High-polymer solutions

In describing the configurations of a polymer molecule in terms of the 'equivalent chain' of N elements, each of length l, it has been usual to simplify the problem by assuming the equivalent chain to have position only and zero volume. The weights of the various configurations of such a 'random flight' chain are different from those of a real chain in which there exists an interaction potential between any pair of chain elements. These differences are particularly important in the theory of solutions of chain molecules, since they are responsible for the deviation of the osmotic pressure from van't Hoff's law. In this paper the average dimensions of a chain with interactions are calculated by a statistical method. For $\langle s^{2}\rangle $, the average square distance of the elements from the centre of gravity, the result is $\langle s^{2}\rangle $ = (Nl$^{2}$/6) [1-0$\cdot $857($\beta _{1}$/l$^{3}$) N$^{-\frac{1}{2}}$], (i) where $\beta _{1}$ is the 'excluded volume' integral for free chain elements. For large N this reduces to the well-known result for a random flight chain. Similar results are obtained for other average dimensions. The possibility of checking (i) from experimental determinations of $\langle s^{2}\rangle $ for chain molecules using the light-scattering technique is examined, and it is shown that a very accurate knowledge of the chain-length distribution in the fractions used will be required if the influence of the second term in (i) is to be detected in this way. A natural extension of the statistical method is used to calculate the pair distribution function F$_{2}$ (X$_{12}$) governing the probability of occurrence of the centres of gravity of two chains in equal volume elements separated by the distance X$_{12}$. This function is needed to calculate the second coefficient A$_{2}$ in the osmotic pressure expansion $\pi $ = RT[M$^{-1}$c + A$_{2}$c$^{2}$ + $\cdots $]. Here M is the molecular weight of the solute and c the concentration. For random flight chains F$_{2}$ is unity for all values of X$_{12}$; A$_{2}$ is zero and the osmotic pressure follows van't Hoff's law. Values of F$_{2}$ different from unity, and hence finite values of A$_{2}$ are only obtained if there are interactions between chain elements. The first approximation to F$_{2}$ is F$_{2}$(X$_{12}$) = exp {(9/2$\pi $)$^{\frac{3}{2}}$ ($\beta _{1}$/l$^{3}$) N$^{\frac{1}{2}}$ exp (- 9X$_{12}^{2}$/2Nl$^{2}$)}. The theory predicts a rather complicated dependence of A$_{2}$ on z, the degree of polymerization and the log-log plot of A$_{2}$ against z is curved. Over a limited molecular weight range A$_{2}$ may be approximated by a formula of the form A$_{2}$ = Cz$^{-\epsilon}$, (ii) where C is constant for a given polymer-solvent system. $\epsilon $ depends upon z and lies between -$\infty $ and $\frac{1}{2}$. If A$_{2}$ is positive, $\epsilon $ goes from 0 to $\frac{1}{2}$ as z goes from 0 to $\infty $ and A$_{2}$ decreases slowly with z. For systems in which A$_{2}$ is negative, $\epsilon $ goes from 0 to -$\infty $ as z goes from 0 to $\infty $ and $|A_{2}|$ increases extremely rapidly with z. There are complications if the solutions are not homogeneous with respect to chain length, but it is shown that, with well-fractionated samples, little difficulty should arise if z is replaced by the number average $\langle z\rangle _{n}$. The theory is illustrated by applying it to some recently published data on the systems: polystyrene-butanone, polystyrene-toluene, and polyisobutylene-cyclohexane.