W. H. Stockmayer

The Dimensions of Chain Molecules Containing Branches and Rings

Formulas for the mean square radii of various branched and ringed polymer molecules are developed under the usual assumptions regarding the statistics of chain configuration. For branched molecules, the mean square radii vary less rapidly with molecular weight than for strictly linear molecules, while for systems containing only rings and unbranched chains the variation is more rapid than for the linear case. These results show that in principle the quantity of branches or of rings can be determined from light‐scattering measurements.

Intramolecular Reaction in Polycondensations. I. The Theory of Linear Systems

For three types of linear polycondensing systems, equilibrium molecular size distributions, including rings as allowable species, are derived. Average molecular weights and amounts of ring and chain fractions are calculated therefrom. The fractions of rings are shown to increase with dilution, and with molecular weight. It is shown that beyond a critical dilution it is sometimes possible to obtain 100 percent yield of rings by driving the condensation to completion. Detailed calculations are made for two important cases corresponding to condensations of the decamethylene glycol‐adipic acid type: (1) for equimolar amounts of the two monomers, and (2) unequal amounts, with one monomer type completely reacted.

Monte Carlo Calculations on the Dynamics of Polymers in Dilute Solution

A method is described for simulating the dynamical behavior of a linear polymer in dilute solution, subject to random collision with solvent molecules. Equilibrium distributions of various chain dimensions may be obtained by periodic inspection of the chain. Relaxation phenomena in such chains may also be studied. Results are given for equilibrium distribution and relaxation behavior of the end‐to‐end length, for chains of 8, 16, 32, and 64 beads. The equilibrium chain dimensions are in satisfactory accord with the calculations of Wall and his collaborators, while the relaxation times are close to those predicted with the aid of the hydrodynamic theory of Rouse and Zimm.

Light Scattering in Multi‐Component Systems

A general equation is developed for the fluctuation of refractive index in multi‐component systems, thus permitting the interpretation of turbidities for such systems in the absence of angular dissymmetry. Applications to several special cases of interest are presented.

Distribution of Chain Lengths and Compositions in Copolymers

The instantaneous distribution of chain compositions and chain lengths in vinyl copolymers is obtained in a simple form valid for long chains. The compositions of chains of a given length are normally distributed about the mean value, with a standard deviation which can be calculated from experimentally observable quantities. The distribution of chain lengths intimately resembles that for simple polymers.

Second Virial Coefficients of Polar Gases

An equation is given for the classical second virial coefficient of a polar gas in terms of the parameters appearing in an intermolecular potential energy which includes London and dipole attraction and inverse‐power repulsion. The equation is successfully fitted to the data for H2O and NH3 with allowance for the small quantum correction, and the derived values of the London constant are in good accord with theoretical estimates. The defects of the model and its relation to the concept of the hydrogen bond are briefly discussed.

Excluded Volume in Polymer Chains

This discussion is concerned with the excluded volume effect in polymer chains, i.e., the mathematical problem of random flights with correlations between the flights, which is of importance in relating the mean size of a chain to its structure. A general formulation of the problem is set up and compared with the theory of Markoff processes. It is shown that although an integral equation of the Markoff form for the distribution function exists, it contains a complicated function whose exact evaluation is difficult. Since the same function occurs in the problem of the osmotic second virial coefficient, it is natural to relate the excluded volume effect to the second virial coefficient. The results of preliminary numerical calculations for the relation between the mean square radius of the chain and the second virial coefficient, in good agreement with experiment, are given. It is concluded that the volume effect is certainly not negligible for real chains, except when the second virial coefficient is zero,...

DILUTE SOLUTIONS OF BRANCHED POLYMERS

In an earlier era of high polymer chemistry, branching not infrequently served as a whipping boy to explain deviations from an expected physical behavior * 9 often without any independent or clear-cut evidence for the occurrence of chemical reactions leading to branched molecular structures. Today, it is clear from the general nature of the chain transfer reaction that high-conversion vinyl polymers (at least if prepared by free-radical initiation) contain more or less highly branched molecules. A sounder and more quantitative attack on the question of branching is therefore mandatory on both experimental and theoretical fronts. From the theoretical point of view, two general problems related to branching then confront us. The one concerns the mechanism and kinetics of the branchproducing reactions and ultimately the distribution of molecular weights and degrees of branching in samples prepared under specified conditions, and is illustrated by other papers3v4 in this symposium. The other concerns the effect of branching on observable physical properties, and forms the subject of this paper, It will be apparent from the previous contributions that in experimental studies of branching these two problems are not so easily separable. Our remarks are confined almost entirely to the properties of dilute solutions. It is not intended thereby to imply that the properties of polymers in bulk are insensitive to branching; on the contrary, the known or possible influences of branching on bulk properties furnish a strong practical impetus for thorough study of the subject. I t seems likely, however, that a firm theoretical understanding can be more easily and quickly attained for the properties of dilute polymer solutions than for those of the bulk materials. The properties to be discussed in the following sections are derivable from measurements of osmotic pressure, light scattering, sedimentation, diffusion and viscosity. The type of molecular structure considered is that of rather long branches, such as might occur through chain transfer reactions or copolymerization with polyfunctional monomers. Shorter, regularly spaced branches on a main backbone, such as the alkyl side groups in polyvinyl stearate, will not be treated.

Excluded Volume Effect of Linear Polymer Molecules

A new closed expression for the excluded volume effect of linear polymer molecules is developed with the aid of an equivalent ellipsoid model, in which the polymer chain with fixed ends is replaced by a uniform distribution of unconnected segments within an ellipsoid whose dimensions are chosen to give the correct principal radii of gyration of the chain. The linear expansion factor α for a chain of N segments is given by (α3−α)/N12={(1+13α−2)−32}−1(4/3)5/2(3/2π)32(β/a3) where a is the effective bond length and β is the binary cluster integral of a segment pair. In contrast to Florys familiar equation, the ratio (α5—α3)/N½ is an increasing function of α instead of a constant. The new equation is in excellent agreement with the Monte Carlo calculations of Wall and Erpenbeck for a diamond lattice chain and also with existing viscosity data for various polymer solutions, provided that the expansion factor is evaluated from the equation [η]/[η]θ=α2.43 in accord with the theories of Kurata‐Yamakawa and Ptitsy...

Intramolecular Reaction in Polycondensations. II. Ring‐Chain Equilibrium in Polydecamethylene Adipate

Decamethylene adipate polymers prepared in the absence of solvent undergo a reversible decrease in solution viscosity when heated for long times in chlorobenzene. This phenomenon is shown to be due to ester interchange reactions which establish equilibrium between ring and chain molecules. Quantitative agreement with the equilibrium theory of the previous paper is obtained if the effective length of a chain link is taken as 4.5±0.2A, which is of the order of magnitude to be expected from hindered rotation and steric effects. Measurements of ring‐chain equilibrium thus appear to offer a useful method for characterizing the stiffness of relatively short and flexible chains.