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Molecular Simulation of Polymer Melts and Blends: Methods, Phase Behavior, Interfaces, and Surfaces

Peter Virnau, Kurt Binder, Hendrik Heinz, Torsten Kreer, and Marcus Müuller

1.1 Introduction

Understanding thermodynamic properties, including also the phase behavior of polymer solutions, polymer melts, and blends, has been a long-standing challenge [1-7]. Initially, the theoretical description was based on the lattice model introduced by Flory and Huggins [1-7]. In this model, a flexible macromolecule is represented by a (self-avoiding) random walk on a (typically simple cubic) lattice, such that each bead of the polymer takes one node of the lattice, and a bond between neighboring beads of the chain molecule takes a link of the lattice. For a binary polymer blend (A,B), two types of chains occur on the lattice (and possibly also "free volume" or vacant sites, which we denote as V). The model (normally) does not take into account any disparity in size and shape of the (effective) monomeric units of the two partners of a polymer mixture. Between (nearest neighbor) pairs AA, AB, and BB of effective monomers, pairwise interaction energies, , and , are assumed. Thus, this model disregards all chemical detail (as would be embodied in the atomistic modeling [8-10], where different torsional potentials and bond-angle potentials of the two constituents can describe different chain stiffness).

Despite the simplicity of this lattice model, it is still a formidable problem of statistical mechanics, and its "numerically exact" treatment already requires large-scale Monte Carlo simulations [11-13]. Consequently, the standard approach has been [1-7] to treat this Flory-Huggins lattice model in mean-field approximation, which leads to the following expression for the excess free energy density of mixing [4]:

Here , and are the volume fractions of monomers of type A, B and of vacant sites, respectively. Every lattice site has to be taken by either an A monomer, a B monomer, or a vacancy, and for simplicity the (fixed) lattice spacing is taken as our unit of length. and are chain lengths of the two types of polymers (we disregard possible generalizations that take polydispersity into account [5]). Thus, the first three terms on the right-hand side of Eq. (1.1) represent the entropy of mixing terms, while the last three terms represent the enthalpic contributions (, and are the phenomenological counterparts of the pairwise interaction energies , and , respectively). Note that in the entropic terms the (translational) entropy of a polymer is reduced by a factor in comparison to a corresponding monomer because of chain connectivity. In deriving this simple expression for the entropy, the fact that polymer chains on the lattice cannot intersect either themselves or other chains has not been explicitly taken care of: the excluded volume constraint is only taken into account via the constraint that a lattice site can be taken by at most one monomer, but only the average occupation probabilities , and not the local concentrations , of a lattice site i enter: while or 0 and or 0, , . By we denote a thermal average of an observable in the sense of statistical mechanics at a given temperature, , that is:

where the sums are extended over all configurations ("microstates") of the considered statistical system, is the corresponding energy function (the "Hamiltonian" of the system [4, 9, 10]), and its partition function.

From these definitions it should be clear that in the exact expression for the enthalpy one should expect terms of the type:

where is the number of nearest neighbors of a site on the lattice, rather than . The latter expression results, of course, if this correlation function is factorized, . This neglect of correlations in the occupancy of lattice sites would become accurate in the limit , but turns out to be rather inaccurate for the simple cubic lattice, which has only. Moreover, as far as unmixing of a polymer blend is concerned, only interchain contacts and not intrachain contacts contribute (strongly attractive intrachain interactions can cause contraction or even collapse of the random coil configurations).

We shall not discuss Eq. (1.1) further for the general case, but rather focus on the two most important special cases, namely incompressible blends and incompressible polymer solutions. Taking one can reduce Eq. (1.1) to a simpler expression [1-4], where , :

where in the mean-field approximation the Flory-Huggins parameter is related to the pairwise energies by:

As an example for the predictions that follow from Eqs. (1.4) and (1.5), we note that the stability limit ("spinodal curve") of the homogenous phase is given by the vanishing of the second derivative of with respect to :

which yields the equation:

Equation (1.7) describes the spinodal curve in the plane of variables . The maximum of the spinodal curve for such a binary incompressible mixture yields the critical point, that is:

For the simplest case of a symmetric mixture , this reduces to , and .

The case of an incompressible polymer solution results if we interpret B as a solvent molecule in Eq. (1.4) by putting [or alternatively put in Eq. (1.1) and reinterpret as solvent molecule]. However, while for polymer mixtures in the state of dense melts incompressibility is often a reasonable first approximation, for polymer solutions in some cases such an assumption is inadequate, for example, if one uses supercritical carbon dioxide as a solvent for the polymers [7, 14].

When one tries to account for real polymer systems in terms of models of the type of Eqs. (1.1)-(1.8) the situation is rather unsatisfactory; however, when one fits data on the coexistence curve or on , the latter quantity being experimentally accessible via small angle scattering, one finds that one typically needs an effective -parameter that does not simply scale proportional to inverse temperature, as Eq. (1.5) suggests. Moreover, there seems to be a pronounced -dependence of , in particular for . Near , on the other hand, there are critical fluctuations (which have been intensely studied by Monte Carlo simulations [11-13, 15] and also in careful experiments of polymer blends [16-18] and polymer solutions [19]). Sometimes in the literature a dependence of the parameter on pressure [18] or even chain length is reported, too. Thus, there is broad consensus that the Flory-Huggins theory and its closely related extensions [20] are too crude as models to provide predictive descriptions of real polymer solutions and blends. A more promising approach is the lattice cluster approach of Freed and coworkers [21-23], where effective monomers block several sites on the lattice and have complicated shapes to somehow "mimic" the local chemical structure. However, this approach requires rather cumbersome numerical calculations, and is still of a mean-field character, as far as critical phenomena are concerned. We shall not address this approach further in this chapter.

A very popular approach to describe polymer chains in the continuum is the Gaussian thread model [24-26], and if one treats interactions among monomers in a mean-field-like fashion this leads to the so-called "self-consistent field theory" [27-33] of polymers. This theory is an extension of the Flory Huggins theory to spatially inhomogeneous systems (like polymer interfaces or microphases-separated copolymer systems), with respect to the description of the phase diagrams of polymer solutions and blends. However, it still lacks chemical detail and is on a mean-field level; hence we shall not dwell on it further here.

An alternative approach that combines the Gaussian thread model of polymers with liquid-state theory is known as the polymer reference interaction site model (PRISM) approach [34-38]. This approach has the merit that phenomena such as the de Gennes [3] correlation hole phenomena and its consequences are incorporated in the theoretical description, and also one can go beyond the Gaussian model for the description of intramolecular correlations of a polymer chain, adding chemical detail (at the price of a rather cumbersome numerical solution of the resulting integral equations) 37, 38. An extension to describe the structure of colloid-polymer mixtures has also become feasible 39, 40. On the other hand, we note that this approach shares with other approaches based on liquid state theories the difficulty that the hierarchy of exact equations for correlation functions needs to be decoupled via the so-called "closure approximation" [34-38]. The appropriate choice of this closure approximation has been a formidable problem [34-36]. A further inevitable consequence of such descriptions is the problem that the critical behavior near the critical points of polymer solutions and polymer blends is always of mean-field character.

There have been many other attempts to base the description of polymer solutions, melts, and blends on liquid-state theory (e.g., [41-44]) and we shall not...