Flory-Huggins lattice

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.

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... 

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) ... 

Fint is the free energy of non-Coulomb interactions of monomer units. Finl can be expressed, for example, in terms of the Flory-Huggins lattice theory [21]. In the general case, when network is immersed in solvent which includes 1 different components some of which can be polymeric with the degree of polymerization Pi(Pi 1, i = L 2,... k), Fim in the Flory-Huggins theory has the following form [21-22] ... 

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. 

Let us consider, as an example, a polymer chain on the Flory-Huggins lattice. If the i-th monomer has got onto the pore wall, for the (i, i — l)-th bond there are only 5 possible states instead of 6 in the unrestricted volume. For the entropy losses, AS = In 6/5, to be compensated, we must set for the interaction energy of the i-th monomer 48)... 

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. 

A thermodynamic approach was put forward by one of us (10), based on the Flory-Huggins lattice theory of a polymer solution the chemical potentials of each monomer must be equal in each phase copolymerization increment causes a little change in the chemical potential in the particles diffusion of monomers from the water phase will reequilibrate the system and in turn diffusion from droplets to water phase takes place. For instance, expression from monomer 1 in the particles is ... 

On the other hand, the Flory-Huggins lattice theory (Flory, 1953) predicts for polymer solutions ... 

Here b is the effective lattice spacing of the underlying Flory-Huggins lattice [199-206], and the coefficient k(( >) of the gradient energy term for a symmetric mixture (effective monomeric units have linear dimensions oA=oB=b, chain lengths are NA=NB=N) is [186,206,207]... 

Note that on the mean field level at %c a second-order transition is predicted for f=H2, while taking fluctuations into account renders the transition first order [192,210,211], as also found experimentally [231]. Although the nature of surface effects on this transition is quite different for the first-order case [6] than for the second-order case [12], we discuss mostly the second-order case here. The constant pc in Eq. (43) is the density of the chains (pc=l/N if a Flory-Huggins lattice with lattice spacing unity is invoked), and the constants e0 and u0 can be derived [197] from the random phase approximation as... 

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... 

A. Sariban and K. Binder (1988) Phase-Separation of polymer mixtures in the presence of solvent. Macromolecules 21, pp. 711-726 ibid. (1991) Spinodal decomposition of polymer mixtures - a Monte-Carlo simulation. 24, pp. 578-592 ibid. (1987) Critical properties of the Flory-Huggins lattice model of polymer mixtures. J. Chem. Phys. 86, pp. 5859-5873 ibid. (1988) Interaction effects on linear dimensions of polymer-chains in polymer mixtures. Makromol. Chem. 189, pp. 2357-2365... 

Despite the drawbacks, Flory-Huggins lattice theory was a major step forward towards understanding the thermodynamics of polymer solutions and is the basis of many other theories. Since it was first proposed, other workers have elaborated on it to improve on the assumptions involved. Some of these more refined theories will be given brief consideration, but in general are beyond the scope of the book. 

The Flory-Huggins Lattice Model and the Mean-Field... 

Fig. 3. Schematic illustration of the Flory-Huggins lattice model for a polymer mixture. Lattice sites taken by (effective) monomers are indicated by full dots lattice sites taken by vacancies are denoted by empty circles. Chains of type A are indicated by thick bonds between the monomers, and B chains by wavy bonds. Nearest neighbor nonbonded interactions between monomers of the same kind (eAA or eBb) are shown as full straight lines and those between monomers of a different kind (eAB) by broken lines. Interactions between monomers and vacancies (or solvent molecules, respectively), eAV and ebv, could be introduced as well but will be assumed here to be zero throughout...

Here

coarse-graining over many cells of the lattice to construct a smoothly varying field q>(f) defined in continuous space. 

Fig. 25a. Qiexistence curves of the Flory-Huggins lattice model for several choices of N according to the Flory (1) approximation n[(l + m) /(I — m)] = Nxm, + m), ct Eq. (14). ...

Flory-Huggins Lattice Theory and Related Theories... 

The Flory-Huggins lattice model assumes that a polymer chain consists of a number of equivalent segments. The extension to polymer solvent interactions assumes that the polymer solution consists of a three-dimensional lattice and each lattice site is occupied either by a polymer segment or by a solvent molecule. Flory and Huggins calculated the entropy change of mixing as... 

Considering a binary blend composed of homopolymer A and B with volume fraction /a and /b = 1 - /a and assuming monomers of equal molecular size, the Flory-Huggins lattice model delivers an expression for the Gibbs free energy of... 

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