J . Phys. Chem. 1989, 93, 2194-2203
2194
FEATURE ARTICLE Lattice Theories of Polymeric Fluids Karl F. Freed* and M. G . Bawendi The James Franck Institute and Department of Chemistry, The University of Chicago, Chicago, Illinois 60637 (Received: August 12, 1988)
We describe the systematic evaluation of the free energy of a set of self-avoiding and mutually avoiding polymer chains on a lattice, where nonbonded nearest neighbors interact through attractive van der Waals energies. A simple algebraic representation is presented for the exact partition function, and a cluster expansion is introduced for evaluating corrections to the zero-order Flory-Huggins approximation. Diagrammatic representations of the cluster expansion bear a strong similarity to Mayer cluster expansions for real fluids, but chain connectivity and excluded-volume constraints introduce additional mathematical complexities. We provide a summary of calculationsof the molecular origins of the entropic portion of the Flory x parameter, the composition and temperature dependence of heats of mixing of polymer blends, and the cross-link dependence of x for polymer networks.
I. Introduction Polymers are essential components in all biological systems and in a wide variety of plastics and other advanced technological materials. Polymers can be found in solutions, as liquids, glasses, crystalline materials, micelles, liquid crystals, and gels. They may be studied in static or flowing systems and under equilibrium or highly nonequilibrium conditions. Although there was considerable early resistance to the idea that polymers are randomly shaped long flexible objects rather than ones with simple rodlike or colloidal structures,’ the rich variety of phenomena associated with polymers is now accepted to be a direct consequence of the extended nature and internal flexibility of polymer chains. Early theoretical studies of polymers’ centered on the description of the properties of polymers in dilute solutions,2on the one hand, or on polymers in concentrated solutions or the liquid state (called the melt) on the other hand. Meyer3 was the first to suggest that the entropy of mixing of long-chain polymers with small solvent molecules could be calculated by using a lattice approximation in which the monomers of the polymer occupy the same kind of sites as solvent molecules. This model is depicted in Figure 1 with two polymer chains on a square lattice, where the polymers are represented by sequentially bonded sets of monomer units such that no two monomers occupy the same lattice site. All sites not occupied by polymers depict either solvent molecules in the case of concentrated polymer solutions, or they are taken to be empty and thus model free volume in treatments of melts. Figure 1 presents the polymer chains as being completely flexible, and for pedagogical illustration the figure uses a square planar lattice where a bond may either continue in the same direction as the preceding bond (a “trans” bond) or proceed in an orthogonal direction (a “gauche” bond). Polymers are thus represented by this model as mutually avoiding and self-avoiding random walks on a lattice. Computation of the systems’ entropy proceeds by the usual statistical mechanical definition relating entropy to the total number of configurations available to the system. The lattice model also includes attractive van der Waals interaction energies between nonbonded nearest neighbors, and these are depicted by wavy lines in Figure 1. A polymersolvent system has polymer-polymer, polymer-solvent, and solvent-solvent at( I ) Flory, P. J. Principles of Polymer Chemistry; Cornell University: Ithaca, NY, 1953. (2) Yamakawa, H. Modern Theory of Polymer Solutions;Harper & Row: New York, 1971. (3) Meyer, K. H . 2. Phys. Chem. A b f . B 1939, 44, 383.
0022-3654/8912093-2194$01.5010 , J
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tractive interactions which are written as epp, tp, and tSs,respectively. The introduction of the interaction energies into the lattice model of polymers, in principle, enables computations of enthalpies and therefore of all thermodynamic properties of polymeric systems. The counting problem posed by the enumeration of all configurations available to self and mutually avoiding polymers on a lattice has been formidable.’ Fowler and Rushbrooke: Chang,5 and Miller6 gave partial solutions for polymers that occupy only a few lattice sites. However, the mean-field treatments of Flory’*’+’ and Hugginslo for long, linear, flexible polymers represented a major breakthrough in providing what has become probably the most widely used theory of the thermodynamic properties of polymer systems. These lattice models have played an important role in our understanding and in developing statistical mechanical theories of, for example, polymer solutions,’ gelation,” the polymer glass transition,I2 liquid ~rystals,’~-’’ rubber e l a s t i ~ i t y , ’ and ~.~~ the segregation of two or more polymer species.20v21 Standard Flory-Huggins mean-field approximations replace the strict constraint of single occupancy of each lattice site by site occupancy probability arguments.’ For example, the Helmholtz free energy of mixing @ for two kinds of polymers in the liquid phase (called a blend) emerges in the well-known form involving a combinatorial entropy and an energy of mixing
(4) Fowler, R. H.; Rushbrooke, G. S . Trans. Faraday Soc. 1937,33,1272. (5) Chang, T. S. Proc. Cambridge Philos. SOC.1939, 35, 265. (6) Miller, A. R. Proc. Cambridge Philos. SOC.1942, 38, 109. (7) Flory, P. J. J . Chem. Phys. 1941, 9, 660. (8) Flory, P. J. Proc. R . SOC.London, A 1956, 234, 60. (9) Flory, P. J. Proc. Natl. Acad. Sci. W.S.A. 1982, 79, 4510. (10) Huggins, M. L. J . Chem. Phys. 1941,9, 440; J . Phys. Chem. 1942, 46, 15 1 ; Ann. N . Y.Acad. Sci. 1943, 44, 43 1. ( 1 1) Kirkpatrick, S . Rev. Mod. Phys. 1973, 45, 574. (12) Gibbs, J. H.; DiMarzio, E. A. J . Chem. Phys. 1958, 28, 373; Ibid. 1958, 28, 807; J . Polym. Sci. Parr A 1963, I , 1417. (13) DiMarzio, E. A. J . Chem. Phys. 1961, 35, 658. (14) Cotter, M. A.; Martire, D. E. Mol. Cryst. Liq. Cryst. 1969, 7, 295. Cotter, M. A. Mol. Cryst. Liq. Cryst. 1976, 35, 33. (IS) Alben, R. Mol. C r y s f .Liq. Cryst. 1971, 13, 193. (16) McCraken, F. L. J . Chem. Phys. 1978, 69, 5419. ( 1 7 ) Baumgartner, A. J . Phys. (Paris) Left. 1985, 46, L659. (18) Ciferri, A.; Flory, P. J. J . Appl. Phys. 1959, 30, 1498. (19) Allen, G.; Tanaka, T. Macromolecules 1977, IO, 426. (20) Scott, R. L. J . Chem. Phys. 1949, 17, 279. (21) de Gennes, P. G. J. Phys. (Paris) Lett. 1977, 38, L441; J . Polym. Sci. (Phys.) 1978, 16, 1883.
0 1989 American Chemical Society
Feature Article
Figure 1. Two polymer chains with 16 monomers ( N = 16) on a square lattice (z = 4). Nearest-neighbor monomers interact with a n energy c (squiggly lines).
where N , is the total number of lattice sites, 4i is the fraction of sites occupied by species i (usually called the segment fraction), and Ni is the number of lattice sites occupied by a chain of type i. The interaction parameter xyp is obtained from the nearest attractive neighbor van der Waals energies ti, as
where z is the number of nearest neighbors to a given lattice site (called the lattice coordination number). In practice, xy? in (1.1) is treated as a phenomenological parameter, and the free energy of mixing then becomes independent of any lattice parameters. Flory arrived at (1.1) by sequentially placing uncorrelated, but connected, monomers on the lattice, whereas Huggins used a more sophisticated counting scheme which begins to account for the short-range correlation^.^^ The approach of Huggins differs from that of Flory by having a small additional contribution to the entropy of mixing that depends explicitly on the lattice coordination number z. Lack of knowledge of the appropriate value of the z for realistic polymer systems and the greater simplicity of the Flory one-parameter theory led to the widespread use of the Flory form [(1.1)], which has been termed Flory-Huggins theory to respect the independent contributions of Huggins. The mean-field expression (1.1) displays a blend as having a very small entropy of mixing because of the high molecular weight of the polymer, Le., the large number of segments Ni on a single polymer. In addition, estimates of the interaction energies tu from molecular polarizabilities lead to the expectation that x12in (1.2) is generally positive, giving a rather unfavorable heat of mixing. Consequently, the Flory-Huggins prediction (1.1) implies that long-chain, flexible polymers would not tend to mix in the liquid state, and this is generally found to be the case. However, blends are useful as precursors of a variety of composite materials, so there has been an enormous amount of experimental work to find polymers that mix (blend) in the liquid state and to find some principles guiding the determination of which polymers form blends. Unfortunately, the standard Flory-Huggins theory of (1 .l) and (1.2) and its straightforward generalizations do not explain why some polymer systems form stable blends and what types of molecular modifications are required to enhance the blending characteristics of particular polymers. The Flory formulation in (1.2) displays the interaction parameter x12as independent of composition, proportional to TI, and energetic in origin. However, when xI2is treated as a phenomenological parameter, comparisons with experiment show x12 to depend on polymer concentration, to contain both energetic and entropic contributions, and to depend on pressure.1s22-26O f t e n (22) Polymer Handbook; Brandup, J., Immergut, E. H., Eds.; Wiley: New York, 1975; IV-131 and references therein. (23) Eichinger, B. E.; Flory, P. J. Trans. Faraday Soc. 1968, 64, 2035, 2053, 2061, 2066. (24) Flory, P. J. Discuss. Faraday Soc. 1970, 49, 7. (25) Scholte, Th. G. J . Polym. Sci. A - 2 1970, 8, 841. (26) Murray, C. T.; Gilmer, J. W.; Stein, R. S. Macromolecules 1985, 18, 996.
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2195 the entropic contribution to xI2greatly exceeds the enthalpic one. The empirical findings strongly conflict with the original model leading to (1.1) and (1.2), and therefore, they imply errors in either the lattice model, the mean-field approximation of Flory, or both. The improved counting scheme of Huggins provides an entropic contribution to x12that is too small in magnitude to explain the experimental results. Koningsveld and Kleitjens have applied Huggins-type counting arguments to the energetic term xI2in (1.I) and thereby describe a composition-dependent heat of mixing. However, observed composition dependences in heats of mixing exhibit much richer forms than predicted by the improved counting methods of Koningsveld and Kleitjen~,~’ and other^,^^^^^ again implying either errors in the lattice model, the improved mean-field counting schemes, or both. Because of several of these deficiencies of the Flory-Huggins theory, Flory developed what is now called the equation of state theory of the statistical thermodynamic properties of polymer system^.^^,^^,^' The approach uses a combination of statistical mechanical models with thermodynamic phenomenology: It utilizes the combinatorial entropy of mixing in (1.1) and a simple one-dimensional statistical mechanical model to describe the entropic contribution from the presence of free volume in a form that is perhaps more realistic than provided by using voids to model free volume in the lattice models. However, the equation of state theories are still forced to introduce a phenomenologicalparameter corresponding to the entropic contribution to the interaction energy term x,a parameter of completely uncertain molecular origins. Composition dependences in heats and entropies of mixing are modeled by Flory following the work of Prigogine and c o - ~ o r k e r s ~ ~ by considering the different polymers in a blend to interact through those parts of the molecule that lie on the vaguely defined “surface” of the randomly shaped polymer. However, the relevant surface fractions are incalculable, so they are relegated to additional phenomenological parameters. Recent analyses33show that the introduction of a similar type and number of phenomenological parameters into the mean-field lattice theories leads to results that are roughly comparable and therefore operationally equivalent to those of the equation of state theories. Our interest here lies in developing a systematic theory of polymer melts, blends, concentrated solutions, gels, etc. (we term these systems polymeric fluids) that is capable of explaining the molecular origins of the large observed entropic contribution to the phenomenological interaction parameter xI2and of explaining the pressure, temperature, and composition dependences of this phenomenological parameter. As noted above, the gross discrepancies between Flory-Huggins mean-field theory and experimental observations lead to the incontrovertible conclusion that either the lattice model is in error, the mean-field approximation is inadequate, or both. We now briefly digress to mention certain aspects of the theory of polymers in dilute and semidilute solutions to assess whether or not the lattice model is most likely the culprit. The standard lattice model of self-avoiding and mutually avoiding polymers with nearest-neighbor nonbonded van der Waals interactions (Figure 1) has been widely used in conjunction with Monte Carlo simulations to study the properties of polymers in dilute and semidilute solutions.34 The Monte Carlo simulations have provided a thorough test of the ability of the lattice model to represent the properties of long-chain, flexible polymers in dilute and semidilute solutions, and recent advances in computer speed and capacity have begun to enable the application of Monte Carlo simulations to the more concentrated regime^.^^,^^ Lattice model (27) Koningsveld, R.; Kleitjens, L. A. Macromolecules 1971, 4, 637. (28) Kurata, M.; Tamura, M.; Watari, T. J . Chem. Phys. 1955, 23, 991. (29) Guggenheim, E. A. Proc. R. SOC.London, A 1944, 183, 203. (30) Patterson, D.; Delmas, G. Zbid. 1970, 49, 98. (31) Sanchez, I. C.; Lacombe, R. H. Macromolecules 1978, 11, 1145. (32) Prigogine, I.; Trapeniers, N.; Mathot, V. Discuss. Faraday SOC.1953, 15, 93. (33) Panayioutou, C. G. Macromolecules 1987, 20, 861. (34) Baumgartner, Binder, K., Eds. Applications of the Monte Carlo
Method in Statistical Physics; Springer: New York, 1984; and references therein.
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Monte Carlo simulation^^^ correctly predict a wide range of subtle polymer properties, such as power law dependences on molecular weight, polymer concentration, etc., and thereby describe all of the general equilibrium phenomena that are observed for polymers in dilute and semidilute solutions. Continuum space Monte Carlo ~imulations'~ with more realistic models, such as hard spheres and Lennard-Jones nonbonded interactions, are in general agreement with the lattice Monte Carlo simulations, with the only changes occurring in various model-dependent constants. The robustness of the lattice model for describing the dilute and semidilute solution properties of polymers suggests that this model should also be of great utility for the polymeric fluids and therefore that the deficiencies of Flory-Huggins theory may lie with the mean-field approximation of Flory. It is, of course, possible that the transition from the dilute and semidilute solutions to the concentrated limit of polymeric fluids introduces additional physical features that are no longer correctly represented by the lattice model. Such a possibility must be borne in mind when developing improved approximations to the lattice model description of polymeric fluids. We note that analytical theories, which agree both with Monte Carlo simulations and with experiment, are currently available for treating the equilibrium properties of polymers in dilute and semidilute ~olution.~'We therefore seek to employ the lattice model to provide an equally accurate representation of the equilibrium statistical mechanics of polymeric fluids. Considerable guidance with this goal is available from the enormous progress that has been made with the understanding of the statistical thermodynamics of fluids composed of small molecules.38 Integral equation methods have been developed to a high degree of precision and show that the dominant factor controlling the structure and thermodynamics of simple molecular fluids is provided by a description of their packing. The attractive interactions are of secondary importance and can be added perturbatively once the basic structural packing effects are included. Hence, hard-core models have been widely used as reference systems in the treatment of simple molecular fluids. Although the polymeric fluids are considerably complicated by the presence of chain connectivity and flexibility, our understanding of simple molecular fluids leads us to anticipate that it is likely that packing considerations are also primary in determining the thermodynamic properties of polymeric fluid under a wide variety of situations. While the lattice model of monatomic fluids is well understood (it is a special case of the standard king model), much less success has been obtained with the development of lattice models of molecular fluids. Although Fowler and R u ~ h b r o o k eChang,5 ,~ and Miller6 give partial solutions for the problems of dimers on a square-planar lattice, it is not until Fisher,39Kasteleyn,a Lieb,4' and Ferdinand42 that a complete solution is presented for this problem, but only in the very special limit where the lattice is completely covered by the dimers. Remove one dimer, creating two voids, and their highly specialized exact solution provides no results. While the lattice model of monatomic fluids is a rather poor representation of experiments mostly because voids and molecules are taken as having the same size, lattice models of polymeric fluids, which do not suffer from this problem, are likely to be better. Our interest here is in understanding the molecular origins for the observed entropic contribution to the x parameter, its composition dependence, and the composition and temperature dependence of the heats of mixing, properties whose observed variations conflict with predictions of Flory-Huggins theory. We (35) Dickman, R.; Hall, C. K. J . Chem. Phys. 1986,85,3023. Dickman has recently sent as a preprint in which he compares similar Monte Carlo simulations for a cubic lattice ( z = 6) with our theory, and the agreement is better than that in Figures 5 and 6 (with z = 4) because of the higher dimensionality (and therefore higher z ) . (36) Sariban, A.; Binder, K. J . Chem. Phys. 1987, 86, 5859. (37) Freed, K. F. Renormalization Group Theories of Macromolecules; Wiley-Interscience: New York, 1987. (38) Chandler, D.; Weeks, J . D.; Andersen, H. C. Science 1983, 220, 787. (39) Fisher, M. E. Phys. Reo. 1961, 124, 1664. (40) Kasteleyn, P. W. Physica 1961, 27, 1209. (41) Lieb, E. H . J . Math. Phys. 1967, 8, 2339. (42) Ferdinand. A. E. J . Math. Phys. 1967. 8. 2332.
Freed and Bawendi proceed with the development of a general scheme for solving the lattice model in the anticipation that some form of lattice model suffices for these purposes. This paper describes a systematic method for the evaluation of the free energy (and therefore all equilibrium thermodynamic properties) of polymeric fluids as a cluster expansion in which the Flory-Huggins mean-field approximation is recovered in zero ~ r d e r . ~ ) - The ~ ' cluster expansion is arranged as an expansion in the inverse of the lattice coordination number and in the Mayer f functions
AI = exp(t'-') - 1
(1.3)
where the t'l depend on the species occupying sites i and j . Although our original derivation^^^-^' of this cluster expansion have required the use of mathematical methods developed in analogy with ones used in field theory and particle p h y s i ~ s , several ~~-~~ results5' deduced from those field theoretic methods now enable us to present a rather simple algebraic derivation of the cluster expansion that does not necessitate either the use or knowledge of these field theoretic methods. The field theoretic approach may be helpful in deriving alternative types of approximation schemes for strongly interacting systems, but here we provide the new algebraic derivation of the cluster expansion treatment of the lattice model that has been applied by us to a wide variety of properties of polymeric f l ~ i d s . ~ ~ - ~ ' The next section presents a simple algebraic derivation of the cluster expansion for evaluating the thermodynamic properties from the lattice model of polymers. The derivation is given for the simple case of flexible linear polymers, but we also discuss the generalizations of the model to polymers in which the monomers are taken to have internal structures and therefore to occupy several lattice ~ i t e s . ~ ~Such - ~ O a generalization is an important aspect of modeling the properties of real polymers in which the monomers, the solvent molecules, and voids generally have different sizes and shapes. Section I11 introduces a diagrammatic representation of our Mayer-like cluster expansion for packing entropies and describes some of our recent r e s ~ l t s . ~ We ~,~~,~~ emphasize a molecular theory of the entropic contribution to the phenomenological x parameter, its composition dependence, and the composition and temperature dependence of the heats of mixing. In addition, we discuss the composition and temperature dependence of the effective interaction parameter that is deduced on the basis of extrapolations to zero angle of small-angle neutron scattering experiments on polymer blends. 11. Packing Entropies in the Lattice Model
The packing entropy is generally obtained by counting the number of configurations available to the polymeric system. The packing entropy of a fluid with polymers of a single type and with voids is expressed in terms of the number of ways of placing np polymer chains, each of length N , on a lattice with Nl sites. Our purpose here is to express this mathematically well posed counting problem in a purely algebraic form that can be evaluated by successive approximations to any desired degree of accuracy. Let us begin by considering the trivial problem of placing a single dimer on the lattice, and then generalize to the highly nontrivial (43) Freed, K. F. J . Phys. A 1985, 18, 871. (44) Bawendi, M. G.; Freed, K. F.; Mohanty, U. J . Chem. Phys. 1986.84, 7036. (45) Bawendi, M . G.;Freed, K. F. J . Chem. Phys. 1988, 88, 2741. (46) Bawendi, M. G.; Freed, K. F.; Mohanty, U. J . Chem. Phys. 1987,87, 5534. (47) Nemirovsky, A. M.; Bawendi, M. G.; Freed, K. F. J . Chem. Phys. 1987, 87, 7272. (48) Freed, K. F.; Pesci, A. I. J . Chem. Phys. 1987, 87, 7342. (49) Pesci, A. I.; Freed, K. F. J . Chem. Phys., in press. (50) Bawendi, M. G.; Freed, K. F. J . Chem. Phys. 1986, 85, 3007. (51) Bawendi, M. G.; Freed, K. F. J . Chem. Phys. 1987, 86, 3720. (52) Ramond, P.Field Theory, A Modern Primer; Benjamin/Cummings: Reading, MA, 1981. (53) Itzykson, C.; Zuber, J.-B. Quantum Field Theory; Mc-Craw-Hill: New York, 1980. (54) Freed, K. F. Ado. Chem. Phys. 1972, 22, 1.
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2197
Feature Article problem of np dimers on the lattice, before turning to the general case of N-mers. Some definitions and conventions are first required: The position of the ith lattice site with respect to an arbitrary origin of coordinates is designated as r,. We employ the shorthand notation 6,, = 6,1,r,, where as usual 6,, = 1 if i and j' correspond to the same lattice site and 6, = 0 otherwise. Only primitive Bravais lattices (with one monomer, solvent molecule, or void per unit cell) are considered; more general ones have at least two monomers, solvent molecules, for voids per unit cell and therefore require more sophisticated b o ~ k k e e p i n g . ~The ~ vectors to the z different nearest neighbors of a lattice site are written as a@,where p ranges from 1 to z. A single dimer must be composed of two segments at nearest-neighbor lattice sites. Designate the lattice sites as il and i2. The quantity 6r,,,r,2:, 611r12+8 mathematically constrains i l and i2to be nearest neiggbors, connected by a bond in the direction p. The total number of possible configurations available to this single dimer is represented as W1,2) =
1
7
2
x x 4,,12+8
11212
(2.1)
8=1
where the overall factor of is present because both ends of the dimer are equivalent and where the summation over p ranges over all directions of the dimer bond. The inequality il # i2 is somewhat superfluous as the 6 function restricts i2to be different from i,, but it is introduced as an excluded-volume constraint that is useful in the more general cases below. If we employ periodic boundary conditions, for simplicity, the sum over lattice sites in (2.1) produces a factor of Nl, whilethe summation over p in (2.1) gives the lattice coordination number z. Consequently, the total number of configurations available to this single dimer is trivially NIz/2. Now consider np indistinguishable dimers on a lattice. There 611,12+Bof (2.1) for each of the dimers on is a factor like the the lattice. In addition, we have the more sophisticated excluded-volume constraint that all monomers must occupy different lattice sites. For bookkeeping purposes the np dimers are labeled m = 1, ...,n,, the segments on a particular dimer are designated as y = 1, 2, and the position of an individual segment is therefore written as iym. These considerations lead us to express the total number of ways of placing the dimers on the lattice as
where the factor of (nP!)-I enters because of indistinguishability of the dimers. The individual terms in the product of (2.2) simply specify all possible orientations of the individual dimer bonds, while the great complexity of the problem is introduced through the outside summation over all possible lattice sites, subject to the excluded-volume constraint of single occupancy of any lattice site. Now let us pass on to the counting problem for a single A'-mer on the lattice. Begin with the expression (2.1) for a single dimer on a lattice in order to place the first bond of the polymer on the lattice. Monomers are successively added one at a time within the two constraints of single site occupancy and chain connectivity. Thus monomer y + I , numbered from one end of the chain, must occupy a vacant site nearest neighbor to monomer y . For each added monomer the above restrictions translate into a factor of xiyTl 6, and into appropriate restricted sums over all possible lattice sites as in (2.1) and (2.2). The number of ways of placing an N-mer on the lattice then emerges as W(1,N) =
c
il#i2#i3# #iN
N-I
2
n [c
a=l
~l,.J.,,+/9,1
&=I
(2.3)
equivalence of the two chain ends in the sequential numbering of the monomers. Purely random walk models of the polymer replace the restricted summations in (2.3) by a totally unrestricted sum over all il,i2,i3, ..., iN. The Kronecker 6 functions remove all the summations over lattice sites apart from the first one, leaving the trivial summations over the pa. The sums over pa yield a factor of z for each of the N - 1 bonds, and there are Ni starting positions for i,. Hence, simple counting produces the trivial result W( 1,N) = NlzN-'/2 for random walk chains. When the chains are self-avoiding, however, the summation constraints introduce the enormous mathematical complexity of describing excluded volume in a single polymer chain. This difficult problem is not discussed here as it is relevant to polymers in very dilute solutions where mean-field approximations of the type described here are inadequate to describe the strong influence of long-range correlations that are well handled by other approaches such as the renormalization group method.37 The generalization from (2.2) and (2.3) to the case of np N-mers on the lattice is now straightforward. It is merely necessary to introduce a factor like that in the summand of (2.3) for each of the individual chains and then to apply excluded-volume constraints within and between monomers on all polymer chains. This leads to the analytical representation of the exact number of configurations available to np N-mers as
The outer summations in (2.2)-(2.4) require the monomers to occupy different lattice sites. Equation 2.4 presents W(np,N)as a purely algebraic exact representation of the enormous counting problem posed by enumerating the number of configurations to np N-mers on a lattice. It might be suspected that (2.4) is merely a transcription of our ignorance into a complicated algebraic formalism. However, the lattice field theory for describing the packing entropies of polymers on a lattice derives (2.4)s1and thus contains the same information. The cluster expansions obtained using field theoretic method^^'-^^ should also be derivable directly from the algebraic expression of (2.4). We can thus use the available results of the lattice field theory to guide us in directly obtaining from (2.4) the Flory-Huggins combinatorial entropy as a leading approximation along with a systematic cluster expansion for corrections to the Flory-Huggins mean-field approximation. Although details are not given here, it is possible to generalize (2.4) to a collection of polymers with some specified distribution of chain lengths N - 1 and to introduce various branched-chain architectures. Before turning to the derivation of the Flory-Huggins meanfield approximation and its systematic corrections, it is convenient to introduce lattice Fourier transforms into (2.4). The Kronecker 6 functions in (2.4) can be written as sums over the wave vectors q in the first Brillouin zone of the reciprocal lattice using the standard formula = N r l Z exp[iq-(r, - r, - a,)] 9
with qx = 2an,/a, and n, = 0, 1, 2, ..., NI1l3- 1, etc. Inserting (2.5) and the definition of the nearest-neighbor structure factor z
into (2.4) yields an expression which serves as the starting point . for all subsequent derivations and discussions, namely
The index a sequentially labels the ( N - 1) individual bonds of the single polymer chain, and the factor of accounts for the exp[-iqam*(rimm- rim+lm)ll(2.7) ( 5 5 ) Freed, K. F.; Bawendi, M. G . Unpublished results.
Each of the 6 functions in (2.4) is transformed by (2.5), so that
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The Journal of Physical Chemistry, Vol. 93, No. 6, 1989
the vectors 4,” are associated with the a t h bond between monomers on the mth chain. The leading or mean-field approximation is obtained by retaining only the q = 0 contributions in (2.7). Physically, this approximation suppresses all effects of correlations that arise because of position dependences of the restricted summations in (2.7). Thus, when placing the nth monomer on the lattice, the mean-field approximation discards specific information concerning the location of the other (n - 1) monomers already on the lattice. Mathematically, the approximation retains only the q = 0 contribution from each sum over q-vectors in (2.7), yielding the mean-field expression WMF(n,,N) =
1
1
c I? ;fil
- -I
2 9 np. r , l Z , , + I N * m = ~
cf(O)/NI)l
(2.8)
U=I
Since the summand no longer depends on the location of the lattice sites, the evaluation of (2.8) proceeds very simply as follows: Using the identity f ( 0 ) = z from (2.6), the product over (Y yields (zN,-I)”-I, while the product over m gives ( ~ N ; l ) ” p ( ” - ~ The ). restricted sums can also be evaluated because the summand is independent of the site positions. This sum has Nl sites available for ill, NI - 1 sites for i z l , ..., for an overall contribution of (Nl!)/[(Nl- n f l ! ] . Hence, the mean-field approximation produces the final overall result
W(l)(n,,N) = - [ n p ( N -
I)/2“p(np!)I
,,+
c
61,
I?
cf(O)/NI)I
m=l a = ]
c If(S)/NIl
9fO
(2.13) where the prime on the product designates omission of the bond connecting sites i and j . The Kronecker 6 , , in (2.1 1) indicates that a pair of the npl” sites are identical, while all npl” - 2 others have excluded-volume constraints among themselves and with i =j. The evaluation of (2.13) proceeds in a similar manner to (2.8). Because of the removal of one summation through the d,,, there are only Nl - 1 sites available to the first site summation, so (2.13) yields W’)(n,,N) =
- [ n p ( N - 1)/2.P(np!)][(N, - l)!/(N1- n p ) ! ] x ( ~ / N ~ ) ” p ( ” - ~ ) -If(q)/Nl] l (2.14) q+o
The summation over q # 0 in (2.14) proceeds usingf(0) = z and the identity
U(q)= 0
(2.15)
9
to give The expression in (2.9) gives a packing entropy identical with that of Flory apart from the fact that Flory prohibits immediate self-reversals so the factor of z in (2.9) is replaced by z - I . It is possible to formulate the algebraic and field theoretic approaches to eliminate self- reversal^,^^ but introducing this local correlation yields a leading approximation that arbitrarily retains only certain correlations, and this considerably increases the complexity of developing a systematic cluster expansion for the remaining corrections. Since the mean-field approximation omits all of the correlations in the system that arise from the q # 0 contributions in (2.8), corrections to the mean-field approximation are simply evaluated by systematically retaining terms where individual q’s are allowed to range over all of the first Brillouin zone but q = 0. An easy way to develop the formulation proceeds by grouping the corrections according to the number of nonzero q-vectors present. The first correction, for example, consists of allowing any of the q-vectors to range over the whole first Brillouin zone. (Readers who are interested in the mathematical details may skip directly to section IV where applications are discussed.) We can choose the q-vector in the leading correction to label any of the n,(N- 1) bonds in the system, and contributions from each of the qumare identical. Thus, the first correction is
W1)(np,N)=
x I? {TiI ~ ( o ) / N , Ix~
[n,(iv- 1)/2np(n,!)]
jlI+..,+jNV m=1
C
If(q)/N11 exP[iq.(ri - rj)l (2.10)
9+0
where the indices i and j represent lattice sites occupied by any two consecutive monomers on the same chain. The evaluation of the summation over all ill, ..., i N n p in (2.10) follows trivially as in (2.8) apart from those for i and j which may be evaluated first and which can be rewritten as (2.11) c C h ( i J ) = C c h ( i j ) - ch(i,i) Ifi i
i j
i
Substituting (2.1 1) into (2.10) enables us to carry out the first summation over i in the double sum term of (2.1 1 ) using the well-known lattice sum Cexp(iq.rj) = NI6,,, (2.12) i
The in (2.10) along with (2.12) imply that the contribution from the double sum in (2.1 1) vanishes identically, leaving the leading correction as
Xf(q) = U(q) - z = -2 9fO
(2.16)
q
[Equation 2.15 follows from use of the relation Cqexp(-iqr,) = Nld,,,,in (2.6).] Consequently, (2.15) may be reexpressed using (2.10) as
*‘)(np,N) = [np(N- ~ ) / ( N-I I ) l p F ( @ 7 (2.17) The next corrections to the mean-field approximation consist of those terms containing 2, 3, ... nonzero q-vectors. The analysis proceeds in a similar manner to that given above for both the mean-field portion and the leading correction, with some increased complexities in the counting and in evaluating a few of the lattice sums. Each of these computations involves some small counting problems, a feature which should not come as a great surprise because the method is, in effect, replacing an intractable counting problem by a sequence of smaller, simpler counting problems. Rather than pursuing the tedious analysis of evaluating the corrections, we turn instead to a description of the physical content of such an expansion in terms of the number of nonzero q-vectors. This insight is gained through introduction of a diagrammatic r e p r e s e n t a t i ~ nof~the ~ ~ correction ~ terms to the partition function for the Flory-Huggins packing entropy. 111. Diagrammatic Representation for Correctioss to Flory-Huggins Theory of Packing Entropies The mean-field contribution in (2.8) ignores all correlations within the system by having the summand independent of the specific positions {r,) of the monomers on the lattice. This independence corresponds to the Flory approximation of estimating excluded volume on the basis of the average fraction of available sites rather than on the precise configuration of monomers surrounding a given lattice site. When q-vectors in (2.7) are allowed to range over the whole first Brillouin zone, explicit dependence on the positions {r,}of the monomers on the lattice reappears and consequently provides corrections arising due to correlations in monomer positions that are induced by packing and chain connectivity. The q-vectors are associated with individual bonds between monomers, and the leading correction term in (2.10) involves a single correlated bond with all other bonds treated in the uncorrelated mean-field approximation. We designate this correlated bond by two circles connected by a straight dashed line, where the circles designate the two lattice sites and the line the bond between them. (See diagram A of Figure 2.) The next correction contains two nonzero q-vectors that must be associated with two correlating bonds in the system. The two bonds may lie on different chains as in diagram D of Figure 2,
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2199
Feature Article
B
A I
/*,
0'
O , \
\*'
/
? I
/
A
E
D
C
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number of lattice sites N,, and another term dB which is dependent on the type of lattice, but not on the chain architecture. Thus, a diagram containing n monomers is rewritten as
N"" A b
M
exp[-iqs.(ril - riT+,)1If(qa)/zl
P P
\Ab N
Figure 2. Diagrams used to calculate entropic corrections to the meanfield packing entropy of linear chains to order z - ~and Nl.
or they may lie on the same polymer chain. In the latter case the two correlating bonds may lie sequentially along the chain, as in diagram B of Figure 2, or they may lie nonsequentially along the chain, Le., be separated by at least one uncorrelated monomer unit. Diagram C of Figure 2 represents the nonsequential bonds as being connected by a wavy line. The two bond diagrams in B-D correspond to correction terms in which there are exponentials in rj involving the depicted lattice sites and in which the summation has excluded-volume constraints on these lattice sites. Hence, this term describes packing and chain connectivity induced correlations between all monomers of the two bonds. The three q-vector diagrams then follow obviously as in diagrams E, F, and J of Figure 2 which display some of the three-bond diagrams. No additional diagram elements are required to represent these three-bond and all higher order diagrams; the dotted lines, filled circles, and wavy lines suffice. An analysis of the two-bond and higher bond diagrams indicates that each diagram has a value that can be written as the product of the mean-field partition function of (2.9) and an expansion in powers of 2-l. The mean-field approximation can be shown to be exact in the limit z a,and this makes it useful to consider a systematic treatment of corrections to mean field in terms of an expansion in powers of 8 . The computation of corrections through order z - ~requires retention of all contributions from diagrams with up to 2n correlating bonds4* A large number of interesting physical effects emerge first in order z - ~ . Their description, therefore, requires retention of contributions with up to four correlating bonds that are depicted in Figure 2. It is possible to provide straightforward rules for passing from the diagrams to expressions for corrections to the mean-field approximation. We indicate this process by the schematic equation47
-
where DB represents the value associated with a diagram of B bonds such as those presented in Figure 2. Each diagram has a combinatorial coefficient yD arising from the number of ways of selecting the nonzero q-vectors in (2.7) from the possible n,(N - 1) vectors qamfor the bonds in the system. The factor y is, in general, the only part of the diagram that is dependent on the chain architecture when the theory is generalized to allow arbitrary branched structure and therefore to permit monomers to extend over several lattice sites.47 In evaluating y it is to be assumed that the chain has a definite direction corresponding to the fact that we have originally numbered the monomers sequentially from a = 1 to N along the chain. The quantity y is equal to the number of distinct ways of extracting the nonzero q-vectors from all possible n,(N - 1) q-vectors. The leading correction (2.8) has y = n,(N- I). Values of y for diagrams F and I are Y~ = np(np - 1)(N - 1)(N - 2) and y, = n,(N - 4)(N - S), respectively. The (DB] in (3.1) further factor into one term, which depends only on the number of monomers in the diagram and on the total
(3.2)
The expression for dB is obtained as follows: Assume that the diagram contains n monomers connected by m bonds. Recalling that a particular direction along the chain has been chosen for each bond, the lattice sites in the diagram are labeled sequentially as i, through in, while the bonds connecting them are labeled sequentially ql through qm. Each bond with label q6 that connects lattice sites i, and i,+l contributes a factor to dB of (3.3)
The example in (2.10) has only a single nonzero q-vector. Excluded-volume constraints still apply to all the lattice sites il through in that are present in diagram DE. Therefore, there is an overall summation such that the i, range over all lattice sites with the omission of configurations where two or more of the i, represent the same lattice site. In addition, there is a summation Cq,,,,,,qmfO that ranges over the first Brillouin zone excluding all q = 0 terms. Just as in the example of the leading correction in (2.10), the constrained summation over lattice sites is evaluated by rewriting it in terms of a series of unconstrained summations over all lattice sites. In order to represent this process, it is convenient to let i, be replaced by x and to let 6ix,iy,iz= 6(x,y,z,...) be unity if i, = iy - i, = ... and be zero otherwise. Then this conversion from constrained to unconstrained summations has the general form Of50
C
h( 1,...,n) =
I # ...# n
C
h(1, ...,n) -
l,.,.,n
I)!
C
I , ...,n
II . .I [&(I,...,m,)...6(C mj+l,... C mj)
h(1, ...,n) + ...
j= I
j= 1
+ ...I
X
+ (-l)n-l(n - l)! E 6(l, ...,n)h(l, ...,n)
(3.4)
1 % . ,n
where 1 is the number of groupings of equated lattice sites, m, is the number of sites equated in thejth grouping, and the sums of Kronecker 6 functions in [...I represent all possible permutations of the same types of groupings. For example, the quantity [6( 1, 2) + ...I represents n ! / [ 2 ! ( n- 2)!] different terms pairing up two lattice sites in all possible ways. The transformation in (3.4) can be represented diagrammatically4' as a process of contracting the n lattice sites in the original diagram dB in all possible ways, thereby generating a rather large number of contracted diagrams. However, the unrestricted lattice sums provide a simplification because of the identity (2.12) and the restriction to q # 0 on the lattice sums. If a contraction leads to diagram in which a particular lattice site i is neither contracted with another site nor is connected to more than one bond and thus appears as a dangling end, the summation over this lattice site i gives q = 0 as in (2.12). However, the constraint q # 0 in the overall sum means that the net contribution from such a term vanishes. Hence, it is only necessary to consider those contractions from (3.4) that produce contracted diagrams with no dangling ends. The contracted diagrams of this type are summarized in Figure 3 for original diagrams containing up to four bonds. Contracted diagrams are written as where B is the number of bonds in the original diagram and c is a sequential counting index. For example, the leading order contribution from (2.10) is transformed by (2.1 1) to the double sum and the single contraction in the latter equation. The double sum has two dangling ends, so this contribution vanishes, whereas the single contraction corresponds to the contracted diagram Rl,,. The constrained summations over q # 0 are readily evaluated by adding and subtracting the q # 0 contribution and by use of , representation of dB the identity (2.8). The coefficients y ~the as a linear combination of the and the values of Rec are given in ref 47 for all diagrams with B I4, where the chains of N bonds
Freed and Bawendi
2200 The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 .,E
I T -
4-
Figure 3. Some leading contracted diagrams (R)as described in the text. Nonvanishing contracted diagrams are always closed and have no dangling ends. There is only one one-bond contracted diagram RI,’. Two bonds produce three different diagrams: Rz,l, R2,2,and R2,. The connected diagrams R2,’ and R2,3have the leading orders of N I 3and of N,’, respectively, while the disconnected R2,2begins with order N t .
can have arbitrary branched structures. The tabulated results in ref 47 are useful for readers who would like to apply the diagrammatic rules to several examples. The theory provides a power series in z-] for corrections to the mean-field partition function of (3.1). The entropy is computed by the standard formulation
,
S=IO
-
Polymer Voiume Fraction, 9
Chain insertion probability -N-I In p vs volume fraction 6 for athermal chains (epp = 0) on a square lattice ( z = 4) with N = 10 monomers. (-) is the prediction of the Flory approximation, (- --) is the is the prediction through prediction of the Huggins theory, and order z-2 from our cluster expansion. The + are the Monte Carlo data points of Dickman and Hall (ref 35). Figure 4.
(-e-)
1.3
.9
.e
(3.5) a
However, one complication emerges here that is absent in all other cluster expansions and many-body expansions of which we are aware. In all of these other cases diagrams with disconnected pieces (portions of diagrams that are not connected to other portions by bonds) have values that factor into the product of the values of the individual disconnected pieces. Such a factorization does not occur with the diagrams of Figure 2, in part because of the excluded-volume constraints on the summation over lattice sites. Thus, diagrams with p disconnected portions can have contributions which vary with the size of the system as NIP. Such terms cannot contribute to the entropy of ( 3 . 9 , which must be proportional to Nl in the thermodynamic limit of NI m. It is, however, straightforward to collect the expansion of the logarithm in (3.5) in powers of the D, into cumulantsS7that are individually Although this proportional to at most Nl in the limit Nl lack of factorization of disconnected diagrams requires some additional algebra, the final entropy is an extensive function, as required.
-c: . s 5.4 .3 .2
Tolymer Volume Fraction, p
-+
-
IV. Comparisons with Monte Carlo Simulations and with Experiment Before turning to a comparison of the lattice theory with experiment, it is imperative that we check on the adequacy of our sole approximation to the lattice model that arises from the expansion in powers of z-’. This is best accomplished by comparing the lattice calculations with Monte Carlo simulations of exactly the same model, so that we may separate deficiencies of the mathematical approximations from any possible deficiencies of the lattice model. Fortunately, Dickman and Hall3s have performed Monte Carlo simulations of the packing of linear polymer chains on a square-planar lattice ( z = 4),corresponding identically with the simple model of linear chains and voids described in sections I1 and 111. They compute the chain insertion probability p(n,N) that is defined as the probability that a randomly generated self-avoiding chain of length N , randomly placed into a config(56) Ashcroft, N . W.; Mermin, N. D. Solid State Physics; Saunders: Philadelphia, 1976. ( 5 7 ) Kubo, R. J . Phys. SOC.Jpn. 1962, 17, 1100.
Chain insertion probability -N-’ In p vs volume fraction 4 for athermal chains (tpp = 0) on a square lattice ( z = 4) with N = 20 monomers. (-) is the prediction of the Flory approximation, (- - -) is the prediction of the Huggins theory, and (-*-) is the prediction through order zV2from our cluster expansion. The + are the Monte Carlo data points of Dickman and Hall (ref 35). Figure 5.
uration of n polymers also of length N , does not overlap or intersect any of the n chains. The insertion probability p ( n , N ) is written in terms of the n-polymer partition functions W(n,N)of (2.4) as p ( n , N ) = (n + l)W(n+l,N)/Wn,N)W(l,N)
(4.1)
and, hence, the logarithm of the insertion probability is related to the chemical potential of the chains (and hence to their entropy in the present case). Figures 4 and 5 display the logarithm of the chain insertion probability as a function of the volume fraction 4 = n N / N l of polymers on a square-planar lattice with 10 and 20 monomers, respectively. The Flory theory is presented as the solid line, while Huggins approximation is given as the dashed line. It is evident that the Huggins approximation is quite superior to that of Flory, which is commonly referred to as the FloryHuggins approximation. Our lattice theory calculation to order z - ~is represented as the dot-dash line that is even superior in accuracy to the Huggins approximation. W e note that the same lattice calculation applies to cubic lattices in which z = 6, so the lattice calculations are anticipated to be more accurate in three than in two dimension^.^^ Computational difficulties have pre-
The Journal ofPhysica1 Chemistry, Vol. 93, No. 6,1989 2201
Feature Article cluded Dickman and Hall from carrying out the Monte Carlo simulations for N = 20 and 4 > 0.65. It is possible that an additional correction in, say, z - ~might be necessary to accurately describe simulations for these higher volume fractions on square lattices, but the corrections may not be necessary for cubic lattices because of the larger z . Similar trends exist between the Flory, Huggins, lattice theory approximations and the Monte Carlo simulations for the compressibility factor.45 We now turn to the major question of understanding the molecular origins of the entropic contribution to the Flory x parameter. In order to do so it is necessary to generalize the theory to describe polymer blends (mixtures of polymers in the liquid state). This is readily accomplished4’by simply appending species labels to each of the np polymer chains in (2.4). The bookkeeping and the analysis of diagrams in which there are chains of several species is somewhat t e d i o ~ s , but ~ ~ the . ~ ~basic procedure follows that outlined here. The free energy of mixing for a binary blend is written in the standard form [cf. (1.1)] (AFmix/NlkBT) = (41/M1) In 41+ (42/M2) In 42
+ ~4142 (4.2)
where Mi denotes the number of sites occupied by a chain of type i , and x is computed from the theory. Interaction energies are not present, so x in (4.2) is purely entropic. The calculations of ref 48 for linear chains on hypercubic lattices give Xentr(linear,linear)= Z-’(Mi-’ - M 2 )
-’
- 72Z-2$~2(M1-1- M2-1)4+ o ( Z - 3 ) (4.3)
When the chains are long ( M i >> I ) , the entropic contribution to x in (4.3) is rather small. The two species of linear chains differ only in their chain length (interaction energies are absent), and the entropic x in (4.3) is only an end correction. The Huggins c ~ u n t i n gscheme ] ~ ~ ~ likewise ~ produces x as an end correction that is too small in magnitude to explain experimental values of the entropic x. In addition, experiments by Ito et aLS8and by Batess9 observe the entropic x to be linear functions of for poly(ethy1ene oxide)-poly(methy1 methacrylate) blends and for 1,4-p0lybutadiene-] ,2-polybutadiene block copolymers and their blends with 1,2-polybutadiene homopolymers, respectively. This linear dependence departs from (4.3). A consideration of the monomeric structures of two abovementioned blends quickly leads to the conclusion that the monomers of the two species in the blend have differing sizes and shapes, and consequently, the real experimental systems strongly violate the model assumption that all monomers occupy identical lattice sites with identical volumes. Thus, it is important to generalize the lattice model to enable consideration of polymers, whose monomers extend over several lattice sites and which therefore have differing sizes and shapes for the monomers of different species.47The Flory approximation’ does not distinguish between these different chain architectures since it ignores all correlations in the chain. However, when there are branch points in the chain, correlations involving three bonds emanating from a common junction are different from those in linear chains.4749 Thus, it is necessary to consider at least three-bond diagrams to first include the effects of monomer structure, and these first contribute in order z-*. While space precludes our description of how the theory is generalized to treat chains in which the monomers cover several lattice sites or to study chains with complicated branched architect~res,4~-”~ such as present in networks, we motivate how this is done by considering the simple example of a single tetrafunctional unit, Le., a lattice site at which four bonds are joined. Such a situation corresponds to the number of configurations 4
W(l,tetra) =
i#i,,#
2
II C ...#i4 a = l @.=I
G~,~,,+@,
(4.4)
where extensions of the theory to chains containing the monomer ( 5 8 ) Ito, H.;Russell, T. P.; Wignal, G . Macromolecules 1987, 20, 2213. ( 5 9 ) Bates, F. S. Macromolecules 1987,20, 2221.
‘.
O
r
7
--
-a-b
__
b - e
_ _ _ _c -
e
-
c
...
0
b
2
4
6
8
10
MOLECULAR WElGTH RATIO (MiIM2) Figure 6. Computed entropic x parameter for blends as a function of the ratio M I / M 2of the molecular volumes of the two polymer species.49All curves use M I = 100 and z = 12. The solid line models a blend of polyethylene (monomers of type a) and poly( 1,2-dimethylethylene)(type b) in which half of the monomers occupies one and two (comb geometry) lattice sites, re~pectively.~ The dashed line models a type b-poly(dimethylsiloxane) (type e) blend. The poly(dimethylsi1oxane)chain has two side groups (each covering one lattice site) on alternate main chain sites. Monomers of type c correspond to poly(tetramethylethy1ene).
units of (4.4) and other monomer structures is given in ref 47-49. Other extensions are provided in ref 50 and 51 to rods and to semiflexible chains, respectively, using the original field theoretic formulation. A consideration of the diagram rules and the results of those references enables transcription of the general approach into the present simple algebraic formulation of the packing entropy. When both chains in the blend are taken to have the same monomer architecture and therefore to only differ in the chain length, the entropic contribution to x is found4* to be given by (4.3), with Mi again designating the total number of lattice sites covered by a single chain. However, when the two chains have different architectures, xentris computed as being equal to the quantity given in (4.3) plus the correction term 6 x that has the form48
where the functionf(x) = u - bx - c/x has coefficients a, b, and c that depend on the two chain architectures and that are presented in ref 48 for 10 different blends. When M I = M 2 , f ( x ) - ~ u o ~ , providing a stabilization to mixing. Note that 6x in (4.5) is a linear function of 4’ in conformity with the available experimental data on entropic x parameters. Figure 6 shows how 6x varies with M1/M2for chains whose monomers have differing structure^.^^ The possibility of significantlyaiding mixing by appropriate choices of M l / M 2 and monomer structure is apparent. In summary, the lattice calculations provide the first molecular derivation48 of the entropic contribution to the x parameter of blends as arising from the packing together of chains whose monomers have different sizes and shapes. The lattice model calculations provide explicit dependences off(x) on the monomer shapes, but perhaps the greatest deficiency of the available calculations is the treatment of all chains as being completely flexible. Methods exist for introducing the differences between trans and gauche e n e r g i e ~ ~(chain ’ . ~ ~ semiflexibility), but these have yet to be implemented for the calculation of the entropic x parameter for blends.
-
V. Introduction of van der Waals Interaction Energies
By analogy with the Mayer theory of nonideal gases,60it is clear
2202
Freed and Bawendi
The Journal of Physical Chemistry, Vol. 93, No. 6 , 1989
that we may include interaction energies by introducing into (2.4) a factor of n,,,(l +A,), where the Mayerffunction is defined in ( 1.3) and where the depends on the particular species occupying sites i and j . The evaluation of such contributions is complicated by considerations of chain connectivity. Nevertheless, the formalism in section I1 enables us to provide an explicit algebraic expression for the interaction energies by first noting that the interactions are taken to apply only between nearest-neighbor lattice sites. Therefore, it is merely necessary to insert the factor
into (2.4), where the aiJ+@ accounts for the nearest-neighbor character of the interactions. The expansion of the product in (5.1) generates a standard Mayer cluster expansion in powers of theAj. The Kronecker 6’s are just of the same form as the bond constraints in (2.4). An extension of the spirit of the mean-field approximation in section I1 would require that we introduce (2.5) into (5.1) for each of the delta functions and retain only the q = 0 contributions as the leading order. However, implementation of this procedure is rendered difficult because of the product nature of (5.1). Therefore, we instead use the expanded form of (5.1) and consider evaluation of the partition function in powers off,. The logarithm of the partition function is then rearranged in terms of cumulants5’ to provide an expansion of the free energy in powers of the Mayer f functions. Let us consider the simple case of a polymer blend in which there are only polymer-polymer interactions. Then, the leading term from (5.1) introduces a factor of Ci>,C~=16,j+~, wheref = exp(t) - 1 with t the polymer-polymer interaction in units of kT. The is already contained in the sums over the polymer sites in (2.4), apart from the need to introduce a factor of (1/2) to avoid double counting of the interactions. The evaluation of this term proceeds almost identically with the treatment of the single bond leading correction to the entropy. Mean field implies that we take the q = 0 contribution from all 6 functions in (2.4) and from the additional energy factor. Because the two interacting sites i and j need not necessarily be bonded to each other, the number of ways of selecting these two sites is (n+V)’ - n$? The Eagives a factor of z , and an overall factor of remains from the use of (2.5) in this leading term. Collecting these factors together and performing the mean-field type counting of section 11, this leading energetic contribution to the partition function emerges as
xi>,
€-diagrams =
Figure 7. Diagrammatic expansion for the energy diagrams. Only the first few diagrams are shown as an illustration.
evaluating the entropy corrections can be applied to the interaction terms if the q = 0 terms are first separated from the q # 0 contribution^.^^ The latter are then evaluated just as the entropy corrections, where the interaction lines are treated just as correlating bonds. The q = 0 contributions from a diagram can be shown to provide values proportional to those of the corresponding entropy diagrams that are obtained by simply removing the interaction lines. The q # 0 contributions from the interaction lines produce some diagrams having no counterpart among the entropy diagrams. However, the rules for forming contractions and for evaluating these diagrams are the same as those presented in section 111. It is possible to formulate the theory in terms of more realistic interaction potentials for nonlattice polymers.61 The zero-order term is a reference monatomic fluid of the monomers with a random distribution of uncorrelated bonds. The cluster expansion corrects for the presence of bonds (additional correlations) between monomers. As will be described elsewhere,6I the off-lattice theory proceeds in formally the same manner, but it requires the nontrivial evaluation of correlation functions in the reference monatomic monomer fluid. While some of the numerical results must differ between the lattice and off-lattice formulations, the general qualitative results must be the same. This heightens the utility of the immensely simpler computational nature of the lattice version.
VI. Computations of Heats of Mixing and Its Composition, Molecular Structure, and Temperature Dependence While the presence of contribution in t2 to the partition function of section V automatically implies that heats of mixing of blends, polymer solutions, etc. must be temperature dependent, it is imTaking logarithms and considering the thermodynamic limit of )VI m provides the leading contribution to F / N , k T as ~ f 4 ~ / 2 k T . portant to evaluate these quantities to determine whether they are of appropriate order of magnitude. For this purpose we now which is just of the simple Flory-Huggins form upon expansion consider the interaction energies e;, to be in energy units rather o f f t o order e. Subsequent terms in the expansion of (5.1) lead than in units of k T . An incompressible model of a blend requires to contributions to the free energy proportional t o p and higher the use of three interaction parameters til, t22, and q2. One of powers, and these terms obviously must be responsible for the these interaction energies may be chosen as the zero of energy, experimental observation of a temperature dependence of T leaving two independent interaction energies whose magnitudes multiplied by the effective enthalpic x parameter in blends. The correspond to differences in van der Waals energies and are existence of such a non-Flory-Huggins temperature dependence therefore of the order of 10-40 K. When the blend is incomto the x parameter follows immediately from the structure of (5.1) pressible (no voids), it is possible to show that the free energy of and (2.4) without the need for postulating models of preferential mixing depends only on the interactions between different types of monomers. It is possible to introduce a diagrammatic representation of the energetic corrections produced by having the factor (5.1) introbut in the more realistic general case of chains with free volume duced into (2.4).45*46,49W e represent the individual Mayer J there are three independent interaction parameter^.^^ (We can, functions as curved lines connecting the interaction sites that are of course, more generally consider the different components of represented with crosses. Some of the leading order diagrams are a monomer to interact with different energies,49but for simplicity represented in Figure 7. Interaction sites may coincide with we retain the model of a blend with just t i l , t 1 2 ,and tZ2.) Figure bonded lattice sites, i.e., terms from the entropy portion of (2.7) 8 presents lattice model computations of the heat of mixing of that contain q # 0 contributions. Substitution of (2.5) into (5.1) a binary blend with typical van der Waals energies. The curves shows that the interaction terms have contributions from both q display a roughly parabolic composition dependence which is in = 0 and q F 0 . The diagrammatic techniques introduced for accord with a great body of experimental literature on heats of -+
(60) Mayer, J . E.; Mayer. M. G . Statistical Mechanics; U’iley. Xew York, 1940
( 6 1 ) Freed, K F J Chem Phys , i n press
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2203
Feature Article 0
.
0
0
0.2
0
0.4
l
0.6
function degree of swelling.64 The polymer network is a single large cross-linked molecule to which the lattice theory can also be applied. The interaction energy may be written in the form
C
0.8
1
VOLUME FRACTlON ( 0,) Figure 8. Composition and temperature dependence of heat of mixing of models49 of polyethylene-polypropylene blends. The polypropylene monomer extends over three lattice sites with one side group methyl on alternate main chain methylene units using M I = Mz = 100 and z = 12. The interaction produces e = 68K with component 1 polyethylene.
mixing of binary blends.62 In addition, there is a considerable temperature dependence whose magnitude is again fairly consistent with ranges observed experimentally. The eF"K are found to be symmetrical functions of composition when both species have monomers with the same structures, but different monomer structures for the two species lead to figures like that in Figure 8 with an asymmetric minimum, as well as ones for which the heat of mixing is unfavorable over certain ranges of composition but is favorable over others.49 Computations using accessible experimental energies yield curves for the heats of mixing that are similar to those observed e ~ p e r i m e n t a l l y . ~It~ is - ~thus ~ clear that the lattice model reproduces the important qualitative features of the composition and temperature dependence of the heat of mixing as well as their quantitative values. A more thorough quantitative test of the theory would require input of van der Waals energies for monomers or for parts of monomers, but before such an analysis can be pursued, it would be useful to extend the theoretical methods to incorporate the semiflexibilityof the bonds. Methods for including these trans-gauche energy differences have already been incorporated into both the field theory form of the cluster expansion and the direct algebraic a p p r ~ a c h ~described ',~~ here. However, only the leading corrections for semiflexibility ~ ' a more thorough treatment is have so far been e ~ a l u a t e d , and required before comparisons can be made with experiment. Another interesting application of the lattice theory is to the description of the heats of mixing of polymer networks as a ~~
~
(62) Walsh, D. J.; Higgins, J. S.; Rostami, S. Macromolecules 1983, 16, 388. (63) Masegasa, R. M.; Prolongo, M. G.; Hirota, A. Macromolecules 1986, 19, 1478.
where @p is the volume fraction of polymer, @s is the volume fraction of solvent (dJs = 1 - dJp), and x is thereby defined as the interaction parameter. The lattice theory predicts64 that the effective interaction x is a function of the cross-length density n,. The leading contributions in the cross-length density are linear in n,, and corrections of order n: enter only rather high order in the cluster expansionM(of order z4), so the quadratic contributions (in n:) should not be observable at the experimental low cross-link densities. These preliminary qualitative predictions of the lattice model are in excellent accord with recent experiments by McKenna and co-~orkers,6~ who find a linear dependence of the x parameter on cross-link density. From the free energy it is also possible to compute the effective interaction parameter xSCat that is measured in extrapolation of small-angle neutron scattering data to zero angle. The lattice model calculations predict49a composition and temperature dependence of these effective interaction parameters that are of the forms and magnitudes experimentally observed. One interesting example arises from the experiments of Bates and co-workers6 using isotopic H / D blends of either poly(ethylethy1ene) or poly(vinylethylene). In these cases the effective interaction parameter xSCat is almost entirely enthalpic in origin, and the van der Waals interaction energies may be estimated from the monomer interthat action energies.66 The calculations obtain49a value of xSCat is greater in magnitude than the Flory-Huggins approximation xFH = zc/2kT, but that is slightly lower than the experimentally deduced values. Better accuracy will presumably be available when semiflexibility and group van der Waals energies are introduced into the computations. Again, the lattice model theory is in general semiquantitative agreement with experimental observations, and it remains now to design experiments with specifically tailored molecular systems in order to more precisely test the limitations of the lattice model in describing the properties of blends, concentrated solutions, networks, and other dense polymeric systems. Acknowledgment. Portions of this work have been done in collaboration with U. Mohanty, A. M. Nemirovsky, and A. I. Pesci. We are particularly grateful to Dr. Pesci for providing Figures 6 and 8. This research is supported, in part, by NSF Grant DMR86-14358 (condensed matter theory program). M.G.B. gratefully acknowledges an ATT Bell Laboratories Ph.D. scholarship. Registry No. Polyethylene, 9002-88-4; poly( 1,2-dimethyIethylene), 32167-46-7; poly(tetramethylethylene), 3 1669-03-1; polypropylene, 9003-07-0. (64) Freed, K. F.; Pesci, A. I. Submitted for publication. (65) McKenna, G. B.; Flynn, K. M.; Chen, Y. Polym. Commun. 1988.29, 272. (66) Bates, F. S.; Muthukmar, M.; Wignall, G. D.; Fetters, L. J. J . Chem. Phys. 1988, 89, 535.
