = [Gxx (0, r) − Gyy (0, r)] + [Gxx (0, r) + 2 Gxy (0, r) + Gyy (0, r)] σ σ k cos 2ϕ q sin 2ϕ = + . πσ r2 πσ r2
(1.27)
It is clear then, by comparison to the permanent response in Eq. (1.16), that a linear (2) mock term,6 −Qij hij (0), in ∆H correctly captures a saddle-like contact line deformation (2)
η2 R2 cos(2ϕ − 2α2 ) if the Wilson coefficient Qij is chosen as cos 2α2 sin 2α2 (2) 4 Qij = πR ση2 . sin 2α2 − cos 2α2
(1.28)
This illustrates how the contact line irregularities in the full theory are encoded into the Wilson coefficients in the effective theory. It is easy to see how this procedure generalizes to linear terms of arbitrary order in derivatives if one has higher order “multipoles” in the contact line deformation. Redundant terms. One more remark would be appropriate at this point, even though it ¯ (2) hii (r a ) or Q ¯ (6) hijkkll (r a ) is not specific to the linear order. A source term of the form Q ij (i.e. with contracted derivatives on the field) may be conceivable. However, note that in the cumulant expansion (1.21), all occurrences of the field will eventually end up in a two-point correlator multiplying others due to Wick’s theorem (see Appendix A.1). But since hhii (r a )h(r b )i ∼ Gii (r a , r b ) = 0 for a 6= b, such terms will not affect the interaction free energy.7 Therefore, such terms—those involving a field occurrence acted upon by the Euler-Lagrange derivative—are redundant in the effective theory.8 (2)
5
The matrix Qij was chosen traceless. Even if it had a trace (i.e. a part ∼ δij ) the response (1.24) would not bear any sign of it away from the point r = r a , since δij ∂i ∂j G(r a , r) = ∇2 G(r a , r) = −δ(r −r a ). (2) Furthermore, Qij was chosen symmetric without loss of generality; it multiplies ∂i ∂j h on the worldline, (2)
(2)
(2)
which is symmetric. Any asymmetric Qij can then be redefined as (1/2)(Qij + Qji ) while preserving the (2) Qij ∂i ∂j h.
product 6 From here on, we adopt the notation ∂i f = fi whenever convenient. 7 They do seem to affect the self-energies since Gii (r a , r a ) 6= 0. But the self-energies are themselves unphysical artifacts of a continuum field theory. More on that in the next section, specifically in Technical Note 1.4, and also Appendix C. 8 A more general argument—which is too technical for the purpose of this paragraph—involves proving that such terms can be eliminated by a canonical transformation which leaves physical observables (more technically, the S-matrix) invariant [Geo91].
16
1 Interactions of rigid particles on a film
1.2.2 Induced sources After having discussed the above, it will be easy to see that quadratic worldline terms of the form h2i (r a ), h2ij (r a ), etc. represent induced sources in the theory. These arise from the restrictions placed on the function h(r) along the boundaries of the objects, much like the induced polarization on a conducting surface under an external field in electrostatics. There is, however, some restriction on the form of the quadratic terms in our problem, as opposed to none on the form of linear terms. This is due to symmetries. If we are writing the linear terms of the effective theory for a particle that permanently deforms the surface at its boundary, this deformation will likely look different after the coordinate transformations parity (r → −r) and rotation. Therefore we do not seek invariance of linear terms under these transformations. On the other hand, the quadratic terms encode not permanent deformations but those induced by a background, as mentioned above. Since the boundary shape, a circle, is invariant under mirror reflection and rotation, the induced response to the same background should be indistinguishable between transformed coordinates. Keeping in mind that the quadratic terms will be of the form ∂ n h(r a )∂ m h(r a )—where ∂ n is a shorthand for ∂i1 ∂i2 . . . ∂in —we see that parity invariance excludes terms with n − m odd; such terms transform into their negative under r → −r. Rotation invariance, on the other hand, requires the product ∂ n h(r a )∂ m h(r a ) to be scalar; all indices contracted without needing a tensor coefficient.9 Recalling that terms which involve a field occurrence with the Euler-Lagrange derivative (−∂i ∂i ) are redundant, all the contractions must be across the two factors ∂ n h(r a ) and ∂ m h(r a ), which implies n = m. Therefore, the most general form of quadratic term for circular objects will be i2 1 1 (ℓ) h ℓ C ∂ h(r a ) = C (ℓ) h2i1 i2 ...iℓ (r a ) 2 2
(1.29)
with the factor 1/2 being merely a religious choice. As we mentioned above, what makes these terms induced sources is their behavior under a background. To see this, we write h → hbg + h and plug it into Eq. (1.29). The resulting linear term Z (ℓ) ℓ bg ℓ C ∂ h (r a )∂ h(r a ) = d2 r δ(r − r a )C (ℓ) ∂ ℓ hbg (r a )∂ ℓ h(r) Z = d2 r (−∂)ℓ δ(r − r a )C (ℓ) ∂ ℓ hbg (r a )h(r) (1.30) captures a point-like source ρ(r) = −C (ℓ) ∂ ℓ hbg (r a )(−∂)ℓ δ(r − r a ) 9
(1.31)
To see this, consider a simple term, Cij hi hj , under the rotation, ri′ = Rik rk . Since hi = ∂i h = = Rki ∂k′ h, the term transforms into Cij Rik Rjl hk hl . If this term is to be equal to the original, then Ckl = Cij Rik Rjl must hold, i.e. Cij itself is rotationally invariant (Cij = Cδij ) and therefore Cij hi hj = Chi hi . This can be generalized to higher derivatives. ∂i rk′ ∂k′ h
17
1 Interactions of rigid particles on a film whose magnitude is determined by the coefficient C (ℓ) and the strength of the local features of the background. Therefore this is a localized polarization induced by the background and hence the name polarizability for the coefficient C (ℓ) . We will not begin labeling the polarizability with a subscript a until more than one particle is considered at a time. Matching We can now write what field is generated by this induced polarization in response to the applied background by taking a convolution as before: Z 1 h(r) = d2 r′ ρ(r ′ )G(r ′ , r) σ Z 1 = − C (ℓ) ∂ ℓ hbg (r a ) d2 r′ (−∂)ℓ δ(r ′ − r a )G(r ′ , r) σ 1 = − C (ℓ) ∂ ℓ hbg (r a )∂ ℓ G(r a , r) . (1.32) σ We did calculate the induced response of the full object to a set of backgrounds in Eq. (1.15), and we can use those full theory results to fix the polarizabilities C (ℓ) . The contraction ∂ ℓ hbg (r a )∂ ℓ G(r a , r) = hbg i1 ...iℓ (r a )Gi1 ...iℓ (r a , r)
(1.33)
may look formidable but we show in Appendix B.1 that it can be performed remarkably easily if the Cartesian coordinates r = (x, y) are transformed to complex coordinates z = (z, z¯) = (x + iy, x − iy). In Section 1.1.2 we hadPplaced the particle at the origin and computed its response to the ℓ background hbg (r, ϕ) = ∞ ℓ=0 bℓ r cos ℓϕ (phase angles are unnecessary because of circular symmetry). Referring to Eq. (B.7) for the index contraction, the induced response (1.32) from this background in the effective theory is found as h(r) = −
∞ X ℓ=1
C (ℓ)
2ℓ ℓ!(ℓ − 1)! bℓ cos ℓϕ b0 log r . − C (0) ℓ 4πσ 2πσ r
(1.34)
The counterpart of the same (set of) observable(s) is Eq. (1.15) in the full theory, and therefore matching fixes the polarizabilities as C (ℓ) =
4πR2ℓ σ 2ℓ ℓ!(ℓ − 1)!
,
ℓ>0
(1.35)
and C (0) = 0. For the latter, recall that the field of a monopole source is not physically possible for a system of infinite extent and therefore the disks have to be free in their overall vertical motion so that monopole responses are out of the question.
18
1 Interactions of rigid particles on a film Finally, the culmination of all that was discussed in Section 1.2 is the effective theory, explicitly written as Z ∞ XX 1 (ℓ) 2 1 (ℓ) 2 2 C h (r a ) − Qa,i1 ...iℓ hi1 ...iℓ (r a ) (1.36) Heff [h] = σ d r (∇h) + 2 2 a i1 ...iℓ a ℓ=1
with the polarizabilities given in Eq. (1.35) and the tensors Q(ℓ) being nonzero only when the contact line shape has a nontrivial multipole of order ℓ. It is worthwhile to remind ourselves that this is not an approximation, but an exact reformulation of the energy functional of a surface which is subject to constraints due to the presence of finite-sized disk-like particles stuck to it.
1.3 The interaction free energy: fluctuation-induced Now that the effective theory (1.36) is determined, we do not have to worry about the boundary conditions of the particles to compute the free energy of the system; the presence of the localized permanent and induced sources takes care of that. The induced sources feature in both the interactions produced by the ground state of the surface and those induced by thermal fluctuations, whereas permanent sources are relevant for only the former. Therefore we will first restrict to the case without permanent sources present, i.e. the particles are all perfectly flat, and work out the fluctuation-induced interactions. Then, the effective theory is characterized by ∆H =
∞ XX 1 a
ℓ=1
2
Ca(ℓ) h2i1 ...iℓ (r a ) ,
(1.37)
and we need to compute the free energy *" #q + ∞ ∞ X XX 1 (ℓ) 2 1 βF = − −β C h (r a ) . q! 2 a i1 ...iℓ a q=1
ℓ=1
(1.38)
c
Let us begin slowly, with the trivial first cumulant (q = 1), and go step by step. The first cumulant is the q = 1 term, βFq=1 = β
∞ XX 1 a
ℓ=1
2
Ca(ℓ) hhi1 ...iℓ (r a )hi1 ...iℓ (r a )ic .
(1.39)
It is not hard to see that these terms never involve the relative positions of the particles, and thus do not encode interactions, but let us be more rigorous. At this order in cumulants, the two-point function10 hh(r ′ )h(r)i and the connected twopoint function hh(r ′ )h(r)ic are identical because hh(r ′ )h(r)ic = hh(r ′ )h(r)i−hh(r ′ )i hh(r)i, 10
The derivatives, which are with respect to spatial coordinates, can go inside or outside the thermal average, which is with respect to field configurations. It may help to imagine a field defined over a lattice and differences instead of derivatives.
19
1 Interactions of rigid particles on a film
Figure 1.4: These graphs depict how the Wick contractions (1.43a)–(1.43c) in the second cumulant 2 can be visualized, respectively. Each (two-legged) vertex stands for two factors ∂ ℓ h(r a ) of the field coming from a power of ∆H. Then the links represent the Wick connections. Observe that the two connected diagrams are identical as far as the connection of vertices is concerned even though the legs are connected differently. In such a case, we do not draw two diagrams, but consider one of them as a Wick contraction of a certain multiplicity, in this case 2.
and hh(r)i = 0 since, in the unperturbed ensemble, h(r) is a Gaussian distributed random variable with zero mean. The two-point function is simply (see Appendix A.1)
h(r)h(r ′ ) = (σβ)−1 G(r, r ′ ) , (1.40)
where we remind that G(r, r ′ ) is the Green function for −∇2 in 2 dimensions, c.f. Eq. (1.25). Therefore, the first cumulant becomes βFq=1 =
∞ (ℓ) XX Ca a
ℓ=1
2σ
∂i1 ∂i1 . . . ∂iℓ ∂iℓ G(r a , r a ) ,
(1.41)
that is, it consists entirely of “self-interaction”. Denoting G(r, r) = G(0), we can see that the terms are divergent as, ∂i ∂i G(0) = −δ(0), ∂i ∂i ∂j ∂j G(0) = −∇2 δ(0), etc. At this point we can argue that, divergent though they might be, these terms are mere constants and hence can be swept under a redefinition of βF = 0 like we did for the free energy of the unperturbed surface (see Technical Note 1.1). However, note that there is a more rigorous argument (details in Technical Note 1.4 and even more in Appendix C) explaining why these apparent divergences are not physical (or not present). The second cumulant is where the positions of different particles begin to get mixed and yield interaction terms. Now we have 1 X X 1 (ℓ) (m) βFq=2 = − β 2 C C hhi1 ...iℓ (r a )hi1 ...iℓ (r a )hj1 ...jm (r b )hj1 ...jm (r b )ic . (1.42) 2! 4 a b a,b ℓ,m
We show in Appendix A.1 that in a Gaussian ensemble such as what we have, Wick’s theorem states that many-point correlators can be written as the product of two-point correlators—termed Wick contractions—summed over all possible pairings. The possible
20
1 Interactions of rigid particles on a film Technical Note 1.3: Connected vs. disconnected The first Wick contraction (1.43a) is an example of a term that has no place in a cumulant due to disconnectedness. Observe that in this contraction, the pairings are such that there are no connections between field factors coming from the two different instances of ∆H in ∆H2 (note the different summation labels a and b), hence the connections result in two “disconnected” clusters.
This can also be observed in the first graph of Fig. 1.4. Generally, at higher cumulant order, ∆Hq>2 c , there may be q or less disjoint clusters. The figure below depicts some examples for q = 8: the first is a connected diagram, whereas the rest are disconnected.
When we are computing cumulants, and therefore connected many-point functions, such Wick contractions featuring disjoint clusters do not contribute. This is the difference between moments and cumulants, mentioned earlier in Technical Note 1.1.
Wick contractions of the four-point function (omitting the derivatives) are repeated below: hh(r a )h(r a )h(r b )h(r b )i → hh(r a )h(r a )i hh(r b )h(r b )i
(1.43a)
hh(r a )h(r a )h(r b )h(r b )i → hh(r a )h(r b )i hh(r a )h(r b )i
(1.43b)
hh(r a )h(r a )h(r b )h(r b )i → hh(r a )h(r b )i hh(r a )h(r b )i .
(1.43c)
Fig. 1.4 shows how these contractions can be visualized by diagrams. The last two Wick contractions are the ones that are connected (see Technical Note 1.3) and they are identical to each other. In such a case, we say we are dealing with one Wick contraction that is of multiplicity 2. For completeness and clarity, let us write the result explicitly, restoring the derivatives: hhi1 ...iℓ (r a )hi1 ...iℓ (r a )hj1 ...jm (r b )hj1 ...jm (r b )ic = 2 hhi1 ...iℓ (r a )hj1 ...jm (r b )i2 .
(1.44)
Using Eq. (1.40) we find the second cumulant (1.42) becomes βFq=2 = −
X X Ca(ℓ) C (m) a,b ℓ,m
b 4σ 2
Gab i1 ...iℓ j1 ...jm
2
,
(1.45)
where we have used a new notation Gab = G(r a , r b ). Let us now explicitly compute some interactions from the second cumulant. Note that the second cumulant mixes the coordinates of only two particles at a time, hence only contains pair interactions. The longest-ranged interaction is between induced dipoles, since
21
1 Interactions of rigid particles on a film (1)
C2
C1
(1)
C1
(a)
(b)
C2
(2)
C2
(1)
(1)
C1
(2)
(c)
Figure 1.5: Properly labeled diagrams depicting the dipole-dipole (a) and quadrupole-dipole (b–c) interactions. The dashed lines represent the worldlines12 of the particles helping to visualize vertices coming from the same ∆Ha . −2 the propagator between two multipoles of order 1, ∂i ∂j G(r a , r b ) ∼ rab , has the slowest possible decay with increasing separation. Therefore, in an asymptotic expansion in particle spacings, we would begin with the dipole-dipole interaction between, say, particles 1 and (1) (1) 2. This interaction is the term in Eq. (1.45) proportional to C1 C2 which is determined easily as (1)
βF dip−dip = −
(1)
C1 C2 2σ 2
G12 ij
2
.
(1.46)
Even though the second cumulant is too simple to appreciate or necessitate the use of Feynman diagrams, we show the properly labeled diagram for this interaction in Fig. 1.5(a). We recall that the polarizabilities are given in Eq. (1.35) and the propagator product 2 2 12 ri rˆj ) [see Eq. (1.26)] with r = (G12 ij ) is easily found by noting Gij = −(1/2πr ) (δij − 2ˆ r 1 − r 2 and 2 (δij − 2ˆ ri rˆj )2 = δij − 4ˆ ri2 + 4ˆ ri4 = 2 − 4 + 4 = 2 .
(1.47)
The result is βF dip−dip = −
R12 R22 , |r 1 − r 2 |4
(1.48)
in agreement with Refs. [LOD06, LO07] where the interaction was first computed. The leading correction to this interaction is between dipole and quadrupole moments,13 12
Worldlines are simply positions in our problem since there is no time dependence. We still stick to the term worldline, though, because of its generality. 13 Refs. [LOD06, LO07] also give this interaction, as the leading term of the situation where one of the disks is restricted in its tilt motion while the other is not. We have not made that distinction here; if (1) particle a can freely tilt, one simply sets Ca = 0.
22
1 Interactions of rigid particles on a film depicted in Figs. 1.5(b–c), and is similarly found to be14 (2) (1) (1) (2) 2 C1 C2 C1 C2 12 2 − G G12 ijk ijk 2 2 2σ 2σ 2 4 4 2 R R + R 1 R2 =−4 1 2 . |r 1 − r 2 |6
βF dip−quad = −
(1.49)
Higher order multipoles will clearly produce potentials of faster decay with increasing separation, as there will be more derivatives on the propagator. It will also turn out that higher cumulants do not produce terms of competing order with these two interactions. Hence the pair interaction can be written as 8 R12 R24 + R14 R22 R R12 R22 +O 8 , (1.50) βFpair = − 4 −4 6 r |r 1 − r 2 | |r 1 − r 2 | where R8 /r8 is a shorthand for the set of smallness parameters R12 R26 /|r 1 −r 2 |8 , R14 R24 /|r 1 − r 2 |8 and R16 R22 /|r 1 − r 2 |8 . We will keep using such shorthands. Feynman diagrams and rules So far we have not used Feynman diagrams as a computational tool but merely a visualization. However, as the perturbation expansion is cranked out, it becomes impossible to write out calculations algebraically. One then uses Feynman diagrams with a set of welldefined rules to evaluate the algebraic expressions they stand for. Here, we will explain the Feynman rules and work an example to establish a certain level of fluency with diagrams required for the rest of the thesis. The example consists of three terms, Figs. 1.6(a–c), picked from the fourth cumulant (q = 4); hence four vertices in each. The vertices are distributed along two, three and four worldlines in different diagrams. These are examples of two-, three-, and four-body interactions. An n-body interaction is one that mixes the coordinates of n particles at a time in a way that cannot be broken down additively into parts that may contain the coordinates of less than n particles. 2 (ℓ) Clearly, at every vertex, see Fig. 1.6, sits a term ∼ Ca ∂ ℓ h(r a ) . Each “leg” ∂ ℓ h(r a ) of a vertex is linked to that of another vertex. These depict the Wick contractions of a particular 2q-point function in the (connected) average h(−β∆H)q ic . It is easily seen that the connectivity of the vertices is immune to swapping the two links at each vertex. Since one may carry out this operation at any of the q vertices and still achieve the same connectivity, a total of 2q equivalent Wick contractions can be found; the Wick multiplicity
14
(1)
(2)
2 Note for example that in C1 C2 (G12 ijk ) , one of the derivatives, say ∂i , act at r 1 while the other two at r 2 . It would over-clutter the notation to make this explicit. But beware, inadequate care about this can lead to superfluous minus signs when the total number of derivatives is odd.
23
1 Interactions of rigid particles on a film Technical Note 1.4: Self-interactions Note that beside the pair interactions we extracted from the second cumulant (1.45), there seem to be self-interactions (a = b) as well. These are marked by divergent propagations from one perturbation to itself, ∂ n G(r a , r a ) = ∂ n G(0), as was also encountered in the first cumulant, and similarly, they can be argued away by a shift in βF = as they are constants. We mentioned earlier that even though these selfinteraction divergences are harmless, this is not the proper way to deal with them; at higher cumulant order, one can imagine such divergences beginning to multiply not mere constants but functions of particle positions. The rigorous explanation of why these divergences can be dropped involves renormalization group (RG) theory. Recall that our effective theory can be viewed as derived from the full theory by renormalizing it until the particle sizes become infinitesimal. The resulting RG flow of the Wilson coefficients must, therefore, produce so-called “counterterms” whose function is to ensure that physical observables obtained from the renormalized theory are still finite [Zee03]. The polarizabilities (1.35), having been derived by matching physical observables, are actually “renormalized” couplings, and it is “implied” that the effective theory accompanies these with the necessary counterterms. The upshot of all this is, seemingly divergent self-energy contributions to physical observables are actually removed by counterterms. On these grounds, we drop self-energies on sight when we compute physical quantities and not bother with verifying that they are removed by counterterms. This means we drop diagrams that link points on the same worldline, of the sort shown below.
A slightly more enlightening discussion of this can be found in Appendix C, where some counterterms and the resulting removal of a divergence are explicitly worked out.
for the diagram is 2q .15 Before moving on to evaluating the diagrams, we must make one last note. Recall that ∆H is a sum of infinitely many terms; these end up as the vertices in the diagrams. Diagrams such as those in Fig. 1.6 appear after ∆H is raised to the power q and then thermally averaged. Each diagram is the average of only one term among the many in the multinomial expansion of (∆H)qQ . Hence, the averages depicted by the diagrams carry a multinomial factor in front; q!/ v pv ! where v labels each distinct vertex in ∆H and pv are number of times each repeats in the diagram. The prescription to convert diagrams to algebraic expressions (i.e. the “Feynman rules”) is as follows: 15 The second cumulant is the only exception to this. In higher cumulants, one can swap the connections at a vertex, keeping the connections at the other vertices intact. So each “vertex-flip” is independent of another one carried out at a different vertex. But in the second cumulant, the two vertices are connected to each other and nothing else. Swapping the connections at one of them is swapping the connections at the other. So the two possible vertex-flips are not independent; you get a multiplicity of 2 from only one. We have indeed used a Wick multiplicity of 2 before [see Fig. 1.4 as well as Eq. (1.44)].
24
1 Interactions of rigid particles on a film (3)
(2)
(1)
(1)
C2
C2
C4
C3
(3)
(3) C2
C3
(2)
C2 (1)
(1)
C1
C1
(a)
(2)
C2 (1)
(1)
C1
C1
(b)
(c)
Figure 1.6: Example diagrams from the fourth cumulant. We apply the Feynman rules to evaluate each, as an illustration. These depict, respectively, a pair interaction, a triplet interaction, and a quadruplet interaction. (ℓ)
1. Vertices Each vertex contributes a factor of (−β/2)Ca to the value of the diagram. 2. Links Every link is a propagator between the points r a and r b carrying ℓ partial derivatives at r a and m at r b : β −1 σ −1 Gab i1 ...iℓ j1 ...jm . 3. Numerical factors The value of the diagram should be multiplied by the following nuq merical Q factors: (i) the Wick multiplicity 2 (for q > 2, and 2 for q = 2), and (ii) −1/ v pv !. The latter is the product of the aforementioned multinomial coefficient and the trivial factor −1/q! in front of the qth term in the cumulant expansion. Notice that the factors of β in the first two steps cancel off, so that βF has no temperature dependence, or F ∼ kB T . Similarly, the q factors of σ −1 in front of the propagators do not survive either, since each of the q polarizabilities are proportional to σ.16 Briefly, let us talk about general features we can right away see, without explicitly evaluating the diagrams in Fig. 1.6. From the multipole orders involved at the four vertices, we see that the diagrams all have 2(1+2+1+3) = 14 derivatives acting on four propagators. Since the propagator G(r, r ′ ) in this problem is such that ∂ n G(r, r ′ ) ∼ |r − r ′ |−n , each diagram affords 14 powers of inverse distance. Also, the combination of polarizabilities produce 14 powers of particle radii [see Eq. (1.35)]. Therefore, using the loose notation we invented in Eq. (1.50), these diagrams are all of order R14 /r14 , i.e. 14 factors of the √ sort Ra Rb /rab —with r ab = r b − r a . These factors are our power-counting parameters, essential for the concept of many scales. They encode how much short distance (high energy) physics there is in the long distance (low energy) observables, hence the form r−1 /R−1 . Now, on to explicit evaluation. The pair interaction depicted in Fig. 1.6(a) is the part of the fourth cumulant that is (1) 2 (2) (3) proportional to C1 C2 C2 . In the multinomial expansion of (∆H)4 , this particular 1
16
If one absorbs β and σ into a redefinition of the field φ = (βσ) 2 h, it is not even necessary to make this point anymore, and it also removes a lot of clutter. But I find myself resisting this, much like every statistical physicist resists using “natural thermal units”, i.e. kB = 1.
25
1 Interactions of rigid particles on a film b ˆab · r ˆbc =r cos ϕac
c rbc b ϕac
r ab a
b
b Figure 1.7: Illustration of the vertex angle ϕac .
4 combination of polarizabilities repeats 2,1,1 = 4!/2! 1! 1! times. Hence, using the Feynman rules we described earlier, we can write (1) 2 (2) (3) C1 C2 C2 4 12 21 12 βF [Fig. 1.6(a)] = − 2 Gijk Gjkl Glmnp G21 mnpi . 2! 1! 1! (2σ)4
(1.51)
The application of the Feynman rules is complete. But for illustration, let us further evaluate this interaction by substituting the polarizabilities and the propagator: βF [Fig. 1.6(a)] = −6
R14 R210 . 14 r12
(1.52)
It is beside the point here to discuss how to perform the index contraction. Appendix B.1 shows how it basically does itself if it is written in complex coordinates. Note that we could predict this up to the prefactor by power-counting. The triplet interaction depicted in Fig. 1.6(b) is the part of the fourth cumulant that is (1) (2) (3) (1) proportional to C1 C2 C2 C3 , so no vertex repeats more than once; pv = 1. One then finds (1)
(2)
(3)
(1)
C1 C2 C2 C3 4 12 23 32 2 Gijk Gjkl Glmnp G21 mnpi (2σ)4 2 10 2 2 R1 R2 R3 =12 cos ϕ13 , 7 r7 r12 23
βF [Fig. 1.6(b)] = −
(1.53)
2 is the angle between r − r and r − r , or the exterior angle formed at r where ϕ13 2 1 3 2 2 when successively traversing the points r 1 , r 2 , r 3 (see Fig. 1.7). Again, the scaling in R/r could be seen before evaluation. The quadruplet interaction of Fig. 1.6(c) should be straightforward at this point: (1)
(2)
(3)
(1)
C1 C2 C3 C4 4 12 23 34 2 Gijk Gjkl Glmnp G41 mnpi (2σ)4 R12 R24 R36 R42 2 3 = − 12 cos 3ϕ41 + 4ϕ14 3 r3 r4 r4 . r12 24 43 31
βF [Fig. 1.6(c)] = −
26
(1.54)
1 Interactions of rigid particles on a film (3)
C2
(1)
C1
C2
C1
(1)
C2
(2)
C2
(3)
C1
(1)
C2
(2)
C1
(1)
C1
(1)
(1)
Figure 1.8: Diagrams of the fluctuation-induced pair interaction of order R8 /r8 . All diagrams have a total of 8 derivatives acting on the links.
Hopefully, this section illustrated the Feynman diagrams and power counting adequately. One can probably see at this point that when it comes to high order cumulants and multibodies, it is not the evaluation of the diagrams that challenges one the most, but the bookkeeping of all the diagrams that contribute to a specific multibody at one or more orders in R/r consistent with the accuracy one is seeking.
1.3.1 Pair interactions In the previous section, we computed the leading pair interaction, namely the dipole-dipole interaction, and the dipole-quadrupole correction [see Fig. 1.5 and Eq. (1.50)]. We can proceed similarly to obtain further corrections to increase the accuracy of the interaction free energy between two rigid disks at finite separation. The next order is R8 /r8 and to achieve that, we need diagrams with a total of 8 derivatives at the vertices. These are shown in Fig. 1.8. Note that the fourth cumulant begins to contribute at this order. We evaluate the diagrams, in the order shown, employing the Feynman rules as (1)
(3)
(3)
(1)
C1 C2 C1 C2 21 21 2 G12 2 G12 ijkl Gjkli − ijkl Glijk 2 (2σ) (2σ)2 (1) 2 (1) 2 (2) (2) C1 C2 C1 C2 12 21 21 12 21 − 2 Gijkl Gklij − 24 G12 ij Gjk Gkl Gli . (2σ)2 2! 2! (2σ)4
βF (8) = −
(1.55)
Reading off the polarizabilities from Eq. (1.35) and the propagator products from Eq. (B.11), we have βF (8) = − 3
R16 R22 R14 R24 R14 R24 6R12 R26 + 19R14 R24 + 6R16 R22 R12 R26 − 3 − 9 − = − . (1.56) r8 r8 r8 2r8 2r8
Let us compute one more order for illustration purposes. Diagrams of the order R10 /r10
27
1 Interactions of rigid particles on a film (4)
C2
(1)
C1
C2
C1
(1)
C2
(3)
C2
(4)
C1
(2)
C2
(2)
C1
(1)
C2
(3)
C1
(2)
C2
(1)
C1
(1)
C2
(1)
C1
(1)
(1)
C1
(2)
Figure 1.9: Diagrams of the fluctuation-induced pair interaction of order R10 /r10 .
...
Figure 1.10: Some unlabeled pair interaction diagrams. These are connections between vertices, as many as the cumulant order, distributed along the worldlines of the two particles involved. Notice that there is only one way to connect the second and fourth cumulant interactions, whereas with six or more vertices, there are more possibilities.
are shown in Fig. 1.9 and they can be similarly evaluated as βF
(10)
(1)
(4)
(4)
(1)
(2)
(3)
(3)
(2)
C1 C2 C C 21 21 2 G12 = − 1 22 2 G12 ijklm Gjklmi − ijklm Gmijkl (2σ) (2σ)2 C1 C2 C1 C2 21 21 (1.57) 2 G12 2 G12 ijklm Gklmij − ijklm Glmijk 2 (2σ) (2σ)2 (1) 2 (1) (2) (1) 2 (1) (2) C2 C2 4 12 21 12 21 C1 C1 C1 4 12 21 12 21 C2 − 2 Gijk Gjkl Glm Gmi − 2 Gij Gjkl Gklm Gmi , 2! (2σ)4 2! (2σ)4 −
which yields R18 R22 R14 R26 R16 R24 R14 R26 R16 R24 R12 R28 − 4 − 24 − 24 − 2 − 2 r10 r10 r10 r10 r10 r10 2 8 4 6 6 4 8 2 4R1 R2 + 26R1 R2 + 26R1 R2 + 4R1 R2 =− . r10
βF (10) = − 4
(1.58)
One could go on, and it is quite straightforward (but not effortless, of course) to do so. In what follows, we will see that writing the complete asymptotic series for the interaction between two rigid disks is not all that difficult. The complete pair interaction Pair interactions are given by those terms in Eq. (1.38) that mix only a pair of particle labels, say a = 1 and a = 2, and are self-energy-free. These are diagrams with two
28
1 Interactions of rigid particles on a film worldlines. A few of them are depicted in Fig. 1.10. Since self-links are not allowed, no odd-numbered cumulant contributes, i.e. it is not possible to draw diagrams similar to those of Fig. 1.10 with an odd number of vertices distributed along two worldlines. Stitches. Another feature that Fig. 1.10 displays is that beginning with the sixth cumulant, there is not only one way to connect the vertices; more that one Wick contraction, each with a multiplicity of 2q , are possible. However, this does not pose a difficulty because the multipole orders at the vertices are summed over. Take the two sixth cumulant diagrams in Fig. 1.10 for example. Explicitly written, their contribution to the free energy is as follows: (ℓ )
(ℓ )
(ℓ )
C2 2 C2 4 C2 6 =
−1 X 3! 3!
, and
(1.59a)
.
(1.59b)
ℓ1 ,...,ℓ6 (ℓ )
(ℓ )
(ℓ )
C1 1 C1 3 C1 5 (ℓ )
(ℓ )
(ℓ )
C2 2 C2 6 C2 4 =
−1 X 3! 3!
ℓ1 ,...,ℓ6 (ℓ )
(ℓ )
(ℓ )
C1 1 C1 3 C1 5
Note that the Feynman rule about multinomial factors was already applied here, although partially; the pair (1,2) occurs 6!/3! 3! times in (β∆H)6 . That takes care of the 6-fold sum over particles but for this section, we will leave the 6-fold sum over multipole orders intact and write it explicitly. Any remaining multinomial occurrences are thereby contained, and the only numerical factor left from the Feynman rules to worry about is the Wick multiplicity. The point here is, the right-hand sides of Eqs. (1.59a) and (1.59b) are identical term by term. Hence, these different “stitches” do not evaluate differently. After one is considered, the rest merely result in a multiplicative factor: how many of them there are altogether. Fig. 1.11 depicts all the different stitchings of the sixth cumulant pair interactions. A straightforward counting exercise shows, at cumulant order q = 2s, the number of stitches is given by gs ≡ s! (s − 1)!/2 for s > 2 and g1 = 1. As a result, the pair free energy
29
1 Interactions of rigid particles on a film
Figure 1.11: All the g3 = 3! 2!/2 = 6 distinct Wick contractions of six-vertex pair diagrams are shown.
can be written as (ℓ )
(ℓ )
C2 2
C2 2 βF = −
g1 X 1! 1!
−
ℓ1 ,ℓ2
(ℓ )
g2 X 2! 2!
(ℓ )
C1 1
ℓ1 ,ℓ2 , ℓ3 ,ℓ4
(ℓ )
C2 4
− (ℓ )
C1 1
(ℓ )
(ℓ )
C2 2 C2 4 C2 6 g3 X 3! 3!
ℓ1 ,ℓ2 , ℓ3 ,...
(ℓ )
C1 3
+ ... (ℓ )
(ℓ )
(1.60)
(ℓ )
C1 1 C1 3 C1 5
As far as the evaluation of the diagrams is concerned, the propagator product represented by the links is the intimidating part. One has an arbitrary number of partial derivatives spread over an arbitrary number of Green functions to contract. This is a daunting task in Cartesian coordinates, to say the least, but fortunately not in complex coordinates (see Appendix B.1). We derive Eq. (B.11) for any propagator product that will be encountered in the cumulant expansion, even for multibody interactions. Using the formula (B.11) for the propagator product, Eq. (1.35) for the polarizabilities, gs = s! (s − 1)!/2 and 22s for the Wick multiplicities,17 the pair free energy (1.60) can be recast as βF = −
∞ 2s 2(ℓ +ℓ +...) 2(ℓ2 +ℓ4 +...) X 1 X Y (ℓi + ℓi+1 − 1)! R 1 3 R 1
s=1
s
{ℓ} i=1
ℓi+1 ! (ℓi − 1)!
2
r2(ℓ1 +ℓ2 +ℓ3 +...)
(1.61)
with ℓ2s+1 ≡ ℓ1 . The multipole sums are over ℓi > 1 when the disks are fixed horizontally, whereas if they are free to tilt, they range over ℓi > 2. We presented this full asymptotic series in Ref. [YRD12]. For horizontally fixed disks of identical radii (R1 = R2 = R) one obtains −βF =
1 4 31 60 697 900 13955 40612 262966 + 6 + 8 + 10 + 12 + 14 + + + + . . . , (1.62) 4 x x 2x x 3x x 4x16 3x18 5x20
17
Neither 22s nor s! (s − 1)!/2 is the correct expression for the Wick multiplicity or gs , respectively, when s = 1, but these oddities of the second cumulant cancel out when the numbers are multiplied: 22 is twice the correct Wick multiplicity, whereas the number of stitches 1! 0!/2 is half of what it should be.
30
1 Interactions of rigid particles on a film
10
r0.05(P )/R − 2
102
|f (r)| [kB T /R]
101 100
1 8.2 P
0.1
10−1
45
10−2
10
20
50
100
200
P
10−3 10−4 10−5
2
3
4 r/R
5
6
7
8
9 10
Figure 1.12: Comparison of truncated expansion (solid curve) to numerical solution of Ref. [LO07] (dotted curve). The magnitude of the force f = −∂F/∂r is plotted. The dashed line is the lowest order (dipole-dipole) interaction and it can be seen to be inadequate when r . 4. Shown in the inset is how the maximum power of R/r included in a truncated expansion of βF depends on the (dimensionless) surface-to-surface separation d = r/R − 2 where the truncated series is off by 5%.
where x is defined as x = r/R. To highlight the efficiency of computations within this formalism, it might be worthwhile to note that the first five terms were calculated without computer aid. That said, Eq. (1.62) is very easily put on a computer to calculate higher orders. In Fig. 1.12, we plot the magnitude of the force f (r) = −∂F/∂r, truncated after O(R71 /r71 ) (solid curve). The dashed line represents the leading order (dipole-dipole) interaction. The dotted curve is based on the data of Lehle and Oettel [LO07], which they obtained by having a computer calculate the partition function numerically by discretizing the functional integrals. Our truncated asymptotic series is, by definition, as close to the correct answer as the condition R ≪ r is good. Therefore it does not correctly capture the near-contact (r ≈ 2R) behavior when it is truncated. The inset to Fig. 1.12 demonstrates how far one needs to carry out the asymptotic expansion before truncating it so that it is still adequate at close separations. Plotted is the (unitless) surface-to-surface distance d = r/R − 2 where a truncated expansion of βF (r) is off from the correct value by 5%, against the highest power of r−1 included in the expansion of βF . As closer separations are considered, a trend
31
1 Interactions of rigid particles on a film of P ≈ 8.2/d0.05 is observed, implying for example that accuracy within 5% is achieved by truncating βF after O(R82 /r82 ) when the edges of the two disks are separated by a tenth of their radii. Monopoles. Recall that we excluded induced monopoles from our theory and computation on the grounds that, with the assumed surface Hamiltonian, those cost infinite energy, hence are impossible. One way to regularize this divergence of the monopole energy, hence allowing them, is to consider a “damped” theory, `a la Yukawa, and take the undamped limit after computing the interaction. We do this in Appendix E and find the same monopole-monopole interaction predicted by Lehle and Oettel [LO07] via the same regularization. Near-contact asymptotics In problems where the field equation, the boundary conditions etc. are more complicated than those considered in this chapter, one generally does not expect to calculate the cumulant expansion in its entirety. However, in this specific case, it was possible to obtain an exact and complete expansion: Eq. (1.61). While finding a closed form for the whole series, or at least the expansion coefficients, is a difficult task, considerable simplifications turn out to be possible. The 2s-fold sum over multipoles of Eq. (1.61) has the form of a so-called binomial cycle [Rio68] and it is possible to perform all but one of the sums adapting from Ref. [Rio68] (see Appendix D). Defining u = R2 /r2 for the case of identical radii, we show in Appendix D that Eq. (1.61) reduces to ∞ ∞ X X 2k f2s−2 (u) u2s+k F −k, 1; −2k; βF = − (1.63) f2s−1 (u) k sf2s−1 (u)f2s−2 (u) s=1 k=0
where F(a, b; c; x) is the hypergeometric function [AS74] and the polynomials fq (u) obey [c.f. Eqs. (D.23) and (D.24)]
fq (u) =
⌋ ⌊ q+1 2
X i=0
q+1−i χq+2 − 1 (−u)i = 2 i (χ − 1)(χ2 + 1)q
(1.64)
√ where ⌊·⌋ denotes integer part and χ(u) = (1 − 1 − 4u − 2u)/2u. This already amounts to a big practical improvement over Eq. (1.61), because Eq. (1.63) can be much more efficiently expanded in powers of inverse distance. Proximity force approximation. The fluctuation-induced force between two very closely spaced objects can be computed to lowest order using the Derjaguin or proximity force approximation (PFA) [Der34]. This method relies on the notion that, when two surfaces are
32
1 Interactions of rigid particles on a film very close, they can be approximated as a series of “parallel plates”. Using the fluctuationinduced force between two infinite parallel lines [LK91, LK92], it was found that the leading short distance behavior of the pair interaction is βF = −
π2 √ 24 d
(1.65)
in Ref. [LO07], with the dimensionless surface-to-surface separation d = (r − 2R)/R. By the help of Eq. (1.63), we are able to extract this divergence ∼ d−1/2 near contact. For arbitrarily short separations, both summations in Eq. (1.63) need to be performed up to infinity. Since different cumulant orders differ in terms of the contained powers of inverse distance at long separations (small powers of r−1 ), we expect that the leading divergence will be caused by the k → ∞ tail of the inner sum. In this limit, the coefficients of the series expansion of the hypergeometric function F(−k, 1; −2k; x) are easily checked to approach the coefficients of (1 − x/2)−1 . That is, x −1 . lim F(−k, 1; −2k; x) = 1 − k→∞ 2
(1.66)
Thanks to the closed-form expression for fq (u) above, one can show that in the interval 0 < u < 1/4 (i.e. from asymptotically separated to osculating disks, recalling u = R2 /r2 ), the ratio f2s−2 (u)/f2s−1 (u) < 2, which is the condition for the hypergeometric function in Eq. (1.63) to converge as k → ∞. One can then substitute Eq. (1.66) to isolate the leading divergence in each cumulant of order q = 2s in Eq. (1.63) as the k → ∞ tail of the inner sum and obtain the resulting divergence in βF as P ∞ 2k k X u2s ∞ k=0 k u . (1.67) βF∞ ∼ − f2s−2 (u) s=1 sf2s−2 (u) f2s−1 (u) − 2
The symbol “∼” reminds us that this only captures the leading order asymptotic divergence. P 2n n − 12 After noting that ∞ , and the surface-to-surface separation d n=0 n u = (1 − 4u) 2 between the disks satisfies u = 1/(d + 2) , we will re-express Eq. (1.67) for d ≪ 1 using Eq. (1.64) in order to obtain the leading divergence in βF∞ : ∞
π2 1 X 1 ζ(2) √ √ , = − βF∞ ∼ − √ = − 4 d s=1 s2 4 d 24 d
(1.68)
in agreement with Eq. (1.65). The crudeness of isolating the divergent part only allows one to obtain the leading order proximity asymptotics. In a recent preprint [Rot11], Rothstein used conformal field theory to exactly compute the partition function of a bosonic field on a plane with two holes.
33
1 Interactions of rigid particles on a film This is isomorphic to the problem of two disks on a film studied in this chapter, and the interaction can in fact be written in closed form (it involves a Dedekind η-function). Its expansion at large distances reproduces our EFT series (1.61), while an expansion at contact leads to 4d 96 − π 2 √ 1 π2 βF = − √ + log 2 − d + O(d) . (1.69) π 576 24 d 4
1.3.2 Multibody interactions We discussed earlier that not all forces are pairwise additive, and hence in the presence of more than one particles, the total force cannot be written as a sum over all pair forces. Examples of interaction terms that were triplet- and quadruplet-wise additive were encountered when we evaluated the diagrams of Fig. 1.6. In this section, we will compute the leading order of these interactions. Subleading orders can be worked out, as necessary, the way described for the interactions of Fig. 1.6, however accounting all N -body diagrams for N > 2 is far more tedious an exercise. In practice, one must determine by power-counting those interactions that will contribute up to the desired accuracy level and work them out. Odd cumulants. At this point, it may not be difficult to guess that the leading order triplet interaction is among three induced dipoles and is therefore proportional to −2 −2 −2 23 31 G12 ij Gjk Gki ∼ r12 r23 r31 . However, the prefactor of this interaction turns out to be identically zero. The reason is that whenever there is an odd number of Green functions with their indices contracted with those of the adjacent Green functions, no nonzero terms are left. This is most easily seen in complex coordinates and keeping with the custom of not using them in the main text in this chapter, the reader is referred to Appendix B.1. The end result is that there is no (fluctuation-induced) interaction originating from odd cumulant orders. Before moving on, note that this is a property of the harmonic Green function for the surface considered in this chapter and it will not hold for other surfaces. Also note that if the particles were anisotropic, the polarizabilities would be tensors and the index contractions would involve them as well as the propagators, and this property would not hold. Leading triplet interaction With the odd cumulants out of the equation, we look for the leading triplet interaction in the fourth cumulant. The diagrams are shown in Fig. 1.13(a) depicting interactions between dipole moments. It is easily seen that the three diagrams are just cyclic permutations of each other in terms of particle label. Applying the Feynman rules, one finds lead βFtri
(1) (1) 2 (1) C1 C2 C3 4 12 23 32 21 =− 2 Gij Gjk Gkl Gli + cyc. perm. of (123) . 1! 2! 1! (2σ)4
34
(1.70)
1 Interactions of rigid particles on a film (1)
(1)
(1)
(1)
C3
C3
C3
C3 (1)
(1)
C2
C2 (1)
C2
(1)
C2 (1)
(1)
C1
(1)
(1)
C1
C1
C1 (a)
(1)
(1)
C4
(1)
C4
(1)
C4
(1)
C3
(1)
C3
C3
(1)
(1)
C2
(1)
C2
(1)
(1)
C1
C2 (1)
C1
C1 (b)
Figure 1.13: (a) Leading triplet interaction diagrams. (b) Leading quadruplet interaction diagrams.
The propagator product can still be done by hand, or one may use Eq. (B.11), and after substituting the polarizabilities (1.35) as well, we obtain lead βFtri =−
R12 R24 R32 R12 R22 R34 R14 R22 R32 − 4 4 − 4 4 . 4 r4 r12 r23 r31 r31 r12 23
(1.71)
This interaction is always (i.e., regardless of the geometrical arrangement of the particles) attractive, in the sense that decreasing distances lowers the free energy. Curiously, it has no dependence on the angles of the triangle (except implicitly through the side lengths rab ). However, we know this to be a special case, since we worked out a higher order correction as an example [see Fig. 1.6(b) and Eq. (1.53)], which did not display this behavior. Leading quadruplet interaction The fourth cumulant is also where the leading quadruplet interaction resides, within the diagrams of Fig. 1.13(b). The resulting interaction is similarly found to be (1)
(1)
(1)
(1)
C1 C2 C3 C4 4 12 23 34 41 24 43 31 13 32 24 41 2 Gij Gjk Gkl Gli + G12 ij Gjk Gkl Gli + Gij Gjk Gkl Gli 4 (2σ) " 2 − ϕ4 + ϕ3 − ϕ1 ) 2 − ϕ3 + ϕ4 − ϕ1 ) cos(ϕ14 cos(ϕ13 23 41 32 24 31 42 + = − 2R12 R22 R32 R42 2 r2 r2 r2 2 r2 r2 r2 r12 r24 23 34 41 43 31 12 # 3 − ϕ2 + ϕ4 − ϕ1 ) cos(ϕ12 34 21 43 . (1.72) + 2 r2 r2 r2 r32 24 41 13
quad βFlead =−
35
1 Interactions of rigid particles on a film Note that this interaction depends on 42 = 6 different distances and 43 × 3 = 12 different angles,18 and therefore it is not easy to map out the dependence exactly or find the maximally attractive/repulsive configuration(s). However, we can see that the arguments of the cosines can all be made zero for collinear and rectangular arrangements of particles, where the vertex angles are either obvious or cancel in pairs. Thus, such arrangements are maximally attractive.
1.4 The interaction free energy: elastic In the previous section, we computed interactions initiated by thermal fluctuations. All the sources involved were induced sources, and thus the surface was fluctuating around a ground state which is flat. In this section, we will consider deformed ground states. We showed in Section 1.1.2 how contact line irregularities deform the film, acting as permanent sources. This type of situation naturally arises from an irregular three-phase contact line between a colloid and the fluid-fluid interface it is trapped at [SDJ00, KDD01] (and Refs. [vNSH05, LAZY05, LCJB+ 10] for anisotropic objects). In a somewhat loose analogy with electrostatics in the presence of conducting spheres, these interactions could be viewed as charged conductors interacting with their images across the other conductors, and the images of the images, etc. (at increasing cumulant order). In the macroscopic world, it is actually possible to observe bits of our favorite morning cereal deform the surface of milk and aggregate.19 We showed earlier [see Section 1.2.1 and Eq. (1.36)] that permanent sources at a point r = r a can be captured in our effective theory as a collection of linear sources X (ℓ) Qa,i1 ...iℓ hi1 ...iℓ (r a ) . (1.73) ∆Haperm = − ℓ
The number of multipoles necessary to include depend on how the contact line is actually deformed. In the diagrams, these terms will be vertices where only one link may be attached. Therefore, since diagrams must be connected, Wick contractions must begin and end at these vertices whenever they are involved, and thus there may either be two or none (this would be the previous case of fluctuation-induced interactions) involved. As a result, the diagrams for the ground-state interactions of this section are open, with permanent sources at the ends (see Fig. 1.14 for examples). Feynman rules. The modification of the Feynman rules for the addition of these onelegged vertices is quite straightforward. Obviously, each of the 2 such vertices out of q 18
Three angles for each triangle that can possibly be drawn with the corners at the particles. The well-known Cheerios effect [VM05] is somewhat inappropriate here, since it stems from a uniform contact-line rise (or depression depending on the balance between the weight of the particles and its wetting by the fluid). Such a monopole deformation, as mentioned before, requires an infinite amount of energy under our assumption of a purely tension-dominated film. 19
36
1 Interactions of rigid particles on a film (2)
Q2
(2)
(2)
Q2
(2)
Q2
Q3
(2)
prtcl. 1 + l prtcl. 2 (2)
Q1
(a)
(2)
C2
cyc. + perm. (123)
(2)
C1
Q1 (b)
(c)
Figure 1.14: (a) The second cumulant interaction between two permanent quadrupoles. (b) The third cumulant pair interaction of one permanent quadrupole with the quadrupole source it induces on another particle. (c) The third cumulant triplet interaction of one permanent quadrupole with the quadrupole on another particle which was induced by yet another permanent quadrupole.
contribute a factor βQa,i1 ...iℓ . And since there are q − 2 two-legged vertices, the Wick multiplicities for these open diagrams are given by 2q−2 . Power-counting. With one less link, i.e. a factor of β −1 σ −1 ∂ n G, among the same number of vertices (= power of β) compared to the fluctuation induced diagrams, the ground state interactions will amount to a free energy βF that scales as βσL2 , or F ∼ σL2 , where L is a typical length that describes the strength of permanent deformations. That is, this part of the interaction shows no dependence on temperature, but on the “elasticity” of the medium; this is the reason we refer to these as elastic interactions. Given that the fluctuation-induced interactions of the previous section scaled as F ∼ kB T , we have a total free energy of the form F = a0 + a1 T , which allows us to identify a1 = ∂F/∂T = −S and a0 = E; entropy and energy. We will therefore denote the elastic interactions of this section with the symbol E, rather than F . Furthermore, since the fluctuation-induced interactions are pure entropy, we will refer to them as entropic in the next chapter. Note that this simple breakdown into an entropy linear in T and temperature-independent energy would not be possible in the presence of higher-than-quadratic terms in the theory.
1.4.1 Interaction between permanent quadrupoles In this section, we will take up the example considered by Stamou et al. [SDJ00], a saddleshaped (quadrupole) contact line. This is the lowest relevant multipole in the contact line irregularities of a colloid under zero vertical force and horizontal torque at an interface; the freedom of the particle to move up and down as well as tilt prevents such deformations from occurring. Therefore, the lowest multipole order for induced sources in this problem is also ℓ = 2.
37
1 Interactions of rigid particles on a film
α
x
Figure 1.15: A saddle-shaped particle is depicted, where its principal axis of positive curvature determines its orientation angle α. (2)
We have already worked out Qij in Section 1.2.1 [see Eq. (1.28)] for a quadrupolar contact line of shape hcl (R, ϕ) = ηR2 cos(2ϕ − 2α) (see Fig. 1.15) around the center of the particle: cos 2α sin 2α (2) 4 . (1.74) Qij = πR ση sin 2α − cos 2α The leading interaction originates from the second cumulant as a direct interaction between the two permanent quadrupoles, Fig. 1.14(a). The changes to the Feynman rules in the presence of one-legged vertices are obvious and this interaction is easily found as η1R14 R24 η2 1 (2) (2) , (1.75) E (4) = − Q1,ij (r 1 )G12 ijkl Q2,kl (r 2 ) = −12πσ cos(2α1 + 2α2 ) σ r4 where the pair was assumed to lie on the x axis to declutter the expression. The same potential was found by Stamou et al. [SDJ00]. The functional dependence of the interaction implies both an attractive force and vertical torques on the saddle-like particles. Maximal attraction occurs when the argument of the cosine is zero, i.e. when α1 = −α2 (mod π). Accordingly, they may align their principal axes with the joining line but this situation is a set of measure zero among all maximally attracting orientation possibilities. We have seen earlier that, between colloids that are free to tilt and fluctuate vertically, the leading fluctuation-induced interaction is of order (R/r)8 , which can potentially only compete with the interaction (1.75) at close separations.
38
1 Interactions of rigid particles on a film Technical Note 1.5: Preferred backgrounds for contact line multipoles (ℓ)
There is a somewhat useful interpretation of the linear worldline terms, −Qi1 ...iℓ hi1 ...iℓ , as follows. Going back to Section 1.1.2, we see that for each possible permanent deformation, there is a choice of background that induces the exact opposite. We can call this the “preferred background” p(ℓ) (r) for that multipole, which satisfies hbg (R, ϕ) = hcl (R, ϕ). With p(ℓ) (r) defined as such, it is possible to combine the linear and quadratic worldline terms h i2 1 (ℓ) 2 1 (ℓ) (ℓ) Ca hi1 ...iℓ (r a ) − Qa,i1 ...iℓ hi1 ...iℓ (r a ) = Ca(ℓ) hi1 ...iℓ (r a ) − pi1 ...iℓ (r a ) 2 2
up to an irrelevant constant. For high order multipoles, this interpretation of the permanent sources (ℓ) makes fixing the Wilson coefficients Qi1 ...iℓ a trivial matter.
On superposition. Although we have restricted to a toy problem where the only permanent deformation is quadrupolar, it should be obvious how one would handle higher order multipoles of the contact line irregularity (see Technical Note 1.5) for accuracy at closer separations. However, one should note that the true interaction potential is not merely a superposition of direct interactions between permanent sources, similar to Fig. 1.14(a), but includes scatterings from induced sources (image charges in the loose electrostatic analogy), such as Fig. 1.14(b). The former is equivalent to assuming the total shape of the surface is given by a superposition of the deformations caused by each particle independently of every other, i.e. neglecting their boundary conditions, which is why it misses induced sources. Such a superposition approximation is captured by the second cumulant only, and holds up to the order the next cumulant begins to contribute. The first interaction where a superposition approximation fails is of the order (R/r)8 , where the third cumulant contributes the diagram in Fig. 1.14(b). This evaluates to the repulsion (2)
E (8) =
4 4 C1 (2) 21 (2) 12 2 4 2 4 R1 R2 Q G G Q + {1 ↔ 2} = 18πσ(η R + η R ) , 1 1 2 2 2! 1! σ 2 2,ij ijkl klmn 2,mn r8
(1.76)
which is the lowest order pair interaction that involves a scattering from an induced source. Curiously, it has nothing to do with how the saddles are oriented. Elastic vs. entropic. We have seen that the strength of these elastic interactionspis given by σ(ηR2 )2 , while that of entropic interactions was ∼ kB T . The length scale kB T /σ determines the amplitude, ηR2 , of the contact undulation where the two interactions are comparable. at room temperature, this amplitude is found p For a typical fluid-air interface −10 20 −21 −2 10 Nm/ (10 N/m) ∼ 10 m. Deviating from flatness by only about the to be 20
At room temperature (300K), kB T ≈ 4 pN·nm. The tension of a typical fluid-air interface is a few 10 N/m near room temperature [Dea98], with the ubiquitous water-air interface, in particluar, having a tension of 72 N/m at 25◦ C. −2
39
1 Interactions of rigid particles on a film size of an atom, this is the most perfectly-flat contact conceivable. Consequently, the competition between elastic and entropic interactions at a fluid interface is heavily biased in favor of the elastic part under most circumstances unless the temperature is high or the surface tension is exceptionally low. In the latter case though, surface fluctuations may be too large to be suitably described by a linear theory. However, incorporating nonlinearities is conceivable once the formalism is general and transparent, which EFT is. Also note that the next chapter on interactions between membrane inclusions deals with a situation where the competition between the elastic and thermal energy scales in the problem is on more even footing. Multibody interactions When induced sources are involved, multibody interactions enter the picture as well. At order (R/r)8 one observes the first multibody interaction, which is depicted in Fig. 1.14(c). This triplet interaction is computed as (8)
(2)
Etri =
C2 (2) (2) Q G12 G23 Q + cyc. perm. of (123) σ 2 1,ij ijkl klmn 3,mn cyc. (123)
=36πσ
X η R4 R4 R4 η a a b c c b cos(4ϕac − 2αa + 2αc ) , 4 r4 rab bc
(1.77)
a,b,c
which can be attractive or repulsive depending on the geometry of the triangle formed by the particles as well as the orientation of the saddles. For example, in an equilateral configuration with the convex principal axes of the saddles aligned with the bisector at b = 2π/3 at each vertex and the orientation angles, α = π/6, each vertex, one has ϕac 1 α2 = −π/6, α3 = π/2. Then, the triplet interaction turns attractive as η1 η2 η2 η3 η3 η1 (8) 4 4 4 Etri = −18πσR1 R2 R3 . (1.78) 4 r4 + r4 r4 + r4 r4 r23 31 31 12 12 23 An interesting situation with this interaction occurs when the particles are on the same b = 0 (mod 2π) and the torques on the particles will tend to make each pair line. Then 4ϕac of saddles 90◦ off. However, this is not possible; the system is frustrated. When two pairs are off by 90◦ , the remaining pair have to be parallel. Interesting though it may be, higher order interactions will likely “smooth out” this oddity. The O(R8 /r8 ) interactions demonstrate that, if the particles possess higher multipole order contact line irregularities, merely taking these into account within a superposition approximation does not capture every possible correction. Contributions such as Figs. 1.14(b) and 1.14(c) will have been forsaken right from the beginning.
40
2 Interactions of rigid particles on a membrane In this chapter, we will investigate interactions on a different type of surface; one whose behavior is determined not by tension but by bending elasticity. We refer to such a surface as a membrane since biological membranes constitute the typical example. A lipid membrane is a fluid bilayer of surfactant molecules (lipids) interfacing two aqueous environments [Saf94, NPW89, LS95]. Dissolved in this bilayer, or attached to it, there exist inhomogeneities; various phases of the lipid mixture, macromolecules such as proteins serving as channels, receptors, etc. [AJL+ 83, BEM91] The lateral distribution of the inhomogeneities is of interest for membrane-remodeling processes (endocytosis, exocytosis, tubulation) where the cooperative action of multiple inclusions is believed to be required [TSHDC99, FRT+ 01, IER+ 05, VPS+ 06, MG05, RIH+ 07, Koz07, AG09]. One possible form of interaction between inhomogeneities that may affect the lateral distribution is the surface-mediated interactions we take up in this thesis. As before, we consider hard inclusions of finite size and circular footprint on the membrane. We will follow a similar route to the previous chapter on interactions on a film: we begin with describing the surface and the effect of the particles, after which we proceed with deriving the correct effective theory and computing fluctuation-induced (entropic) as well as elastic interactions. Unlike the previous surface choice, though, relevant elastic energy scales are not enormous compared to kB T . Hence the two interactions compete on more even ground.
2.1 System and effective theory 2.1.1 Surface energetics of the membrane The energy functional of a membrane was first given by Helfrich [Hel73] as Z hκ i Hmem = dA (K − K0 )2 + κ ¯ KG + σ . 2
(2.1)
The bending rigidity κ measures the quadratic energy cost of the total extrinsic curvature K of the membrane to vary from its preferred value K0 . For simplicity, we will restrict to the case of zero spontaneous curvature K0 . Gaussian curvature KG is similarly penalized by the modulus κ ¯ . Due to the Gauss-Bonnet theorem (see Technical Note 2.1), this term is a constant for our problem and hence will be dropped. Finally, we will alsopassume that the tension σ of the membrane is negligible. Equivalently, the length ξ −1 = κ/σ is 41
2 Interactions of rigid particles on a membrane Technical Note 2.1: Gauss-Bonnet theorem For a multiply connected and open surface with no handles, the Gauss-Bonnet theorem states that [Kre91] Z Z dA KG = 2π(1 − m) − ds Kg , ∂S
S
where KG is the Gaussian curvature of the surface S, Kg the geodesic curvature of the curve(s) ∂S and m the number of disconnected boundaries S has except for the outer boundary. For the case of rigid inclusion boundaries as we have in this chapter, the geodesic curvature of the boundaries (the curvature towards a normal vector to the curve that is parallel to the surface) does not change, meaning the right hand side of the equation is constant. Therefore, the integral of the Gaussian curvature is constant.
assumed to be by far the largest length in the problem. After these simplifications we are left with Z κ (2.2) Hmem = dA K 2 . 2 Typical values for the bending rigidity κ of a lipid bilayer are around 20kB T near room temperature [LS95]. We will again forgo the covariant expression in favor of the Monge parameterization (normal displacement h(x, y) about a flat base plane xy) and also assume a nearly-flat surface such that |∇h| ≪ 1. In the Monge gauge, the curvature tensor Kij —whose trace is the total extrinsic curvature K by definition—is given by1 ! ∂j h . (2.3) Kij = ∂i p 1 + (∇h)2 With the near-flatness assumption applied as well, we have Z 2 1 Hmem [h] = κ d2 r ∇2 h . 2 S
(2.4)
Denoted by S is the domain of the base plane where the membrane has uniform bending rigidity κ, i.e. it is free of inclusions. As kinks in the surface are punished by an infinite energy penalty, the field h is subject to continuity conditions on its normal gradient and itself at the boundaries of S between the membrane and the inclusions. Equilibrium field configurations are stationary points of the Hamiltonian (2.4) consistent with the boundary conditions prescribed on the boundaries. A small variation of the field gives Z (2.5) δHmem [h] = κ d2 r ∇2 h∇2 δh + O(δh2 ) . S
1 Please note that Eq. (2.3) is not a proper tensor equation. It merely gives the elements of the covariant tensor K ij = g ik Kkj .
42
2 Interactions of rigid particles on a membrane Since ∇2 h∇2 δh = ∇ · (∇δh∇2 h) − ∇ · (δh∇∇2 h) + δh∇4 h, one has Z Z Z 2 2 4 ˆ · ∇δh∇2 h . ˆ · ∇∇ h + κ dℓ n dℓ δh n δHmem [h] = κ d r δh∇ h − κ
(2.6)
∂S
∂S
S
Stationarity, δHmem [h] = 0, of an equilibrium shape h(r) requires the above to vanish point-wise, due to arbitrariness of the variation δh(r). Hence, from the bulk term we obtain the Euler-Lagrange equation as ∇4 h = 0 .
(2.7)
The boundary terms vanish for boundary conditions that prescribe the value of h(r ∈ ∂S) ˆ · ∇δh(r ∈ ∂S) (which is the relevant case, as we will soon see). For other boundary and n conditions, the idea is the same but there are subtleties involved [CH53].
2.1.2 Boundary conditions In the film problem, the particles prescribed what the surface height may look like at their rim. Along with regular behavior at infinity, this information was sufficient for a well-posed boundary value problem. Here, the field equation is of higher order and the situation is slightly different, which is what we will discuss here. In addition to the continuity of height across the rim, continuity of slope will enter the boundary conditions (the Hamiltonian (2.4) does not allow kinks). In a sense, the shape of the particle does not matter just at the contact line but also in an infinitesimal neighborhood toward the interior. More quantitatively, the shape around the contact line can again be written in the form of a Taylor expansion (we restate Eq. (1.7) here for convenience): (1)
(2)
hcl (r) = η (0) + ηi ri + ηij ri rj + . . .
(2.8)
but with the predication that it is valid in an infinitesimal but big enough annular region receding from the boundary such that a normal gradient can be taken. The continuity of this normal gradient will impose a “contact angle”. Two particles that look different, but similar enough (near their boundaries) that they impose the same contact height and angle on the surface are considered equivalent. The role of each coefficient η (ℓ) is the same as before: η (0) and η (1) encode the rigid motions (height and tilt) whereas η (>2) capture irregularities of the contact height/angle. If all η (>2) are zero, then, in a small neighborhood of its rim, the particle is within the (1) plane z = η (0) +ηi ri . Note that, we will be interested in this flat case and also a particular (2) curved case where only ηij 6= 0 and η (>3) = 0, even though Eq. (2.8) represents the most general boundary condition. Finally, the boundary condition at infinity is a bit different in the case of a membrane. We will see that, even if we require square-integrability [in the sense of finite energy (2.4)], the general solution to the biharmonic equation ∇4 h = 0 contains functions that grow as r → ∞. These violate the very reasonable expectation that a disturbance localized to a
43
2 Interactions of rigid particles on a membrane finite region should not be felt infinitely far away from it. For eliminating such components, we require the asymptotic flatness condition, ∇h(r) → 0 as r → ∞, in line with most of the literature that deals with such systems. Response of a single particle We will now do the same exercise we did for the film, namely, solving the field equation around an isolated (circular) particle in a deformed background. As was the case before, the results of this exercise will be utilized later in the chapter. Let us begin by stating the most general solution to the biharmonic equation ∇4 h = 0 in polar coordinates. Since this is a bit of a crowded expression, it is more economical to write it exploiting complex numbers. We have, n ˜1 r ln k1 r + a ˜′1 r3 eiϕ h(r) = a0 ln k0 r+˜ a′0 r2 ln k˜0 r + Re a1 r−1 + a +
∞ X ℓ=2
o aℓ r−ℓ + a ˜′ℓ rℓ + a′ℓ r−ℓ+2 + a ˜′ℓ rℓ+2 eiℓϕ .
(2.9)
The coefficients with index > 1 are all complex, to concisely account for any phase angles. We stated the most general solution (2.9) for completeness. We are only interested in the part of it that makes sense in an unbounded domain (∇h(r) → 0 for r → ∞ as mentioned above). All the offending terms are marked by a coefficient that carries a tilde. Let us rewrite the relevant part of the solution for clarity: resp
h
n
(r) = a0 ln k0 r + Re a1 r
−1 iϕ
e +
∞ X ℓ=2
o aℓ r−ℓ + a′ℓ r−ℓ+2 eiℓϕ .
(2.10)
Note that the terms with unprimed coefficients are harmonic functions.2 Our task is to apply a background, i.e. hout = hbg + hresp , and fix the coefficients by enforcing the boundary conditions hbg (R, ϕ) + hresp (R, ϕ) = hcl (R, ϕ) ,
(2.11a)
∂r hbg (R, ϕ) + ∂r hresp (R, ϕ) = ∂r hcl (R, ϕ) .
(2.11b)
In the vicinity of the circular contact curve r = Rˆ r , we rewrite the contact height (2.8) as hcl (r) = Re
(
∞ X
ηℓ rℓ + ηℓ′ rℓ+2 eiℓϕ
ℓ=0
2
)
.
(2.12)
Here, k0 is an annoying relic of the fact that there is some arbitrary choice of units we have to set for the lengths, as well as the even more annoying fact that a vertical shift in coordinates is equivalent to it. In the next section, with our choice of Green function, we will be effectively choosing that as well. So, consider it as a constant.
44
2 Interactions of rigid particles on a membrane Technical Note 2.2: Boundary conditions for the disk-on-membrane BVP Here are the boundary conditions hresp (R) = hcl (R) − hbg (R) and ∂r hresp (R) = ∂r hcl (R) − ∂r hbg (R), respectively: ℓ=0:
a0 ln k0 R = η0 + η0′ R2 − b0 − b′0 R2
a0 R−1 = 2η0′ R − 2b′0 R , ℓ=1:
a1 R−1 = η1 R + η1′ R3 − b1 R − b′1 R3
−a1 R−2 = η1 + 3η1′ R2 − b1 − 3b′1 R2 , ℓ>2:
aℓ R−ℓ + a′ℓ R−ℓ+2 = ηℓ Rℓ + ηℓ′ Rℓ+2 − bℓ Rℓ − b′ℓ Rℓ+2
−ℓaℓ R−ℓ−1 + (−ℓ + 2)a′ℓ R−ℓ+1 = ℓηℓ Rℓ−1 + (ℓ + 2)ηℓ′ Rℓ+1 − ℓbℓ Rℓ−1 − (ℓ + 2)b′ℓ Rℓ+1 . Recall that all the coefficients with index larger than zero are complex. Both the real and imaginary parts of the equations should hold, corresponding respectively to the cosine and sine parts of Eqs. (2.10), (2.12) and (2.13).
Only two radial functions suffice, as this function need only specify the height and the slope of the field independently, and not more. (Also, note that the rigid height and tilt fluctuations of the particle are encoded in the coefficients η0 and η1 , respectively.) Similarly, the background can be chosen as (∞ ) X bg ℓ ′ ℓ+2 iℓϕ h (r) = Re bℓ r + bℓ r e . (2.13) ℓ=0
We can now begin to solve the boundary value problem. The boundary conditions (2.11a) and (2.11b) as applied to hresp (r), hbg (r) and hcl (r) are collected in Technical Note 2.2 For ℓ = 0, we find that a0 = 2R2 (η0′ − b′0 ). We also see that, since the only undetermined coefficient is a0 , the system of two equations is over-determined unless something is a free parameter; that would be η0 . In other words, since we have no right to forbid backgrounds of the sort b0 + b′0 r2 (from thermal fluctuations or other inclusions), we need the vertical motion of the particle to be unconstrained. Next, the condition for ℓ = 1 tells us that a1 = R4 (b′1 − η1′ ) and similarly that we need η1 (this time, the tilting degree of freedom) to be unconstrained if we want a well-posed problem. Finally, the ℓ > 2 equations yield aℓ = − (ℓ − 1)R2ℓ (ηℓ − bℓ ) − ℓR2ℓ+2 (ηℓ′ − b′ℓ ) a′ℓ =ℓR2ℓ−2 (ηℓ − bℓ ) + (ℓ + 1)R2ℓ (ηℓ′ − b′ℓ ) .
and
(2.14a) (2.14b)
The resulting response can be found by plugging these coefficients into Eq. (2.10). Clearly, it consists of two pieces as before: a permanent deformation and an induced response,
45
2 Interactions of rigid particles on a membrane given as 2 ′ hresp perm (r) =2R η0 ln er + Re
+
n
− R4 η1′ r−1 eiϕ
∞ h i X (1 − ℓ)R2ℓ ηℓ − ℓR2ℓ+2 ηℓ′ r−ℓ eiℓϕ
(2.15)
ℓ=2
∞ h i o X ℓR2ℓ−2 ηℓ + (ℓ + 1)R2ℓ ηℓ′ r−ℓ+2 eiℓϕ , + ℓ=2
and n 2 ′ (r) = − 2R b ln er + Re R4 b′1 r−1 eiϕ hresp 0 ind + +
∞ h X
ℓ=2 ∞ h X ℓ=2
i (ℓ − 1)R2ℓ bℓ + ℓR2ℓ+2 b′ℓ r−ℓ eiℓϕ
(2.16)
i o −ℓR2ℓ−2 bℓ − (ℓ + 1)R2ℓ b′ℓ r−ℓ+2 eiℓϕ .
Recall that the coefficients are complex to account for phase angles. Moreover, note that we set k0 = e in the argument of the logarithm. This is for compatibility with later expressions (also see footnote 2).
2.1.3 Effective theory We will neither try to motivate why there would be surface-mediated interactions between inclusions, nor the essential philosophy of effective field theory here. All of the analogous discussion of the previous chapter (which, hopefully, was satisfactory) is applicable to this case, save a number of technicalities. It is those technicalities what the rest of this chapter is for. We would like to write a worldline effective theory for a collection of circular inclusions on a membrane, analogous to Eq. (1.36), where the worldline terms (that we will soon find) will augment the free membrane Hamiltonian Z 1 H[h] = κ d2 r (∇2 h)2 . (2.17) 2 R2 The easiest route to take is pointing out the differences of this problem to the previous. Now, the Euler-Lagrange equation is the biharmonic equation ∇4 h = 0 instead of the harmonic equation ∇2 h = 0. Correspondingly, the Green function that embodies how deformations propagate on this surface is the inverse of the operator ∇4 , G(r, r ′ ) =
1 (r − r ′ )2 log(r − r ′ )2 , 16π
46
(2.18)
2 Interactions of rigid particles on a membrane such that ∇4 G(r, r ′ ) = δ(r − r ′ ).3 We know that when we form our effective theory in terms of localized derivatives of the field, this encodes propagations from the inclusions proportional to a derivative of the Green function. We can see from Eq. (2.10) that the physically allowed propagations from a single inclusion involve at least two derivatives of the Green function (2.18). Therefore, there will be no worldline terms that involve less than two derivatives on the field. Another disparity from the film problem due to the change of field equation is the criterion for redundant terms. Recall that we did not include any terms involving hii in our effective theory, since ∂i ∂i was the Euler-Lagrange derivative for that problem and we argued that there is no use crowding the theory with such terms. Here, ∂i ∂i ∂j ∂j is the Euler-Lagrange derivative, and terms involving hiijj are redundant, but those that involve hii are meaningful. This increases the number of terms we need to write for the membrane, even though the symmetries of each particle are the same. All considered, the entire series of quadratic worldline terms (induced sources) for a disk at the origin turn out to be ∞
∆HO =
1 X (ℓ) 2 ˚(ℓ) hi ...i hkki ...i + ˚ ˚(ℓ) h2 C hi1 ...iℓ + C C , 1 kki1 ...iℓ−2 1 ℓ−1 ℓ−1 2
(2.19)
ℓ=2
where we have omitted the particle label and also the arguments (0) of the fields, as well as labeled ∆H with a subscript to denote that it is the derivative expansion of one particle at the origin.4 Note that the number of “ring”s over the polarizability symbol is the same as the number of Laplacians that occur in its associated term. We will defer dealing with linear terms (permanent sources) until we compute interactions that involve them. In the presence of a source distribution ρ(r), the Euler-Lagrange equation analogous to that in Technical Note 1.2 is ∇4 h = ρ/κ. Under an applied deformation, h → hbg + h, the terms in ∆H give rise to the induced sources [see Eqs. (1.30-1.31)] ρ(r) = (−)ℓ
∞ h X ℓ=2
+
− C (ℓ) hbg i1 ...iℓ (0)∂i1 . . . ∂iℓ +
˚(ℓ) bg C h (0) ∂k ∂k ∂i1 . . . ∂iℓ−1 2 i1 ...iℓ−1
(2.20)
i ˚(ℓ) bg C δ(r) . hkki1 ...iℓ−1 (0)∂i1 . . . ∂iℓ−1 − ˚ C˚(ℓ) hbg (0) ∂ ∂ ∂ . . . ∂ i i k k 1 ℓ−2 kki1 ...iℓ−2 2
The convolution of these sources with the Green function then yields the field that emanates
3 There is some arbitrariness in choosing G(r, r ′ ). One can add a term proportional to r2 and the new function still satisfies ∇4 G(r, r ′ ) = δ(r − r ′ ). Note that such a shift in the Green function can equivalently be seen as a scale introduced in the argument of the logarithm. Therefore, this choice of Green function is akin to a choice of length units where you are allowed to put (r − r ′ )2 in the logarithm’s argument by itself. 4 ˚(2) = 0. Due to insufficient derivatives on the field, a nonzero term C ˚(2) hi hijj is not We also set C ˚(2) = 0 or separate the C (2) and ˚ ˚(2) terms from the sum in Eq. (2.19). allowed. We either had to set C C
47
2 Interactions of rigid particles on a membrane from the induced sources: Z 1 d2 r′ ρ(r ′ )G(r ′ , r) h(r) = κ ∞ h (ℓ) X ˚(ℓ) bg C C =− hbg h (0)Gkki1 ...iℓ−1 (0, r) i1 ...iℓ (0)Gi1 ...iℓ (0, r) + κ 2κ i1 ...iℓ−1 +
ℓ=2 ˚(ℓ) C
2κ
hbg kki1 ...iℓ−1 (0)Gi1 ...iℓ−1 (0, r) +
(2.21)
i ˚ C˚(ℓ) bg hkki1 ...iℓ−2 (0) Gkki1 ...iℓ−2 (0, r) . κ
Apparently, the fourth order Euler-Lagrange equation of the membrane is going to make things a little crowded; and that includes the effective theory. In the film case, using Cartesian coordinates did not really impede our calculations. We only needed to convert into complex coordinates to finally evaluate things. Here, it turns out to really pay off to rewrite the effective theory in complex coordinates to begin with. (Appendix B describes the underlying mathematics and applies it to derivations that were used in the previous chapter.) This is easily done thanks to the simple form of the metric (B.2) to find, ∞
∆HO =
1 X (ℓ) ℓ ¯ℓ Γ ∂ h∂ h +˚ Γ(ℓ) ∂ ℓ−1 h ∂¯ℓ ∂h 2 ℓ=2
¯ +˚ ¯ ∂¯ℓ−1 ∂h . ˚(ℓ) ∂ ℓ−1 ∂h +˚ Γ(ℓ) ∂¯ℓ−1 h ∂ ℓ ∂h Γ
(2.22)
Note that 4∂ ∂¯ = g ij ∂i ∂j = ∇2 .5 The new polarizabilities are still real, and can be expressed as a linear combination of the old ones (but this will not be needed). In the same way, the induced response (2.21) becomes ∞
−1 X h (ℓ) ℓ bg ¯ℓ h(z) = Γ ∂ h (0)∂ G(0, z) + ˚ Γ(ℓ) ∂ ℓ−1 hbg (0)∂¯ℓ ∂G(0, z) 2κ
(2.23)
ℓ=2
i ¯ bg (0) ∂¯ℓ−1 G(0, z) + ˚ ¯ bg (0) ∂¯ℓ−1 ∂G(0, z) + c. c. , ˚(ℓ) ∂ ℓ−1 ∂h +˚ Γ(ℓ) ∂ ℓ ∂h Γ
where “c. c.” denotes complex conjugate. Matching
The reason why we wrote down the induced response (2.23) is of course to fix the polarizabilities by comparing to the full theory response. We have already computed the latter, Eq. (2.16) , in the previous section as an exercise. The background that we applied then can be recast in the following more convenient form for our current purpose: ∞ 1 X ℓ h (z) = bℓ z + b′ℓ z ℓ+1 z¯ + c. c. 2 bg
ℓ=0
5
A tip of the hat to Robert Haussmann for the ∂, ∂¯ notation.
48
(2.24)
2 Interactions of rigid particles on a membrane Technical Note 2.3: Derivatives of the biharmonic Green function For reference, we present the derivatives of the biharmonic Green function, G(z, z ′ ) =
1 (z − z ′ )(¯ z − z¯′ ) ln[(z − z ′ )(¯ z − z¯′ )] , 16π
that are encountered in the computations here: 1 ln e2 (z − z ′ )(¯ z − z¯′ ) , 16π (n − 2)! z¯′ − z¯ ∂nG = (n > 2) , 16π (z ′ − z)n−1 −1 ¯ = (n − 3)! (n > 3) . ∂ n−1 ∂G 16π (z ′ − z)n−2 ¯ = ∂ ∂G
and, of course, their complex conjugates. One should watch out for negative signs when differentiating with respect to the primed coordinate.
The induced response (2.23) can now be rewritten explicitly by plugging in this background (derivatives of the Green function are compiled in Technical Point2.3). One obtains,6 1 z 1 ˚(2) 1 1 (2) Γ b2 − ˚ ln(e2 z z¯) Γ b0 2κ 16π z¯ 2κ 16π ∞ 1 X h (ℓ) bℓ ℓ! (ℓ − 2)! z bℓ−1 (ℓ − 1)! (ℓ − 2)! −1 − Γ +˚ Γ(ℓ) 2κ 2 16π z¯ℓ−1 2 16π z¯ℓ−1 ℓ=3 b′ (ℓ − 1)! (ℓ − 3)! −1 i b′ ℓ! (ℓ − 3)! z ˚ ˚(ℓ) ℓ−2 + c. c. + Γ +˚ Γ(ℓ) ℓ−1 2 16π z¯ℓ−2 2 16π z¯ℓ−2
h(z) = −
(2.25)
After noting z = reiϕ and some rearranging of terms, this response can be written in the same form as Eq. (2.16), for straightforward matching: ( ˚ ˚(2) ˚ ˚(3) Γ Γ ′ h(z) = − b0 ln er + Re b′ r−1 eiϕ 8πκ 16πκ 1 " # ∞ ˚(ℓ+1) X ˚ ˚(ℓ+2) Γ Γ ′ + (2.26) ℓ!(ℓ − 1)!bℓ + (ℓ + 1)!(ℓ − 1)!bℓ r−ℓ eiℓϕ 32πκ 32πκ ℓ=2 " # ) ∞ X −˚ Γ(ℓ+1) −Γ(ℓ) ℓ!(ℓ − 2)!bℓ + (ℓ + 1)!(ℓ − 2)!b′ℓ r−ℓ+2 eiℓϕ . + 32πκ 32πκ ℓ=2
6
¯ bg . Consider yourself warned. It is easy to lose a factor of 2 when taking ∂ ∂h
49
2 Interactions of rigid particles on a membrane Comparison to Eq. (2.16), reproduced below for convenience, n 2 ′ hresp (r) = − 2R b ln er + Re R4 b′1 r−1 eiϕ 0 ind + +
∞ h X
ℓ=2 ∞ h X ℓ=2
i (ℓ − 1)R2ℓ bℓ + ℓR2ℓ+2 b′ℓ r−ℓ eiℓϕ
(2.27)
i o −ℓR2ℓ−2 bℓ − (ℓ + 1)R2ℓ b′ℓ r−ℓ+2 eiℓϕ ,
determines the complete set of polarizabilities to be Γ(ℓ) =
32πκR2ℓ−2 (ℓ − 1)!(ℓ − 2)!
,
˚ Γ(ℓ) = (ℓ − 2)Γ(ℓ)
˚(ℓ) = (ℓ − 2)2 Γ(ℓ) and ˚ Γ
(2.28)
˚(2) = Γ(2) /2. for ℓ > 2 with the exception of ˚ Γ Thus, a free membrane theory augmented by a derivative expansion (2.22) at each particle’s worldline, with the polarizabilities as given above, is the effective theory for a membrane occupied by several rigid disks as described in the previous section (with η (>2) = 0).7 We can now go on with computing the free energy (or any other observable) of the system.
2.2 Entropic interactions Recall that in the previous chapter we identified the fluctuation-induced interactions to be purely entropic. Therefore we refer, from now on, to these interactions as such. Here, we will right away dive into drawing and evaluating diagrams, since it is by now familiar business. However, there will be certain differences. Let us begin discussing these by considering an example.
2.2.1 Diagrams and rules Fig. 2.1 shows some terms involved in a triplet interaction originating from the fourth cumulant. The first diagram is not to be evaluated but holds a symbolical meaning: it represents all diagrams between the particular multipole orders and connection topology indicated. The fact that there are many such diagrams is simply due to the fact that the derivative expansion (2.22) contains four different vertices per multipole order.8 The right hand side of the equal sign shows a few combinations. (Don’t mind the doublelinks and rings yet.) Even though we have not written down the Feynman rules yet, the 7 Of course, as previously done, this theory is missing its counterterms but we will not bother with writing them. We will just set any self-interaction diagram to zero, as explained in Technical Note 1.4 and Appendix C. 8 Multipole order and derivative order are used interchangeably in this thesis. People who are accustomed to the usage of the term multipole exclusively for angular dependence (which happens to be the case in electrostatics) may prefer the latter, but the author of this text is comfortable with it.
50
2 Interactions of rigid particles on a membrane (4)
(2) (2)
(4)
Γ3
(4)
(2)
=
Γ2 (2)
Γ2
(3)
Γ2 (2)
Γ2
(3)
(3) ˚ Γ1
(4)
Γ3
Γ2 (2)
Γ2
(3) ˚ Γ1
+
(4)
Γ3 ˚ ˚(2) Γ 2
(2)
Γ2 (2)
(2)
+
(4)
Γ3
Γ2
Γ3 (2)
+
Γ1
+
(4)
Γ3
(2)
Γ2
(3)
˚ ˚(3) Γ 1
Γ1
+
˚ ˚(2) Γ 2 (2)
Γ2
+ ···
(3) ˚ Γ1
Figure 2.1: The first diagram symbolically stands for all fourth cumulant triplet diagrams among multipole orders of 3, 2 and 4 on the worldlines r 1 , r 2 and r 3 , respectively. Shown to the right of the equal sign are a few of these diagrams. How the complete list is found, as well as what the double links and rings denote, is the subject of this subsection and explained in the main text.
power-counting is obvious: links count as a power of 2 (see Eq. (2.18) and/or Technical Note 2.3) of the separation between worldlines and each derivative acting on a link reduces the power by one. Therefore, all the diagrams in Fig. 2.1 are proportional to a total power 2(3 + 2 + 4 + 2) − 4 × 2 = 14 of inverse separation. Rings. The new features in the diagrams to be evaluated are the rings marking legs of the vertices and the double-links. Let us first talk about the rings. These mark Laplacians on the legs of the vertices. Recall that the worldline terms with single-“rung” polarizabilities ¯ acting on one of its field factors while a term with a doublehave a Laplacian (∼ ∂ ∂) rung polarizability has a Laplacian in each of its field factors. Especially for single-rung vertices, the ring is useful; it shows which leg is the one carrying the Laplacian. Not only does this aid the evaluation of diagrams but also helps identify diagrams that are not necessary to draw: Any diagram that connects two rung legs has a ∇4 (∼ ∂ 2 ∂¯2 ) acting on the propagator connecting them, whose very definition is ∇4 G(r, r ′ ) = 0 (for r 6= r ′ , which they are—recall that self-links are discarded). Therefore some diagrams containing connections between rung and double-rung vertices simply drop out. Double links. The next new feature is that there are two different kinds of links in the diagrams. The origin of this variety is the simple fact that the vertices in the derivative expansion (2.22) have unidentical legs: One field factor is mostly differentiated with respect
51
2 Interactions of rigid particles on a membrane ¯ (and, perhaps, one ∂). (Yes, to z (except, perhaps, one z¯) while the other has mostly ∂’s (2) ˚ term is an exception—and the only one—we will come to that.) We adopt the the ˚ Γ convention that a leg which carries mostly z derivatives is a “plain leg” while one that ¯ kill the propagator, carries mostly z¯ derivatives is a “double leg”. And since too many ∂ ∂’s plain and double legs are not to be linked.9 Hence the appearance of plain and double links. Also note that as a consequence of all this, the links alternate between plain and double at each vertex. ˚(2) vertex poses a slight exception: its legs are identical to each other Exception 1: The ˚ Γ ¯ (∂ ∂h). The most obvious consequence of this is that one can swap the connections at a ˚ ˚(2) vertex and achieve another diagram equal to the original. So, these special vertices Γ afford a Wick multiplicity of 2 each. Such a swap at any other vertex would make the new ˚(2) vertices at both diagram vanish due to the ∂-∂¯ alternations—unless it is connected to ˚ Γ ends. This is related to another exception of these vertices, which will be discussed shortly. Other contractions. Before going on to write down the Feynman rules that prescribe how to convert a diagram to an algebraic expression, we should discuss one more thing to make sure we draw all possible diagrams. Fig. 2.1 explicitly shows only one possible contraction among a given set of vertices: The connections begin with a plain link from the bottom vertex and proceed counter-clockwise in an alternating succession of double and plain links. Once the first link is made, the rest follows unambiguously (at least if ˚(2) vertices, as mentioned above). Of course, one can begin with a double there are no ˚ Γ link and achieve a different diagram [compare Figs. 2.2(a)–(b)]. Clearly, this diagram is the complex conjugate of the former, and ensures that the sum of all diagrams is real. In the film case, the vertices had identical legs, and the 2q ways of arranging the links (while conserving the vertex connectivity) were 2q identical Wick contractions. Here, there are only two ways to arrange the links and the results are complex conjugates of each other. ˚(2) vertices, explained below. That is, with the exception of ˚ Γ ¯ legs are of neither kind—plain or double—by themException 2: Note that the ∂ ∂h selves, but can act as either, depending what kind they are linked to (and these cannot be connected to their own kind because of too many Laplacians). That is, the link will ˚(2) leg is connected to the plain leg of another vertex, have all ∂’s (save one) when a ˚ Γ ¯ otherwise. In particular—and this is the root of the exception—both legs of and all ∂’s ˚(2) vertex can participate in the same kind of link, both plain or both double, the same ˚ Γ ˚(2) thereby breaking the alternating sequence of plain and double links. This oddity of ˚ Γ vertices may allow other Wick contractions (other, not perfectly alternating, plain-double sequences) into the diagram expansion. Figs. 2.2(c)–(d) show the two additional diagrams to those in Figs. 2.2(a)–(b) that must be considered due to this. One can see that if the ˚(2) vertex, the diagram will not close propplain-double alternation is broken at only one ˚ Γ
9 One may worry that this may fail if the involved vertices already have a low number of derivatives. This is a straightforward check; it does not fail.
52
2 Interactions of rigid particles on a membrane (4)
(4)
Γ3
˚ ˚(2) Γ 2 ˚ ˚(2) Γ 2
(4)
Γ3 ˚ ˚(2) Γ 2
˚ ˚(2) Γ 2 (3)
(4)
Γ3
Γ3 ˚ ˚(2) Γ 2
˚ ˚(2) Γ 2 (3)
˚ ˚(2) Γ 2 ˚ ˚(2) Γ 2
(3)
(3)
Γ1
Γ1
Γ1
Γ1
(a)
(b)
(c)
(d)
Figure 2.2: Different Wick contractions between a given set of vertices are shown. All these diagrams are part of the “. . . ” in Fig. 2.1. (a) This diagram is contracted in the same way as those explicitly shown on the right hand side of Fig. 2.1. (b) This diagram is another Wick contraction of the previous diagram, and clearly is the complex conjugate of it. It would be the other Wick contraction were it ˚(2) vertices. (c) This diagram shows another Wick contraction between these not for the “special” ˚ Γ ˚(2) vertices; notice how the links do not alternate vertices which is made possible by the presence of two ˚ Γ between plain and double at these vertices. (d) This is the complex conjugate to the previous diagram.
erly; a plain and double leg will end up being connected, which will kill the diagram.10 Therefore an even number (including zero) of these “alternation defects” is required for a nonzero contraction. If one has n such vertices where the alternation can break, and it must happen an even number of times, the total number of Wick contractions made ˚(2) vertices in this manner is the answer to this question: possible by the presence of ˚ Γ Given n marbles, how many handfuls of marbles consisting of an even number (including zero) can one pick? The straightforward answer (i.e. without bothering with potential simplifications) is n0 + n2 + n4 + . . . Feynman rules
Let us wrap up everything we said above: Given a particular connection between a particular collection of multipoles, we draw all diagrams having every possible polarizability of the indicated order at each vertex. This is equivalent to considering all possible collections of rings at the legs of every vertex. Any link is allowed at most one ring on it. One connects the vertices in an alternating succession of plain and double links and in the absence ˚(2) vertices, there are only two ways of doing this. The presence of ˚ ˚(2) allow more of ˚ Γ Γ plain-double successions (details above) and additionally afford a Wick multiplicity of 2 each, since connections can be flipped (details above). 10 Another way to look at it—and this may be how the reader interprets the diagrams in Fig. 2.2—is as follows: When a plain (double) link coming into one of these “odd” vertices also leaves the vertex plain (double), it means the legs of the next vertex have been swapped. (Normally, this results in too many ˚(2) vertex is involved, which has too few derivatives to begin with.) Laplacians on that link, but not if a ˚ Γ ˚(2) vertex is met Then the connections will have to alternate between plain and double, until another ˚ Γ where the broken alternation can be fixed and the diagram will close properly.
53
2 Interactions of rigid particles on a membrane (2)
(2) = (2)
˚ ˚(2) Γ 2
(2)
Γ2
Γ2 +
(2)
+ ˚ ˚(2) Γ 1
Γ1
(2)
Γ1
Figure 2.3: The quadrupole-quadrupole interaction between rigid membrane inclusions. Recall that ˚ ˚(2) vertices afford a multiplicity factor of 2 each, in the Γ(2) = 0 by construction. The presence of ˚ Γ diagrams they appear.
After one draws all the relevant diagrams, the following Feynman rules are applied to evaluate them: 1. Vertices Each vertex contributes a factor of (−β/2) times the polarizability labeling the vertex. Unrung and double-rung vertices of order ℓ carry ℓ derivatives on each leg, whereas a single-rung vertex of order ℓ has ℓ − 1 derivatives on one leg, and ℓ + 1 on the other (the one marked with a ring since this is the leg carrying a Laplacian). This is important for counting derivatives for the next step. 2. Links Links are propagators (βκ)−1 G(za , zb ) between the worldlines they connect, differentiated at each end by the derivatives carried by the legs; see above rule for counting. All the derivatives on a plain link without a ring are ∂’s while they are all ¯ on a double link. The presence of a ring on a plain link means all derivatives are ∂’s ¯ and the other way around for a double link with a ring. ∂’s except for a single ∂, 3. Numerical factors There are no Wick multiplicities other than theQexception of a factor ˚(2) vertex. The other numerical factor is simply −1/ of 2 per ˚ Γ v pv ! as before, pv is the number of times a vertex of type v is repeated in the diagram.
2.2.2 Pair interactions We are now ready to compute interaction free energies. We begin with the longestranged one, which was calculated via different approaches several times [GBP93a, GBP93b, GGK96a, GGK96b, PL96, DF99a, DF99b, HW01, BDF10]. We will then improve on this pair interaction by quite a few orders. Unfortunately, it was not possible to obtain a complete asymptotic expansion like in the film case. Nevertheless, there is a clear recipe to go up to arbitrary accuracy. In the following subsection, we present some multibody interactions. The leading pair interaction, which is the longest-ranged interaction in the system, takes place between two quadrupoles, as depicted in Fig. 2.3. Employing the Feynman rules
54
2 Interactions of rigid particles on a membrane (3)
(2)
(4)
(3)
(2)
(2)
(2)
(2)
(3)
(2)
(3)
(4)
(2)
(2)
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.4: Symbolic diagrams for the interactions of order r−6 and r−8 . (a)–(b) These are responsible for an interaction proportional to r−6 . (c)–(f) These are responsible for an interaction proportional to r−8 .
discussed above, we can easily evaluate this interaction as follows: (2) (2) ˚ ˚(2) Γ(2) Γ1 Γ2 4 Γ 4 1 2 ¯ ¯ ¯3 ∂ G(z , z ) ∂ G(z , z ) − 2 ∂ 3 ∂G(z 1 2 2 1 1 , z2 )∂ ∂G(z2 , z1 ) (2κ)2 (2κ)2 (2) ˚(2) Γ1 ˚ Γ2 3 ¯ −2 ∂ ∂G(z1 , z2 )∂¯3 ∂G(z2 , z1 ) . (2.29) (2κ)2
βF = −
Note the factors of 2 in the last two terms: they originate from the Wick multiplicity due ˚(2) vertices. The last step is just to read off the polarizabilities from to the presence of ˚ Γ Eq. (2.28) and take the derivatives11 (see Technical Note 2.3). Let us show the calculation explicitly, as this is our first time: 32πR12 16πR22 −1 −1 32πR12 32πR22 2! z¯ 2! z − 4 16πz 3 16π¯ z3 2 16πz 2 16π¯ z2 2 2 2 2 16πR1 32πR2 −1 −1 R R − = −6 1 4 2 . 2 2 2 16πz 16π¯ z |z|
βF (4) = −
(2.30)
We labeled the free energy with a superscript showing the power of r−1 (or |z|−1 ). As we mentioned before, we are not by a long shot the first to compute this asymptotic attraction between rigid disks on a membrane. But we are set to make more accurate predictions for finite separations. We will explicitly show the calculations for corrections of order R6 /r6 and R8 /r8 .12 Relevant diagrams (in symbolic form) are collected in Fig. 2.4. Further corrections were obtained by a computer following the same program; we will quote these after the calculations. 11
Two of the four derivatives are acting on the Green function at each position. No negative sign will be missed even if one computes all derivatives with respect to one argument of the Green function. But we remind that when the number of derivatives on one worldline is odd and the other even, a minus sign has to be placed by hand afterwards. √ 12 Recall that by R/r we mean all relevant parameters of the form Ra Rb /rab .
55
2 Interactions of rigid particles on a membrane (3)
(3)
(2)
Γ2 +
= (2)
Γ1
(3) ˚ Γ2
(3)
Γ2
(3) ˚ Γ2
+
+ (2)
˚ ˚(2) Γ 1
˚ ˚(3) Γ 2 +
(2)
Γ1
Γ1
(2)
Γ1
Figure 2.5: Here, the quadrupole-octupole interaction of Fig. 2.4(a) is shown in expanded form, used in Eq. (2.31) in the order shown here.
The interaction of order R6 /r6 originates from the diagrams in Figs. 2.4(a)–(b). The power-counting can easily be verified: 2(2 + 3) − 2 × 2 = 6. We will compute Fig. 2.4(a) which is “written out” in Fig. 2.5. Following the Feynman rules, we have (2) (3) (2) (3) ˚ ˚(2) Γ(3) Γ1 Γ2 5 ¯5 Γ Γ1 ˚ Γ2 5 ¯ ¯4 4 ¯ ¯4 1 2 Fig. 2.5 = − ∂ G∂ G − 2 ∂ ∂G∂ ∂G − ∂ ∂G∂ G (2.31) 2 2 (2κ) (2κ) (2κ)2 (3) (2) ˚(3) 2 4 ˚ ˚(2)˚ Γ Γ Γ ˚ Γ ¯ ∂¯4 ∂G = (−18 − 2 + 6 + 6 − 2) R1 R2 . − 1 22 ∂ 4 G∂¯5 ∂G − 1 22 ∂ 4 ∂G (2κ) (2κ) r6
The prefactors originating from each diagram in Fig. 2.5 were written separately, and in the order they appear in the figure, for clarity. Nonetheless, one can also see that some diagrams are repulsive, even though they are overwhelmed by the attractive ones. After adding Fig. 2.4(b), which is just the above with the particle labels switched, we have βF (6) = −10
R12 R24 + R14 R22 . r6
(2.32)
Since this correction involves scattering from higher-than-quadrupole induced sources, point-particle approaches that employ a pointwise localized curvature constraint [DF99a, DF99b, BF03, MM02] are bound to miss this term. This is evidenced by the claim in Refs. [DF99a, BDF10] that the next correction to the asymptotic result (2.30) scales as r−8 . The yet-shorter-ranged interaction of order R8 /r8 follows from Figs. 2.4(c)–(f). The diagrams comprising Figs. 2.4(c), (d) and (f) are shown in Figs. 2.6, 2.7 and 2.8, respectively. (Fig. 2.4(e) is just Fig. 2.4(c) with particle 1 and 2 swapped.) Since we have enough practice applying the Feynman rules, we will skip the evaluation steps and quote the final results. Fig. 2.4(c) (or Fig. 2.6) evaluates as Fig. 2.4(c) = (−48 − 3 + 24 + 24 − 12)
R12 R26 R2 R6 = −15 1 8 2 . 8 r r
(2.33)
The second-to-last step gives the value of each diagram in Fig. 2.6, in the order shown.
56
2 Interactions of rigid particles on a membrane (4)
Γ2
(4) ˚ Γ2
(2)
˚ ˚(2) Γ 1
Γ1
Γ2
Γ1
(4)
(2)
(4) ˚ Γ2
˚ ˚(4) Γ 2
(2)
(2)
Γ1
Γ1
Figure 2.6: The expansion of the quadrupole-hexadecapole interaction of Fig. 2.4(c), used in Eq. (2.33) in the order shown here.
From this result, we can also deduce that Fig. 2.4(e) = −15
R16 R22 . r8
(2.34)
The octupole-octupole interaction of Fig. 2.4(d) (or Fig. 2.7) is found to be Fig. 2.4(d) = (−144 + 36 + 36 − 9 + 36 − 9 + 36 − 9 − 9)
R14 R24 R14 R24 = −36 , r8 r8
(2.35)
while the fourth cumulant interaction among quadrupoles (Fig. 2.4(f) or Fig. 2.8) evaluates as R14 R24 1 1 1 1 R14 R24 = −18 . (2.36) Fig. 2.4(f) = 2 −4 − 1 − 1 − 1 − 1 − − − − 4 4 4 4 r8 r8 Shown inside the parentheses are the values for each diagram in the first two rows of Fig. 2.8. The factor of 2 outside accounts for the last two rows—the complex conjugates of the diagrams in the first row.13 We now have all the ingredients of the pair interaction βF (8) . Adding up Eqs. (2.33), (2.34), (2.35) and (2.36), we have βF (8) = −
15R12 R26 + 54R14 R24 + 15R16 R22 . r8
(2.37)
Notice that the precise dependence of the interaction on the particle radii follows from the polarizabilities’ dependence on them, but it cannot be simply guessed without properly treating the inclusions as finite-sized objects.
13
Each individual diagram is real valued for pair interactions. In multibody interactions, on the other hand, individual diagrams are complex valued and the Wick contractions that are their complex conjugates make the final result real.
57
2 Interactions of rigid particles on a membrane (3)
(3)
(3)
(3)
Γ2
Γ2
Γ2
(3) ˚ Γ2
(3)
(3) ˚ Γ1
(3) ˚ Γ1
˚ ˚(3) Γ 1
Γ1
Γ2
Γ1
(3)
(3) ˚ Γ2
(3) ˚ Γ2
(3) ˚ Γ2
˚ ˚(3) Γ 2
(3) ˚ Γ1
Γ1
(3)
(3) ˚ Γ1
Γ1
(3)
Figure 2.7: The expansion of the octupole-octupole interaction of Fig. 2.4(d), used in Eq. (2.35) in the order shown here.
Repulsion? In computing these interactions, we have encountered some repulsive diagrams. This is interesting since there was no chance of ending up with an extra minus sign in the film case, but it is evidently a possibility here. One may then ask, are there any orders that are repulsive? Or, is it possible to tweak the particle sizes to get repulsion? Unfortunately, we do not have a definite answer. All orders of the pair interaction we computed (some are presented below) are attractive. The only insight we have right now is that repulsive diagrams are those that contain an odd number of Laplacians (rings) on the links (this is how the derivatives of the Green function work out; see Technical Point 2.3). Saying anything conclusive about the sign of the resulting interaction to all orders requires one to understand the relative numbers and magnitudes of these diagrams to the rest.14 Computational results The increasing number of diagrams make it evident that it would be a tall order to go after a complete series expansion for the pair interaction like we did for the film case. There are two factors that make this a much tougher task: the set of worldline terms being more crowded and varied, and the fact that the derivatives of the Green function are not of just
14 A favorite argument for why entropic interactions should be attractive goes like this: When constraints are close, fluctuations are restricted in a more compact region than they would be if the constraints were far apart. Hence it is more preferable that the constraints are close. This would seem to hold regardless of what the kernel is—film, membrane, whatever—and mean that we won’t find repulsion. Even if that turns out to be true eventually, the reasoning is not infallible: We will see later that non-pairwise additive interactions do not necessarily prefer the particles to come closer.
58
2 Interactions of rigid particles on a membrane
Figure 2.8: The list of diagrams in the fourth cumulant pair interaction of Fig. 2.4(f). The polarizability symbols were omitted; they are all of order 2 and the rings at the vertices unambiguously indicate their type. The bottom two rows are the complex conjugates of those in the top two rows. Note the plainplain-double-double sequence (as opposed to plain-double alternation) in the last two diagrams in the ˚(2) vertices as mentioned before. second and fourth rows: These are possible due to the ˚ Γ
one functional form.15 But even if it is substantially time-consuming (and error-prone) for us to go after higher order corrections, it is not for a computer, because we precisely know what program to follow: • Draw pair interaction topologies with increasing vertex count. (We already did that for the film case—recall “stitches”.) • Put every conceivable combination of multipole orders at the vertices. For each combination, the power of 1/r is going to be twice the sum of multipole orders minus (twice) the number of links. • Mark the legs of the vertices with rings in every imaginable way, as long as there is at most one on any link (thereby tracing over all four vertex types in Eq. (2.22) at 15 The only way to see that writing the theory in complex coordinates is helpful is to actually try using real coordinates. It is a big mess...
59
2 Interactions of rigid particles on a membrane the same time). • Pick any one Wick contraction (single-double link sequence) and find the prefactor using the Feynman rules. • Since each individual Wick contraction is real in the case of pair interactions (because there is only one separation z and the real axis can be defined to include it), every other Wick contraction has the same value. So, the result of the previous step is ˚˚(2) multiplied by the count of Wick ncontractions. In the presence of n vertices of Γ n n n type, the number is 2 × 2 × 0 + 2 + 4 + . . . . The first factor of 2 is due to ˚(2) -related exceptions, as complex conjugates, while the others stem from the two ˚ Γ previously discussed. Following this recipe,16 we were able to obtain many higher order corrections to the pair interaction between rigid disks. After about 107 hours of computing on a regular desktop machine, the following terms of the series were found: 6R12 R22 10R14 R22 + 10R12 R24 15R16 R22 + 54R14 R24 + 15R12 R26 + + r4 r6 r8 10 2 8 8 2 6 4 20R1 R2 + 152R1 R2 + . . . 25R1 R2 + 327R1 R24 + 684R16 R26 + . . . + + r10 r12 12 2 10 4 8 6 30R1 R2 + 600R1 R2 + 2106R1 R2 + . . . + r14 14 2 12 35R1 R2 + 993R1 R24 + 5230R110 R26 + 8854R18 R28 + . . . + (2.38) r16 12 6 10 8 40R116 R22 + 1528R114 R24 + 33760 3 R1 R2 + 28840R1 R2 + . . . + r18 10 10 45R118 R22 + 2227R116 R24 + 21836R114 R26 + 78939R112 R28 + 598406 5 R1 R2 + . . . + r20 16 4 18 2 20 50R1 R2 + 3112R1 R2 + 39174R1 R26 + 190050R114 R28 + 406446R112 R210 + . . . + r22 24 24 +O(R /r ) .
−βF =
The ellipses in the numerators denote the remaining terms with uneven powers of the particle radii. Obviously, the prefactor of R1m R2n is equal to that of R1n R2m . Let us also state what this series looks like in the special case of R1 = R2 = R after we define x = r/R: −βF =
20 84 344 1388 5472 21370 249968 1628876 1277664 6 + 6 + 8 + 10 + 12 + 14 + 16 + + + . 4 x x x x x x x 3x18 5x20 x22
The magnitude of the force, f (r) = −∂F/∂r, derived from this truncated series is plotted in Fig. 2.9 as well as its leading asymptotic term for comparison. We observe that, when 16
In MATLAB.
60
2 Interactions of rigid particles on a membrane
102
|f (r)| [kB T /R]
101 100 10−1 10−2 10−3 10−4 2
3
4
5
6
7
8
9 10
r/R Figure 2.9: The magnitude of the pair force, f (r) = −∂F/∂r, between two identical (R1 = R2 = R) rigid disks derived from the truncated series (2.38) is plotted (solid line). For comparison, the leading order term is shown as well (dashed). The disks touch when r/R = 2, and smaller separations are inaccessible and meaningless (shown in gray).
the surface-to-surface separation between the disks is one radius√(i.e. when r = 3R), the asymptotic result underestimates the force by about a factor of 10 ≈ 3. The proximity force theorem [Der34] was used by Ref. [LZMP11] to find the leading behavior near contact, r ≈ 2R, as ∼ (r/R − 2)−3/2 for the force, or ∼ (r/R − 2)−1/2 for the potential, just like in the film case. Here, however, we were unable to retain a sufficient number of terms in the truncated free energy expansion to make this trend more appreciable to the eye.
2.2.3 Multibody interactions There are, of course, non-pairwise additive interactions present in the system if there are more than two particles on the surface. These interactions (and the pair interactions, just as well) can be viewed as the result of a thermally induced source polarizing another, and the field of that polarization inducing a source on yet another object, and so forth. As such, one can predict their dependence on the inter-particle separations based on the multipole orders involved; this is the power-counting we keep doing. But multibody interactions will also depend on (sometimes implicitly, but often explicitly) the angles present in the geometrical configuration of the particles. The dependence is in terms of simple trigonometric functions
61
2 Interactions of rigid particles on a membrane (2)
(2) (2)
(2)
(2) (2)
(2) (2)
(2)
(2) (2)
(a)
(2) (2)
(b)
(2) (2)
(c)
(d)
Figure 2.10: Symbolic diagrams for the leading entropic triplet and quadruplet interactions on a membrane. The lists of diagrams comprising (a) and (b) are displayed in Figs. 2.11 and 2.14, respectively. (2)
(2)
Γ3
(2)
˚ ˚(2) Γ 1
(2)
Γ1
(2)
(2)
˚ ˚(2) Γ 1
˚ ˚(2) Γ 3
Γ3
(2)
˚ ˚(2) Γ 2
Γ2
Γ2
Γ1
(2)
Γ3 (2)
˚ ˚(2) Γ 2
Γ2
(2)
˚ ˚(2) Γ 3
Γ3
(2)
Γ1
Γ2
(2)
Γ1
Figure 2.11: The expansion of the symbolic diagram of Fig. 2.10(a) is shown. Notice, that all six diagrams can be found by complex conjugation and cyclic permutations of any one of them.
as we will see. Given that an N -particle configuration involves N2 distances and 3 N3 angles (3 times the number of triangles), a full expansion of a general N -body interaction will not be very accessible. One should consider the distances involved in a given configuration and decide what the relevant interactions are. Nevertheless, we will compute the leading triplet and quadruplet interactions, as we did for rigid disks on a film. These are shown symbolically in Fig. 2.10. Contrary to the film case, derivatives of the Green function here can be multiplied (and the derivative indices contracted) an odd number of times and have a nonzero value. Indeed, the leading triplet interaction of Fig. 2.10(a) originates from the third cumulant as the number of vertices suggests. Leading triplet interaction When explicitly written, the symbolic diagram of Fig. 2.10(a) comprises those shown in Fig. 2.11. The diagrams are the same up to complex conjugation and cyclic permutations and the total interaction is therefore easily evaluated as lead βFtri
˚ ˚(2) Γ(2) Γ(2) Γ 3¯ ¯ ¯4 = 1 2 3 3 2 ∂ 3 ∂G(z 1 , z2 )∂ G(z2 , z3 )∂ ∂G(z3 , z1 ) + c. c. + cyc. (2κ)
(2.39)
˚(2) . We then plug in the polarizabilities Note the factor of 2 arising from the presence of a ˚ Γ from Eq. (2.28) and perform the derivatives. For the latter, we will use the variables defined
62
2 Interactions of rigid particles on a membrane
zb
c
=
rb
c
e iϕ
bc
c
b ϕac
zab = rab eiϕab a
b
b b is the difference . It is clearly seen that the angle ϕac Figure 2.12: Illustration of the vertex angle ϕac ϕbc − ϕab between the phases of the two complex numbers zbc and zab .
as follows: zb − za := zab := rab eiϕab . We then have, lead βFtri =2R12 R22 R32
z23 2 3 z2 z12 z¯23 31
+ c. c. + cyc. =
cyc. (123) R2 R2 R2 X b cos(2ϕac =4 21 22 23 r12 r23 r31 a,b,c
c − 2ϕba ).
cyc. (123) 2 2 2 R R R X cos(2ϕab 4 21 22 23 r12 r23 r31 a,b,c
− 4ϕbc + 2ϕca )
(2.40)
b = −ϕ + ϕ . It can be observed easily from Fig. 2.12 In the last step, we have used ϕac ab bc b that ϕac is the difference between the phase angles of the two complex numbers.
Odd cumulants. Even though odd cumulants do not vanish for the biharmonic propa˚(2) , links alternate gator, we can still observe a curious property. Recall that, except ˚ Γ between double and plain at each vertex. With an odd number of vertices, this will not ˚(2) vertices where the plain-double alternabe possible unless there is an odd number of ˚ Γ tion does not have to happen. Hence, diagrams from odd cumulants contribute, but their numbers are still severely reduced. b is constant (they add up to 2π), we can Since the sum of the three exterior angles ϕac illustrate the angular dependence of the leading triplet interaction (2.40) as a ternary plot (see Fig. 2.13). Repulsion and attraction for a triplet (and higher N -tuplets) is understood in the sense that the geometrical configuration of particles tends to “scale up” or “scale down”, respectively. Fig. 2.13 shows a maximal repulsion (dark spots of magnitude 12 in the plot) for an equilateral configuration of particles (center of plot) or a collinear arrangement (corners of plot). This demonstrates that fluctuation-induced forces are not always attractive. Consider, for example, isosceles triangles with apex angle 0 < α < π (this would be tracing either one of the three bisectors of the ternary plot). The triplet
63
12
π
0
2 Interactions of rigid particles on a membrane
2
21
ϕ 13
ϕ3
6
0
π
0 π
1 ϕ32
0
−6
P b c Figure 2.13: Ternary plot depicting the strength, 4 cyc. cos(2ϕac − 2ϕba ), of the leading entropic triplet interaction (2.40). Dark and bright correspond, respectively, to repulsion and attraction.
interaction then becomes lead βFtri =4
R12 R22 R32 2 r 2 r 2 [1 − cos(3α)] , r12 23 31
(2.41)
which is attractive in the regions 0 6 α < π/9 and 5π/9 < α < 7π/9, with the most attractive isosceles configuration having an apex angle of α = 2π/3. The strongest attraction, considering all possible triangles (i.e., the whitest spots on the ternary plot), is a very “narrow” triangle of internal angles 0, π/3 and 2π/3; two of the points are basically on top of each other. This is practically a pair configuration. Recall that in the previous subsection, we saw that pairwise interactions seemed to be exclusively attractive. It is interesting that a triplet finds its maximally attractive arrangement when it looks like a pair. Leading quadruplet interaction The leading quadruplet interactions are depicted symbolically in Figs. 2.10(b)–(d). We will evaluate Fig. 2.10(b) and obtain the rest by permuting the particle labels appropriately. The expansion of Fig. 2.10(b) is shown in Fig. 2.14. That is a lot of diagrams to evaluate. But they can be related by cyclic permutations thereby reducing necessary calculations: The first diagram is invariant under cyclic permutations, hence it can be written as 1/4 times a sum over all of its cyclic permutations. The remaining diagrams on the top row are already the four cyclic permutations of one diagram. The first diagram, when its four cyclic permutations are taken, repeats itself and the third diagram twice. So their sum can be written as 1/2 times a sum over the cyclic permutations of the first diagram. Same goes for the second and fourth diagrams. Finally,
64
2 Interactions of rigid particles on a membrane (2)
(2)
(2)
(2)
(2)
Γ2
(2)
˚ ˚(2) Γ 1
Γ1
(2)
(2) Γ2
˚ ˚(2) Γ 1
˚ ˚(2) Γ 4 (2)
(2)
Γ2 (2)
Γ1 ˚ ˚(2) Γ 4
(2)
Γ3
Γ3 ˚ ˚(2) Γ 2
(2) Γ2
(2)
˚ ˚(2) Γ 1
(2)
Γ2 Γ1
Γ4 ˚ ˚(2) Γ 3
(2)
Γ3
(2)
Γ1 (2)
Γ4 ˚ ˚(2) Γ 3
˚ ˚(2) Γ 2
(2)
Γ2
Γ4 ˚ ˚(2) Γ 3
Γ3
Γ3
˚ ˚(2) Γ 4
(2)
Γ4
Γ4 (2)
Γ3
(2)
(2)
Γ4
˚ ˚(2) Γ 2
and complex conj.
(2)
Γ1
Γ1
Figure 2.14: The diagrams that comprise the quadruplet interaction of Fig. 2.10(b) are shown. Note ˚(2) vertices that complex conjugation is for all diagrams, not only the second row. Also recall that ˚ Γ carry an additional factor of 2.
one can write the sum of all the diagrams in Fig. 2.14 as (2) (2) (2) (2)
1 Γ1 Γ2 Γ3 Γ4 4 ∂ G(z1 , z2 )∂¯4 G(z2 , z3 )∂ 4 G(z3 , z4 )∂¯4 G(z4 , z1 ) 4 (2κ)4 ˚ ˚(2) Γ(2) Γ(2) Γ(2) Γ 4 ¯ ¯4 ¯3 − 1 2 34 4 2∂ 3 ∂G(z 1 , z2 )∂ G(z2 , z3 )∂ G(z3 , z4 )∂ ∂G(z4 , z1 ) (2κ) ˚(2) Γ(2)˚ ˚(2) (2) Γ 1˚ 3¯ 1 2 Γ3 Γ4 ¯ ¯3 ¯3 − 4∂ 3 ∂G(z 1 , z2 )∂ ∂G(z2 , z3 )∂ ∂G(z3 , z4 )∂ ∂G(z4 , z1 ) (2.42) 2 (2κ)4 (2) ˚(2) (2)˚ ˚(2) Γ2 Γ3 Γ 1 Γ1 ˚ 3¯ 4 ¯ ¯3 ¯3 4∂ 3 ∂G(z − 1 , z2 )∂ ∂G(z2 , z3 )∂ ∂G(z3 , z4 )∂ ∂G(z4 , z1 ) 2 (2κ)4 + cyc. + c. c.
Fig. 2.14 = −
After inserting the polarizabilities and taking the derivatives, one has Fig. 2.14
R12 R22 R32 R42
z¯12 z23 z¯34 z41 z23 z¯34 −4 2 3 3 2 3 3 3 3 z12 z¯23 z34 z¯41 z12 z¯23 z34 z¯41 1 1 1 1 − 2 z 2 z2 z 2 − 2 z2 z 2 z 2 2 + cyc. + c. c. 2 z12 ¯23 ¯ ¯ 34 41 12 23 ¯34 z41
=−4
(2.43)
Note that we need to add the interactions of Figs. 2.10(c)–(d) to this. One can check that the permutations of (1234) afforded by the three diagrams Figs. 2.10(b)–(d) along with the cyclic permutations in their corresponding algebraic expressions [such as Eq. (2.42)]
65
2 Interactions of rigid particles on a membrane exhaust all possible permutations of (1234) up to an inversion (i.e. 1234 → 4321). However, since the links have no directionality, an inversion leaves the diagrams invariant. So we can introduce the “missing” inversions to the sum, so as to have a sum over all permutations of (1234), and divide the result by 2 to fix the over-counting introduced by adding the inversions. Hence, the leading quadrupole interaction can be written as the sum of all the terms that are shown explicitly in Eq. (2.43), and all of their permutations with respect to particle labels, and divided by 2, which is perm. (1234)
lead βFquad =−
1 X Ra2 Rb2 Rc2 Rd2 b d b d 2 r 2 r 2 r 2 8 cos(4ϕac + 4ϕca ) + 8 cos(4ϕac + 2ϕca ) 2 rab bc cd da a,b,c,d
b d b d + cos(2ϕac + 2ϕca ) + cos(2ϕac − 2ϕca ) .
(2.44)
We simplified this expression using the fact that the sum of the exterior angles of a polygon b = −ϕ + ϕ . is an integer multiple of 2π. Also recall that zab = rab eiϕab and ϕac ab bc 4 Unfortunately, with all the 3 3 = 12 angles in a four particle configuration, it is not easy to obtain a map of repulsive and attractive regions. Suffice it to say that in a regime where O(R8 /r8 ) pair interactions are non-negligible, chances are, this quadruplet interaction is relevant as well (at least if none of the separations rab is much larger/smaller than the rest). The computational efficiency of the EFT formalism is one of its highlights and this can be considered a demonstration.
2.3 Elastic interactions The interactions of the previous section were initiated by polarizations around the inclusions that were induced by the thermal undulations of the membrane. Here, we will consider interactions that are initiated by permanent sources; those present even in the absence of thermal fluctuations. These interactions have to do with the energy-minimizing shape of the surface, determined by its curvature-elastic properties; hence the name, as before. Examples to particles that enforce a permanent deformation on a membrane include BAR domain proteins such as amphiphysins and endophilin, or dynamin [ZK06], which are for instance involved in membrane tubulation [IER+ 05]. Membrane-mediated interactions between such curved inclusions have been investigated many times in the nearly-flat limit that we adopt here [GBP93a, KNO98, WKH98, DF99a, DF99b, MM02] as well as within a fully covariant geometrical framework [MDG05a, MDG05b]. As with the fluctuation-induced interactions, the elastic interactions have mostly been investigated at an asymptotically separated pair level, with the exception of Refs. [DF99b, KNO98]. We will present improvements and corrections to the results found in these references. Again, similarly to the case of interactions on a film, we note that permanent sources will have the form of linear terms in the effective theory, thereby giving rise to open interaction diagrams terminating at both ends with permanent sources (see Fig. 2.16, for example).
66
2 Interactions of rigid particles on a membrane
α
x
Figure 2.15: Two permanent sources of second derivative order are depicted. On the left is an axisymmetrically-curved (cup-like) particle, whereas on the right is a saddle-like particle.
Therefore, as far as dependence on the energy scales κ and kB T of the problem are concerned, the interaction free energy scales with one more power of κβ due to lack of one link compared to the entropic interaction diagrams with the same number of vertices. As a result, Felas ∼ κ while Fentr ∼ kB T , and accordingly, we will denote elastic interactions with E instead of F . We have already taken a thorough look at permanent deformations emanating from irregularly-shaped inclusions. The preferred shape of an inclusion near its contact line, Eq. (2.12), gave rise to the deformation (2.15). Here, we will specialize to two possible shapes (or a combination of them): cup-like and saddle-like inclusions (see Fig. 2.15). Let us see how we can describe such objects in our effective theory. The preferred shape of such particles is a special case of Eq. (2.12): hcl (z) = η0 + η0′ z z¯ +
1 η¯2 z 2 + η2 z¯2 , 2
(2.45)
where η0 and η0′ are real, and η2 is complex. We also remind that η0 encodes the free height of the particle, while the remaining coefficients are fixed numbers related to the rigid shape near the boundary. Eq. (2.15) gives the deformation that propagates from such a particle at the origin as η z¯ η¯ z R4 η η¯2 2 2 2 + − + . (2.46) h(z) = η0′ R2 ln(e2 z z¯) + R2 z z¯ 2 z¯2 z 2
We observe that we can rewrite this by employing the derivatives of the Green function as ¯ h(z) =16πR2 η0′ ∂ ∂G(0, z) + 16πR2 η2 ∂ 2 G(0, z) + η¯2 ∂¯2 G(0, z) ¯ + 8πR4 η2 ∂ 3 ∂G(0, z) + η¯2 ∂¯3 ∂G(0, z) . (2.47)
67
2 Interactions of rigid particles on a membrane Let us define the quantities ˚(2) = −16πR2 κη0′ Q
,
Q(2) = −16πR2 κη2
and
˚(4) = −8πR4 κη2 . Q
(2.48)
˚(2) is real.) Then, we can rewrite Eq. (2.47) as (Note that Q ˚(2) Q(2) 2 Q ¯ ∂ ∂G(0, z) − ∂ G(0, z) − κ κ ˚ ˚(4) ¯ (4) Q Q ¯ − ∂¯3 ∂G(0, z) ∂ 3 ∂G(0, z) − κ κ
h(z) = −
¯ (2) Q ∂¯2 G(0, z) κ .
(2.49)
We recognize these terms as those originating from point-like permanent sources at the origin, corresponding to the worldline terms perm ˚ ¯ − Q(2) ∂ 2 h − Q ¯ −Q ˚(2) ∂ ∂h ¯ (2) ∂¯2 h − Q ˚(4) ∂ 3 ∂h ¯ (4) ∂¯3 ∂h , = −Q ∆HO
(2.50)
where we have suppressed the argument (0) of the field instances, and labeled ∆H by an “O” for origin. You will notice that the sources Q marked by a ring carry a Laplacian acting on the field, in line with our earlier convention. One word regarding notation is in order. We will soon want to label the curvature constants η0′ and η2 with particle numbers. To avoid too many labels on these symbols, let us introduce J := 2η0′
and
Se2iα := 2η2 ,
(2.51)
where J and S are both real, in which case, the permanent sources (2.48) will become ˚(2) = −8πR2 κJ Q
,
Q(2) = −8πR2 κSe2iα
,
˚(4) = −4πR4 κSe2iα . Q
By writing down the curvature tensor ∂i ∂j hcl (0) of the preferred shape (2.45), J + S cos 2α S sin 2α hcl (0) = ij S sin 2α J − S cos 2α
(2.52)
(2.53)
one easily sees that J is its trace, and therefore the total extrinsic curvature. On the other hand, S is the magnitude of each eigenvalue of its traceless part, measuring the “saddleness” of the inclusion, and the angle α measures how much the convex-bent principal axis departs from the x axis. With the sources given in Eq. (2.52), Eq. (2.50) tells us how to extend the polarizable worldline terms (2.22) of a rigid disk to account for a cup- or saddle-like curvature, corresponding, respectively, to nonzero J and nonzero S. Now we can proceed with computing the interactions initiated by these permanent deformations.
68
2 Interactions of rigid particles on a membrane (2)
(4)
(2)
(2)
(2)
(4)
(2)
(a)
(b)
(c)
(d)
(2)
(2)
(2)
(2) (e)
(2)
(2)
(3)
(3)
(2)
(f)
(2) (g)
(2)
(2)
(2)
(2)
(2) (h)
Figure 2.16: Symbolic diagrams of elastic pair interactions that will be worked out in this section. (a) The pair interaction of order R2 /r2 between curved inclusions. This is a “Coulombic” interaction between two point-like curvature sources. (b)–(c) These interactions are similar, but involve sources of different multipole order, and hence scales differently; namely, as R4 /r4 . (d)–(e) These are the first contributions of the third cumulant. They are analogous to an electrostatic situation with a point charge in the presence of a conductor. With a total of 8 derivatives over 2 links, these interactions scale as (R/r)8−2×2 = (R/r)4 . (f)–(g) Similar interactions to those of (d)–(e), but this time the induced source involved is of higher multipole order, giving rise to a scaling of R6 /r6 . (h) The remaining interaction that scales the same way with distance, which originates from the fourth cumulant. Note that one could imagine an interaction between the permanent sources of derivative order 2 and 4 on a particle, through an induced source of order 2, which scales the same way. At this order, though, we will restrict to cup-like inclusions to keep the number of diagrams manageable. In that case, S = 0 and the only ˚(2) . permanent source is Q
2.3.1 Pair interactions Inclusions of general curvature The longest ranged interaction between two inclusions of general curvature (J 6= 0 and S 6= 0) is a direct interaction between two permanent sources of derivative order 2 (Q(2) ˚(2) ), depicted in Fig. 2.16(a) symbolically, and in Fig. 2.17 explicitly. This interaction or Q is quite easily evaluated as E (2) = −
(2) (2) (2) ˚(2) ˚(2) (2) Q1 Q2 4 Q Q ¯ − Q1 Q2 ∂ 3 ∂G ¯ + c. c. ∂ G − 1 2 ∂ 3 ∂G κ κ κ
(2.54)
Here, we suppressed the argument (z1 , z2 ) of the Green function. Assuming, without loss of generality, that the joining line between the particles is on the real axis (z12 = z = r ∈ R),
69
2 Interactions of rigid particles on a membrane (2)
¯ (2) Q 2
˚(2) Q 2
˚(2) Q 2
Q2
(2)
¯ (2) Q 2
(2)
¯ (2) Q 1
Q1
(2)
¯ (2) Q 1
˚(2) Q 1
˚(2) Q 1
Q2
Q1
Figure 2.17: This is the list of diagrams embodied in Fig. 2.16(a) for the elastic interaction between ¯ (2) because two permanent quadrupoles. Note that there can be no interaction between Q(2) and Q they have plain and double legs, respectively. Recall that such legs, when connected, kill the propagator between them by a biharmonic operator, ∂ 2 ∂¯2 .
we have −1! 2! z¯ − κ8πR12 S1 e2iα1 8πR22 J2 16πz 3 16πz 2 −1! + c. c. − κ8πR12 J1 8πR22 S2 e2iα2 16πz 2 R2 R2 = − 8πκ 1 2 2 2S1 S2 cos(2α1 + 2α2 ) − S1 J2 cos 2α1 − J1 S2 cos 2α2 . r
E (2) = − κ8πR12 S1 e2iα 8πR22 S2 e2iα2
(2.55)
Up to explicit dependence on the particle sizes, the above interaction was worked out in Ref. [DF99b].17 With perfect saddles (Ja = 0), maximal attraction occurs when they are oriented such that α1 +α2 = 0 (mod π). When the mean curvatures Ja are of the same sign, the inclusions tend to align their principal axes of smaller curvature (in absolute value) with the joining line (i.e., αa = π/2 for Ja > 0 and αa = 0 for Ja < 0), and attract.18 Note that the absence of a “direct” interaction between two symmetrically-curved inclusions (a term ∼ J1 J2 ) can be explained intuitively as follows: The field emanating from one 2 12 symmetric source has the form δij G12 ij = ∇ G . Incident on another symmetric curvature 4 12 = 0. Of course, this corresponds 2 source, this interaction vanishes as δij ∇ G12 ij = ∇ G ˚(2) and Q ˚(2) in Fig. 2.17 because it would have two to the absence of a diagram between Q 1 2 rings on one link. The next correction is of order R4 /r4 and follows from the diagrams of Figs. 2.16(b)–(e). We will begin with the last one of these. Fig. 2.16(e) is composed of the diagrams shown in Fig. 2.18. In the order displayed, the diagrams evaluate as (see figure caption for notes 17 Note that the signs of the angles differ from the expressions given in Ref. [DF99b], since we use a slightly different convention for them. All angles are measured counter-clockwise here. Furthermore, Ref. [DF99b] defines the orientation of an inclusion in terms of its principal axis with smaller absolute curvature. This results in a discrepancy of π/2 in the angles α depending on the sign of J. 18 If the mean curvatures are of opposite signs, one must simultaneously solve J2 S1 sin 2α1 = J1 S2 sin 2α2 and cot 2α1 + cot 2α2 = J1 /2S1 sin 2α1 for the orientation.
70
2 Interactions of rigid particles on a membrane
˚ ˚(2) Γ 2
(2)
Γ2
(2)
Q1
¯ (2) Q 1
¯ (2) Q 1
(2)
Q1 (2)
(2)
Q1
(2)
˚(2) Q 1
Q1
¯ (2) Q 1
(2)
Γ2
Γ2
¯ (2) Q 1
¯ (2) Q 1
(2)
Q1
(2)
Γ2
˚(2) Q 1
˚ ˚(2) Γ 2
˚ ˚(2) Γ 2
˚(2) Q 1
˚(2) Q 1
Figure 2.18: Diagrams of the interaction in Fig. 2.16(e). These depict the interaction between a ˚(2) permanent quadrupole and the quadru-polarization it induced on another particle. Recall that ˚ Γ vertices are special in that its legs can be plain or double. Also, watch out for multiplicity factors in ˚(2) ’s always bring a factor of 2 since they can be flipped, and (ii) some diagrams the evaluation: (i) ˚ Γ repeat vertices twice, so there is a factor of 1/2! associated with each. Moreover, the very last diagram has a multiplicity of 2 as well. That is because the two-legged vertex can flip, even though it is not a ˚ ˚(2) ; they are linked to Q ˚(2) ’s which can be both plain or double. (Or one can just write down the Wick Γ contractions and see.)
regarding evaluation) (2)
˚˚(2)
(2) ¯ (2) Γ2 (2) ¯ (2) Γ2 ¯ ∂¯3 ∂G Fig. 2.16(e) =Q1 Q ∂ 4 G∂¯4 G + Q1 Q 2 ∂ 3 ∂G 1 1 2 2
2κ 2κ (2) ˚ ˚ ˚˚(2) (2) 2 Γ2 (2) 2 Γ2 3¯ 3¯ ¯ + Q1 2 ∂ ∂G∂ ∂G + Q1 2 ∂¯3 ∂G∂¯3 ∂G 2! 2κ2 2! 2κ2 (2) (2) ¯ ∂¯4 G + Q(2) Q ˚(2) Γ2 ∂ 4 G∂¯3 ∂G ˚(2) Q ¯ (2) Γ2 ∂ 3 ∂G +Q 1 1 1 1 2κ2 2κ2 (2) ¯ ∂¯3 ∂G . ˚(2) 2 Γ2 2 ∂ 3 ∂G + Q 1 2! 2κ2
(2.56)
Along with Fig. 2.16(d), we have 2
2
˜ (4) = 4πκ R1 R2 5R12 S12 + R12 S12 cos 4α1 − 4R12 J1S1 cos 2α1 + R12 J12 E r4 +5R22 S22 + R22 S22 cos 4α2 − 4R22 J2S2 cos 2α2 + R22 J22 .
71
(2.57)
2 Interactions of rigid particles on a membrane
˚(4) Q 2
˚ ¯ (4) Q 2
Q2
(2)
¯ (2) Q 2
(2)
¯ (2) Q 1
˚(4) Q 1
˚ ¯ (4) Q 1
Q1
Figure 2.19: The diagrams that comprise the interactions Figs. 2.16(b)–(c) between the short (4 derivatives) and long-ranged (2 derivatives) features of two saddles.
The terms proportional to J12 and J22 constitute the repulsion between conical inclusions, (4) Econ = 4πκ
R12 R22 2 2 2 2 , R J + R J 1 1 2 2 r4
(2.58)
that was discussed in Refs. [GBP93a, KNO98, WKH98, DF99a, DF99b, MM02]. Note the simple interpretation of each term in this interaction [Figs. 2.16(d)–(e)] as that between a permanent quadrupole and the quadrupole (among other orders) it induced around another particle. The anisotropic interaction (2.57) was proposed in Ref. [DF99b] as well, except that it was mistaken for the total pair interaction proportional to r−4 . This is not true, which is why we did not call the left hand side of Eq. (2.60) E (4) . The remaining interactions that ˚(4) and contribute at this order are those of Figs. 2.16(b)–(c), which involve the sources Q ˚ ¯ (4) . The field of these sources decay as the fourth derivative of the Green function, rather Q than 2 derivatives as with the Q(2) ’s. While this means that the second order permanent source dominates the fourth order source at asymptotic distance from the particle, the latter is relevant for corrections to the asymptotically correct result 2.55. A point curvature defect type of description as in Ref. [DF99b] does not suffice in this regard, since it only captures the dominant source. Let us compute the missing term now. Figs. 2.16(b)–(c) amount to the four diagrams displayed in Fig. 2.19, which simply evaluate as (2) ˚(4) (2) ˚(4) Q1 Q 2 ¯ − Q2 Q1 ∂ 5 ∂G ¯ + c. c. ∂ 5 ∂G κ κ R2 R2 = + 24πκS1 S2 (R12 + R22 ) 1 4 2 cos(2α1 + 2α2 ) . r
Fig. 2.16(b)–(c) = −
(2.59)
This combines with Eq. (2.57) to yield the total pair interaction of order R4 /r4 : R12 R22 2 2 R1 S1 cos 4α1 + R22 S22 cos 4α2 − 4R12 J1S1 cos 2α1 − 4R22 J2S2 cos 2α2 r4 + R12 (5S12 + J12 ) + R22 (5S22 + J22 ) + 6S1 S2 (R12 + R22 ) cos(2α1 + 2α2 ) . (2.60)
E (4) =4πκ
72
2 Interactions of rigid particles on a membrane (3)
Γ2
˚(2) Q 1
˚(2) Q 1
(3)
˚(2) Q 2
Γ2
˚(2) Q 1
˚(2) Q 1
˚(2) Q 2
˚(2) Q 2
(3)
˚(2) Q 2
(3)
Γ1
Γ1
(2)
Γ2
˚(2) Q 2
˚(2) Q 1
Γ1
(2)
(2)
Γ2
˚(2) Q 2
˚(2) Q 1
Γ1
(2)
Figure 2.20: Elastic pair interactions of order R6 /r6 between axisymmetrically curved inclusions. These diagrams are expanded forms of Figs. 2.16(f)–(h). Notice, in the first four diagrams corresponding to ˚(2) ’s it is linked to—giving Figs. 2.16(f) and (g), the quadratic vertex can be flipped—thanks to the Q rise to a Wick multiplicity of 2.
For identical inclusions (|S1 | = |S2 | and |J1 | = |J2 |), this interaction is always repulsive, regardless of the orientation of the particles, if J1 = J2 . However, when J1 = −J2 , the inclusions tend to align their axes of larger absolute curvature and attract provided |Sa | < |Ja | < 3|Sa |. It is also worthwhile noting that without the last term in Eq. (2.60), which is not possible to capture by an ad hoc point particle approximation, this attractive feature disappears. Axisymmetrically curved inclusions Another example of a term which requires a systematic treatment of the inclusion’s finite size is provided by the next order, O(R6 /r6 ), which is the highest order we will consider. We also restrict to axisymmetrically curved inclusions, i.e. S1,2 = 0, for this order, to keep the number of possible diagrams manageable. The interactions that contribute to this order are shown symbolically in Figs. 2.16(f)– (h), and explicitly in Fig. 2.20. We will again begin with the last one. The interaction of Fig. 2.16(h) is easily evaluated as (2)
(2) (2) (2) Γ1 Γ2 ¯ 4 G∂ 3 ∂G ¯ ∂ 3 ∂G∂ 4κ3
˚ Q ˚ Fig. 2.16(h) = −Q 1 2
+ c. c. = −16πκJ1 J2 R14 R24 r−6 ,
(2.61)
which is also reported in Ref. [DF99a], where a point-particle approximation was made. However, this translates to permanent and induced sources being captured only as good as they seem from infinitely far away; no source or polarizability of order larger than 2. Hence, Figs. 2.16(f)–(g) are missed due to the absence of an octupole order polarizability, which contribute at the same order with (3)
(3)
Γ2 4 ¯ ¯4 ¯ ∂¯4 ∂G + c. c. ˚(2) 2 Γ1 2 ∂ 4 ∂G 2 ∂ ∂G ∂ ∂G + Q 2 2! 2κ2 2! 2κ2 R4 R4 =8πκ(J12 + J22 ) 1 6 2 . (2.62) r (2) 2
˚ Figs. 2.16(f)–(g) = Q 1
73
2 Interactions of rigid particles on a membrane (2)
(2)
(2)
(2)
(3)
(2)
(2)
(2) (2)
(2) (2)
(2) (a)
(2)
(2)
(2) (b)
(2)
(c)
(2) (d)
(2) (e)
Figure 2.21: The curvature-elastic triplet interaction diagrams in symbolic from. Cyclic permutations of the particles (worldlines) are implied. (a) O(R4 /r4 ) (b)–(e) O(R6 /r6 ). Expanded forms of (a)–(b) and (c)–(e) can be found in Figs. 2.22 and 2.23, respectively.
Altogether, we find the interaction of order R6 /r6 to be E (6) = 8πκ(J1 − J2 )2
R14 R24 . r6
(2.63)
This order is always repulsive and vanishes for identical inclusions, contrary to the earlier prediction, Eq. (2.61), which suggested it could be attractive as well as repulsive, depending on the relative signs of the curvatures.
2.3.2 Triplet interactions −2 −2 The leading triplet interactions, which scale as r12 r23 etc., were shown in Ref. [DF99b] for inclusions of general anisotropic curvature. In our formalism, these result from the diagram in Fig. 2.21(a). However, we will restrict to the isotropic curvature case (conical inclusions) for the following discussion. The interaction can be derived from the first two diagrams in Fig. 2.22 as (2)
(4) ¯ ¯3 ˚(2) Q ˚(2) Γ2 ∂ 3 ∂G(z E{123} =Q 1 , z2 )∂ ∂G(z2 , z3 ) + c. c. + cyc. 1 3 2κ2 −1! −1! =κ8πR12 J1 8πR32 J3 16πR22 2 2 + c. c. + cyc. 16πz12 16π¯ z23 cyc. (123)
=8πκ
X J R2 R2 R2 J a c c a b b , cos 2ϕcb 2 r2 rca ab
(2.64)
a,b,c
a = −ϕ + ϕ (see Fig. 2.12). Both originating from the third cumuwhere we remind, ϕcb ca ab lant, this interaction is a close relative of the pair interaction (2.58). Here, a permanent quadrupole is interacting with one that was induced on an intermediate particle by another permanent quadrupole [see Fig. 2.21(a)]. Ref. [KNO98] computes membrane-mediated interactions between axisymmetric inclusions based on the energy cost of introducing a rigid inclusion in the curvature field of
74
2 Interactions of rigid particles on a membrane
˚(2) Q 3
˚(2) Q 3 (2)
˚(2) Q 3 (2)
Γ2
(3)
Γ2
˚(2) Q 1
˚(2) Q 3 (3)
Γ2
˚(2) Q 1
˚(2) Q 1
Γ2
˚(2) Q 1
Figure 2.22: Elastic triplet interactions of order R4 /r4 (left) and R6 /r6 (right) between axisymmetric inclusions that originate from the third cumulant. These diagrams constitute the expanded form of Figs. 2.21(a)–(b). Cyclic permutations of particles (worldlines) must be added to complete the interaction.
conical particles far away. These authors recognize the non-pairwise additive nature of this energy but their presentation does not make it entirely clear “how much” of the true multibody character is captured in their treatment. In fact, their subsequent discussion of special cases (such as symmetric five particle arrangements) could leave the casual reader with the impression that a full N -body result might have been obtained. We will now show that the presentation in Ref. [KNO98] comprises only the pair and triplet interactions among permanent and induced quadrupoles. √ Defining γa ≡ Ra Ja and ζab ≡ zab / Ra Rb [see second to last step of Eq. (2.64)], the combined pair and triplet interactions, Eqs. (2.58) and (2.64), among N conical inclusions can be rewritten in the same form as in Ref. [KNO98]: E
(4)
=4πκ
N X a=1
X b6=a
X γc γb γb2 + 2 z 2 2 ζ¯2 zab ¯ab ζca ab c6=b6=a c6=a
!
2 N X X γb . = 4πκ ζ2 a=1 b6=a ab
(2.65)
Hence, even though it involves N particles, this interaction consists solely of pair- and triplet-wise additive parts. A true N -body interaction involves N − 2 induced sources: the deformation of one curved inclusion polarizes an intermediate inclusion, whose response polarizes yet another one, and so on, until the field of the last induced source interacts with the curvature of another inclusion. The first subleading correction to the triplet interaction (2.64) stems from the diagrams depicted in Figs. 2.21(b)–(e). The first of these interactions is given in expanded form as the last two diagrams of Fig. 2.22, which easily evaluates as (3) (2) Γ2 ¯ ¯4 ∂ 4 ∂G(z 1 , z2 )∂ ∂G(z2 , z3 ) 2κ2 cyc. (123) X J R2 R4 R2 J a c c a b b cos 3ϕcb . =16πκ 3 r3 rca ab a,b,c (2)
˚ Q ˚ Fig. 2.21(b) =Q 1 3
75
+ c. c. + cyc.
(2.66)
2 Interactions of rigid particles on a membrane
˚(2) Q 3 (2)
˚(2) Q 3
(2)
Γ3
˚(2) Q 3
(2)
Γ3
(2)
(2)
˚(2) Q 1
(2)
Γ2
(2)
˚(2) Q 1
Γ1
Γ2
(2)
Γ2 (2)
Γ3 (2)
Γ2
Γ1
(2)
Γ3
(2)
Γ2
˚(2) Q 1
˚(2) Q 3
Γ2
˚(2) Q 1
˚(2) Q 1
˚(2) Q 1
˚(2) Q 1
˚(2) Q 1
Figure 2.23: Elastic triplet interactions of order R6 /r6 between axisymmetric inclusions that originate from the fourth cumulant. These diagrams constitute the the expanded form of Figs. 2.21(c)–(e). Cyclic permutations of particles (worldlines) must be added to complete the interaction.
Figs. 2.21(c)–(e) are given explicitly in Fig. 2.23. The first of these is found as (2) (2) (2) Γ1 Γ2 3¯ ¯ ¯4 ∂ 3 ∂G(z 1 , z2 )∂ G(z2 , z1 )∂ ∂G(z1 , z3 ) 4κ3 cyc. (123) X R2 R2 R2 a Ja Jc Ra2 a4 b2 c cos 2ϕcb . 16πκ rab rac a,b,c (2)
˚ Q ˚ Fig. 2.21(c) = − Q 1 3 =−
+ c. c. + cyc.
(2.67)
Noting that Fig. 2.21(d) is merely an anti-cyclic permutation of Fig. 2.21(c), we can directly obtain it from the above result by summing, instead of cyclic permutations of (123), over anti-cyclic permutations of (123). Then the two of them together yield perm. (123)
Figs. 2.21(c)–(d) = −16πκ
X
a,b,c
Ja Jc Ra2
Ra2 Rb2 Rc2 a cos 2ϕcb , 4 r2 rab ac
(2.68)
where the summation is now over all permutations of (123). Lastly, we evaluate Fig. 2.21(e) (see last two diagrams of Fig. 2.23) as ˚(2) Fig. 2.21(e) = − Q 1
(2) 2 Γ(2) 3¯ 2 Γ3 ¯ ¯4 ∂ 3 ∂G(z 1 , z3 )∂ G(z3 , z2 )∂ ∂G(z2 , z1 ) + c. c. + cyc. 2! 4κ3
cyc. (123)
= − 8πκ
X
a,b,c
Ja2 Ra2
Ra2 Rb2 Rc2 b c 2 r 2 r 2 cos(2ϕac − 2ϕba ) . rab bc ca
(2.69)
The total correction to the elastic triplet interaction of order R6 /r6 then becomes—adding
76
2 Interactions of rigid particles on a membrane up Eqs. (2.66), (2.68) and (2.69)
(6)
E{123} = 4πκ
perm. (123)
X
a,b,c
Rc2 Ra4 Rb2 Rc2 Ra4 Rb2 a a cos 3ϕ − 4J J cos 2ϕcb a c cb 3 r3 4 r2 rca rab ab ac 2 4 2 2 Rc Ra Rb b c − Ja 2 2 2 cos(2ϕac − 2ϕba ) . rab rbc rca
2Jc Jb
(2.70)
The first term is the first correction beginning to incorporate the finite sizes of the inclusions through higher-than-quadrupole polarizabilities, c.f. Fig. 2.21(b). The rest of the terms involve two scatterings from polarizable sources, c.f. Figs. 2.21(c)–(e). In other words they incorporate the energy cost of introducing a curved inclusion in the presence of two rigid inclusions.
77
3 Outlook We have made some simplifying assumptions throughout this work. However, none of them is essential for the formalism to work, in principle. Here, we will briefly discuss and sketch possible generalizations and extensions.
3.1 Anisotropic particles Even though we restricted to inclusions that have a circular footprint, the extension to anisotropic boundaries is straightforward. Let us discuss what this entails, using the film case as an example. The final form of the effective theory will not be different for the membrane. Induced sources Under the assumption of circular symmetry, the worldline terms for induced sources had the rotationally invariant form C (ℓ) h2i1 i2 ...iℓ . If the boundary of the particle is not circular, on the other hand, these terms assume the more general form (ℓ)
hi1 ...iℓ Ci1 ...iℓ j1 ...jℓ hj1 ...jℓ .
(3.1)
Yet, this is not the whole story. Recall the reason why terms with an uneven number of derivatives, of the form ∂ ℓ h∂ m h with ℓ 6= m, were absent: The excess derivatives would have to contract among each other, becoming Laplacians, and render the term redundant since the Laplacian is the Euler-Lagrange derivative of the problem. However, when rotational invariance is forgone and the derivatives on the field factors contract over a polarizability tensor as above, the excess derivatives do not contract onto each other. Therefore, the most general form for quadratic terms is (ℓ)
hi1 ...iℓ Ci1 ...iℓ j1 ...jm hj1 ...jm .
(3.2)
Thus, there will be a lot more terms to consider for the effective theory, and some tensor operations will have to be done for evaluation. As far as the matching procedure for the tensor polarizabilities is concerned, the full theory boundary value problem is going to be harder due to the shape of the boundary. These factors make easy analytical computations to as high orders as we have done in the main text somewhat unlikely. But it should be obvious that this difficulty is by no means due to the EFT formalism. As the formalism is systematic, the difficulty increases systematically with the complications in the nature of the problem.
78
3 Outlook Permanent sources For permanent sources, we had linear terms of the form (ℓ)
Qa,i1 ...iℓ hi1 ...iℓ .
(3.3)
These already transform nontrivially under rotations, and are of the most general form. Hence, nothing changes as far as evaluation of interactions is concerned. Again, one will have to solve a boundary value problem in unpleasant coordinate systems to fix the tensors. But one can still predict what interactions are present, up to numerical prefactors.
3.2 Mixed Hamiltonians In the previous two chapters, we focused on surfaces whose energy density is purely due to tension, or bending elasticity. We did not consider a capillary surface (except in Appendix E), Z h ρg i 1 (3.4) H[h] = σ d2 r (∇h)2 + h2 , 2 σ or a membrane under finite tension, 1 H[h] = κ 2
Z
d2 r
h
i 2 σ ∇2 h + (∇h)2 . κ
(3.5)
The main feature pof these “mixed” Hamiltonians is a characteristic length scale: the capilinterface and g the lary length λ = σ/ρg (with ρ being the density difference across thep gravitational acceleration) for the first, and a crossover length ξ −1 = κ/σ across which the features of the surface are dominated by tension or bending elasticity for the second. As a result, the Green functions for the two problems become G(r, r ′ ) = and G(r, r ′ ) =
1 K0 r − r ′ /λ , 2π
−1 r − r ′ + log ξ r − r ′ , ξ K 0 2πξ 2
(3.6)
(3.7)
where K0 (z) is the modified Bessel function of the second kind of order zero. This brings about an immediate consequence: the series expansions for physical observables will not involve (only) powers of distances and radii but complicated functions as well. This is again a manifestation of the fact that the original problem got more complicated, not of a shortcoming of the EFT description; the propagations on the surface are not power-like anymore. As a result, one cannot order their expansions as a power series expansion, and comparing magnitudes of their terms ahead of calculation (thereby allowing
79
3 Outlook systematic preemptive truncation) goes out of the window. Keeping in mind that we have not studied this case extensively, let us say that one will likely have to keep all terms that involve Bessel functions whose argument and index are comparable. Nevertheless, we can write the effective theory for such a case. Let us discuss how one would go about doing that for the capillary Hamiltonian. This case is slightly more interesting than the membrane under tension. Also note that we will only deal with induced sources here. Finding the effective theory boils down to answering the question: what worldline terms can one write? For the film (λ → ∞), we had ∞
∆H =
1 X (ℓ) 2 C hi1 ...iℓ 2
(3.8)
ℓ=1
for a rigid disk. One immediate question is whether we need to include any terms that involve a ∇2 , now that the Euler-Lagrange equation is not ∇2 h = 0 anymore, but ∇2 h = λ−2 h. The answer is a no, but not a straightforward one. Consider the term hijkk hij , for instance. Using the Euler-Lagrange equation, this can be rewritten as λ−2 h2ij , which can be lumped into the C (2) term by a redefinition of the polarizability.1 So, terms like hijkk hij , h2ikk , hikk hikkll , etc. are still redundant. But there are those that are not. Recall that if the inclusion is free to execute rigid body motions, worldline terms that involve zero or one derivative drop out. However, even if one begins with an effective theory without these terms before the redefinition of terms described in the previous paragraph, such terms will appear after the redefinition. Take h2kk for example. This term becomes λ−4 h2 under the redefinition and modifies the monopole term. Or, even if the inclusion can tilt and therefore the coefficient of the term h2i starts out zero, terms such as hikk hi or h2ikk will ruin that. This is one point that one needs to pay attention to, and as a result, the most general worldline Hamiltonian for a rigid disk on a capillary surface becomes ∞
∆H =
1 X (ℓ) 2 C hi1 ...iℓ , 2
(3.9)
ℓ=0
(note the lower limit of the sum) regardless of its allowed rigid body motions. We conclude by stating that the above is not an issue for the membrane under tension and the worldline Hamiltonian is of exactly the same form as the tensionless membrane if the inclusion is allowed to bob and tilt. This is easy to see. The Euler-Lagrange equation for this case is ∇4 h = ξ 2 ∇2 h; a redefinition cannot reduce the number of derivatives on a worldline term further than two.
1
A way to convince yourself is the following. Since ∇2 G = λ−2 G by definition of the Green function, vertices with legs such as hijkk , hikkll , etc. “emit” deformations of the form Gijkk = λ−2 Gij , Gikkll = λ−4 Gi , etc. Instead of all these different vertices having almost the same response, one lumps them into one vertex.
80
3 Outlook
3.3 Soft inclusions One may want to study a surface modified by the presence of flexible inclusions or domains. This changes absolutely nothing for the effective theory apart from the values of the polarizabilities. The boundary value problem one must solve now involves two regions, inside as well as outside the domain. The five continuity equations across two adjacent membrane domains was given in Ref. [M¨ ul07] using covariant geometry. First, let us define the quantities Ξ1 ≡κ (K − K0 ) + κ ¯ Kk ,
(3.10a)
Ξ2 ≡κ∇⊥ K − κ ¯ ∇k K⊥k , h 2 i 1 2 2 − Kk − K0 Ξ3 ≡ κ K⊥ +κ ¯ K⊥k −σ . 2
(3.10b) (3.10c)
Here, the perpendicular and parallel symbols denote directions with respect to the boundary between the domains, which is a circle of radius R around the origin for our case. K⊥ , Kk and K⊥k are components of the curvature tensor in the basis aligned with those directions, K = K⊥ + Kk is the total extrinsic curvature and K0 is the spontaneous curvature of the surface. κ, κ ¯ and σ are the bending modulus, the Gaussian bending modulus and the surface tension, respectively, as before. In terms of these quantities, the continuity equations are Kkin =Kkout , in K⊥k Ξin 1 in Ξ2 Ξin 3
out =K⊥k , out =Ξ1 , =Ξout 2 − out =Ξ3 +
(3.11a) (3.11b) (3.11c) γKk ,
(3.11d)
γKg ,
(3.11e)
where γ and Kg are the line tension and geodesic curvature along the boundary, respectively. In line with the assumptions of this thesis, we write the involved quantities in linearized Monge gauge (and polar coordinates) and assume zero spontaneous curvature to find (3.12a) Kk = ∇2 − ∂r2 h , K⊥k =R−2 (R∂r − 1) ∂ϕ h , Ξ1 = (κ + κ ¯ )∇2 − κ ¯ ∂r2 h ,
Ξ2 =κ∂r ∇2 h − κ ¯ R−3 (R∂r − 1) ∂ϕ2 h .
(3.12b) (3.12c)
(3.12d)
In the linear regime Ξ3 = −σ, and the last boundary condition (3.11e) becomes a simple statement of lateral stress balance, σout − σin = γ/R .
81
(3.13)
3 Outlook analogous to the Young-Laplace equation [DGBWQ04]. Both the inside and outside regions are surfaces with finite rigidity and tension. p The 4 2 general solution for the height profile is then the solution of ∇ −ξ h = 0 where ξ = κ/σ. We would like to find out the response of both regions under a background hbg (r) = f (r) cos nϕ. Those solutions that behave well at the origin and infinity are hin (r) =
∞ X
n=0 out
h
bg
ain rn + Ain In (ξi r) cos nϕ
(r) =h (r) +
∞ X
n=0
, and
aon r−n + Aon Kn (ξo r) cos nϕ ,
(3.14a) (3.14b)
where In (z) and Kn (z) are modified Bessel functions of the first and second kind, respectively. Plugging these into the boundary conditions (3.11a)–(3.11e), the equations that must be satisfied by the coefficients ani,o and Ani,o can be found. For each multipole order n, this is a linear equation in the four unknowns, where the coefficients involve algebraic expressions of R, κi,o , κ ¯ i,o , ξi,o and γ as well as Bessel functions (and derivatives) of the arguments Rξi,o . Of course, this is not a homogeneous system of linear equations, the “forcing” comes from derivatives of the background at r = R. We will not quote the expressions here.2 But one can see how, after identifying the coefficients of the outside solution, the polarizabilities will come out as complicated functions of the parameters of the problem. One interesting problem to look at would be the case of domains that are ever so slightly different from the surface in terms of rigidity. This small contrast affords another smallness parameter to the series expansions on top of those related to distance. Then, one can justify truncating the cumulant expansion after second order and potentially find closed-form expressions thanks to all the discarded terms.
2
Frankly, the author cannot figure out how to fit them on a page.
82
Summary and Conclusion This concludes our discussion of the application of an effective field theory formalism to the problem of surface-mediated interactions between particles at soft surfaces. Since the capital merit of this work is conceptual and formal, most of the discussion takes place where it is most relevant within the main text. However, we will very briefly sum up. The main point of this work is that after formulating the problem via a fluctuating field, it really pays off to further exploit the most central idea beneath such a formulation: separation of scales. This directly translates into the primary advantage of the formalism, which is the separation of computing (long-wavelength) physical observables from the boundary conditions around the particles. Indeed, the entanglement of these two is the main reason for the (usually severe) approximations adopted in the literature. One point we were adamant to remind the reader was that the effective theory is neither a model nor an approximation, but a reformulation of the original problem. Owing to the systematic nature of this reformulation, one gains a handle on the full solution by counting powers of relevant parameters (which was a ratio between particle sizes and distances throughout this text). This allows for a controlled means of truncating the results consistent with the desired level of accuracy. For the cases we took up in the text, we were able to rewrite the complete effective theory for the surfaces as modified by the presence of particles. However, in general, one will have to forsake some features of the short-distance physics. In this framework, one can decide what these features are, ahead of any calculation. Another crucial feature of the formalism is that computations are very efficient. As a result, we were able to find the interactions to great accuracy, especially for a pair of particles. Non-pairwise additive interactions were also relatively unexplored in the literature. We have computed such multibody contributions to the interaction free energy and shed light on them where previous results either did not exist, were incomplete or even misunderstood. Of course, the free energy of the modified surface (modified by the presence of particles) is not the only observable one may want to calculate using this formalism. For example, the two-point correlator, which is the Fourier transform of the power spectrum, can be used to fit the elastic properties of the surface. Given a geometric arrangement of particles, and hence the corresponding effective Hamiltonian, one can calculate the two-point function in the ensemble of the modified surface to extract its new elastic properties. Admittedly, we made simplifying assumptions and restricted to special cases. To recapitulate, we worked in the limit of weak deformations to keep the bulk theory linear, assumed a surface of infinite extent, restricted to particles of circular footprint and considered bulk Hamiltonians that were dominated by either tension or bending elasticity (and not both).
83
Summary and Conclusion However, these are not essential for the applicability of the approach. Thus, even though we have not explicitly addressed situations where these simplifications are lifted, we tried to sketch what each would entail, exploiting the transparency of the formalism. The most desirable and perhaps challenging extension of this work would be uniting its power with the fully covariant approach developed by M¨ uller, Deserno and Guven [MDG05a, MDG05b] for the description of the surface. Being transparent and straightforward, EFT is probably the best candidate to allow tractable computation of interactions between compact objects on general non-planar surfaces within a covariant framework. Although, note how close a fluctuating curved space sounds to quantum gravity; it will not be a cakewalk.
84
Appendix A Gaussian Integrals In this appendix, we will review some aspects of Gaussian integrals that are relevant for the main discussion. We begin by reminding the reader that a multivariate Gaussian integral has the form +∞ 1 n Z Y K −2 − 21 qi Kij qj dqa e (A.1) I0 = = det 2π a=1−∞
if the “kernel” Kij is symmetric and positive-definite (and hence invertible and diagonalizable). It can easily be shown by transforming to the eigenbasis of Kij , where it is diagonal, by a unitary transformation (Jacobian = 1), thereby factoring the integral into n decoupled single-variable Gaussian integrals. We will not do that here. More generally, a Gaussian integral with a linear term +∞ n Z Y 1 dqa e− 2 qi Kij qj +Ji qi IJ =
(A.2)
a=1−∞
is evaluated by “completing the square” in the exponent:1 1 1 1 1 −1 −1 Jj + Ji Kij Jj − qi Kij qj + Ji qi = − qi Kij qj + Ji qi − Ji Kij 2 2 2 2 1 1 −1 −1 −1 qi − Jk Kki =− Jl + Ji Kij Jj . Kij qj − Kjl 2 2
(A.3)
−1 The integration over the shifted variables qi − Jk Kki is just I0 . Thus one finds,
1 K − 2 1 Ji Kij−1 Jj IJ = det e2 . 2π
(A.4)
In field theory, the discrete indices are replaced by continuous ones, such as a coordinate x, or more generally r in d dimensional space, and the vector/list qi of variables becomes QR a function φ(r). The continuum limit of the integration dq is denoted by a so-called i R functional integral Dφ; instead of an integral over all states of a vector qi we have an integral over all states of a field φ(r). The dot products in the integrand also become 1
−1 is not 1/Kij , but (K −1 )ij . Beware, Kij
85
Appendix A Gaussian Integrals integrals, over the continuous index r. The functional counterpart of IJ can therefore be written as Z Z Z 1 d d ′ ′ ′ d d r d r φ(r)K(r, r )φ(r ) + d r J(r)φ(r) . (A.5) ZJ = Dφ exp − 2 For example, in the chapter about interactions on a film, we have K(r, r ′ ) = −βσ∇2 δ(r − r ′ ) and in the case of a membrane, K(r, r ′ ) = βκ∇4 δ(r −r ′ ). Of course, the dimensionality is always 2. The evaluation of a functional integral is simply a generalization of the discrete case.2 Analogously to Eq. (A.4), one has Z 1 d d ′ −1 ′ ′ ZJ = Z0 exp d r d r J(r)K (r, r )J(r ) . (A.6) 2 For the case of a film, for example, we have K −1 (r, r ′ ) = (βσ)−1 G(r, r ′ ) where G(r, r ′ ) is the Green function for the operator −∇2 , satisfying −∇2 G(r, r ′ ) = δ(r − r ′ ). One rightfully asks, how is Z0 =
Z
1 Z K −2 1 d d ′ ′ ′ d r d r φ(r)K(r, r )φ(r ) = det Dφ exp − 2 2π
(A.7)
calculated? Putting aside any question about what the determinant means for this case in the first place, without some manner of regularization, this diverges. Fortunately, averages over the probability distribution Z 1 1 d d ′ ′ ′ d r d r φ(r)K(r, r )φ(r ) (A.8) exp − P [φ(r)] = Z0 2 automatically have this divergence divided out. This is more explicit in the following section. But what about Z0 ? Even though we are not interested in the Helmholtz free energy, βF0 = − log Z0 of an unperturbed surface, let us shed some light on the matter. One has, 1 K 1 K βF0 = − log Z0 = − log det = Tr log . 2 2π 2 2π 2
(A.9)
Well, perhaps it is not that simple. It is not at all obvious (or even true) that the density of phase space points (i.e. the functional integration measure) is constant. This is a very sophisticated issue in which the author is not competent enough to provide much enlightenment. However, consider this: In the weak-deformation limit studied in this thesis where the theory is linear, we are only looking at a small neighborhood of the phase space around the flat state of the field. It is a reasonable conjecture that the phase space is roughly uniform in this small patch of phase space. If, however, nonlinear corrections are also considered, then the window on the phase space is enlarged, and one has to account for the intrinsic curvature of phase space in the integration measure [NPW89]. As far as the author is aware of, there is no consensus on what the correct measure is.
86
Appendix A Gaussian Integrals For example, r ′ ) = −(βσ)∇2 δ(r − r ′ ), then in wave vector space it is diagonal: R if2 K(r, ′ ′ ′ 2 ′ K(k, k ) = d r d r e−ik·r eik ·r K(r, r ′ ) = σβ(2πk)2 δ(k − k′ ).3 Then, log(K/2π) has the eigenvalue spectrum log(2πσβk 2 ), and summed over all k, it would yield Z Z 1 K 1 1 d2 k 2 βF0 = Tr log = log(2πσβk ) = dk k log(2πσβk 2 ) . (A.10) 2 2π 2 (2π)2 4π This integral exhibits an ultraviolet divergence, i.e. as k → ∞. This is a consequence of the continuum approximation. It may, for example, be fixed by cutting off the integral over wave vectors at short wavelengths where the continuum description is known to be inappropriate—an atomic spacing, perhaps.
A.1 Gaussian averages and Wick’s theorem Computing physical observables in field theory invariably involves lots of many-point correlation functions R 1 Z e− 2 φKφ (A.11) hφ(r 1 )φ(r 2 ) . . . φ(r 2n )i = Dφ φ(r 1 )φ(r 2 ) . . . φ(r 2n ) Z0 1
R
where Z0−1 e− 2 φKφ is the probability of each microstate φ(r) in the unperturbed ensemble; a Gaussian distribution with zero mean. We will explicitly compute the two-point and fourpoint averages, which should illustrate the general rule for computing 2n-point averages, namely Wick’s theorem. The neatest way to compute these averages is to temporarily add a linear term in the exponent, see Eq. (A.5) above, and drop down instances of the field by successive application of δ/δJ(r a ).4 For instance, the two-point function can be written as follows: 1 δ 2 ZJ hφ(r 1 )φ(r 2 )i = . (A.12) Z0 δJ(r 1 )δJ(r 2 ) J=0
According to this equation, it is clearly the term quadratic in J(r) in the power series expansion of Eq. (A.6) that contributes to the two-point function. So, Z δ2 1 d d ′ −1 ′ ′ d r d r J(r)K (r, r )J(r ) hφ(r 1 )φ(r 2 )i = 2 δJ(r 1 )δJ(r 2 ) J=0 1 −1 1 −1 = K (r 1 , r 2 ) + K (r 2 , r 1 ) = K −1 (r 1 , r 2 ) (A.13) 2 2 since we are dealing with symmetric operators. R ′ ′ ′ ′ d2 r d2 r′ e−ik·r eik ·r (−σβ)∇2 δ(r − r ′ ) = d2 r (−σβ)(−k′ )2 e−ik·r eik ·r = σβk2 (2π)2 δ(k − k′ ). 4 Since ZJ is a functional of the function J(r), we take functional derivatives. The derivative Z δ ∂ d ′ Jj q j = q i . d r J(r ′ )φ(r ′ ) = φ(r) is analogous to the partial derivative δJ(r) ∂Ji 3
R
87
Appendix A Gaussian Integrals Now let us compute the four point average δ 4 ZJ 1 . hφ(r 1 )φ(r 2 )φ(r 3 )φ(r 4 )i = Z0 δJ(r 1 )δJ(r 2 )δJ(r 3 )δJ(r 4 ) J=0
(A.14)
Clearly, it is the quartic term Z 1 1 dd r dd r′ dd ρ dd ρ′ J(r)K −1 (r, r ′ )J(r ′ )J(ρ)K −1 (ρ, ρ′ )J(ρ′ ) 2! 22
(A.15)
in the expansion of ZJ this time that matters. The possible distributions of the 4 derivatives on the 4 J’s produce 4! = 24 terms of the sort (2! 22 )−1 K −1 (r 1 , r 2 )K −1 (r 3 , r 4 ). However, these 24 terms are not all distinct. Note, for instance, that (2! 22 )−1 K −1 (r 2 , r 1 )K −1 (r 3 , r 4 ) and (2! 22 )−1 K −1 (r 4 , r 3 )K −1 (r 2 , r 1 ) are the same as the aforementioned term since (i) K −1 is symmetric and (ii) it does not matter in which order you multiply two (or more) matrix elements. Because of these two reasons, each pairing—such as r 1 r 2 r 3 r 4 in the 3 terms just mentioned—repeats 22 2! times; each factor corresponding, respectively, to (i) and (ii). What’s left is the 4!/2! 22 = 3 distinct pairings, each occurring once, yielding hφ(r 1 )φ(r 2 )φ(r 3 )φ(r 4 )i =K −1 (r 1 , r 2 )K −1 (r 3 , r 4 )
+ K −1 (r 1 , r 3 )K −1 (r 2 , r 4 ) + K −1 (r 1 , r 4 )K −1 (r 2 , r 3 ) . (A.16)
This is an illustration of Wick’s theorem which states that in a (zero-mean) Gaussian ensemble, 2n-point averages5 can be written as the product of n 2-point averages summed over all pairings of the 2n points. The pairings are called Wick contractions in field theory and are sometimes represented with the notation hφ(r 1 )φ(r 2 )φ(r 3 )φ(r 4 )i → hφ(r 1 )φ(r 2 )i hφ(r 3 )φ(r 4 )i hφ(r 1 )φ(r 2 )φ(r 3 )φ(r 4 )i → hφ(r 1 )φ(r 3 )i hφ(r 2 )φ(r 4 )i hφ(r 1 )φ(r 2 )φ(r 3 )φ(r 4 )i → hφ(r 1 )φ(r 4 )i hφ(r 2 )φ(r 3 )i . For averages of 2n field occurrences, it is not hard to see that there will generally be (2n)!/n! 2n =: (2n − 1)!! such connections, or Wick contractions. When the field occurrences have derivatives attached to them, as they do in this thesis, one just pays attention to which derivatives (especially their indices) end up in which 2point correlator. Since thermal averages are over field configurations, and derivatives are over spatial coordinates, the two can be done in any order. At least if you are a physicist, that is.
5
It should be easy to see that all odd-point averages are zero. Just think of the integral
88
R
2
dx e−x x2n−1 .
Appendix B Index contractions in complex coordinates In this appendix we will go into details of how to perform the index contractions encountered in computing induced fields and propagator products by transforming to complex coordinates, z = x + iy and z¯ = x − iy. We first note that in Cartesian coordinates, covariant and contravariant elements of tensors are identical, i.e the metric is the identity. Therefore we have not distinguished between them and wrote all indices as subscripts in the main body of the article. However, now that we are considering a change of coordinates, we will use a covariant notation instead. For example, the response (1.32) of a disk on a film to a background will read C (ℓ) bg h (r a )Gi1 ...iℓ (r a , r) σ i1 ...iℓ C (ℓ) i1 j1 g . . . g iℓ jℓ hbg =− i1 ...iℓ (r a )Gj1 ...jℓ (r a , r) , σ
h(r) = −
(B.1)
where g ij is the (inverse) metric tensor. From the embedding x = (1/2)(z + z¯), y = (1/2i)(z − z¯) the tangent vectors ~ez = (1/2)(1, −i) and ~ez¯ = (1/2)(1, i) follow. Then the metric tensor gij = ~ei · ~ej and its inverse are found as 1 0 1 0 1 . (B.2) , g ij = 2 gij = 1 0 2 1 0
Since the metric is constant, the Christoffel symbols vanish, and therefore covariant derivatives are just partial derivatives (and hence also commute).
B.1 Harmonic kernel The reason why it is useful to go to complex coordinates in the case of the film, where the integration kernel of the Hamiltonian is the harmonic operator −∇2 , is as follows. The index contractions needed for our computations all involve partial derivatives of the harmonic Green function, which is 1 log(z − z ′ )(¯ z − z¯′ ) , 4π in complex coordinates. It is easily seen that G(z, z ′ ) = −
∂z ∂z¯G(z, z ′ ) = 0 ,
89
(z 6= z ′ ) ,
(B.3)
(B.4)
Appendix B Index contractions in complex coordinates which is equivalent to the harmonic property of the Green function since ∇2 = g ij ∂i ∂j = 4∂z ∂z¯. This is a useful property because, when expressed in complex coordinates, all index combinations involving an alternation of indices on G vanish from the contraction, leaving only terms that involve either 1 (n − 1)! ∂nG = ∂z n 4π (z ′ − z)n
,
n>0
(B.5)
(note the order of z and z ′ in the denominator) or its complex conjugate. This property and the simple form of the metric tensor (B.2) simplify things greatly, allowing us to express products involving an arbitrary number of indices and factors. The computation of the induced deformation, Eq. (1.32) or Eq. (B.1), is now as follows: The background hbg = bℓ rℓ cos ℓϕ can be rewritten in complex coordinates as hbg = (bℓ /2)(z ℓ + z¯ℓ ). Using Eqs. (B.2) and (B.4) we find C (ℓ) bg h (0)Gi1 ...iℓ (0, z) σ i1 ...iℓ C (ℓ) i1 j1 =− g . . . g iℓ jℓ hbg i1 ...iℓ (0)Gj1 ...jℓ (0, z) σ C (ℓ) ℓ ℓ bg =− 2 ∂z¯h (0)∂zℓ G(0, z) + c. c. σ
h(r) = −
(B.6)
where “c. c.” denotes the complex conjugate of whatever precedes. Note that the index contraction, which generally has 2ℓ terms, was reduced to only two terms, owing to the property ∂z ∂z¯G = 0. Now, using Eq. (B.5) for ℓ > 0, the result we used in Eq. (1.34) can be found: ℓ C (ℓ) ℓ bℓ ℓ! (ℓ − 1)! 1 1 (ℓ) 2 ℓ!(ℓ − 1)!bℓ cos ℓϕ h(r) = − 2 + . (B.7) = −C σ 2 4π 4πσ z ℓ z¯ℓ rℓ Note that one has to pay attention to which argument of the Green function is differentiated so as to avoid sign errors. Propagator products. The other place where we encounter index contractions is in propagator products such as ∂ ℓa ∂ ℓb Gab ∂ ℓb ∂ ℓc Gbc ∂ ℓc ∂ ℓd Gcd ∂ ℓd ∂ ℓa Gda ,
(B.8)
where ∂ ℓ is a shorthand for ∂i1 ∂i2 . . . ∂iℓ . To make the underlying algebra more transparent, let us first look at this product of four propagators, before we generalize to an arbitrary number. Similarly to the derivation of Eqs. (B.6) and (B.7), and defining zab = zb − za , this product can be written as (B.8) =2ℓa +ℓb +ℓc +ℓd ∂zℓa ∂zℓb G(zab )∂z¯ℓb ∂z¯ℓc G(zbc )∂zℓc ∂zℓd G(zcd )∂z¯ℓd ∂z¯ℓa G(zda ) + c. c.
90
(B.9)
Appendix B Index contractions in complex coordinates
zb
c
=
rb
c
e iϕ
bc
c
b ϕac
zab = rab eiϕab a
b
b b is the difference . It is clearly seen that the angle ϕac Figure B.1: Illustration of the vertex angle ϕac ϕbc − ϕab between the phases of the two complex numbers zbc and zab .
Now we substitute from Eq. (B.5)—as well as define Λab ≡ (−2)ℓa (ℓa + ℓb − 1)!/4π temporarily for convenience—to find (B.8) =
Λab Λbc Λcd Λda ℓa +ℓb ℓb +ℓc ℓc +ℓd ℓd +ℓa zab z¯bc zcd z¯da
=2
Λab Λbc Λcd Λda ℓa +ℓb ℓb +ℓc ℓc +ℓd ℓd +ℓa rab rbc rcd rda
+ c. c. b c d a cos(ℓb ϕac − ℓc ϕbd + ℓd ϕca − ℓa ϕdb ),
(B.10)
b := −ϕ + ϕ is the angle between r − r and r − r (see where zab =: rab eiϕab and ϕac a c ab bc b b Fig. B.1). It is obvious how this result generalizes to the product of an arbitrary number q of propagators: ai i q Y (−2)ℓi (ℓi + ℓi+1 − 1)! ei(−) ℓi ϕai−1 ai+1 2 Re (B.11) ℓ +ℓi+1 4π raii ai+1 i=1
with ℓq+1 = ℓ1 . We use this formula to evaluate many interactions in Section 1.3, most noticeably in Eq. (1.61) where a pair interaction is considered and thus the angles are all π radians. Vanishing odd cumulants. Lastly, we would like to revisit a statement we made in Section 1.3.2, namely that the odd cumulants in the free energy expansion (1.21) do not contribute to the fluctuation-induced interaction for the surface tension Hamiltonian. This is easily seen to hold by observing that it is not possible to write a nonzero expression analogous to (B.9) for an odd number of propagators, due to the property (B.4) and the off-diagonal form of the metric. As it rests on Eq. (B.4), or ∇2 G = 0, this property of all odd numbered cumulants vanishing is special to the harmonic free surface Hamiltonian. Since the Green functions are not only contracted among each other but also the tensorial Wilson coefficients in the case of elastic interactions, this does not apply there.
91
Appendix C Renormalization of polarizabilities and counterterms It was claimed in Technical Note 1.4 that (implicit) counterterms in the effective theory ensure that physical observables, such as the interaction free energies, are finite—or that the contribution of diagrams featuring self-interactions is eliminated by counterterms. Here we will demonstrate this claim. Recall that our point particle effective theory can be thought of as a scaled, coarsegrained and renormalized version of the full theory, where the particle sizes finally became infinitesimal. Therefore the eventual coupling constants (here the Wilson coefficients) result from a certain renormalization group flow. At each step of the flow, divergent contributions creep in from the coarse-graining of short wavelength physics.1 In our case, the divergences are power-like, ∂ n G(λr, λr ′ ) ∼ λ−n as λ → 0, which means the renormalization group flow is trivial—there is no physical information in the divergences. As such, these divergences are just the counterterms in the renormalized Hamiltonian, that cancel (“counteract”) the divergences that may artificially appear in physical observables calculated in the theory caused by the continuum approximation.2 The polarizabilities (1.35), having been derived by matching physical observables, are actually “renormalized”—or “dressed”, to use a somewhat popular field theory term. A particular worldline term with its renormalized coefficient along with all the divergent counterterms with the same dependence on the field make up the “bare” coupling. We will illustrate these concepts by computing the thermally averaged response hh(r)i of one particle (at the origin) to a background, in the effective theory of a rigid disk on a film. When we did the same (in a somewhat simpler way) in Section 1.2.2 for matching, the polarizabilities were renormalized. Now we do it in a way that will manifest the divergences. The effective theory for one particle at the origin is as follows ∆H[h] =
1 X (ℓ) 2 CB hi1 ...iℓ (0) , 2
(C.1)
ℓ
1 The reason why the coarse-graining reflects on the couplings in a divergent way is the continuum assumption; an (uncountable) infinity of degrees of freedom in the tiniest amount of space. It is not the case, for example, in block spin renormalization on a lattice. 2 Phrased in this rigor, what we did with the purely numerical self-energy divergences of the first and second cumulants was to drop them in recognition of suitable counterterms originating from the renormalization of the free part of the theory—not the Wilson coefficients in that trivial case.
92
Appendix C Renormalization of polarizabilities and counterterms where the subscript B indicates that the polarizability is bare; all the counterterms ∼ h2i1 ...iℓ (0) are encoded in it. In other words, denoting the renormalized polarizabilities with (ℓ)
CR , ∆H[h] =
1 X (ℓ) 1 X (ℓ) 2 (ℓ) CR hi1 ...iℓ (0) + CB − CR h2i1 ...iℓ (0) , 2 2 ℓ
(C.2)
ℓ
with the second term being the counterterm. To write the one point average, consider the partition function Z R 2 ZJ = Dh e−H[h]−∆H[h]+ d r J(r)h(r) ,
(C.3)
where it was assumed that β = 1 for convenience. Then, the thermal average of the field away from the particle, computed in the effective theory, follows as Z δ 1 hh(r)ieff = = Dh e−H[h]−∆H[h] h(r) . (C.4) ln ZJ δJ(r) ZJ=0 J=0 On the other hand,
D E R 2 ln ZJ = ln Zfree e−∆H[h]+ d r J(r)h(r)
free
(C.5)
where the subscript “free” indicates that the quantities pertain to the unperturbed ensemble, i.e. ∆H = 0 and J(r) = 0. We will also set σ = 1 for convenience so that hh(r)h(r ′ )i = G(r, r ′ ). Next, we introduce the change of variables h → hbg + h. The free thermal average P (ℓ) over h(r) is unaffected, and ∆H[h] goes to ∆H[hbg ] + ∆H[h] + ℓ CB hbg i1 ...iℓ (0)hi1 ...iℓ (0) assuming the particle is at the origin. Upon the change of variables, Z bg ln ZJ = ln Zfree − ∆H[h ] + d2 r J(r)hbg (r) P (ℓ) R 2 − ℓ CB hbg i1 ...iℓ (0)hi1 ...iℓ (0)−∆H[h]+ d r J(r)h(r) , + ln e free
(C.6)
which means, according to Eq. (C.4) (with h → hbg + h), ∞
hh(r)ieff
δ X 1 = δJ(r) q! q=1
−
X
(ℓ)
CB hbg i1 ...iℓ (0)hi1 ...iℓ (0)
ℓ
1 X (ℓ) 2 − CB hi1 ...iℓ (0) + 2 ℓ
93
Z
2
d r J(r)h(r)
q
c,free J=0
. (C.7)
Appendix C Renormalization of polarizabilities and counterterms The terms of interest are those linear in J, and thus linear in h(r), because of the single J derivative evaluated at J = 0. At every cumulant order q, connected Wick contractions terminating at one end on the single occurrence of h(r) must then terminate at the other end on the linear induced source term, bouncing off q − 2 occurrences of ∆H in the middle. In other words, among the q −1 links, q −2 (i.e. all except the one terminating at h(r)—the “observation point”) are self-interactions, G(0, 0). Note that G(0, r) is spherically symmetric around r = 0. Hence, tensor elements Gi1 ...in (0, 0) are zero unless all of their indices are contracted among each other; the contrary would imply directionality. This property will allow substantial simplification. For actual calculations, it turns out to be more practical to transform the effective Hamiltonian into complex coordinates as, ∆H =
1 X (ℓ) ℓ ΓB ∂z h(0)∂z¯ℓ h(0) , 2
(C.8)
ℓ
with Γ(ℓ) = 2ℓ+1 C (ℓ) . The linear induced charge term in Eq. (C.7) then breaks up into two: 1 (ℓ) ℓ bg 1 (ℓ) ℓ bg (ℓ) ℓ ℓ CB hbg i1 ...iℓ (0)hi1 ...iℓ (0) = ΓB ∂z h (0)∂z¯h(0) + ΓB ∂z¯h (0)∂z h(0) . 2 2
(C.9)
With complex derivatives, the spherical symmetry of G(0, 0) mentioned above translates into ∂zn ∂z¯m G(0, 0) = δmn ∂zn ∂z¯n G(0, 0). This means two things: (i) if one end of a self-link is connected to a ∂zℓ h leg, the other is necessarily connected to a ∂z¯ℓ h, and (ii) we do not have to worry about self-links connecting different multipole orders. As a result, per multipole order ℓ, there are 2(q − 2)! Wick contractions: (q − 2)! beginning with ∂zℓ hbg (0) and (q − 2)! with ∂z¯ℓ hbg (0). Hence, accounting for the vertex factors and the multinomial coefficient in the expansion of the power q as well, the average response (C.7) can be written as hh(r)ieff
!q−2 (ℓ) ∞ (ℓ) X ΓB ℓ bg ΓB ℓ ℓ (q − 2)! q X =− ∂z ∂z¯G(0) ∂ h (0)∂zℓ G(0, z) + c. c. − q! 2 2 z¯ 1, 1 q=2 ℓ ∞ n o q X (ℓ) X (ℓ) ΓB (C.10) =− −2−1 ΓB ∂zℓ ∂z¯ℓ G(0) Re ∂z¯ℓ hbg (0)∂zℓ G(0, z) . ℓ
q=0
This is a physical observable, and the whole point is that whatever divergences the bare (ℓ) polarizabilities ΓB have in them, they should combine with the divergent self-interactions ∂zℓ ∂z¯ℓ G(0) in such a manner that the value of the observable is eventually finite. As we noted before, the induced response we calculated in Section 1.2.2 gave us the “dressed” value for the polarizability. In complex coordinates (and with σ = 1), it reads n o X (ℓ) ΓR Re ∂z¯ℓ hbg (0)∂zℓ G(0, z) . (C.11) h(r) = − ℓ
94
Appendix C Renormalization of polarizabilities and counterterms Therefore, with Eq. (C.10) in perspective, it is required that (ℓ)
(ℓ)
ΓR = ΓB
∞ X q=0
(ℓ)
(ℓ)
−2−1 ΓB ∂zℓ ∂z¯ℓ G(0)
q
.
(C.12)
(ℓ)
We solve for ΓB in terms of ΓR by substitution of the ansatz (ℓ)
(ℓ)
ΓB = ΓR
∞ X q=0
h iq (ℓ) αq(ℓ) ΓR ∂zℓ ∂z¯ℓ G(0)
(C.13)
(ℓ)
into Eq. (C.12). One can in principle find all the αq by ensuring Eq. (C.12) holds order ℓ (ℓ) by order in the divergence ∂zℓ ∂z¯ℓ G(0) = 2−2ℓ ∇2 G(0). A few of them are, α0 = 1 (ℓ)
(ℓ)
(ℓ)
(trivially), α1 = 1/2, α2 = 1/4, α3 = 1/4, ... More rigorously, one can show that after some manipulation, Eq. (C.12) can be recast as ∞ h ip X (ℓ) 1= ΓR ∂zℓ ∂z¯ℓ G(0) p=0
{ri |
P∞
X
i=0 (i+1)ri =p+1}
r0 + r1 + . . . α0r0 α1r1 . . . (−2)−1+r0 +r1 +... r0 , r1 , . . .
(ℓ)
(C.14)
(ℓ)
to find all the αq satisfying it. Powers p = 0, 1, 2, 3 yield the few αq we quoted earlier. Let us recap the exercise we just did. We knew that if we probed the couplings (polarizabilities) by a low energy excitation, we would be oblivious to what might be going on at very short length scales; the particle interacting with its own structure. That is why the scattering response should reflect the renormalized (dressed, physical) values for the polarizabilities. We stipulated that if we computed the same observable using the bare values, we must obtain the same finite response. This allowed us to obtain the bare polarizabilities in terms of the renormalized polarizabilities, or equivalently the counterterms in Eq. (C.2). We can illustrate, graphically, how the scattered response on the left-hand side of Eq. (C.10) came out finite despite all the self-interactions. We rewrite Eq. (C.10) in graphical form as follows: hbg
h(r) −
=
1 2
+
1 4
− ...
(C.15)
Shaded vertices denote bare polarizabilities. A graphical representation of Eq. (C.13), on the other hand, gives these in terms of the renormalized polarizabilities as =
+
1 2
+
95
1 4
+ ... ,
(C.16)
Appendix C Renormalization of polarizabilities and counterterms (ℓ)
where the coefficients are the values of αq we found earlier. Combining the latter with the former, we can see that the observed response is due solely to the renormalized vertex, hbg
h(r)
=
+
−
1 2
+
1 4
1 2
+
−
1 4
+ ...
1 2 1 2 1 2
−...
(C.17)
+ ...
and the self-interactions are cancelled by the diagrams generated by counterterms. The same happens for the vertices in the expansion of any observable, such as the cumulant expansion of the free energy. For example, if bare polarizabilities are used in vertices, the second cumulant contains many divergent terms, as follows
=
+
+
+
+
+··· ,
(C.18)
which can be seen to be of the same form as the self-interaction diagrams that originate from higher cumulants. We will not do the tedious exercise of verifying that these extra diagrams contributed by the counterterms do indeed cancel self-interaction diagrams among renormalized vertices originating from higher cumulants, similarly to Eq. (C.17). But it was on these grounds that we omitted self-interactions and counterterms together in our main calculations.
96
Appendix D Binomial cycle simplification of the entropic pair free energy on a film Here, the simplification of the full asymptotic expansion of the entropic pair free energy on a film is described. What is presented is a slightly adapted and expanded variation of the treatment found in the section titled “Cycles of binomial coefficients” in John Riordan’s book on combinatorial identities [Rio68]. In Section 1.3.1, we found that the fluctuation-induced pair interaction between two horizontally fixed rigid disks on a film has the asymptotic expansion βF = −
∞ 2s 2(ℓ +ℓ +...) 2(ℓ2 +ℓ4 +...) X 1 X Y (ℓi + ℓi+1 − 1)! R 1 3 R 1
s=1
s
{ℓ} i=1
ℓi+1 ! (ℓi − 1)!
2
r2(ℓ1 +ℓ2 +ℓ3 +...)
,
(ℓ2s+1 = ℓ1 ) .
(D.1)
It turns out that the 2s-fold sum involving the cyclic product of binomial coefficients in each cumulant q = 2s can be simplified. Let us define new summation labels ni = ℓi − 1 so that ni > 0. Then, ni + ni+1 + 1 ℓi + ℓi+1 − 1 (ℓi + ℓi+1 − 1)! = = , (D.2) ni ℓi − 1 ℓi+1 ! (ℓi − 1)! and the cumulant expansion can be rewritten as 2 2 2 ∞ X R22 R1 R2 R1 1 R12s R22s , C2s , , ,..., 2 βF = − s r4s r2 r2 r2 r
(D.3)
s=1
where Cq (u1 , u2 , . . . uq ) =
∞ X
n1 =0
···
∞ X
nq =0
n1 +n2 +1 n1
n2 +n3 +1 n2
···
nq +n1 +1 nq
n
un1 1 · · · uq q .
(D.4)
It will be possible to reduce the q-fold sum into just one by manipulating continued fractions. We need to introduce a series of functions first. The function fq (u1 , . . . , uq ) is defined recursively as uq−1 fq (u1 , . . . , uq ) = (1 − uq )fq−1 u1 , . . . , , (D.5) 1 − uq
97
Appendix D Binomial cycle simplification of the entropic pair free energy on a film with f0 = 1. For example,
u1 u3 u2 f4 (u1 , u2 , u3 , u4 ) = (1 − u4 ) 1 − 1 − u3 1 − u2 | {z } 1 − u4 1− 1− u3 f1 (u4 ) 1 − u4 1− {z } | 1−u f2 (u3 ,u4 )
{z
|
f3 (u2 ,u3 ,u4 )
The following relation, then, is easily observed to follow:
}
4
.
u1 fq−2 (u3 , . . . , uq ) fq (u1 , . . . , uq ) =1− , fq−1 (u2 , . . . , uq ) fq−1 (u2 , . . . , uq )
(D.6)
fq (u1 , . . . , uq ) = fq−1 (u2 , . . . , uq ) − u1 fq−2 (u3 , . . . , uq ) .
(D.7)
or equivalently
One can prove another recursion, namely fq (u1 , . . . , uq ) = fq−1 (u1 , . . . , uq−1 ) − uq fq−2 (u1 , . . . , uq−2 )
(D.8)
by mathematical induction: Using the definition (D.5), one easily verifies that the recursion is valid for q = 2 and q = 3. Now we need to show that if the recursion (D.8) is true for q, it also holds for q + 1. To that end, we note that the recursion to be proved would require fq+1 (u1 , . . . , uq+1 ) = fq (u1 , . . . uq ) − uq+1 fq−1 (u1 , . . . , uq−1 ) ,
(D.9)
while the definition (D.5) implies fq+1 (u1 , . . . , uq+1 ) = (1 − uq+1 )fq u1 , . . . ,
uq 1 − uq+1
.
(D.10)
We need to show these two expressions are the same under the assumption that the recursion (D.8) holds. We apply the recursion to the first expression to find uq fq−2 (u1 , . . . , uq−2 ) fq+1 (u1 , . . . , uq+1 ) = (1 − uq+1 ) fq−1 (u1 , . . . , uq−1 ) − 1 − uq+1 = fq−1 (u1 , . . . , uq−1 ) − uq fq−2 (u1 , . . . , uq−2 ) − uq+1 fq−1 (u1 , . . . , uq−1 ) .
Using the recursion one more time, we identify the last line to be equal to the right hand side of Eq. (D.9), hence proving Eq. (D.8). Recursion (D.8) can be rewritten as fq (u1 , . . . , uq ) uq fq−2 (u1 , . . . , uq−2 ) =1− . fq−1 (u1 , . . . , uq−1 ) fq−1 (u1 , . . . , uq−1 )
98
(D.11)
Appendix D Binomial cycle simplification of the entropic pair free energy on a film Now, noting that (1 − x)
−m−2
∞ X n+m+1
=
n
n=0
xn ,
(D.12)
consider the ratio (the shift of indices is for later convenience)
uq−1 fq−2 (u0 , . . . , uq−3 ) −n−2 1− fq−1 (u0 , . . . , uq−2 ) fqn+2 (u0 , . . . , uq−1 ) nq−1 ∞ X nq−1 + n + 1 nq−1 fq−2 (u0 , . . . , uq−3 ) = uq−1 nq−1 +2 nq−1 fq−1 (u0 , . . . , uq−2 ) nq−1 =0 n (u , . . . , u fq−1 0 q−2 )
−2 =fq−1 (u0 , . . . , uq−2 )
=
X
n
nq−1 +n+1 nq−1
n1 +n2 +1 n1
nq−2 +nq−1 +1 nq−2
nq−1 nq−2
nq−1 nq−2 uq−2 uq−1
q−2 (u0 , . . . , uq−4 ) fq−3
n
q−2 fq−2
+2
(u0 , . . . , uq−3 )
.. . =
X
n0 +n1 +1 n0
n0 ,...,nq−1
···
n0 nq−1 . u0 · · · uq−1
nq−1 +n+1 nq−1
(D.13)
This is still not our cycle (D.4) but if we introduce the variable uq , multiply both sides of Eq. (D.13) by unq u−n 0 and sum over n (after which we rename it nq ), we have G≡
n nq ∞ X (u0 , . . . , uq−2 ) uq q fq−1 n
nq =0
n +2
u 0 q fq q
(u0 , . . . , uq−1 )
=
X
n0 +n1 +1 n0
{n}
···
nq−1 +nq +1 nq−1
n1 n n −n u1 · · · uq q u0 0 q (D.14)
whose u00 term is the cycle (D.4); hence it is a generating function. The first relation in Eq. (D.14) is simply the series expansion of G=
fq−1 (u0 , . . . , uq−1 )
uq fq (u0 , . . . , uq−1 ) − fq−1 (u0 , . . . , uq−2 ) u0
−1
.
(D.15)
We now want to extract all the powers of u0 in G so we can identify the u00 term. To this end, we resort to the recursion formula (D.7). Applying the recursion to all three instances of f in Eq. (D.15) we arrive at G = (S − Qu0 )−1 P − Qu0 − Ru−1 0 where P =fq−1 (u1 , . . . , uq−1 ) + uq fq−3 (u2 , . . . , uq−2 ) ,
R =uq fq−2 (u1 , . . . , uq−2 )
and
99
−1
Q =fq−2 (u2 , . . . , uq−1 ) , S =fq−1 (u1 , . . . , uq−1 ) .
(D.16)
(D.17a) (D.17b)
Appendix D Binomial cycle simplification of the entropic pair free energy on a film Expanding the powers of −1 in Eq. (D.16), we find G=
∞ X i=0
S −1−i Qi ui0
∞ X j=0
P −1−j Qu0 + Ru−1 0
j
j ∞ X X j = P −1−j Qj−k+i Rk S −1−i uj−2k+i , 0 k
(D.18)
i,j=0 k=0
so that the term with j − 2k + i = 0 power of u0 is the cycle k ∞ X X 2k − i Cq (u1 , . . . , uq ) = P −1−2k+i Qk Rk S −1−i . k
(D.19)
k=0 i=0
The upper limit of the inner summation may not be obvious at first but can be checked by, for example, inserting a step function inside the sum instead of the original upper limit. Now, we can use k k X 2k − i i X (2k − i)! i i! k! (2k)! x x = k! (k − i)! i! k! (2k)! k i=0 i=0 h i X i k! k (−) (k−i)! i! xi 2k 2k h i = F(−k, 1; −2k; x) , (D.20) = k k i! (−)i (2k)! i=0
(2k−i)!
where F(a, b; c; x) is the hypergeometric function [AS74], so that the cycle becomes ∞ X QR k F(−k, 1; −2k; P/S) 2k , (D.21) Cq (u1 , . . . , uq ) = P2 PS k k=0
and we can use this, along with Eqs. (D.17a)–(D.17b), to evaluate the pair interaction (D.3). Identical radii By writing out the defining recursive relation (D.5) for the functions fq (u1 , . . . , uq ), one actually sees that they obey such a pattern: fq (u1 , . . . , uq ) =1 − (u1 + . . . + uq ) + (u1 u3 + u1 u4 + . . . + u1 uq + u2 u4 + u2 u5 + . . . + u2 uq + . . . + uq−2 uq )
− (u1 u3 u5 + u1 u3 u6 + . . . + u1 u4 u6 + . . . + uq−4 uq−2 uq ) + . . .
(D.22)
That is, its terms are all possible products of ui excluding those that multiply adjacent ui (u1 and uq are not adjacent). It is then a simple matter to show that, for ui = u, fq (u) =
⌋ ⌊ q+1 2
X i=0
q+1−i (−u)i i
100
(D.23)
Appendix D Binomial cycle simplification of the entropic pair free energy on a film where ⌊·⌋ denotes integer part and it is understood that with one argument fq (u) stands for fq (u, u, . . . , u). The software package MATHEMATICA actually recognizes a closed form fq (u) =
χq+2 − 1 (χ2 − 1)(χ2 + 1)q
(D.24)
√ for Eq. (D.23) with χ(u) = (1 − 1 − 4u − 2u)/2u. With identical ui , using recursion (D.8), one finds that P = fq−2 (u) = Q = R/u and S = fq−1 (u). Then, with Eq. (D.3), βF = −
∞ X ∞ X 2k s=1 k=0
k
f2s−2 (u) u2s+k F −k, 1; −2k; sf2s−1 (u)f2s−2 (u) f2s−1 (u)
(D.25)
follows. Before this simplification, each cumulant of order q = 2s was a 2s-fold infinite series, whereas now, they are all one-fold.
101
Appendix E Induced monopoles on a film In our treatment we avoided induced monopole terms in the effective theory, i.e terms of the form 1 X (0) 2 C h (r a ) , (E.1) 2 a a
since the field of these charges—each proportional to log|r − r a |—violates square integrability of its gradient in R2 . Such terms would describe particles with frozen vertical fluctuations. We will discuss a possible workaround for this issue. Consider the regularized free surface Hamiltonian Z 1 H[h] = σ d2 r (∇h)2 + λ−2 h2 (E.2) 2 which approaches the tension Hamiltonian (1.3) in the limit of large λ. For an interface between two fluids subject to gravity (horizontal p in its unperturbed state), λ is the capillary length [DGBWQ04] and is given by λ = σ/gρ, where g and ρ are the gravitational acceleration and mass density difference between the fluids, respectively. The addition of the “mass” term damps correlations over distances larger than λ, hence regularizing the infrared divergence of monopole fields. The Green function for this choice of surface energy is 1 (E.3) K0 r − r ′ /λ , G(r, r ′ ) = 2π instead of (−1/2π) log |r − r ′ |, where K0 (x) is a modified Bessel function of the second kind. After this regularization, we may safely consider particles that are completely pinned. The effective theory of such particles involves the same induced charges as before, Eq. (1.37), with the addition of the monopole (ℓ = 0) terms (E.1).1 The matching of the polarizability coefficients is done similarly by comparing the response of the induced charges to backgrounds of the form hbg = αrℓ cos ℓϕ in the full and effective theories. One finds Rℓ (E.4) hresp full (r) = − K (R/λ) Kℓ (r/λ) cos ℓϕ ℓ 1
With the new choice of kernel, there may seem to be new terms such as h2ii or hijkk hij , etc., but these terms can be eliminated using the equation of motion, (−∇2 + λ−2 )h = 0. This is the business about eliminating redundant terms by “matching on-shell” we mentioned in the very beginning.
102
Appendix E Induced monopoles on a film and hresp eff (r) = −
C ℓ ℓ! Kℓ (r/λ) cos ℓϕ , σ 2πλℓ
respectively, yielding C (ℓ) =
(E.5)
2πRℓ λℓ σ . ℓ!Kℓ (R/λ)
(E.6)
Expanding the Bessel functions, one observes that for ℓ > 0 these polarizabilities converge to those in Eq. (1.35) as λ → ∞, whereas the massless limit of the monopole polarizability is 2πσ γ as λ → ∞ , C (0) → − (E.7) e R log 2λ where γ is the Euler-Mascheroni constant. Pair interaction We observe that the monopole polarizability vanishes like 1/ log(R/λ) in the massless limit of infinite capillary length that we eventually want to take. This means diagrams involving monopoles could vanish as well, unless every factor of 1/ log(R/λ) due to a monopole polarizability is canceled by a similar factor in the numerator. Factors of this form indeed exist: they come from monopole-monopole links in the propagator product, since in the massless limit G(r) = −(1/2π) log(eγ r/2λ). Due to the closed topology of the diagrams, there are enough monopole-monopole links to balance the vanishing monopole polarizabilities only when there are no higher order multipoles in the diagram; replacing one monopole polarizability in the diagram costs two monopole-monopole links. In other words, the only monopole interactions that do not vanish in the massless limit are those with other induced monopoles and nothing else. This elucidates and generalizes the findings of Lehle and Oettel that monopole-dipole and monopole-quadrupole interactions indeed vanish [LO07]. We can now write the pair interaction between two pinned particles, on a capillary surface for which the capillary length tends to infinity. The interaction will consist of Eq. (1.61) due to induced multipoles of order ℓ > 0 and, based on the discussion of the previous paragraph, a part βF mon that is due solely to monopole polarizabilities. Observe that in the latter, propagators do not carry any derivatives and therefore all the cumulants are of the same order in the inter-particle separation. Hence, all cumulants must be summed for the monopole interactions. To evaluate this, one can refer to Eq. (1.60), keeping in mind that there is no sum over multipole orders ℓi now; only the monopoles are taken. Assuming identical particle radii R to declutter expressions, one finds ∞
βF
mon
1X1 =− 2 s s=1
C (0) G(r) σ
!2s
103
∞
1X1 =− 2 s s=1
K0 (r/λ) K0 (R/λ)
2s
.
(E.8)
Appendix E Induced monopoles on a film Owing to the damping of the regularized theory, this series converges for all r (≪ λ → ∞) to 1 K02 (r/λ) mon βF = log 1 − 2 2 K0 (R/λ) 1 1 = log [K0 (R/λ) − K0 (r/λ)] + log [K0 (R/λ) + K0 (r/λ)] + const. (E.9) 2 2 When the massless limit λ → ∞ is taken, the first term on the right hand side of the last equality gives 1 r βF mon = log log , (E.10) 2 R in agreement with Ref. [LO07]. The second term is proportional to log log(λ2 /Rr), which is associated with a force ∼ 1/r log(Rr/λ2 ) → 0 as √λ → ∞. We note that extension to particles of unequal radii results in the change R → R1 R2 in Eq. (E.10).
104
List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15
Monge parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigid particle motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformations emanating from a disk . . . . . . . . . . . . . . . . . . . . . Second cumulant diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuation-induced dipole-dipole and quadrupole-dipole interactions . . . Example diagrams from the fourth cumulant . . . . . . . . . . . . . . . . b . . . . . . . . . . . . . . . . . . . . . . Illustration of the vertex angle ϕac Fluctuation-induced pair interactions of order R8 /r8 . . . . . . . . . . . . Fluctuation-induced pair interactions of order R10 /r10 . . . . . . . . . . . Some pair interaction diagrams . . . . . . . . . . . . . . . . . . . . . . . . Distinct Wick contractions for the sixth cumulant pair interactions . . . . Truncated series versus numerics for the fluctuation-induced pair potential Leading three- and four-body fluctuation-induced interactions . . . . . . . A few elastic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . A saddle-shaped particle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
6 8 11 20 22 25 26 27 28 28 30 31 35 37 38
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19
Example diagrams for the membrane effective theory . . . . . . . . Different Wick contractions for membrane interaction diagrams . . Entropic quadrupole-quadrupole interaction . . . . . . . . . . . . . Symbolic diagrams for the entropic interactions of order R6 /r6 and Entropic quadrupole-octupole interaction . . . . . . . . . . . . . . Entropic quadrupole-hexadecapole interaction . . . . . . . . . . . . Entropic octupole-octupole interaction . . . . . . . . . . . . . . . . Entropic interaction between four induced quadrupoles . . . . . . . Entropic pair force between rigid disks on a membrane . . . . . . . Leading entropic triplet and quadruplet interactions . . . . . . . . Leading entropic triplet diagrams . . . . . . . . . . . . . . . . . . . b . . . . . . . . . . . . . . . . . . Illustration of the vertex angle ϕac Strength of the leading triplet interaction . . . . . . . . . . . . . . Diagrams for the leading entropic quadruplet interaction . . . . . . Two permanent sources of second derivative order . . . . . . . . . Curvature-elastic pair interactions on a membrane . . . . . . . . . Elastic interaction between two permanent quadrupoles . . . . . . Interaction of a permanent quadrupole with its quadrupole image . Direct interaction of the long and short tails of two saddles . . . .
. . . . . . . . . . . . . . . . . . .
51 53 54 55 56 57 58 59 61 62 62 63 64 65 67 69 70 71 72
105
. . . . . . . . . . . . R8 /r8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures pair interactions of order R6 /r6 between axisymmetric particles . . . triplet interactions up to O(R6 /r6 ) between axisymmetric particles . triplet interactions from the third cumulant . . . . . . . . . . . . . . triplet interactions from the fourth cumulant . . . . . . . . . . . . .
73 74 75 76
b . . . . . . . . . . . . . . . . . . . . . . . B.1 Illustration of the vertex angle ϕac
91
2.20 2.21 2.22 2.23
Elastic Elastic Elastic Elastic
106
List of Technical Notes 1.1 1.2 1.3 1.4 1.5
Cumulant expansion . . . . . . . . . . . . . . . . . Euler-Lagrange equation in the presence of sources Connected vs. disconnected . . . . . . . . . . . . . Self-interactions . . . . . . . . . . . . . . . . . . . . Preferred backgrounds for contact line multipoles .
. . . . .
14 15 21 24 39
2.1 2.2 2.3
Gauss-Bonnet theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary conditions for the disk-on-membrane BVP . . . . . . . . . . . . . Derivatives of the biharmonic Green function . . . . . . . . . . . . . . . . .
42 45 49
107
. . . . .
. . . . .
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. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
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. . . . .
. . . . .
Bibliography [AG09]
T. Auth and G. Gompper, Budding and vesiculation induced by conical membrane inclusions, Physical Review E 80, 031901 (2009).
[AJL+ 83]
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland Science, New York (1983).
[AS74]
M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, Dover, New York (1974).
[BBC+ 03]
A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G. Nikolaides, A. Travesset and D. A. Weitz, Grain boundary scars and spherical crystallography, Science 299, 1716–1718 (2003).
[BDF10]
A.-F. Bitbol, P. G. Dommersnes and J.-B. Fournier, Fluctuations of the Casimir-like force between two membrane inclusions, Physical Review E 81, 050903(R) (2010).
[BDFN95]
J. J. Binney, N. J. Dowrick, A. J. Fisher and M. E. J. Newman, The theory of critical phenomena, Clarendon Press, Oxford (1995).
[BE95]
T. W. Burkhardt and E. Eisenriegler, Casimir interaction of spheres in a fluid at the critical point, Physical Review Letters 74, 3189–3192 (1995).
[BEM91]
M. Bloom, E. Evans and O. G. Mouritsen, Physical properties of the fluid lipid bilayer component of cell membranes: a perspective, Quarterly Reviews of Biophysics 24, 293–397 (1991).
[BF03]
D. Bartolo and J.-B. Fournier, Elastic interaction between hard or soft pointwise inclusions on biological membranes, European Physical Journal E 11, 141–146 (2003).
[Bin02]
B. P. Binks, Particles as surfactants – similarities and differences, Current Opinion in Colloid and Interface Science 7, 21–41 (2002).
[BKT+ 09]
M. A. Bucaro, P. R. Kolodner, J. A. Taylor, A. Sidorenko, J. Aizenberg and T. N. Krupenkin, Tunable liquid optics: electrowetting-controlled liquid Mirrors based on self-assembled Janus tiles, Langmuir 25, 3876–3879 (2009).
[BMM01]
M. Bordag, U. Mohideen and V. M. Mostepanenko, New developments in the Casimir effect, Physics Reports 353, 1–205 (2001).
108
Bibliography [BSR00]
A. B. D. Brown, C. G. Smith and A. R. Rennie, Fabricating colloidal particles with photolithography and their interactions at an air-water interface, Physical Review E 62, 951–960 (2000).
[Cas48]
H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proceedings of the Koninklijke Nederlandse Academie van Wetenschappen 51, 793–795 (1948).
[CFHQ02]
D. J. Campbell, E. R. Freidinger, J. M. Hastings and M. K. Querns, Spontaneous assembly of soda straws, Journal of Chemical Education 79, 201 (2002).
[CH53]
R. Courant and D. Hilbert, Methods of mathematical physics, Interscience Publishers, New York (1953).
[CHJR81]
D. Y. C. Chan, J. D. Henry Jr. and W. L. R., The interaction of colloidal particles collected at fluid interfaces, Journal of Colloid and Interface Science 79, 410–418 (1981).
[CJBL+ 11]
M. Cavallaro Jr., L. Botto, E. P. Lewandowski, M. Wang and K. J. Stebe, Curvature-driven capillary migration and assembly of rod-like particles, Proceedings of the National Academy of Sciences 108, 20923–20928 (2011).
[CSRS10]
I. Casuso, P. Sens, F. Rico and S. Scheuring, Experimental evidence for membrane-mediated protein-protein interaction, Biophysical Journal 99, 47– 49 (2010).
[Dea98]
J. A. Dean (ed.), Lange’s handbook of chemistry, 15th ed., McGraw-Hill, New York (1998).
[Der34]
B. V. Derjaguin, Analysis of friction and adhesion IV The theory of the adhesion of small particles, Kolloid Zeitschrift 69, 155–164 (1934).
[DF99a]
P. G. Dommersnes and J.-B. Fournier, Casimir and mean-field interactions between membrane inclusions subject to external torques, Europhysics Letters 46, 256–261 (1999).
[DF99b]
P. G. Dommersnes and J.-B. Fournier, N-body study of anisotropic membrane inclusions: membrane mediated interactions and ordered aggregation, European Physical Journal B 12, 9–12 (1999).
[DGBWQ04] P.-G. De Gennes, F. Brochard-Wyart and D. Qu´er´e, Capillarity and wetting phenomena, Springer, New York (2004). [DKYN95]
C. D. Dushkin, P. A. Kralchevsky, H. Yoshimura and K. Nagayama, Lateral capillary forces measured by torsion microbalance, Physical Review Letters 75, 3454–3457 (1995).
109
Bibliography [DPK01]
K. D. Danov, B. Pouligny and P. A. Kralchevsky, Capillary forces between colloidal particles confined in a liquid film: the finite meniscus problem, Langmuir 17, 6599–6609 (2001).
[DVK+ 92]
N. Denkov, O. Velev, P. Kralchevski, I. Ivanov, H. Yoshimura and K. Nagayama, Mechanism of formation of two-dimensional crystals from latex particles on substrates, Langmuir 8, 3183–3190 (1992).
[DVK+ 93]
N. Denkov, O. Velev, P. Kralchevski, I. Ivanov, H. Yoshimura and K. Nagayama, Two-dimensional crystallization, Nature 361, 26–26 (1993).
[EGJK08]
T. Emig, N. Graham, R. L. Jaffe and M. Kardar, Casimir forces between compact objects: the scalar case, Physical Review D 77, 025005 (2008).
[ER95]
E. Eisenriegler and U. Ritschel, Casimir forces between spherical particles in a critical fluid and conformal invariance, Physical Review B 51, 13717–13734 (1995).
[FRT+ 01]
K. Farsad, N. Ringstad, K. Takei, S. R. Floyd, K. Rose and P. De Camilli, Generation of high curvature membranes mediated by direct endophilin bilayer interactions, Journal of Cell Biology 155, 193–200 (2001).
[Gam09]
A. Gambassi, The Casimir effect: from quantum to critical fluctuations, Journal of Physics: Conference Series 161, 012037 (2009).
[GBP93a]
M. Goulian, R. Bruinsma and P. Pincus, Long-range forces in heterogeneous fluid membranes, Europhysics Letters 22, 145–150 (1993).
[GBP93b]
M. Goulian, R. Bruinsma and P. Pincus, Long-range forces in heterogeneous fluid membranes (erratum), Europhysics Letters 23, 155 (1993).
[Geo91]
H. Georgi, On-shell effective field theory, Nuclear Physics B 361, 339–350 (1991).
[GGK96a]
R. Golestanian, M. Goulian and M. Kardar, Fluctuation-induced interactions between rods on a membrane, Physical Review E 54, 6725–6734 (1996).
[GGK96b]
R. Golestanian, M. Goulian and M. Kardar, Fluctuation-induced interactions between rods on membranes and interfaces, Europhysics Letters 33, 241–245 (1996).
[GLR10]
C. R. Galley, A. K. Leibovich and I. Z. Rothstein, Finite size corrections to the radiation reaction force in classical electrodynamics, Physical Review Letters 105, 094802 (2010).
[GR06]
W. D. Goldberger and I. Z. Rothstein, An effective field theory of gravity for extended objects, Physical Review D 73, 104029 (2006).
110
Bibliography [GS71]
W. A. Gifford and L. E. Scriven, On the attraction of floating particles, Chemical engineering science 26, 287–297 (1971).
[GSF05]
H. Gao, W. Shi and L. B. Freund, Mechanics of receptor-mediated endocytosis, Proceedings of the National Academy of Sciences 102, 9469–9474 (2005).
[GYX99]
B. Gates, Y. Yin and Y. Xia, Fabrication and characterization of porous membranes with highly ordered three-dimensional periodic structures, Chemistry of Materials 11, 2827–2836 (1999).
[HACB03]
T. S. Horozov, R. Aveyard, J. H. Clint and B. P. Binks, Orderdisorder transition in monolayers of modified monodisperse silica particles at the octanewater interface, Langmuir 19, 2822–2829 (2003).
[HBS98]
B. T. Holland, C. F. Blanford and A. Stein, Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids, Science 281, 538–540 (1998).
[Hel73]
W. Helfrich, Elastic properties of lipid bilayers – theory and possible experiments, Zeitschrift f¨ ur Naturforschung C 28, 693–703 (1973).
[HSED98]
A. Hanke, F. Schlesener, E. Eisenriegler and S. Dietrich, Critical Casimir forces between spherical particles in fluids, Physical Review Letters 81, 1885– 1888 (1998).
[HW01]
W. Helfrich and T. R. Weikl, Two direct methods to calculate fluctuation forces between rigid objects embedded in fluid membranes, European Physical Journal E 5, 423–439 (2001).
[IER+ 05]
T. Itoh, K. S. Erdmann, A. Roux, B. Habermann, H. Werner and P. De Camilli, Dynamin and the Actin Cytoskeleton Cooperatively Regulate Plasma Membrane Invagination by BAR and F-BAR Proteins, Developmental Cell 9, 791–804 (2005).
[KDD01]
P. A. Kralchevsky, N. D. Denkov and K. D. Danov, Particles with an undulated contact line at a fluid interface: interaction between capillary quadrupoles and rheology of particulate monolayers, Langmuir 17, 7694–7705 (2001).
[KG99]
M. Kardar and R. Golestanian, The “friction” of vacuum, and other fluctuation-induced forces, Reviews of Modern Physics 71, 1233–1245 (1999).
[KNO98]
K. S. Kim, J. Neu and G. Oster, Curvature-mediated interactions between membrane proteins, Biophysical Journal 75, 2274–2291 (1998).
111
Bibliography [Koz07]
M. M. Kozlov, Biophysics – bending over to attract, Nature 447, 387–389 (2007).
[KPIN92]
P. A. Kralchevsky, V. N. Paunov, I. B. Ivanov and K. Nagayama, Capillary meniscus interaction between colloidal particles attached to a liquid-fluid interface, Journal of Colloid and Interface Science 151, 79–94 (1992).
[Kre91]
E. Kreyszig, Differential geometry, Dover, Mineola, NY (1991).
[Kre94]
M. Krech, Casimir effect in critical systems, World Scientific, Singapore (1994).
[KRS99]
I. Koltover, J. O. R¨adler and C. R. Safinya, Membrane mediated attraction and ordered aggregation of colloidal particles bound to giant phospholipid vesicles, Physical Review Letters 82, 1991–1994 (1999).
[LAZY05]
J. C. Loudet, A. M. Alsayed, J. Zhang and A. G. Yodh, Capillary interactions between anisotropic colloidal particles, Physical Review Letters 94, 018301 (2005).
[LCJB+ 10]
E. P. Lewandowski, M. Cavallaro Jr., L. Botto, J. C. Bernate, V. Garbin and K. J. Stebe, Orientation and self-assembly of cylindrical particles by anisotropic capillary interactions, Langmuir 26, 15142–15154 (2010).
[LK91]
H. Li and M. Kardar, Fluctuation-induced forces between rough surfaces, Physical Review Letters 67, 3275–3278 (1991).
[LK92]
H. Li and M. Kardar, Fluctuation-induced forces between manifolds immersed in correlated fluids, Physical Review A 46, 6490–6500 (1992).
[LO07]
H. Lehle and M. Oettel, Importance of boundary conditions for fluctuationinduced forces between colloids at interfaces, Physical Review E 75, 011602 (2007).
[LOD06]
H. Lehle, M. Oettel and S. Dietrich, Effective forces between colloids at interfaces induced by capillary wavelike fluctuations, Europhysics Letters 75, 174–180 (2006).
[LS95]
R. Lipowsky and E. Sackmann (eds.), Structure and dynamics of membranes, Elsevier, Amsterdam (1995).
[LZMP11]
H.-K. Lin, R. Zandi, U. Mohideen and L. P. Pryadko, Fluctuation-induced forces between inclusions in a fluid membrane under tension, Physical Review Letters 107, 228104 (2011).
[MDG05a]
M. M. M¨ uller, M. Deserno and J. Guven, Geometry of surface-mediated interactions, Europhysics Letters 69, 482–488 (2005).
112
Bibliography [MDG05b]
M. M. M¨ uller, M. Deserno and J. Guven, Interface-mediated interactions between particles: a geometrical approach, Physical Review E 72, 061407 (2005).
[MG05]
H. T. McMahon and J. L. Gallop, Membrane curvature and mechanisms of dynamic cell membrane remodelling, Nature 438, 590–596 (2005).
[Mil01]
K. A. Milton, The Casimir effect: physical manifestations of zero-point energy, World Scientific, Singapore (2001).
[MM02]
V. I. Marchenko and C. Misbah, Elastic interaction of point defects on biological membranes, European Physical Journal E 8, 477–484 (2002).
[MMY+ 12]
R. Murakami, H. Moriyama, M. Yamamoto, B. P. Binks and A. Rocher, Particle stabilization of oil-in-water-in-air materials: powdered emulsions, Advanced Materials 24, 767 (2012).
[M¨ ul07]
M. M. M¨ uller, Theoretical studies of fluid membrane mechanics, Ph.D. thesis, Johannes Gutenberg Universit¨at in Mainz (2007).
[MZ]
U. Mohideen and R. Zandi, private communication.
[Net97]
R. R. Netz, Inclusions in fluctuating membranes: exact results, Journal de Physique I (France) 7, 833–852 (1997).
[Nic49]
M. M. Nicolson, The interaction between floating particles, Proceedings of the Cambridge Philosophical Society 45, 288–295 (1949).
[NO09]
E. Noruzifar and M. Oettel, Anisotropies in thermal Casimir interactions: ellipsoidal colloids trapped at a fluid interface, Physical Review E 79, 051401 (2009).
[NPW89]
D. R. Nelson, T. Piran and S. Weinberg, Statistical mechanics of membranes and surfaces, World Scientific, Singapore (1989).
[Pic07]
S. U. Pickering, CXCVI.-Emulsions, Journal of the Chemical Society, Transactions 91, 2001–2021 (1907).
[PKDN93]
V. N. Paunov, P. A. Kralchevsky, N. D. Denkov and K. Nagayama, Lateral capillary forces between submillimeter particles, Journal of Colloid and Interface Science 157, 100–112 (1993).
[PL96]
J.-M. Park and T. C. Lubensky, Interactions between membrane inclusions on fluctuating membranes, Journal de Physique I (France) 7, 1217–1235 (1996).
[PRR11]
R. A. Porto, A. Ross and I. Z. Rothstein, Spin induced multipole moments for the gravitational wave flux from binary inspirals to third post-Newtonian order, Journal of Cosmology and Astroparticle Physics 03, 009 (2011).
113
Bibliography [Ram03]
W. Ramsden, Separation of solids in the surface-layers of solutions and ‘suspensions’ (Observations on surface-membranes, bubbles, emulsions, and mechanical coagulation) – preliminary account, Proceedings of the Royal Society of London 72, 156–164 (1903).
[RIH+ 07]
B. J. Reynwar, G. Illya, V. A. Harmandaris, M. M. M¨ uller, K. Kremer and M. Deserno, Aggregation and vesiculation of membrane proteins by curvaturemediated interactions, Nature 447, 461–464 (2007).
[Rio68]
J. Riordan, Combinatorial identities, Wiley, New York (1968).
[Rot11]
I. Z. Rothstein, Partition functions on the Euclidean plane with compact boundaries in conformal and non-conformal theories, arXiv:1111.0533v1 (2011).
[Saf94]
S. A. Safran, Statistical thermodynamics of surfaces, interfaces, and membranes, Perseus, Cambridge (1994).
[SDJ00]
D. Stamou, C. Duschl and D. Johannsmann, Long-range attraction between colloidal spheres at the air-water interface: the consequence of an irregular meniscus, Physical Review E 62, 5263–5272 (2000).
[SJ05]
P. Singh and P. P. Joseph, Fluid dynamics of floating particles, Journal of Fluid Mechanics 530, 31–80 (2005).
[SVR06]
Y. Shibata, G. K. Voeltz and T. A. Rapoport, Rough Sheets and Smooth Tubules, Cell 126, 435–439 (2006).
[TSHDC99]
K. Takei, V. I. Slepnev, V. Haucke and P. De Camilli, Functional partnership between amphiphysin and dynamin in clathrin-mediated endocytosis, Nature Cell Biology 1, 33–39 (1999).
[VM05]
D. Vella and L. Mahadevan, The “Cheerios effect”, American Journal of Physics 73, 817–825 (2005).
[vNSH05]
E. A. van Nierop, M. A. Stijnman and S. Hilgenfeldt, Shape-induced capillary interactions of colloidal particles, Europhysics Letters 72, 671–677 (2005).
[VPS+ 06]
G. K. Voeltz, W. A. Prinz, Y. Shibata, J. M. Rist and T. A. Rapoport, A class of membrane proteins shaping the tubular endoplasmic reticulum, Cell 124, 573–586 (2006).
[WK74]
K. G. Wilson and J. Kogut, The renormalization group and the ǫ expansion, Physics Reports 12, 75–200 (1974).
[WKH98]
T. R. Weikl, M. M. Kozlov and W. Helfrich, Interaction of conical membrane inclusions: effect of lateral tension, Physical Review E 57, 6988–6995 (1998).
114
Bibliography [YRD12]
C. Yolcu, I. Z. Rothstein and M. Deserno, Effective field theory approach to fluctuation-induced forces between colloids at an interface, Physical Review E 85, 011140 (2012).
[YSLY11]
P. J. Yunker, T. Still, M. A. Lohr and A. G. Yodh, Suppression of the coffeering effect by shape-dependent capillary interactions, Nature 476, 308–311 (2011).
[Zee03]
A. Zee, Quantum field theory in a nutshell, Princeton University Press, Princeton (2003).
[ZK06]
J. Zimmerberg and M. M. Kozlov, How proteins produce cellular membrane curvature, Nature Reviews Molecular Cell Biology 7, 9–19 (2006).
[ZM00]
K. Zahn and G. Maret, Dynamic criteria for melting in two dimensions, Physical Review Letters 85, 3656–3659 (2000).
115
