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
I
~
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.
2196
The Journal of Physical Chemistry, Vol. 93, No. 6 , 1989
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
2198
Freed and Bawendi
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
/*\ \
o1
/
9 9 9
I
l
l
A A d
J
DE = [(Nl - n)!ldB/(Nl!)
/
\o
/
/*\ \
?I P,
? ? ? ?
aids
b a a
L
K
/
o/
/*\
\
0
'
/*\ \
I \
/
\
ti
'0-a'
I
H
0'
F
/*\ \
o/
\0
G
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.