Entropy of mixing, polymers

As N 00, Xc 0 and the chains become immiscible at all temperatures (see Figure 31.9). The entropy of mixing polymers is so small that ev en an extremely small unfavorable enthalpy of mixing can prevent mixing. For example, even deuterated polybutadiene will not mix with protonated polybutadiene below a critical temperature of 61.5 °C, if N = 2300. Rather than mix, polymer blends often separate into domains of each of the component polymers. 

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. 

A plot of these values is shown in Fig. 8.1. Note the increase in the entropy of mixing over the ideal value with increasing n value. Also note that the maximum occurs at decreasing mole fractions of polymer with increasing degree of polymerization. 

The separation of Hquid crystals as the concentration of ceUulose increases above a critical value (30%) is mosdy because of the higher combinatorial entropy of mixing of the conformationaHy extended ceUulosic chains in the ordered phase. The critical concentration depends on solvent and temperature, and has been estimated from the polymer chain conformation using lattice and virial theories of nematic ordering (102—107). The side-chain substituents govern solubiHty, and if sufficiently bulky and flexible can yield a thermotropic mesophase in an accessible temperature range. AcetoxypropylceUulose [96420-45-8], prepared by acetylating HPC, was the first reported thermotropic ceUulosic (108), and numerous other heavily substituted esters and ethers of hydroxyalkyl ceUuloses also form equUibrium chiral nematic phases, even at ambient temperatures. 

Due to the smallness of the entropy of mixing, most polymer mixtures are at least partially incompatible, and blends contain A-rich and B-rich domains, separated by interfaces. The intrinsic width of these interfaces is rather broad (it varies from w = aJin... 

Polymers undergoing dissolution show much smaller entropies of mixing than do conventional solutes of low relative molar mass. This is a consequence of... 

With the introduction of these relations into Eq. (41), the configurational entropy change (the entropy of mixing solvent and polymer excluded see Chap. XII) relative to the initial state ax = ay = az = l becomes... 

If each solvent molecule may occupy one of the remaining lattice sites, and in only one way, 0 represents also the total number of configurations for the solution, from which it follows that the configurational entropy of mixing the perfectly ordered pure polymer and the pure solvent is given hy Sc —k In Introduction of Stirling s approximations for the factorials occurring in Eq. (7) for fi, replacement of no with Ui+xn[Pg.501]

The entropy of mixing disoriented polymer and solvent may be obtained, according to the original assumptions pertaining to the lattice model, by subtracting Eq. (9) from (8). The result reduces to ... 

The sum of these expressions represents the entropy of mixing. The result obtained by substituting for the free volumes from Eqs. (13), (14), and (15), is identical with Eq. (10). According to this simple derivation, the terms appearing in Eq. (10) represent contributions to the entropy which originate in the greater spatial freedom of the molecules in the solution. By a simple extension of the derivation to a mixture of polymer species, Eq. (12) may be obtained. 

Hence the theoretical configurational entropy of mixing AaSm cannot be compared in an unambiguous manner with the experimentally accessible quantity ASm- It should be noted that the various difficulties encountered, aside from those precipitated by the character of dilute polymer solutions, are not peculiar to polymer solutions but are about equally significant in the theory of solutions of simple molecules as well. 

The physical reason for the inherent lack of incentive for mixing in a polymer-polymer system is related to that already cited in explanation of the dissymmetry of the phase diagram for a polymer-solvent binary system. The entropy to be gained by intermixing of the polymer molecules is very small owing to the small numbers of molecules involved. Hence an almost trivial positive free energy of interaction suffices to counteract this small entropy of mixing. 

Now, let s look at a polymeric system. To begin with, the motion in polymer chains is hindered. The massive size of the polymer itself and the intermolecular forces within the chains create an inflexible system, especially when compared to the aqueous systems with which we are most familiar. Secondly, the entropy of mixing is not actually as great as that seen in typical solution formation. Polymers are inherently highly entropic, so the benefit of mixing them together is modest. Therefore, any two polymers that form a miscible blend depend primarily... 

In Eq. (2), mi is the chemical potential of the solvent in the polymer gel and /al 0 is the chemical potential of the pure solvent. At equilibrium, the difference between the chemical potentials of the solvent outside and inside the gel must be zero. Therefore, changes of the chemical potential due to mixing and elastic forces must balance each other. The change of chemical potential due to mixing can be expressed using heat and entropy of mixing. 

So far we have considered mixtures of atoms or species of similar size and shape. Now we will consider a mixture of a polymeric solute and a solvent of monomers [7, 8], The ideal entropy of mixing used until now cannot possibly hold for this polymer solution, in which the solute molecule may be thousands or more times the size of the solvent. The long chain polymer may be considered to consist of r chain segments, each of which is equal in size to the solvent molecule. Therefore r is also equal to the ratio of the molar volumes of the solute and the solvent. The solute and the solvent can be distributed in a lattice where each lattice site can contain one solvent molecule. The coordination number of a lattice site is z. 

The entropy of mixing of the disoriented polymer and the solvent, the monomer, is obtained by subtracting eq. (9.54) from eq. (9.52) giving... 

Equation (9.55) is the expression for the entropy of mixing of polymer solutions introduced first by Flory [7], Nm and iVp can be related through x m + Xp = 1, which for one mole of molecules (polymers + monomers) gives... 

Figure 9.6 (a) Molar entropy of mixing of ideal polymer solutions for r = 10, 100 and 1000 plotted as a function of the mole fraction of polymer compared with the entropy of mixing of two atoms of similar size, r = 1. (b) Activity of the two components for the same conditions. 

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