Nonideal Polymer Solutions

A basic experimental problem with polymers is that it is not always possible to find different 0-solvents for a given polymer over a range of temperatures of interest. Thus, in practice, it is often necessary to perform some experiments away from the 0-point. Ultimately, it is therefore necessary to consider the statistical mechanics of nonideal polymer solutions (i.e., those with all interactions 1, 2, 3). Of course, we always begin with the simpler ideal case. 

The simplicity of (2.25) is to be contrasted with the complexity of the exact P(R n) for realistic models of flexible chains for all R (and for small n). When dealing with the complicated problems of nonideal polymer solutions, etc., it is therefore customary to replace the real polymer chain by the so-called Kuhn effective random flight chain. An effective chain is one with N (in general different from n) links of size A5 such that N lS.s = L and (R ) is as given by (2.29). This substitution of a real chain by its equivalent chain is often a necessity so that we may separate errors in principle from errors arising from a poor mathematical approximation to the exact P(R n) when dealing with problems which are not exactly soluble. This equivalent chain therefore provides us with reasonable approximations to the properties of real polymer chains, provided the physical properties of interest do not depend heavily upon those chain configurations with i > L or upon chain properties over small distances for which the real chain is stiff. 

We have not discussed the subject of nonideal polymers in any detail apart from the excluded volume problem. Thus no mention is made of the evaluation of the potential of mean force from the monomer-solvent interaction, and subsequently the evaluation of the osmotic pressure. We refer to the treatment of Yamakawa (Ref. 5, Chapter IV) for this subject and mention only that the osmotic pressure of a polymer solution at finite concentrations is represented as a virial expansion in the polymer concentration. " The second, third, etc., virial coefficients represent the mutual interaction between two, three, etc., polymer chains in solution. Thus the functional integral techniques presented in this review should also be of use in understanding the osmotic pressure of nonideal polymer solutions. We hope that this review will stimulate such studies of this important subject. It should also be mentioned in passing that at the 0-point the second virial coefficient vanishes. In general, the osmotic pressure -n is given by the series... 

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

Figure 1. Curve 1 could represent an ideal polymer solution containing A and B but undergoing no association the nonideal counterpart of this is shown in curve 2. An ideal mixed association between A and B, such as described by Equation 1 might be described by curve 3, whereas, curve 4 could represent a nonideal, mixed association.

In polymer science, the ideal form of the thermodynamic equations is preserved and the nonideality of polymer solutions is incorporated in the virial coefficients. At low concentrations, the effects of the cl terms in any of the equations will be very small, and the data are expected to be linear with intercepts which yield values of and slopes which arc measures of the second virial coefficient of the polymer solution. Theories of poly mer solutions can be judged by their success in predicting nonideality. This means predictions of second virial coefficients in practice, because this is the coefficient that can be measured most accurately. Note in this connection that the intercept of a straight line can usually be determined with more accuracy than the slope. Thus many experiments which are accurate enough for reasonable average molecular weights do not yield reliable virial coefficients. Many more data points and much more care is needed if the experiment is intended to produce a reliable slope and consequent measure of the second virial coefficient. 

Nonideal thermodynamic behavior has been observed with polymer solutions in which A Hm is practically zero. Such deviations must be due to the occurrence ofa nonideal entropy, and the first attempts to calculate the entropy change when a long chain molecule is mixed with small molecules were due to Flory [8] and Huggins [9]. Modifications and improvements have been made to the original theory, but none of these variations has made enough impact on practical problems of polymer compatibility to occupy us here. 

When analyzing thermodynamic properties of polymer solutions it is sufficient to consider only one of the components, which for reasons of simplicity normally is the solvent. It is convenient to separate Eq. (3.88) for solvent (i = 1) into ideal and nonideal (or excess) contributions by defining... 

The nonideality of polymer solutions is incorporated in the virial coefficients. Predicting nonideality of polymer solutions means, in reality, predictions of the second virial coefficient, because this is the coefficient which can be measured most accurately. Better solvents generally produce greater swelling of macromolecules and result in higher virial coefficients. 

Nonideality of solutions is discussed in Section 2.2.5. It can be expressed as the deviation of the colligative properties from that of an ideal, i.e., very dilute, solution. Here we will consider the virial expansion of osmotic pressure. Equation (2.18) can conveniently be written for a neutral and flexible polymer as... 

Concentrated polymer solutions show strong nonideality. This is, for instance, observed in the osmotic pressure being very much higher than would follow from the molar concentration. The main variables are the [j value and the volume fraction of polymer, and for polyelectrolytes also charge and ionic strength. 

Thus, there is no temperature difference between them. Exchanging the solvent droplet of one of the probes by a droplet of the solution leads to condensation of solvent vapor due to the lower vapor pressure of the solvent above the solution. Thereby, the released condensation enthalpy increases the temperature of the solution droplet, which simultaneously leads to an increase of the vapor pressure. After reaching the vapor pressure equilibrium of the solution droplet, a relatively stable temperature is obtained at the solution droplet. This temperature is converted by the measuring system to a direct voltage signal and is thereby at the user s disposal as a measuring value. The resulting relative measured value is nearly proportional to the osmolal concentration of the solutions. However, it may be affected by heat loss and the nonideal behavior of the polymer solutions. 

Relation (3.11) was obtained empirically by Raoult and is called Raoult s law. Hence in ideal solutions Raoult s law holds over the entire range of compositions, this being represented by a straight line in the v or pressure composition curve (Fig. 3.1). In reality, most solutions do not obey Eqs. (3.9) and (3.11). Such solutions are called nonideal, or real. Polymer solutions characteristically display sharp negative deviations from ideality, as can be seen in Fig. 3.1. 

Hence, any colligative method should yield the number average molar mass M of a polydisperse polymer. Polymer solutions do not behave in an ideal manner, and nonideal behavior can be eliminated by extrapolating the experimental (F/c) data to c = 0. For example, in the case of boiling point elevation measurements (ebullio-scopy) Equation 9.2 takes the form... 

The deviation from the ideal solution is magnified by N. A small difference of x from 1 /2 shows up as a large nonideality when N is large. Thus the polymer solutions, especially those of high-molecular-weight polymers, can be easily nonideal. When x> i/2, in particular, N(l - 2x) can be easily as large as to cause a dip in the plot of Apip. 

Colligative properties reflect this difference. Figure 31.3 shows the vapor pressure of the solvent benzene over a solution containing rubber, which is a polymer (a) as a function of mole fraction, and (b) as a function of volume fraction. The vapor pressure of a small-molecule solvent over a polymer solution shows nonideal behavior when plotted versus mole fraction x. The Flory-Huggins theory described in the next section shows that a better measure of concentration in polymer solutions is the volume fraction cf>. 

This apparent partial specific volume now contains die effects of nonideal mixing of both the polymer and the solvent. Substituting the specific volumes of polymer solution and solvent by experimentally measured densities, it is readily found that... 

From the viewpoint of statistical mechanics, a similar solution to the one used for a nonideal gas applies. In other words, if the intermolecular potential of a gas is replaced with the elFective potential among solute molecules, which is the interaction transmitted through solvent molecules, the partition ftinction will be in the same form. In the case of a polymer solution, each segment can be regarded as a gaseous molecule. Using this similarity, the theoretical approach developed for a nonideal gas can be applied to polymer solutions. It is easier to understand the excluded volume and 9 temperature fh)m this point of view [1, 2]. 

For the numerical simulation of flowing polymers, several mesoscopic models have been proposed in the last few years that describe polymer (hydro-)dynamics on a mesoscopic scale of several micrometers, typically. Among these methods, we like to mention dissipative particle dynamics (DPD) [168], stochastic rotation dynamics (sometimes also called multipartide collision dynamics) [33], and lattice Boltzmann algorithms [30]. Hybrid simulation schemes for polymer solutions have been developed recenfly, combining these methods for solvent dynamics with standard particle simulations of polymer beads (see Refs [32, 169, 170]). Extending the mesoscopic fluid models to nonideal fluids including polymer melts is currently in progress [30, 159,160,171]. 

Fifty-six isothermal data sets for vapor-liquid equilibria (VLB) have been used for 15 polymer-HSolvent binaries, 11 copolymer-nsolvent binaries and for 30 polymer-polymer-solvent ternaries to study compatibility of polymer blends. The equilibrium solubility of a penetrant in a polymer depends on their mutual compatibility. Equations based on theories of polymer solution tend to be more successful when there is some kind of similarity between the penetrant and the monomer repeat unit in the polymer, e.g., for nonpolar penetrants in polymers which do not contain appreciable polar groups. Expected nonideal behavior has been observed for systems containing hydrocarbons and poly(acrylonitrile-co-butadiene). The role of intramolecular interaction in vapor-liquid equilibria of copolymer-nsolvent systems is well documented for poly(aciylonitrile-co-butadiene) that have higher affinity for acetonitrile than do polyaciylonitrile or polybutadiene. 

Several attempts have been made to find theoretical explanations for the nonideal behavior of polymer solutions. There exist numerous excellent books on this topic (272,387,389-398,618,860). 

Solutions can deviate from ideality because they fail to meet either one or both of these criteria. In reference to polymers in solutions of low molecular weight solvents, it is apparent that nonideality is present because of a failure to meet criterion (2), whether the mixing is athermal or not. 

---