EUROPHYSICS LETTERS OFFPRINT Vol. 64 • Number 5 • pp. 620–626

Anomalous scaling of multivalued interfaces ∗∗∗ ´ and D. Casero A. Bru

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EUROPHYSICS LETTERS

1 December 2003

Europhys. Lett., 64 (5), pp. 620–626 (2003)

Anomalous scaling of multivalued interfaces ´ 1 and D. Casero 2 A. Bru 1

CCMA, Consejo Superior de Investigaciones Cient´ıficas Serrano 115 dpdo, 28006 Madrid, Spain 2 Departamento de Matem´ atica Aplicada, Universidad Complutense de Madrid Avda. Complutense s/n, 28040 Madrid, Spain (received 17 March 2003; accepted in final form 2 October 2003) PACS. 05.60.Cd – Classical transport. PACS. 02.50.Ey – Stochastic processes. PACS. 89.75.Fb – Structures and organization in complex systems.

Abstract. – The main goal of this letter is the scaling analysis of multivalued interfaces showing anomalous height fluctuations. A model of two-fluid transport through porous media is studied by means of dynamical scaling techniques. When the displacement is unstable, the interfaces are multivalued. In this paper we compare three definitions of the height function h(x, t). In our model we prove the kind of scaling to depend on the height function definition, and those anomalous fluctuations not to exist actually, as would be in many models and experiments exhibiting multivalued interfaces.

Disorder in nature has been a challenge for science in the last decades. Since the development of fractal geometry, the dynamics of non-equilibrium growing processes in disordered systems has often been described in terms of the so-called dynamical scaling [1, 2]. In most cases, growth processes develop self-affine rough interfaces. This scale invariance is displayed by the spatial and temporal evolution of the fluctuations of the height func¯ l (t). For instance, we can measure the local intertion h(x, t) around its mean value h ¯ l (t)]2 1/2  or the local height-height correlation function face width w(l, t) = [h(x, t) − h l C(l, t) = [h(x + l, t) − h(x, t)]2 l , where  l denotes spatial average inside a window of size l, and   is an average over windows of size l. The computation of these functions in the early models proposed for growing interfaces, like EW or KPZ [3, 4], led to the so-called FamilyVicsek ansatz [2]: C(l, t) ∼ w2 (l, t) ∼ t2β if t  ts and C(L, t) ∼ L2α if t  ts for a system of size L, where β and α are the growth and roughness critical exponents, respectively, ts ∼ Lz is the global saturation time, and z = α/β is the dynamical exponent. This framework has been successfully used to analyse a number of models. A common aim of scientists has been to classify growth processes in a few universality classes. However, the number of models exhibiting a so-called anomalous scaling behavior has considerably increased over the last decade. Actually, during the last years a number of models have been considered that cannot be accounted for in terms of the Family-Vicsek ansatz [5–14]. Most of them are related with unstable regimes, grooved phases and models of columnar or Laplacian growth. The scaling analysis of some of these models led to the so-called anomalous scaling ansatz (by which we denote a scaling behavior which is not encompassed by the Family-Vicsek ansatz). As

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concerns the interface width function this anomalous scaling means
tα/z , if t  lz , w(l, t) = β ∗ α t l loc , if t  lz . The main property of such scaling is the existence of non-saturated spatial fluctuations growing with a β ∗ exponent at all length scales. Thus, small-scale fluctuations do not saturate at t ∼ lz but instead do so when the system as a whole saturates at t ∼ Lz . These additional fluctuations produce a lack of self-affinity; the spatial scaling of such interfaces has to be described with two different exponents at local and global scales, αloc and α, respectively. Anomalous scaling was proposed to describe the phenomenon of super-roughness, when α > 1, and αloc = 1, in models concerning epitaxial growth under the MBE universality class [5–7]. A grooved phase with large local slopes seems to be responsible for anomalous scaling in these models [7–15]. Super-roughness has also been observed in simulations and experiments of Laplacian growth [9]. However, anomalous scaling also emerges from models with αloc < 1, so, it is referred to as intrinsic anomalous scaling [8, 12, 16]. We have found this type of anomalous scaling by inspecting the dynamics of interfaces in a standard model for transport through porous media, where the critical exponents depend on the control parameter. Anomalous scaling is observed when growth becomes unstable. In contrast to local models, the growth is driven by the global pressure field. Then, large local pressure gradients at the tip of the instabilities inhibit the growth in a large region of the system. There is a non-conserved mass distribution around the growth zone unlike in solidon-solid (SOS) models. So, because of non-locality, the system develops multivalued interfaces. The main goal of this letter is the scaling analysis of such interfaces in terms of the definition of the height function. We will show that anomalous scaling only appears in some cases. An overall description and a dynamical characterization of the model can be found in [17– 19]. We use a square network (1024×1024) of interconnected pores and throats. A fluid with a viscosity µ1 fills the whole network at t = 0. A fluid with a viscosity µ2 is injected through one side (y = 0) at a pressure P above the reference pressure P0 = 0 at the opposite side y = L. Periodic boundary conditions are imposed at x = 0, L. The viscosities ratio, M = µ1 /µ2 , is the control parameter in the absence of capillary forces [18]. We use Darcy’s law to calculate the fluid flow in every throat as well as Poiseuille’s law for conductivities, and mass conservation is imposed. We obtain a weighted Laplace equation for the induced pressure field in a four-point discrete version, which is solved iteratively with the Gauss-Seidel over-relaxation method [17]. In this way, we are able to conduce the displacement in a deterministic fashion, with a given injected volume at every step. Randomness is provided by the volumes of pores and radii of cylindrical throats (whose length is equal to 1), which obey a uniform distribution function. By tuning the control parameter M it is possible to obtain many dynamical behaviors of the system from stable (M  1) to unstable (M  1) displacement. In this paper we will be concerned only with the case of fairly unstable dynamics, M  1 [20]. In fig. 1 we show three typical interfaces of the model from a simulation for M = 10. One can observe the so-called fingers having a characteristic width λf . This is a well-known fact: systems which are unstable because of a density or viscosity contrast develop such fingers whose width depends on the system size [22] corresponding to the more unstable mode characterizing Laplacian-like growth [23]. The typical definition of interface [9, 17, 24, 25], that we call h(1) (x, t), can be written as (1)

hi (x, t) = max[j : Aij = 1]j=1,...,N . i

Here x represents a one-dimensional space variable and Aij is a matrix that codify the invaded domain, that is, Aij = 1, if the pore (i, j) has been reached by the µ2 -fluid, and 0, otherwise.

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Fig. 1 – Several interfaces from simulations at three different times. Also shown are the univalued interfaces using the h(1) (x, t) definition.

Therefore, this definition corresponds to the outer envelope of the invaded domain. One expects this method to provide a good description of the scaling behavior of self-affine interfaces at sufficiently large scales [25]. However, it has been argued that some discrete models do not belong to the expected universality class because of overhanging [26, 27]. In our case, this definition introduces large local slopes, and the interface looks like a Heaviside function with a characteristic step size (see fig. 1). A second definition for the interface height, h(2) (x, t), is given as follows. The x-coordinate of a point at the interface is now parametrized by the arc-length measured along the actual interface [27], while the height is measured from the bottom side again. This definition picks up the whole interface, so the size of the system is not constant. A vertical line would thus be mapped into a straight line with a slope equal to 1. Then, the lateral sides of a finger are mapped into smooth hills with a mean slope close to 1. Finally, we propose a third definition, h(3) (x, t), in an attempt to eliminate the fictitious slopes introduced by the aforementioned definitions. In order to obtain the scaling functions w(l, t) or C(l, t) for a window Λ of length l, we define a new substrate being parallel to the local orientation of the interface. To do so, we compute the normal vector ni at every point in Λ using the nearest-neighbor interface points xl and xr , that is, ni ⊥(xr − xl ). The  normal vector of Λ is defined as n(Λ, l) = i∈Λ ni . This vector allows us to define a reference straight line, say C, so that n(Λ, l)⊥C. The line C is used to define a new height function as h(3) (x, t) = dist(x, C) (the distance being positive or negative, since fluctuations are computed over the mean position of the interface points). The x-coordinate is again measured along the arclength. We expect this method to provide good results at small scales. In fact, we have obtained the same scaling behavior and critical exponents with the three methods for stable displacement (M  1) [19], but this is no longer the case when the system becomes unstable. In fig. 2 we show the temporal evolution of w(l, t) for different window sizes when h(1) (x, t) is used. It is easy to see that small-scale fluctuations w(l, t) with l  L do not saturate gradually, but they instead saturate when w(l, t) with l → L saturates at ts ∼ Lz . These non-saturating fluctuations are, as we have remarked above, the footprint of anomalous scaling [6, 7, 28]. In fact, the procedure followed in the definition of h(1) (x, t) connects, via the jumps showed in fig. 1, points belonging to different regions, i.e., the active and screened ones.

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Fig. 2 – Plot of w(l, t) vs. time for different window sizes (l = 8, 16, 32, 64, 128, 256) using h(2) (x, t). Inset: The same plot using h(1) (x, t).

Therefore, we lump together in small length scales points which grow with very different rates. This is the source of non-saturating fluctuations at all scales. Moreover, we can estimate the value of the anomalous-growth exponent from the slope of the small-scale width fluctuations to be β ∗ ∼ 0.75. The same value is obtained from the nearest-neighbor height-height corre∗ lation function C(l = 1, t) ∼ t2β . The real existence of a non-zero β ∗ exponent for h(1) (x, t) implies a lack of self-affinity (αloc = α) as a result of the enveloping procedure. However, using h(2) (x, t) we obtain a C(l = 1, t) function that saturates at short times. We show the results for w(l, t) corresponding to this definition in fig. 2. In this case, we observe a behavior corresponding to correlations that spread along the interface in a standard fashion: one can see that small-scale fluctuations saturate before the larger ones do. It is worth mentioning that, in both cases, the measured value for the growth exponent β is the same (within statistical errors). This is due to the fact that the active or growing zone is picked up by both definitions. In the same way, it is easy to understand why β ∗ = 0 for h(2) (x, t). When the instability arises, a few fingers are formed. The greater the height of the fingers, the greater the length of the interface. This fact allows the fluctuations produced at the active zone to reach greater length scales and no to be confined at small length scales. Additional features concerning the h(1) (x, t) definition follow. In the inset of fig. 3 we show the square root of the height-height correlation function vs. window size l at several times. The temporal shift is clear. We found systematically a power law behavior with a critical exponent around αloc ∼ 0.5, intrinsically related with the enveloping procedure (using the definition of w(l, t) one can show that αloc = 0.5 for the Heaviside function). Therefore, we should not relate that value with a dynamics corresponding to annealed versions of continuous equations like EW or KPZ. w(l, t) and C(l, t) have been generally used indistinctly because they have analogous power law behavior. Nevertheless, C(l, t) has been widely used in simulations and analytical approaches because, on the one hand, it has a more useful relationship with the power spectrum and, on the other hand, it has greater sensibility to height differences at the interface [10]. According to this, we have found some differences among w(l, t) and C(l, t). The scaling behavior of w(l, t) seems to be consistent with previous works [25]. Indeed, at very small length scales, we find a power law behavior with αloc ∼ 0.8–1.0, but at intermediate scales, we find again αloc ∼ 0.5. Anyway, a clear temporal shift for w(l, t) can be observed as well.

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Fig. 3 – Square root of the height-height correlation function at several times using h(2) (x, t). No temporal shift is observed. Inset: the same plot using h(1) (x, t). The temporal shift is apparent.

We now turn our discussion into the h(2) (x, t) definition. Figure 3 shows plots of C(l, t) vs. l at several times. The local roughness exponent is found to be αloc ∼ 0.8 at length scales smaller than the correlation length. One can fairly see that there is no temporal shift, so that the scaling behavior could be explained in terms of a standard scaling ansatz. A similar scaling behavior is obtained for w(l, t) in this case. Because of the previous discussion, we think that these results ensure the self-affinity of interfaces in terms of h(2) (x, t). However, we expect the αloc exponent to be just an effective value. A “stretching” procedure is performed when the interface is parametrized with the arc-length variable, and is mapped into a pyramidal-like structure. We thus obtain a local roughness exponent lying between the theoretical value of a pure triangle (αloc = 1) and the actual value, that would describe the local fluctuations at our interfaces. In fact, the power spectrum (which scales as S(k, t) ∼ k −(2α+1) , see [1, 2]) of these interfaces shows a super-rough phase at large scales (small k), α ∼ 1.4 (close to the computed value for a pure triangle α = 1.5). At intermediate scales, self-affinity is recovered and a global exponent α ∼ 0.8 is obtained. Most of the models describing growing interfaces are based on local rules. In discrete models, the particle deposition rate plays the role of the driving force. In continuum Langevin-like equations a constant term is used instead. In both cases, the interface is forced to move perpendicularly to the substratum. The theoretical scaling behavior is obtained for interfaces obeying this condition. In our model, the global pressure field remains almost uniform along the interface because M  1, so that the pressure gradient at the interface is pointing to the normal direction at every point. The small-scale fluctuations and relaxation mechanisms are governed by the local fluctuations of the pressure field (because the randomness of the medium) and the flow laws. So, we have to measure local fluctuations in the normal direction to the interface. This is the rationale leading to our third definition h(3) (x, t). As we have stated above, this method provides the same results for stable displacement. For M  1, we expect this definition to be valid just below a critical length lc . For l > lc we believe that the direction of the local pressure gradients inside a window Λ will fluctuate too much to be described just with a single n(Λ, l). It is worth mentioning that, if anomalous fluctuations existed in our model, this method should pick them up. Figure 4 shows w(l, t) vs. l at several times. We highlight again that this method describes fluctuations saturating progressively in time without temporal shift.

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Fig. 4 – Width function w(l, t) vs. l at several times using h(3) (x, t). One can observe power law behavior obeying a standard dynamical scaling.

Once more, a standard scaling behavior is noticed. However, the critical exponents describing the local fluctuations are somewhat different. At very small length scales we obtain systematically a power law behavior with αloc ∼ 1. This is due to the fact that a window containing only a few points is mapped into an almost straight line. Above this cut-off, a new phase comprising almost two decades arises, obeying a non-trivial scaling behavior with αloc ∼ 0.67. We believe that these results describe properly the dynamics of our system. We have seen that growth mostly take place at the tip of the fingers because of the greater pressure gradients. A pinned region is abandoned behind as a quenched interface with local fluctuations governed by the random medium. So, two kinds of regions, pinned and depinned, coexist at the interface, and h(3) (x, t) is suitable to describe their local fluctuations, since it is not affected by the global geometry. In fact, this is not the first time such a roughness exponent appears in systems with similar dynamical rules [29,30]. We expect our model to show, in some limit, some resemblance with other models having pinned and depinned zones. This is a matter of a future work. Finally, in some works [10] a connection is argued between unstable dynamics, anomalous scaling and spatial multiscaling. Once again, we have checked that, in our simulations, multifractality only appears for height functions h(1) (x, t). Computing the q-th-order height-height correlation function Cq (l) = [h(x + l, t) − h(x, t)]q l 1/q for q = 1, 2, 3, 4, we obtain a power law behavior with αq being a decreasing function of q. In our case, α1 ∼ 0.70, α2 ∼ 0.5, α3 ∼ 0.43, α4 ∼ 0.36. And once more, we believe that this result comes from the large jumps introduced by the convolution of our interfaces and the Heaviside function. The same computation using h(2) (x, t) leads to α1 ∼ 0.80, α2 ∼ 0.82, α3 ∼ 0.84, α4 ∼ 0.85. The difference between these values is so small that allows us to rule out any discussion upon the multifractal nature of our actual interfaces. Anomalous scaling has been proved to be a very useful tool in systems where the FamilyVicsek ansatz fails. In our case, we were initially surprised because we did not know any reasonable physical mechanism to account for it. There is no reason for a grooved phase to be selected, as is the case in other models. Fingers arise in unstable displacement because of the large pressure gradients existing at some points of the interface, but fluctuations are expected to spread out in a standard fashion. We have observed that the very definition of h(1) (x, t) is responsible for such anomalous fluctuations. Using h(2) (x, t) we have obtained

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self-affine interfaces, but the unrealistic large-scale geometry of the interfaces affects the value of the critical exponents. If one assumes that our interfaces are self-affine, we expect the local behavior to be enough to account for a suitable scaling description. This is the reason behind x, t). In this manner, we hope to be able to fit some fundamental aspects the definition of h(3) (¯ of fluid transport through porous media into suitable universality classes. ∗∗∗ One of the authors (DC) acknowledges the support by MCyT (Spain), Grant No. BFM20000605. REFERENCES [1] Meakin P., Fractals, Scaling and Growth Far from Equilibrium (Cambridge University Press) 1998. [2] Family F. and Vicsek T., Dynamics of Fractal Surfaces, edited by Family F. and Vicsek T. (World Scientific, Singapore) 1991. [3] Edwards S. F. and Wilkinson D. R., Proc. R. Soc. London, Ser. A, 17 (1982) 381. [4] Kardar M., Parisi G. and Zhang Y., Phys. Rev. Lett., 56 (1986) 889. ´ A., Pastor J. M., Fernaud I., Bru ´ I., Melle S. and Berenguer C., Phys. Rev. Lett., [5] Bru 81 (1998) 4008. ´ pez J. M. and Valentin G., Phys. Rev. E, 58 (1998) 6999. [6] Morel S., Schmittbuhl J., Lo [7] Das Sarma S., Ghaisas S. V. and Kim J. M., Phys. Rev. E, 49 (1994) 122. [8] Ryu C. S., Heo K. P. and Kim I., Phys. Rev. E, 54 (1996) 284. ´ nchez A. and Dom´ınguez-Adame F., Phys. Rev. E, 57 (1998) [9] Castro M., Cuerno R., Sa R2491. [10] Dasgupta C., Kim J. M., Dutta M. and Das Sarma S., Phys. Rev. E, 55 (1997) 2235. [11] Plischke M., Shore J. D., Schroeder M., Siegert M. and Wolf D. E., Phys. Rev. Lett., 71 (1993) 2509. ´ pez J. M. and Rodr´ıguez M. A., Phys. Rev. E, 54 (1996) 2189. [12] Lo [13] Amar J. G., Lam P. L. and Family F., Phys. Rev. E, 47 (1993) 3242. [14] Siegert M. and Plischke M., Phys. Rev. Lett., 68 (1992) 2035. [15] Krug J., Plischke M. and Siegert M., Phys. Rev. Lett., 70 (1993) 3271. ´ pez J. M., Rodr´ıguez M. A. and Cuerno R., Phys. Rev. E, 56 (1997) 3993. [16] Lo [17] Ferer M. and Smith D. H., Phys. Rev. E, 49 (1994) 4141. [18] Chan D. Y. C., Hughes B. D., Paterson L. and Sirakoff C., Phys. Rev. A, 38 (1988) 4106. ´ A. and Casero D., in preparation. [19] Bru [20] It should be remarked that the time at which the instability arise shows strong run-to-run variations [21]. Because of this, we had to analyze our runs one by one and to average the computed critical exponents afterwards. The figures showed through the paper are examples of single runs. ´ pez J. M. and Rodr´ıguez M. A., Phys. Rev. E, 52 (1995) 6442. [21] Lo ´ ndez J. F. and Albarra ´n J. M., Phys. Rev. Lett., 64 (1990) 2133. [22] Ferna [23] Pastor J. M. and Rubio M. A., Phys. Rev. Lett., 76 (1996) 1848. [24] Birovljev A., Maloy K. J., Feder J. and Jossang T., Phys. Rev. E, 49 (1994) 5431. [25] Nolle C. S., Koiller B., Martys N. and Robbins M. O., Phys. Rev. Lett., 71 (1993) 2074. [26] Ko D. Y. K. and Seno F., Phys. Rev. E, 50 (1994) R1741. [27] Maritan A., Toigo F., Koplik J. and Banavar J. R., Phys. Rev. Lett., 69 (1992) 3193. [28] Schroeder M., Siegert M., Wolf D. E., Shore J. D. and Plischke M., Europhys. Lett., 24 (1993) 563. ´ si A. L., Grinstein G. and Mun ˜oz M. A., Phys. Rev. Lett., 76 (1996) 1481. [29] Baraba ´ si A. L., Makse H. A. and Stanley H. E., Phys. Rev. E, 52 [30] Amaral L. A. N., Baraba (1995) 4087.