Mixtures athermal

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

The quantity x is called the Flory-Huggins interaction parameter It is zero for athermal mixtures, positive for endothermic mixing, and negative for exothermic mixing. These differences in sign originate from Eq. (8.39) and reaction (8.A). 

Those involving solution nonideality. This is the most serious approximation in polymer applications. As we have already seen, the large differences in molecular volume between polymeric solutes and low molecular weight solvents is a source of nonideality even for athermal mixtures. 

It is convenient to begin by backtracking to a discussion of AS for an athermal mixture. We shall consider a dilute solution containing N2 solute molecules, each of which has an excluded volume u. The excluded volume of a particle is that volume for which the center of mass of a second particle is excluded from entering. Although we assume no specific geometry for the molecules at this time, Fig. 8.10 shows how the excluded volume is defined for two spheres of radius a. The two spheres are in surface contact when their centers are separated by a distance 2a. The excluded volume for the pair has the volume (4/3)7r(2a), or eight times the volume of one sphere. This volume is indicated by the broken line in Fig. 8.10. Since this volume is associated with the interaction of two spheres, the excluded volume per sphere is... 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

Escobedo, F. A. de Pablo, 1. J., Monte Carlo simulation of athermal mesogenic chains pure systems, mixtures, and constrained environments, J. Chem. Phys. 1991,106,9858-9868... 

Figure 4 Chemical potentials of an athermal mixture with r2 = 4.

Figure 5 Probabilities of 1-1 pairs of athermal mixtures with different chain lengths (Yang et al., 2006a).

We will briefly discuss the molecular dynamics results obtained for two systems—protein-like and random-block copolymer melts— described by a Yukawa-type potential with (i) attractive A-A interactions (saa < 0, bb = sab = 0) and with (ii) short-range repulsive interactions between unlike units (sab > 0, aa = bb = 0). The mixtures contain a large number of different components, i.e., different chemical sequences. Each system is in a randomly mixing state at the athermal condition (eap = 0). As the attractive (repulsive) interactions increase, i.e., the temperature decreases, the systems relax to new equilibrium morphologies. 

Where V2/V1 (r) is the ratio of the molar volume of the polymer to that of the solvent and x is the Flory parameter which depends primarily on the intermolec-ular forces between solute and solvent. According to the original formulation, this parameter is zero for athermal mixtures. However, subsequent work has shown that both the excess entropy and the excess enthalpy contribute to x ... 

Mixtures of hydrocarbons are assumed to be athermal by UNIFAC, meaning there is no residual contribution to the activity coefficient. The free volume contribution is considered significant only for mixtures containing polymers and is equal to zero for liquid mixtures. The combinatorial activity coefficient contribution is calculated from the volume and surface area fractions of the molecule or polymer segment. The molecule structural parameters needed to do this are the van der Waals or hard core volumes and surface areas of the molecule relative to those of a standardized polyethylene methylene CH2 segment. UNIFAC for polymers (UNIFAC-FV) calculates in terms of activity (a,-) instead of the activity coefficient and uses weight fractions... 

In a paper regarding phase transitions in monolayers,13 Cantor and Mcllroy incorporated the bond correlations and intermolecular interactions using an approach similar to that of Ref. 10. Finally, Cantor has incorporated the bond correlations in the generator-matrix formalism to calculate the elastic properties14 of films of athermal surfactant mixtures. 

In another approximation athermal mixtures are discussed with a zero heat of mixing just as ideal solutions, but with a different entropy of mixing, owing to a difference in size of the two kinds of molecules. The most simple expression is that derived by Flory ... 

In a similar way one could write down the expressions for trimers, tetramers, etc., adsorbing from an athermal mixture. The equations become more and more tedious, and we will not give them here. For trimers, one obtains three cubic equations in three unknown volume fractions for tetramers four... 

The comparison of the experimental solubilities [4,5] of Ar, CH4, C2H6 and CsHg in the binary aqueous mixtures of PPG-400, PEG-200 and PEG-400 with the calculated ones is presented in Figs. 1-3 and Table 2. They show that Eq. (4) coupled with the Flory-Huggins equation, in which the interaction parameter x is used as an adjustable parameter, is very accurate. The Krichevsky equation (1) does not provide accurate predictions. While less accurate than Eq. (4), the simple Eq. (2) provides very satisfactory results without involving any adjustable parameters. It should be noted that Eq. (4) coupled with the Flory-Huggins equation with X (athermal solutions) does not involve any adjustable parameters and provides results comparable to those of Eq. (2). 

Finally we may observe that we have defined perfect solutions through equation (20.1) for the chemical potentials, and from this we have established the properties discussed in this paragraph. Conversely, for a solution to be perfect, all these properties must be satisfied simultaneously. Thus it is not sufficient that the mixture can be made without heat effect, and without change in volume. The entropy of mixing must also have the form (20.17). Indeed later on we shall discuss solutions (athermal solutions) for which the deviations from ideality arise entirely from the entropy term. 

Kontogeorgis, G.M. et ah. Improved models for the prediction of activity coefficients in nearly athermal mixtures. Part I. Empirical modifications of free-volume models. Fluid Phase Equilibria, 92, 35, 1994. Coutinho, J.A.P., Andersen, S.I., and Stenby, E.H., Evaluation of activity coefficient models in prediction of alkane SEE, Fluid Phase Equilibria, 103, 23, 1995. 

Problem 3.1 Based on the lattice model for low-molecular-weight mixtures, determine the effect of non-athermal mixing on Raoult s law. 

This is the Flory-Huggins (FH) equation for athermal solutions [5, 6]. The equation formally appears symmetric to all components of a mixture, but in fact it describes the great asymmetry between the polymer and the small solvent in solution. Consider the solution of a small solvent and a polymer that is r times the volume of the solvent. The solvent may well be the monomer, here designated as component 1, and the polymer may be an r-mer as component 2. Setting V2/V1 in Equation (4.367) gives... 

Hildebrand and Rotariu [14] have considered differences in heat content, entro])v and activity and classified solutions as ideal, regular, athermal, associated and solvated. Despite much fundamental work the theory of binary liquid mixtures is still e.ssentiaUy unsatisfactory as can be seen from the. systematic treatment of binar> mi.Ktures by Mauser-Kortiim [15]. The thermodynamics of mixtures is presented most instructively in the books of Mannchen [16] and Schuberth [17]. Bittrich et al. [17a] give an account of model calculations concerning thermophysical properties of juire and mixed fluids. 

The quantity 6 is characteristic of the binary mixture at a given temperature no matter how large an initial concentration is chosen e.g., a mole fraction of 0.99 or 0.5 or even 10" , etc.). It is called separation parameter and can be calculated from the vapour pressure values provided that the mixture in question is athermal, which implies that the heat of mixing is zero ... 

K the mixtures are athermal, as is mostly the case with isotopes, approximate values of d can be calculated from the formula... 

The same effect in Zeolite-guest systems was demonstrated by Auerbach [26] by equilibrium molecular dynamics and non-equilibrium molecular dynamics after experimental work by Cormer [27]. The energy distributions obtained in Zeolite and Zeolite-Na are shown in Fig. 5.6A. At equilibrium, all the atoms in the system are at the same temperature. When Na-Y Zeolite is exposed to microwave energy, however, the effective steady-state temperature of Na atoms is substantially higher than that of the rest of the framework this is indicative of athermal energy distribution. The steady-state temperatures for binary methanol/benzene mixtures in... 

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