Flory-Huggins lattice model

Mixing term random-mixing lattice model (Flory-Huggins quasi-chemical lattice model Prausnitz et al. [36]... 

Due to the shortcomings of the classical Flory-Huggins lattice model, Flory and co-workers abandoned the whole concept of a lattice, and characterized each pure component by three equation of state parameters, V, T and P which may be evaluated from the pure component data, density, thermal expansion coefficient and... 

Polymer simulations can be mapped onto the Flory-Huggins lattice model. For this purpose, DPD can be considered an off-lattice version of the Flory-Huggins simulation. It uses a Flory-Huggins x (chi) parameter. The best way to obtain % is from vapor pressure data. Molecular modeling can be used to determine x, but it is less reliable. In order to run a simulation, a bead size for each bead type and a x parameter for each pair of beads must be known. 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

The lattice model that served as the basis for calculating ASj in the last section continues to characterize the Flory-Huggins theory in the development of an expression for AHj . Specifically, we are concerned with the change in enthalpy which occurs when one species is replaced by another in adjacent lattice sites. The situation can be represented in the notation of a chemical reaction ... 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

Both Flory [143] and Huggins [144] in 1941 addressed themselves to this problem with the initial aim of describing solutions of linear polymers in low molecular weight solvents. Both used lattice models, and their initial derivations considered only polymer length (rather than shape, i.e. branching, etc.) The derivation given here will also limit itself to differences in molecular size, but will be based on an available volume approach. 

From a thermodynamic point of view, the heteropolymer globule in hand represents a subsystem which is composed of a macromolecule involving lu l2 units Mi, M2 and molecules of monomers Mi, M2 whose numbers are Mi,M2. Among these variables and volume fractions a in the framework of the simplest Flory-Huggins lattice model there are obvious stoichiometric relationships... 

Any parameter that influences either the enthalpy or the entropy of the system might change the free energy in such a manner that phase separation occurs. For a system like those studied here, being composed of a polymer and a solvent, the entropy of mixing, AS, has been calculated independently by Flory [53,55,56] and Huggins [57,58] based on the lattice model. It is assumed that each solvent molecule occupies one cell of the lattice. The volume of the smallest unit serves as reference volume. It is stated that the macromolecule can be composed of ad-... 

Gibbs and DiMarzio [47] (GD) first developed a systematic statistical mechanical theory of glass formation in polymer fluids, based on experimental observations and on lattice model calculations by Meyer, Flory, Huggins, and... 

Simulations [73] have recently provided some insights into the formal 5c —> 0 limit predicted by mean field lattice model theories of glass formation. While Monte Carlo estimates of x for a Flory-Huggins (FH) lattice model of a semifiexible polymer melt extrapolate to infinity near the ideal glass transition temperature Tq, where 5c extrapolates to zero, the values of 5c computed from GD theory are too low by roughly a constant compared to the simulation estimates, and this constant shift is suggested to be sufficient to prevent 5c from strictly vanishing [73, 74]. Hence, we can reasonably infer that 5 approaches a small, but nonzero asymptotic low temperature limit and that 5c similarly becomes critically small near Tq. The possibility of a constant... 

Recent developments in the theory of polymer solutions have been reviewed by Berry and Casassa (32), and by Casassa (71). Casassa, who has contributed very largely to these developments, has adopted a statistical mechanical approach using molecular distribution functions, as first outlined by Zimm (72), rather than using a lattice model like that used by Flory, Huggins, and many later workers. 

Another important application of experimentally determined values of the osmotic second virial coefficient is in the estimation of the corresponding values of the Flory-Huggins interaction parameters x 12, X14 and X24. In practice, these parameters are commonly used within the framework of the Flory-Huggins lattice model approach to the thermodynamic description of solutions of polymer + solvent or polymer] + polymer2 + solvent (Flory, 1942 Huggins, 1942 Tanford, 1961 Zeman and Patterson, 1972 Hsu and Prausnitz, 1974 Johansson et al., 2000) ... 

The Flory-Huggins theory begins with a model for the polymer solution that visualizes the solution as a three-dimensional lattice of TV sites of equal volume. Each lattice site is able to accommodate either one solvent molecule or one polymer segment since both of these are assumed to be of equal volume. The polymer chains are assumed to be monodisperse and to consist of n segments each. Thus, if the solution contains TV, solvent molecules and TV2 solute (polymer) molecules, the total number of lattice sites is given by... 

The shortcoming of this expression comes from the fact that in the Flory-Huggins lattice model, chain microstructure and real flexibility mechanisms are not taken into account. 

Thermodynamic descriptions of polymer systems are usually based on a rigid-lattice model published in 1941 independently by Staverman and Van Santen, Huggins and Flory where the symbol x(T) is used to express the binary interaction function [16]. Once the interaction parameter is known we can calculate the liquid liquid phase behaviour. 

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

Several other studies have also been made in an attempt to account theoretically for the phase transition in terms different from those of the Flory-Huggins theory. Otake et al. [55] thus proposed a theoretical model that takes hydrophobic interaction into account in explaining the thermally induced discontinuous volume collapse of hydrogels. In addition, Prausnitz et al. [56] proposed a lattice model, an improvement of which was made to explain the swelling curves of gels consisting of /V,/V -methylenebis(acrylamide) (MBA)-crosslinked copolymers of AAm with [(methacrylamide)propyl]trimethyl-ammonium chloride (MAPTAC) [57],... 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

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