Combinatorial entropy term

The first two terms correspond to the combinatorial entropy terms of Eq. (1) and form the non-interacting part of the structure factor which is just a weighted average of the single-chain structure factors SA(q) and SB(q) of both blend components. SA(q) and SB(q) are characterized by the radius of gyration RgA= aA(NA/6)1/2 and RgB=aB(NB/6)1/2, where aA and aB are the statistical segment lengths of polymer A and polymer B, respectively. The last term of Eq. (4) yields the SANS determined interaction parameter %SANS ... 

The following contributions have to be considered additively, the letters in parenthesis designate the terms in Eq. (1.30) the solute must be fluidized (A), there is a combinatorial entropy term B), there is a solvophobic effect on a self-associated solvent such as water F), a change in non-specific forces D) occurs, and hydrogen bonding or donor-acceptor interactions take place 0 and OH). 

The contribution from the combinatorial entropy term is favorable for miscible systan formation, but the absolute values of the contribution are very small, especially for high molecular weight blends. 

In the model of regular solution (or its extension to polymer mixtures), the Aj2 term is purely enthalpic and Ajj is thus a true constant. In a real mixture, Aj is a function of T, p, and the composition of the mixture, but the utility of Eq. (1) relies on the fact that the dependence of Ajj on these variables is only moderate in most cases. Only in the case of the systems exhibiting LCST behavior is the temperature dependence of Aj2 appreciable. The strong concentration dependence of Xi2> often found with dilute polymer solutions, is not encountered with polymer mixtures. This is mainly due to the fact that the mean-field approximation, as stated earlier, is fairly satisfactory and the entropy of mixing two polymeric components is reasonably well represented by the combinatory entropy term. 

The dependence of Gi in Eq. 20 on the entropic term Fa leads to a connection with other theoretical approaches to Gi for polymer blends. According to equation of states theories a finite entropic term Fa arises from the sample compressibility [33] which also becomes apparent from the Lattice Cluster Theory (LCT) [12]. It has to be mentioned here that the Ginzburg criterion derived on the basis of mean field parameters by Bates et al. [20] and Hair et al. [21] behave inversely proportional to the non-combinatorial entropy term Fa [see also Eq. (4.2) in Dudowicz et al. [12]]. So there still seems to exist some confusion with respect to the effect of Fo on Gi. 

As in a previous case, the genera number of adsorbed initial molecules is 1 mole. If intermolecular bonds are great enough, molecules in the volume cannot transit, and there is no motion at the surface. In this case, we can omit terms connected with combinatorial entropy of molecule disposition in the volume, and, even more so, their difference ... 

Terms Gch-s and Gch-s are defined by combinatorial entropy, depending on the transposition of intermolecular bonds at the molecule, taking into account the geometry of the surface and the chain (some intermolecular bonds, for example, the ones at neighboring chain atoms cannot be performed as a consequence of the structure... 

This is identical to the projected entropy spr, Eq. (9), except for the last term. But by construction, the combinatorial entropy assumes that p0—the overall density—is among the moment densities retained in the moment free energy. The difference scomb — spt = —p0 In p is then linear in this density, and the combinatorial and projection methods therefore predict exactly the same phase behavior. 

It should be pointed out that in both these cases the degree of chlorination differs from PVC by around 10%. By any estimate, the heat of mixing in these cases should be quite unfavourable. For example, an estimate based on solubility parameters and using group contribution gives for PVC (6 = 19.28 J cm ) and chlorinated polyethylene (45wt.-%Cl) (5 = 18.77 J cm" ), hence for a 50/50 mixture AH is -1-0.065 J per cm of mixture. Together with unfavourable equation-of-state terms and a small combinatorial entropy contribution these mixtures would not be expected to be miscible. 

The link between GC quantities and the interacticm parameters of solution theories is readily established (39). In statistical theories of K>lution thermodynamics, the )lute activity is expressed as the sum of two terms, a combinatorial entropy and a noncombinatorial free energy of mixing. In the Flory-Hi ins approximation one has. 

Lichtenthaler et al. (55) determined interaction parameters for 22 solutes in poly(dimethyl siloxane) to test several expressions of the combinatorial entropy of mixing [Eq. (7)]. The magnitude of the interaction parameter is indeed directly dependent on the evaluation of the combinatorial contribution. The combinatorial contribution was computed following both the Flory-Huggins approximation and the multiple-connected-site model recently developed by Lichtenthaler, Abrams and Prausnitz (56). This model, which retains the Flory-Huggins term, also corrects for the bulkiness of the components of the mixture. Interaction parameters were computed through both approximations, showing the sensitivity of the results to the model chosen. 

Fig. 26. Hypothetical simulated spinodal curves for the phase separation of two polymers on heating illustrating the effect of the interaction parameter (X12) the non-combinatorial entropy parameter (Q12) and the ratio of surface areas per unit volume S2/Sj. The curves are all simulated using values of y, = y2 = 1 (Jcm 3K l) r2/r, = 1 V =s 100,000 cm-3mole-1 a, = 5xl0-4, a2 = 4x 10 4K-1. If X12 = —0.6Jcm 3,Q12 = 0, and S2/S, = 1, the curve is much flatter than those in the previous figure. If there is a larger (favourable) X12, say —1.2 and this is balanced by an unfavourable Q12 = —0.0023 J cm 3K l, then the curve is much flatter as these parameters swamp the effect of other terms (2). If XI2 = —0.6, Q12 = 0 and S2/S, is now set at 1.5 then the spinodal curve will be skewed to one side as shown (3)...

In Eq 2.39, ([)j is the volume fraction and Vj is the molar volume of the specimen i . The first two logarithmic terms give the combinatorial entropy of mixing, while the third term the enthalpy. For polymer blends Vj is large, thus the combinatorial entropy is vanishingly small — the miscibility or immiscibility of the system mainly depends on the value of the last term,... 

The first term of Eq 2.48 is the H-F combinatorial entropy. The second term (where the subscript v indicates the free volume fraction) represents the free volume contribution to the entropy of mixing. The third term represents the non-combinatorial contribution [g ((t) ) is the non-combinatorial energy on a molten state of polymer i having the free volume fraction (()y]. The fourth term represents the energetic contribution originating from interaction between unlikely species, i j. Here gj.( (()j ) is the interaction term expressed as polynomial with coefficients that depend on the structure of the polymer chains. These coefficients are computed as double expansions in 1/z ( z is the lattice coordination number), and e jj/kgT (e j is the van der Waals interaction energies between groups a and b). 

The combinatorial term, in accordance with the dictates of the Second Law of Thermodynamics, always opposes flocculation. As the steric barriers interpenetrate, molecules of the dispersion medium are forced out of the overlap volume, thus decreasing the combinatorial entropy. At constant temperature and pressure, this means that must be positive. 

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