Coil molecules virial coefficient

Equation (8.97) shows that the second virial coefficient is a measure of the excluded volume of the solute according to the model we have considered. From the assumption that solute molecules come into surface contact in defining the excluded volume, it is apparent that this concept is easier to apply to, say, compact protein molecules in which hydrogen bonding and disulfide bridges maintain the tertiary structure (see Sec. 1.4) than to random coils. We shall return to the latter presently, but for now let us consider the application of Eq. (8.97) to a globular protein. This is the objective of the following example. 

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. 

One thing that is apparent at the outset is that polymer molecules in solution are very different species from the rigid spheres upon which the Einstein theory is based. On the other hand, we saw in the last chapter that the random coil contributes an excluded volume to the second virial coefficient that is at least... 

The deterioration of the solvent qnality, that is, the weakening of the attractive interactions between the polymer segments and solvent molecules, brings about the reduction in the coil size down to the state when the interaction between polymer segments and solvent molecules is the same as the mutual interaction between the polymer segments. This situation is called the theta state. Under theta conditions, the Flory-Huggins parameter % assumes a value of 0.5, the virial coefficient A2 is 0, and exponent a in the viscosity law is 0.5. Further deterioration of solvent quality leads to the collapse of coiled structure of macromolecules, to their aggregation and eventually to their precipitation, the phase separation. 

The essence of this model for the second virial coefficient is that an excluded volume is defined by surface contact between solute molecules. As such, the model is more appropriate for molecules with a rigid structure than for those with more diffuse structures. For example, protein molecules are held in compact forms by disulfide bridges and intramolecular hydrogen bonds by contrast, a randomly coiled molecule has a constantly changing outline and imbibes solvent into the domain of the coil to give it a very soft surface. The present model, therefore, is much more appropriate for the globular protein than for the latter. Example 3.3 applies the excluded-volume interpretation of B to an aqueous protein solution. 

The virial coefficients reflect interactions between polymer solute molecules because such a solute excludes other molecules from the space that it pervades. The excluded volume of a hypothetical rigid spherical solute is easily calculated, since the closest distance that the center of one sphere can approach the center of another is twice the radius of the sphere. Estimation of the excluded volume of llexible polymeric coils is a much more formidable task, but it has been shown that it is directly proportional to the second virial coefficient, at given solute molecular weight. 

An expression for the apparent excluded volume of a coil can be obtained by comparing the perturbation second virial coefficient [Eq. (3.117)] with the one from excluded-volume theory for compact molecules [Eq. (3.115)]. This... 

The dependence of the second virial coefficient of coil-shaped molecules is difficult to calculate since the excluded volume is a complicated function of the molar mass (see Section 4.4.5). The usual method is to replace the excluded volume of the molecule u in Equation (6-64) by the excluded volume of the chain segment Wseg the molar mass Mi of the molecule is also replaced by the formula molar mass Mu of the chain segment. The molar mass dependence of the excluded volume is expressed in terms of a function h(z) whose coefficients have been evaluated theoretically ... 

The environment can have a decisive influence on the course of a reaction in macromolecular chemistry, altering not only the rate but also the optimum obtainable yield. At equal concentrations, the probability for intramolecular reactions of coil molecules is greater in poor solvents (low second virial coefficient) than in good solvents (high second virial coefficient). Ring-forming reactions therefore occur preferentially in poor solvents. 

Theta solvent n. A solvent, at a particular temperature, in which the polymer is at the edge of solubility and exists in the form of a statistical coil. Long-range forces between polymer molecular segments are balanced by polymer solvent interactions. At these conditions the second virial coefficient becomes zero and entropy is at its minimum. Kamide K, Dobashi T (2000) Physical chemistry of polymer solutions. Elsevier, New York. Flory PJ (1969) Statistical mechanics of chain molecules. Interscience Publishers Inc., New York. Flory PJ (1953) Principles of polymer science. The Cornell University Press, Ithaca, NY. 

Tlieta conditions are of great theoretical interest because the diameter of the polymer chain random coil in solution is then equal to the diameter it would have in the amorphous bulk polymer at the same temperature. The solvent neither expands nor contracts the macromolecule, which is said to be in its unperturbed state. The theta solution allows the experimenter to obtain polymer molecules which are unperturbed by solvent but separated from each other far enough not to be entangled. Theta solutions are not normally used for molecular weight measurements, because they are on the verge of precipitation. The excluded volume vanishes under theta conditions, along with the second virial coefficient. 

As well as controlling chain dimensions, solvent quality affects the thermodynamics of dilute polymer solutions. This is because interactions between polymer chains are modified by the presence of solvent molecules. In particular, solvent molecules will change the excluded volume for a polymer coil, i.e. how much volume it takes up and prevents neighbouring chains from occupying. In a theta solvent, the excluded volume is zero (this holds for the excluded volume for a polymer segment or the whole coil). The solution is said to he ideal if the excluded volume vanishes. Deviations from ideality for polymer solutions are described in terms of a virial equation, just as deviations from ideal gas behaviour are. The virial equation for a polymer solution in terms of polymer concentration is given by Eq. (2.9). The second virial coefficient depends on interactions between pairs of molecules in particular it is proportional to the excluded volume. Therefore, in a theta solvent, = 0. If the solvent is good then Ai > 0, but if it is poor Ai < 0. If the solvent quality varies as a function of temperature and theta (0) conditions are attained, this occurs at the theta temperature. 

The different crosslinking behaviours of photopolymer films formed from different solvents can be explained with the help of different polymer conformations in these solvents. Studies were performed with one selected photopolymer (27). The second virial coefficients (A2), determined by light scattering, revealed the most coiled photopolymer structure in THF solution. This is to be expected since the magnitude A2 (found to be lowest in THF solution) is related to the Flory-Huggins constant Xi widely used in fundamental polymer studies as measure for the polymer-solvent interactions. If A2>0, the interactions between polymer molecules and solvent molecules are more attractive than the solvent-solvent interactions. Such solvents are considered to be thermodynamically good for the polymer. Thus, the low value for THF means that this solvent can be considered to be poor , leading to a more... 

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