Polymer nonideal

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. 

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. 

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. 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

Electrostatic and adsorption effects conspire to make aqueous GPC more likely to be nonideal than organic solvent GPC. Thus, universal calibration is often not obeyed in aqueous systems. Elence, it is much more critical that the standard chosen for calibration share with the polymer being analyzed chemical characteristics that affect these interactions. Because standards that meet this criterion are often not available, it is prudent to include in each analysis set a sample of a secondary standard of the same composition and molecular weight as the sample. Thus, changes in the chromatography of the analyte relative to the standards will be detected. 

It is essential that the solution be sufficiently dilute to behave ideally, a condition which is difficult to meet in practice. Ordinarily the dilutions required are beyond those at which the concentration gradient measurement by the refractive index method may be applied with accuracy. Corrections for nonideality are particularly difficult to introduce in a satisfactory manner owing to the fact that nonideality terms depend on the molecular weight distribution, and the molecular weight distribution (as well as the concentration) varies over the length of the cell. Largely as a consequence of this circumstance, the sedimentation equilibrium method has been far less successful in application to random-coil polymers than to the comparatively compact proteins, for which deviations from ideality are much less severe. 

When monomers with dependent groups are involved in a polycondensation, the sequence distribution in the macromolecules can differ under equilibrium and nonequilibrium regimes of the process performance. This important peculiarity, due to the violation in these nonideal systems of the Flory principle, is absent in polymers which are synthesized under the conditions of the ideal polycondensation model. Just this circumstance deems it necessary for a separate theoretical consideration of equilibrium and nonequilibrium polycondensation. 

Gree-Kubo expression, 102-104 mesoscale simulation of complex systems basic princples, 90-92 real system simulations, 113-114 multicomponent systems, 96-97 nonideal fluids, 136-137 polymers, 122-128... 

Because of the highly nonideal behavior of polymer-solvent solutions, polymer-vapor equilibrium relations account for the third major difference found in stripping operations with polymeric solutions. The appro-... 

From a survey of the literature in chemically modified electrodes [13], one can identify simple phenomenological models that have been very successful for the analysis of a particular aspect of the experimental data. Such models are, for instance, the Dorman partition model [24, 122], the Laviron [158], Albery [159] and Anson models [127] to account for the nonideal peak width, the Smith and White model for the interfacial potential distribution [129], and so on. Most of these models contain one or more adjustable parameters that give some partial information about the system. For example, the lateral interaction model proposed by Anson [127] provides a value for the lateral interactions between oxidized and reduced sites, but does not explain the origin of the interactions, neither does it predict how they depend on the experimental conditions or the polymer structure. In addition, none of these models provide information on the interfacial structure. 

Of the possible types of measurements, heats of micellar mixing obtained from the mixing of pure surfactant solutions are perhaps of the greatest interest. Also of interest is the titration (dilution) of mixed micellar solutions to obtain mixed erne s. While calorimetric measurements have been applied in studies of pure surfactants (6,7) and their interaction with polymers ( ), to our knowledge, applications of calorimetry to problems of nonideal mixed micellization have not been previously reported in the literature. 

Another type of nonideal SEC behavior, which will not be covered in this chapter, is related to the use of mixed mobile phases (multiple solvents). Because solute-solvent interactions play a critical role in controlling the hydrodynamic volume of a macromolecule, the use of mixed mobile phases may lead to deviations from ideal behavior. Depending on the solubility parameter differences of the solvents and the solubility parameter of the packing, the mobile phase composition within the pores of the packing may be different from that in the interstitial volume. As a result, the hydrodynamic volume of the polymer may change when it enters the packing leading to unexpected elution results. Preferential solvation of the polymer in mixed solvent systems may also lead to deviations from ideal behavior (11). 

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.

But Leonard and Ivin point out, Tc is not given by A HlcjA Slc since this takes no account of the partial molar free energies of the monomer and polymer. This can be seen in Fig. 4 which shows that AFlcjRT is positive above 37° C, suggesting that polymerization will not occur above this temperature. Experimentally (Fig. 1) at 37° C approximately 65 % conversion to polymer is obtained and in fact polymerization continues to occur up to about 85° C. Practically, it would seem that either THF behaves ideally enough to justify the use of equation (1) for the derivation of meaningful heat and entropy terms for the derivation of Te or nonidealities fortuitously cancel out. It should be understood that terms derived from equation 1 will include terms for the removal of monomer and addition of polymer to the polymer-monomer mixture. 

Theta solvents. Selection of a poor solvent for a polymer is desirable when making solution property measurements because it permits the use of higher concentrations and minimizes the effects of nonideality. The most suitable choice is a theta solvent (73). Table 12 lists the theta solvents and the corresponding theta temperatures which have been found for PTHF. 

Of course, it is uncommon for the free energy/ to obey (1). In particular, the entropy of an ideal mixture (or, for polymers, the Flory-Huggins entropy term) is definitely not of this form. On the other hand, in very many thermodynamic (especially mean field) models the excess (i.e., nonideal) part of the free energy does have the simple form (1). In other words, if we decompose the free energy as (setting kn = 1)... 

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