Polymer-Solvent Mixtures Flory-Huggins Model

Polymer-Solvent Mixtures Flory-Huggins Model 

The theory sets out to predict AGmix for the formation of polymer solutions based on lattice model. In order to place both the solvent molecules and polymer molecules onto the same pseudolattice, it becomes necessary to consider the polymer molecules to be chains of segments, each segment being equal in size to a solvent molecule. The number, a, of these segments in 

Problem 3.2 Calculate, in terms of the lattice model of the Flory-Huggins theory, the number of segments per polystyrene molecule of molecular weight 290,000 dissolved in (a) toluene and (b) methyl ethyl ketone (MEK) at 25°C. 

The first stage in the development of the Flory-Huggins theory is to derive an expression for A mix when Alfniix = 0- mixing 

The quantity in the first of the square brackets represents the total number of conformations possible for N2 polymer molecules when each one has the complete freedom of the lattice, that is, the lattice is assumed to be empty when adding each polymer molecule. The quantity in the second pair of square brackets represents the fractions of these conformations that are allowed when N2 polymer molecules compete for cells in a lattice of (TN2 cells corresponding to the pure amorphous polymer. The quantity in the third pair of square brackets is the factor by which the total number of conformations of N2 polymer molecules increases on account of the greater spatial freedom provided by dilution with Ni solvent molecules. Evidently, the product of the quantities in the first two pair of square brackets of Eq. (3.44) represents 2 and so Eq. (3.42) simplifies to 

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Polymer-Solvent Mixtures Flory-Huggins Model... 

In the processing of polymers, and also for polyrfier devolatilization (the removal of the solvent from the polymer), it is important to be able to calculate the equilibrium partial pressure of a solvent above solvent-polymer mixtures of different compositions. Calculate the partial pressure of benzene in benzene + polyisobutylene (PIB) mixtures at 298.15 and 312.75 K. In this calculation you can assume that polyisobutylene has a negligible vapor pressure, and that the Floty-Huggins model describes the solution behavior of this polymer + solvent mixture. Do the calculations fdr values of the Flory-Huggins y parameter equal to 0.5 to 1.0. 

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

The most relevant theory for modeling the free energy of binary polymer mixtures is the Flory-Huggins theory, initially employed for solvent-solvent and polymer-solvent mixtures. This theory was independently derived by Flory [4, 5] and Huggins [6, 7]. The key equation (combined from discussions earlier in this chapter on entropy and enthalpy of mixing) is ... 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

To calculate AWm (the enthalpy of mixing) the polymer solution is approximated by a mixture of solvent molecules and polymer segments, and AW is estimated from the number of 1,2 contacts, as in Section 12.2.1. The terminology is somewhat different in the Flory-Huggins theory, however. A site in the liquid lattice is assumed to have z nearest neighbors and a line of reasoning similar to that developed above for the solubility parameter model leads to the expression... 

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... 

The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

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...

A thermodynamic model based on Flory-Huggins polymer-solution theory was developed and coupled with Equation of State model to predict the amount of asphaltene precipitation. The model prediction shows close agreement with the experimental data after regression of asphaltene properties such as molar volume, solubility parameter and molecular weight. The model, however, fails to account for the effect of large changes in the solubility parameters of the oil-solvent mixtures. 

The simple Flory-Huggins %-function, combined with the solubility parameter approach may be used for a first rough guess about solvent activities of polymer solutions, if no experimental data are available. Nothing more should be expected. This also holds true for any calculations with the UNIFAC-fv or other group-contribution models. For a quantitative representation of solvent activities of polymer solutions, more sophisticated models have to be applied. The choice of a dedicated model, however, may depend, even today, on the nature of the polymer-solvent system and its physical properties (polar or non-polar, association or donor-acceptor interactions, subcritical or supercritical solvents, etc.), on the ranges of temperature, pressure and concentration one is interested in, on the question whether a special solution, special mixture, special application is to be handled or a more universal application is to be foxmd or a software tool is to be developed, on munerical simplicity or, on the other hand, on numerical stability and physically meaningftd roots of the non-linear equation systems to be solved. Finally, it may depend on the experience of the user (and sometimes it still seems to be a matter of taste). 

Despite the fact that these solutions represent binary systems, at least three Flory Huggins interaction parameters are involved in their modeling, like with ternary mixtures. Because of the necessity to account for the interaction of the solvent with monomer A and with monomer B, plus the interaction between the polymers A and B, one should expect the need for a minimum of two additional parameters. Experimental data obtained for solutions of a given copolymer of the type A-ran-B with a constant fraction / of B monomers can be modeled [25] by means of (32), with one set of a, v, and parameters. For predictive purposes, it would of course be interesting to find out how these parameters for the copolymer solution (subscripts AB) relate to the parameters for the solutions of the corresponding homopolymers in the same solvent (subscripts A and B, respectively) at the same temperature. 

Later Hildebrand [10] obtained the same result assuming that free volume available to the molecules per unit volume of liquid is the same for the polymer as for the solvent. The heat of mixing is defined as the difference between the total interaction energy in the mixture compared with that of pure components. Based on their lattice theory model, Flory [7,8,9] and Huggins [11,12] obtained the following expression for the heat of mixing ... 

Flory and Huggins, simultaneously but independently of one another, constructed a term representing the excess entropy known as the conformation term or the combinatorial excess entropy. We shall now present their reasoning process for a mixture of small molecules, of a solvent A, and large one-dimensional molecules making up the solute B. The hypothesis at the heart of the model is still the pseudo-lattice, whose mesh is defined as follows the molecule of component A (the smallest molecules) determines the acceptable dimension on each site of the lattice - its base volume. The larger molecules of polymer (component B) are virtually divided into sequences of the same voliune as the small molecule, so that the large molecule contains sequences such that is equal to the ratio of the molar volumes of the two pure components ... 

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