Comparison of a Born-Green-Yvon integral equation treatment of a compressible binary polymer blend on a lattice with recent simulations H. M. Sevian, P. K. Brazhnik,a) and J. E. G. Lipsonb) Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755

(Received 23 April

1993; accepted 17 May 1993)

Predictions of macroscopic and microscopic thermodynamic properties of a polymer blend using the Born-Green-Yvon (BGY) integral equation treatment of a compressible polymer mixture on a lattice are compared to recent simulations of compressible symmetric binary mixtures on a lattice. The theory is consistent with recent simulation studies which indicate that the critical temperature scales linearly with chain length. In addition, we show that the theory is effective in modeling the qualitative trends in the numbers of different types of pair contacts on the lattice, and also present comparisons of binodal and spinodal curves. These comparisons indicate that quantitative dilIerences between the theory and simulation results are likely associated with the manner in which connectivity is treated in the lattice BGY description. We conclude by discussing avenues whereby connectivity may be incorporated more consistently in the development of the theory.

I. INTRODUCTION Much of the existing theoretical work on the structure and thermodynamics of polymer blends begins with the Flory-Huggins lattice model,’ which is tractable and involves only one fitting parameter to describe inter-particle interactions. However, its liberal use of the assumption that all components are randomly mixed means that it is ill-suited to describe systems in which strong interactions are believed to play an important role. To address this and other problems, Lipson2*3 has used a Born-Green-Yvon (BGY) integral equation treatment to describe polymer blends and solutions. Unique to the BGY approach is that the problem can be solved both on a lattice and in the continuum. Published work to date in our group on the continuum problem has focused on the case of a fluid composed of hard-sphere monomers. Taylor and Lipson explored two closures to the BGY equation that improve upon the Kirkwood superposition approximation,5 and via h-bond corrections improve the calculated pair and triplet distribution functions. They are currently extending this approach to polymeric hard-sphere fluids.6 The primary advantages of using a lattice model are that it is much easier to solve than the corresponding continuum formulation, and it lends itself well to analytic solutions. Lipson2’3 has presented BGY results for an incompressible binary mixture on a lattice, and compared them to Flory-Huggins,’ Guggenheim random mixing and quasichemical,’ and lattice cluster8-I0 theories. The BGY theory showed very good agreement with the Monte Carlo simulation results of Madden et aL9 Lipson and Andrews” have extended the lattice work to describe a compressible one-component polymeric fluid on a lattice. They showed @Permanent address: Institute of Applied Physics, Nizhnij Novgorod, Russia, 603600. b)Author to whom correspondenceshould be addressed. 4112

J. Chem. Phys. 99 (5), 1 September 1993

excellent fits of the resulting equation of state to experimentally obtained pressure-volume-temperature (p VT) surfaces for a homologous series of alkanes and several polymeric fluids, and thereby obtained reasonable microscopic fit parameters corresponding to the effective degree of polymerization, lattice site volume, and the nonbonded monomer site-site interaction energy. Lipson and Brazhr&l2 have extended the BGY theory to a compressible multicomponent blend on a lattice. They illustrated thermodynamic gesults for a theoretical binary system exhibiting a closed loop phase diagram. There has been considerable recent activity in the development of theories to describe polymer solutions and blends. Freed and co-workers’-” have derived a “lattice cluster theory” using field theoretic techniques to compute correction terms to the Flory-Huggins theory via an expansion in powers of the reciprocal of the lattice coordination number. This is a powerful approach, but calculation of the expansion coefficients appears to be a challenging task. For a comparison of the thermodynamic properties of a polymer solution predicted by Flory-Huggins,’ Guggenheim,’ lattice cluster,8-‘0 and BGY2,3 theories, the reader is referred to a recent article by Cui and Donohue.13 Extensions to the “lattice fluid” model by Panayiotou and VeraI and by Sanchez and co-workers”-” yield predictions that rely on the assumption of random mixing, which is not necessarily appropriate to polymer blends. A continuum theory of polymers has been developed by Curro and Schweizer,‘8,‘9 who generalize the reference interaction site mode12’ to the situation of flexible long molecules. This theory makes use of the site-site Ornstein-Zemike equation and an appropriate closure to yield the intermolecular structure factor, from which other thermodynamic properties may be derived. However, predictions are sensitive to both the required intramolecular correlation function chosen and the closure employed. In this paper we describe the extension of the BGY

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g/$6.00

0 1993 American Institute of Physics

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theory to include compressible mixtures of polymers. At this point the formalism should be tested against both simulation results and experimental data. In order to accomplish the former, which is the subject of this paper, we compare the BGY theory to simulations of a compressible symmetric binary polymer blend on a lattice. Differences between the theoretical predictions and the lattice simulations can be attributed to the approximations made in solving the lattice BGY theory analytically. The application of the BGY theory to the specific models studied in the simulations is outlined in Sec. II, and the comparisons are presented in Sec. III. In Sec. IV we conclude. The comparison between the lattice BGY theory and experimental data, which is the next step, involves using the BGY theory to predict the equilibrium thermodynamic properties, including phase behavior, of a binary blend. This involves applying the BGY equation of state for a pure polymer fluid to obtain the microscopic parameters for two pure polymer fluids, and obtaining the site-site heteromonomer interaction energy from a minimal amount of information for the blend. This work is currently in progress.

This is the conditional probability of finding a particle of type i as a nearest neighbor of a particle of type j, where i and j can take on three values: A (or B), meaning the site is occupied by an A (or B) segment, or h, meaning the site is occupied by a void or hole. The total number of sites on the lattice is No, with NO = f-/&A + r BNB + Nh,

(2.2)

where r, (or rB) is the degree of polymerization of the polymers of type A (or B), NA (or iVB) is the number of A (or B) polymers in the system, and iVh is the number of voids (with r,= 1 by definition). The volume fractions are defined in the usual manner, as

r&i pi=%.

(2.3)

The Guggenheim concentration variables take into account the nearest neighbor connectivity of chain segments, 4fli

(ii= qANA+qBNB+Nh

’

where II. APPLICATION OF THE BGY THEORY TO SPECIFIC BINARY SYSTEMS ON A LATTICE We consider a lattice of coordination number z, with lattice sites occupied by either an A or B monomer, or a void. The pair probability density, p ( ra ,rb), of finding two particles, a and b, within volume elements dr, and dq, about ra and rb, respectively, may be related to the site-site interaction potentials via the BGY integral equation. Physically interpreted, this equation relates the pair probability to direct interactions between particles a and b, and to indirect interactions mediated through a third particle. As shown previously,2 by dividing the equation through by the pair probability of interest, closing the hierarchy at the three-particle-interaction level with the Kirkwood superposition approximation, assuming pair independence for the remaining integrands on the right hand side of the equation, applying a square-well nearest neighbor interaction between lattice sites, and incorporating the Guggenheim concentration variables7 in order to reflect more accurately the total number of sites available next to a given polymer segment, a simple expression is derived for the pair conditional probability density &e-&j

P(ilj)

= ~Ae-PeAj+{#-P’Bi+~

h

(2.1)

*

qg=rj.Z--2ri+2.

(2.5)

The energetics are described by square-well potentials acting on nearest neighbors only, with interaction parameters and EAB representing the A-A, B-B, and A-B cAA 5 EBB, interaction well-depths, respectively. Interaction potentials between voids and any other site are assumed zero. The internal energy of the system may be derived from an ensemble average over the energies of the sites which, after accounting for nearest neighbor interactions only, reduces to2

where the asterisk (*) indicates a sum over the nonbonded nearest neighbors of i, and the factor of one-half accounts for overcounting. The free energy may be obtained by applying the Gibbs-Helmholtz relation, using a boundary condition of infinite temperature, and assuming in that limit that the entropy dominates the Helmholtz free energy, which is approximated by the Guggenheim estimate for a randomly mixed system.21 Other thermodynamic variables follow.” In particular, the pressure and chemical potential are derived from the standard Maxwell relations as

NAPNB,T hph+zh-+;I 2

6-h qh

CA2

6Ate-

‘%L

1) +‘$B(e +QB-

gAe-@-4A+&J#-PEAB+gh

1)

gg EA(e-oeAB- 1) +cB(e-BEBB+2

&;4e-PEAB+~$-PEBB+~h

and J. Chem. Phys., Vol. 99, No. 5, 1 September

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1) ’

(2.7)

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Sevian, Brazhnik, and Lipson: Binary polymer blend on a lattice

l-r~-;

=ln~A+;h$/,

(q~-r~)

-~ln(~Ae-8E*a+~8e-8B+~~) 1

&A(Cp’AA-rA/qA) 1-5

(qA-rA)

(1-&z)

iTde-BcAB-rA/qA)

~Ae-~~AA+~Be-~~AB+~h+~Ae~~EAB+gge;~

-y

where Y is the volume per lattice site, and the total volume of the system, V=vNc. Note that the term in Eq. (2.8) in curly braces is zero when ~~#l. In recovering the FloryHuggins limit, qA is set equal to rA, and this term becomes important. For all the cases that we study in this paper, the system is fully symmetric. That is, the two components have the same degree of pOlymeriZatiOn, rA = yE, and interact among themselves in the same way, eAA= eBB . Of course, this does not necessarily imply that eAB=eu. We consider here three symmetric cases, which are the symmetric cases most commody simulated: (i) EAB’ -eAA=e, (ii) eAB=e, EAA =0, and (iii) EAB--0, eAA=E. In addition, all the simulations to which we compare the BGY predictions are performed on a simple cubic lattice (z= 6). As an aside concerning the phase behavior in these cases, it may be helpful to consider a comparison between cases (ii) and (iii). At infinite temperature a system favors a single phase. In case (ii), when eAA=0, i.e., no likeparticle interactions, as the temperature is lowered from infinity, the only way to drive phase separation is for the unlike-particle interactions to be repulsive. Therefore, in order to have phase separation in case (ii), l AB must be positive. In case (iii), when EAB=O and there are no unlike-particle interactions, the only way to drive phase separation is for the like-particle interaction to be favorable or attractive. Therefore, phase separation occurs in case (iii) when EAA< 0. This means that kTJe is negative in case (iii). The reader must bear in mind that in most current literature, when the reduced temperature, kTJe, is referred to, what is meant is kTJ ( E ( . We will be explicit about the modulus of E when referring to case (iii). A. Critical temperature polymerization

as a function

of degree

of

EBB+&

'

(2.8)

(%LB,v,&3NA,NB,, - (~),,,~,,(~),,~B,,=o. (2.9)

In the symmetric case, the critical temperature where NA =NB, and at this critical point”

occurs

? (%)NB,,T+ (%$NA,NB,Tro (2.10)

The behavior of the critical temperature, T,, with degree of polymerization, r, may be derived from Eq. (2.10) by taking the derivatives of the chemical potential, Eq. (2.8)) with respect to NA and No and solving for the temperature at which Eq. (2.10) holds true. The resulting expression does not look at first to give a linear relation between T, and r. Rather, for the three cases studied, the expressions are involved equations relating exp( dkT,) and powers of r. Case (i) yields a fourth-order polynomial in y, which is defined by kTJe= l/ln(y + 1). The full expressions are presented in the Appendix. Equations (Al) and (A2) may be expanded in order to obtain a simpler approximation for the variation of the critical temperature with chain length. As explained in the Appendix, first- and second-order (in dkT,) expansions result in various oscillating terms, but the zeroth-order term, which is the slope of the linear relation between the critical temperature and chain length, is the same, kTc -N E Cases Appendix). for kTJe, panded in

For a system with pressure, volume, and temperature fixed, the spinodal condition is given by”

1

( 1-9~) (~-2)~ r. (2.11) z--2(134 (ii) and (iii) both yield quadratics in y (see the Each may be solved exactly for y, and hence and then the resulting expressions may be expowers of l/r. For case (ii), this results in

I

kT,

-= E

U-qd(z-2j2 2[z-2( l-qh)]

For case (iii),

kTc -= E

-22+

(1 -ph)

r+

(9z- lO)z+2( 1 -rp/J2(6-z-222) 4[z-2(1--cph)12 .-

(2.12)

the result of the expansion is

A u-Q)h)k-2)2 2[z-2( l-&)lr-

~+(1-~~)(52/2--9)z-(l-q+)2(2~-3z-6)+o 2b---2(1-q412 J. Chem. Phys., Vol. 99, No. 5, 1 September

1 0r ’ 1993

(2.13)

Sevian, Brazhnik, and Lipson: Binary polymer blend on a lattice

Note that the slope in case (iii) is positive for negative E. Each of the three cases produces a single physical solution for y for the entire range of the void fraction, qh =O+ 1, which must be associated with an upper critical solution temperature since the system exhibits a single phase at high enough temperatures. (For a nonsymmetric system, it is possible to obtain both an upper and lower critical solution temperatures from the BGY theory.12) Obviously, the linear relations between T, and r are valid only for large enough r. However, for the particular parameter choices we study in Sec. III, the linear relation holds even for very short chain lengths. By comparison, Flory-Huggins theory for an incompressible mixture of A and B predicts that21

kTc -= %H

22

(2.14)

(1/J;;;+1/&>2’

where the Flory-Huggins the usual manner,

interaction

energy is defined in

EAA+EBB cm=EAB-

2

*

Thus for the symmetric system, kTJ(eAAB-EAA) =rz/2. Application of Flory-Huggins theory to the ternary (compressible) system (with A, B, and voids), yields for the symmetric case,

1-P PO=-;

I

AE=EAB-EAA.

en(2.17)

Equations (2.14) and (2.16) result from both the mean field nature of the theory and the assumption of random mixing, which is only strictly valid at infinite temperature.

B. Phase coexistence

PZ p(l-e-O”)

ln(l-PO)+~ln~+~~

ply

1

(2.18)

and

1 +

I

1-r-i

(q-r

I

> -T

ln( 1 -p+pe-DO)

15(q-r)$--T P(emP”-r/q) 1 -$+Pe-@

1 ’

(2.19)

where q and r correspond to the particular component, a=~~ for the A component and EBB for B, and q” and p are the concentration variables associated with the pure component. The algorithm is as follows: given a particular 9A for the two-component system, ,ui is calculated by finding the & that yields the same pressure, p=p’, from Eqs. (2.7) and (2.18), respectively, for a pure A system composed of the same number NA of A chains as in the mixture. That 95 is used to compute ,L& from Eq. (2.19), and then the chemical potential of mixing is hpA=pA-/~i. The spinodal is given by Eq. (2.9)) or by the inflection points in the chemical potential curves. C. Temperature

where he is defined analogously to the Flory-Huggins ergy parameter,

4115

variation

of number of pair contacts

Because the BGY theory begins with the conditional probabilities of having two sites as nearest neighbors, it is possible to derive the numbers of different types of pair contacts in the system. The number of B sites next to A Sites, ngA, can be Written down by realizing that qAz iS the number of nearest neighbors of an A chain, so qA.z/rA is the number of nearest neighbors of any one particular A site. rANA is the number of A sites on the lattice, and p( B 1A) is the conditional probability that there is a B site next to a given A site. Therefore,

curves

The binodal may be derived either by numerically fitting a double tangent construction to the Gibbs free energy of mixing, or by finding the two sets of volume fractions at which each of the two chemical potentials of mixing, A/LA and APB, passes through the same value (i.e., by Maxwell construction of the chemical potentials). The second numerical method is the more accurate. In practice, the values of A/JA and Apg are located by finding the intersection point of A/LA vs Apg for the two physical sides of the Maxwell construction. For a symmetric system, at this point, ApA = ApB. In order to find the thermodynamic properties of mixing, it is necessary to know the properties of the pure polymer systems at the same pressure and temperature. For a compressible one-component system on a lattice, the pressure and chemical potential of the pure component are given by4

e-flEAB

(2.20) X(qA/rA)~Ae-~~AA+(qs/r~)~Be-'~As+~h'

Likewise, the number of A sites next to B sites is

The two numbers, ngA and IZAB, should be equal, but are only so in the fully symmetric case at equal concentrations of A and B. This is because in the derivation of the conditional probability, the pair independence approximation is not applied symmetrically, since the BGY equation is first

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4116

divided through by the joint probability. We account for this here by defining an “average” number of A-B contacts as

e-@AB + -peAB+

(qA/rA)vAe

(qB/rB)p$-pEBB+qh

+I

I

MM-

Ii ,.*.

1

1

I

**

.’

-

1 *

(2.22)

NAB is the total number of A-B contacts in a system (or phase) defined by the three volume fractions, qA, Q)B, and Q)h. Similarly, the other heterocontact numbers are given by

0

50

1CQ

1.50

2w

2.m

chain Length(r)

FIG. 1. Comparison of the critical temperature to chain length for the system studied by Deutsch and Binder (Refs. 22, 23) corresponding to our case (i). Their Monte Carlo simulation results (0) show clear evidence of a linear relation (evenly dashed line). BGY theory, Eqs. (Al ) and (A2), also yields a curve ( 0 ) which is fit well by the linear relation, kTJe=L60r+ 1.08 (solid line). Flory-Huggins theory also predicts a linear relation (unevenly dashed line).

e-@AB (qA/rA)qAe-PEAA+

(qB/rB)q#-pEAB-@h

1 +

(qA/rA)qA+

(qB&?)Q)B-@h

1

(2.23)

’

and NBh the same as Eq. (2.23) except with AttB. homocontact numbers are given by

The

&%4A X (qA/rA)qAe-pEAA+

(qB/rB)qg-PEAB+ph

’

(2.24) and NBB the same as Eq. (2.24) except with A+-+B, and (2.25) as expected for a symmetric system, in the infinite temperature limit, NAB-’ NAA .

III. COMPARISON

OF THE THEORY

TO SIMULATIONS

Deutsch and Binder22’23 have shown recently, using the fluctuating bond model simulation technique, that a compressible symmetric binary polymer blend on a lattice exhibits a linear dependence of the critical temperature on chain length, which concurs with the mean field scaling law, Eqs. (2.14) and (2.16), and also with recent experiments on an isotopic blend.24 Although these results are in disagreement with the generalization by Schweizer and Curro17 of the reference-interaction site model,lg which predicts that in a three-dimensional system, T, - &, Yethiraj and Schweizer25 have recently shown that by incorporating a different closure between the total and direct intermolecular site-site correlation functions, they obtain a linear relation.

The symmetric system studied by Deutsch and Binder22P23corresponds to our case (i), that is, EAB= -EM =E. They chose a void fraction of qh=O.5 and determined the critical temperatures for chain lengths r=4 to 256. From their Monte Carlo simulations, they obtain a clearly linear relation between the critical temperature and chain length: kTJez2.15r+ 1.35. For this particular set of parameters, Flory-Huggins theory, both for an incompressible binary mixture, Eq. (2.14)) and for a lattice mixture with avoids, Eq. (2.16), predict that kTJe-6r. The BGY theory prediction is compared to these in Fig. 1. The points plotted along the BGY curve (circles) are calculated by solving Eqs. (Al) and (A2) exactly for the particular parameters of the system. A fit of these points to a straight line yields kTJe=1.60r+1.08. The slope of the fitted line is the same as the slope predicted by the linearization, Eq. (2.11). As can be seen, the BGY curve for this set of parameters is clearly linear, and falls short of the simulation results. We turn next to examining the phase behavior for a symmetric binary blend. Sariban and Binde36 have simulated several of these systems using their fluctuating bond model, with the chain length, rA =r,=r, varying from 4 to 64, the void fraction, qh, varying from 0.2 to 0.8, and energetics corresponding to our cases (ii) and (iii). Figure 2 here is meant to be a comparison to Fig. 6 of Sariban and Binder.26 In our Fig. 2 we have shown binodal coexistence curves (solid lines) and spinodals (dashed lines) for the symmetric system with &=0.6 and corresponding to our case (ii), that is, EAB=E, E~=O. The critical temperatures that BGY theory predicts are consistently about twice those found by simulation. Still, BGY theory comes closer to modeling the simulation than either the Flory-Huggins or Guggenheim theories (refer to Fig. 6 of Sariban and Binder26 for a comparison). An advantage of comparing the theory to simulations is that the numbers of contacts can be counted directly.

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Sevian, Brazhnik, and Lipson: Binary polymer blend on a lattice 40

xl

0 +I

10

10

/ / / 0 -L

12

14

16

k=&I

-I 0.2

I 0.4

I

I 0.8

FIG. 2. BGY predictions for the phase diagrams for the symmetric binary mixture studied by Sariban and Binder (Ref. 26) corresponding to our case (ii), to be compared with their Fig. 6. Solid lines are binodal coexistence lines, dashed lines are spinodals.

Number-of-contact information is what determines the thermodynamics of the system, and so if we can understand the shortcomings of the theory in predicting the numbers of different types of contacts, we can better understand which approximations to reconsider. In order to calculate the BGY predictions for the numbers of different pair contacts in a system at equilibrium, it is necessary first to determine the phase diagram for the system. We then determine the numbers of pair conta+s in each phase, and add the results from each phase. For a symmetric system, since the binodal is symmetric and there are equal numbers of A and B chains, each phase (within the two-phase region of the phase diagram) is composed of half the total system. Thus if N& is the number of pair contacts in phase a, for the symmetric system, the total number of A-A contacts in the system in the two-phase region is [Ng +NB]/2. If the system were not symmetric, we would find the proportions by applying the lever rule. In the one-phase region the symmetric system is composed of a single phase with rpA=qB= (l-&/2. Equations (2.22) through (2.25) give the numbers of pair contacts in the entire lattice of size No. The number of contacts per chain is the total number of contacts divided by the total number of chains in the system, NA+NB, which for the symmetric system is equal to NO( 1 -P~)/Y. In Fig. 3 we have compared the BGY theory predictions for pair contact information to another Sariban and Binderz6 simulation for a system with r=32, qh=0.6, and corresponding to our case (iii), eAAB=O, eAA= E. In their Fig. 10(c), they have plotted the number of A-A ( = B-B) interchain and intrachain contacts per chain, and the number of A-B contacts per chain in the equilibrium system. Because with the present version of BGY theory we have no way of distinguishing between like-monomer inter- and

FIG. 3. Comparison of BGY theory to Fig. 10(c) of Sariban and Binder (Ref. 26): the number of contacts in an equilibrium symmetric binary system corresponding to case (iii). The o’s are the numbers of A-A contacts per chain from simulation and the X’s are the numbers of A-B contacts. The dashed line is the BGY prediction for the number of A-A contacts, Eq. (2.24), and the solid line is the prediction for the number of A-B contacts, Eq. (2.22).

intrachain contacts, we have compared the BGY theory to the total number of like-monomer contacts. Since Sariban and Binder observe a roughly constant number of intrachain contacts over the entire temperature range, it is still possible to make a comparison of the trend in the interchain contact curve. Following the argument in Sec. II, as the temperature increases through the two-phase region, we expect to find more A-B contacts in solution and fewer A-A and B-B contacts, and this is borne out in both the simulations and BGY theory predictions. By comparison, Flory-Huggins and Guggenheim theories predict a constant total number of contacts per chain. The former predicts the number as (1 -ph)rz, which is 76.8, and the latter predicts ( 1 -cph)qz which is 52. Both are overestimates for this system. At the critical point, we expect a change in slope in the pair contact curves, reflecting the second-order phase transition. The critical temperature found by simulation occurs at roughly kTJI E[ z 10.27 The BGY theory, Eq. (A5), predicts a critical ,point at kTJI E] =20.48. Cifra, Karasz, and MacKnight’* have also performed simulations of a compressible symmetric binary polymer blend on a lattice, and report A-B pair contact information. In Fig. 4 we have shown a comparison of BGY theory to the number of A-B pair contacts in the equilibrium system as a function of e/kT, for the symmetric system they simulated: ~7~~=0.0909 and r=20, and corresponding to our case (ii), E~~=E, E~=O. The BGY prediction for the number of A-B contacts is calculated from Eq. (2.22), and is shown as the solid line, while the simulation data are the diamonds. The BGY prediction for the critical temperature, as calculated from Eq. (A4), is kTJe=35.48, or dkTp0.0282 and is shown in Fig. 4. Cifra et al. observe a critical point in their simulation data at dkT,=O.O5, but because are faced with the reciprocal of one significant figure, we can say only that the BGY theory overestimates the critical temperature.

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TABLE I. Critical temperaturespredicted by BGY theory for three cases having the same he and different 6, shown for two different void fractions, rph=0.6 and qh=0.090R Chain length is the same in all, r=20. Case (3 (ii) (iii)

0.0

0.4

02

0.4

c/b=

FIG. 4. Pair contact information for the system studied by Cifra, Karasz, and MacKnight (Ref. 28) corresponding to our case (ii). Their Monte Carlo simulation results for the number of A-B contacts (0) are compared to the prediction from BGY theory, Eq. (2.22) (solid line): Also shown are the BGY predictions for the other types of pair contacts in the lattice solution, Eqs. (2.24) through (2.25). The BGY prediction for the critical temperature is included for reference.The sum total of these is shown as the solid line in the inset, and is to be compared to the expected theoretical total.

W e have also calculated the numbers of other types of contacts: A-A= B-B, A-h= B-h, and the void-void contacts. By plotting the inverse temperature along the abscissa, two features are obvious: the second order phase transition at the critical temperature, and NAB=NAA at dkT=O. In the inset of Fig. 4 we have compared the sum total of all these contacts (solid line) with the actual number of distinct contacts per site on a lattice, which is simply z/2. In the one-phase region, the total predicted by BGY theory is reasonably accurate, while in the two-phase region, the total is overestimated. Since we can only compare the theory to simulation data for one of the four distinct types of pair contacts (and only two in Fig. 3), we cannot draw any conclusions regarding which of the remaining three are overestimated by the theory. IV. CONCLUDING

REMARKS

W e have studied the chain length dependence of the critical temperature for three different cases of interactions. W e have compared the linear approximations to the full relation and, for the systems studied, we find that BGY theory predicts that the critical temperature varies linearly with the chain length, r. For the symmetric system, there are only two energy parameters, which we may rewrite as AEEQ-E~ and a=eAB+eAA. Flory-Huggins theory, both for an incompressible binary blend, Eq. (2.14)) and for a ternary mixture including voids, Eq. (2.16), predict that the critical temperature depends only on he. In Table I we have shown the comparison between the critical temperatures predicted by the BGY theory, Eqs. (Al) and (A2), (A4), and (A5), in each of the three cases for a choice of parameters for which he, ph, and r are the same but S differs. The BGY theory predicts that the critical

EAB

EAA

he

6

e/z

-e/2 0 --E

E E E

0 E --E

0’

qh=0.6, kTJe 12.16 12.40 13.09

q+,=O.O909,kTJc 35.55 35.48 35.61

temperature depends almost entirely on he and very little on 6. This is to be expected since the phase behavior is a result of the properties of mixing, which depend primarily on the relative difference between the like- and unlikeparticle interactions. A further interesting point about the results shown in Table I is that the critical temperature is predicted toroccur at a higher temperature at higher pressure (lower void fraction). This is consistent with the Flory-Huggins prediction for a ternary system incorporating voids, Eqs. (2.16), although experimental results on polymer blends more typically indicate an increase in miscibility with increasing pressure.29 W e have also shown that the BGY theory accurately models the phase behavior of compressible symmetric binary polymer mixtures on a lattice. W e have compared the BGY theory predictions for the numbers of pair contacts in equilibrium solution to simulation results and shown that the BGY theory correctly predicts the trends in the curves. W e can draw conclusions about the shortcomings of the theory only from the comparison presented in Fig. 3, since for this system, which corresponds to case (iii) where eAB=O and eAA=E, we have most of the pair contact information and the critical temperature. Here, the BGY theory overpredicts the critical temperature by a factor of 2. While halving the attractive interaction E between like monomers would more accurately predict the critical temperature, the number of pair contacts would be further underestimated. The problem lies in the treatment of connectivity. The two most serious assumptions made in solving the BGY equation for the conditional probability are the Kirkwood superposition approximation and the assumption of pair independence. (The latter is used after the superposition approximation is applied, for only the pairs associated with indirect interactions between the pair of interest.) When the Kirkwood approximation is applied to a triplet, all three pairs are treated equivalently. However, if two of the particles are nearest neighbors along a chain, this cannot be correct, and the result is an underestimate of the probability of finding, as nearest neighbors, two segments which are connected. This implies that the number of likemonomer interactions is also underestimated. Use of the independence approximation on a connected pair further exacerbates the problem. Therefore, we conclude that improvements to the lattice BGY theory must incorporate a more sophisticated treatment of connectivity. W e are working toward this end.

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ACKNOWLEDGMENTS

We are grateful to K. Binder for providing us with more precise graphs of the simulation data, and for helpful suggestions. We are also especially grateful to M. P. Taylor for many enlightening discussions. This work was generously supported by the Petroleum Research Fund and by NSF DMR-9122337.

APPENDIX

--kTc E

(1-9)~ (~-2)~ z--2(1-q&)

(Al)

fdr)Yk=o

- 1=

(A3)

2(r--l)(l-~h)-~Z

2

x(1-%2)+42, 51222 d(ii)=(

3rY

2--4r+2?-~+il+3d+T+T

l-ph)’

+rz( l-q/J

(4-4r-6z+4rz-r2)

+2r2.

(A4)

The linear approximation to this equation may be derived by a straightforward series of Taylor expansions in powers of l/r, about l/r=O, resulting in Eq. (2.12). For case (iii), the physical solution is y=exp(dkTJ

with

r.

Cases (ii) and (iii) are simpler and yield only quadratics in y, which may be solved exactly. The physical solution for case (ii) is yrexp(dkT,)

We present here the full expressions for the relation between the critical temperature and chain length for each of the three symmetric cases studied. The spinodal condition is given by Eq. (2.9), and for the symmetric case at the critical temperature, qA=qB= (1 -(pJ/2, and Eq. (2.10) holds true. Taking the derivatives of the chemical potential, Eq. (2.8), with respect to NA and iVc, and substituting into Eq. (2.10) results in a complicated equation relating y, defined for each of the three cases by kTJe= l/ In(y+ l), and the chain length, r. Case (i) yields a fourth-order polynomial in y, j.

terms. In any event, since the slope results from the zerothorder term of the initial Taylor expansion in dkT,, we can say that

Id(iii) 1’

2(r-l)(l-qh)-rZ

-l=

?Z(iii)

2

n(iii) = [ -4+6r-22-r~+2?~-?2/2 + Jr(z-2)

(rz-2r+2)‘]

22 1322 22 --~+T+T -2rz(5z-6)

(1 -ah)

-2rz2+?(

-rz+2?z--H/2)

1-qh)2(z-2)

+rz(l-qqJ(-4+6r--212 -A?.

(A5)

The linear approximation to the critical temperature may again be derived by a straightforward series of Taylor expansions, and results in Eq. (2.13).

xtl-qdl, = -r(

-2rz,

1 -(Pi)

X(Z-~)~C(Z+~)(~-~~)-~Z}I,

f4(r)

( l-qh)

(rz-2~+2)~/4.

(A21

Equations (Al) and (A2) may be solved numerically for y with little effort. Henceforth, we assume that there are at least two nearest neighbors of each lattice site, z>2. Since it is too complicated to solve Eq. (Al ) for y, in order to obtain a direct relation between the critical temperature and chain length, we perform Taylor expansions about zero in dkT,. Retaining terms up to second-order in dkT,, solving the resulting quadratic for kTJe, and then expanding the right-hand side in l/r leads to an expression with the zeroth-order term proportional to r, the first-order term independent of r, and higher-order terms proportional to l/J’, n>l. Unfortunately, for reasonable parameter choices, these higher-order terms are large and alternate in sign, but cancel one another to give the critical temperature well-represented by the zeroth- and first-order

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