Theoretical and practical evidence is put forward to show that copolymers can be treated like solutions of small molecules in the interpretation of packing phenomena, and that ideal volumeadditivity of the repeating units in copolymers is frequently realized. On this basis equations are derived for predicting 0, the second-order transition temperature, of binary copolymers from the two second-order transition temperatures of the pure polymers and their coefficients of expansion in the glassy and rubbery states. Previous mechanistic theories of the second-order transition temperature of such copolymers are thus superseded by a general reduction of the problem to the mechanism of thermal expansion. Practical applications to the choice of monomers in producing synthetic rubbers are outlined, and attention is drawn to the importance of second-order transitions in kinetic measurements on the reactions of polymers. Derivation of theoryMeasurements of density, d, preferably expressed as specific volume, V = Ild, are easy to make and give direct evidence of the packing together of molecules in various states. For ordinary small molecules such measurements are interpreted in terms of partial specific volumes of the components (see Lewis & Randall, 1923)~ and the same treatment has been applied to solutions of polymers in small molecules (Heller & Thompson, 1951). If two liquids mix without change of volume, a plot of V against weight fraction c, is linear and the two partial specific volumes are constant. We extend this treatment to the monomeric units of a polymer and show by theoretical and practical evidence that this linearity, reflecting volume additivity of the units, is common both in the rubbery and in the glassy states. We are thus led to a simple law of ideal copolymers, corresponding to that of ordinary ideal solutions, which postulates a constant rubber volume ~V R and a constant glass volume iVG at each temperature for the ith species of repeating unit in all its ideal rubbery and glassy copolymers, these two constants being equal to the specific volume of the pure polymer in these two states. If, then, we consider a copolymer made up of two components whose pure polymers are both rubbers (at the temperature under consideration) the specific volume of the copolymer containing a fraction c2 of component (2) will be given by:A similar argument applies when both pure polymers are glasses, but the situation is more complex when one (which we shall always denote by i = I) is a rubber above its transition temperature O,, and the other is a glass (i = 2) below its transition temperature Oz. Our postulate implies that the state, rubbery or glassy, of the particular copolymer considered determines whether the rubbery or glassy volumes of both components enter into the specific-volume equation:For any temperature T lying between 0, and 8, there will be a composition at which the copolymer has its transition temperature 8 at T. It is clear from the foregoing that at this point the plot of V against c2 will change slope ...

Applications of the theory of branching processes to polymer systems can be so formulated, that all statistical parameters emerge automatically in a form which applies to the soluble part of the system, i.e., to the whole system up to the gel point, and to the sol fraction beyond. In a self-contained presentation, previous work along these lines is here extended to computing configurational statistics of systems arising most directly by condensation processes. These statistics are the mean-square radii of the molecules in such systems, averaged in various ways and useful for theories of light scattering and viscosity. Numerical calculations for random f-functional polycondensation are presented as plots of configurational parameters against conversion for f=3, 4, and 6. The general equations given extend beyond the random case to condensations with ``first-shell'' substitution effect, i.e., in which the rates of making or breaking a given bond between two repeat units depends on how many other bonds these two units carry (to further units). In terms of current theories, the intrinsic viscosity of a system at its gel point is shown to be finite. The spatial averaging of molecular size is based on random flight statistics. The well-known theorem by Kramers used in this connection is rederived on a simple topological basis, and generalized to deal with copolymerization of units of different sizes.

Theoretical and practical evidence is put forward to show that copolymers can be treated like solutions of small molecules in the interpretation of packing phenomena, and that ideal volume-additivity of the repeating units in copolymers is frequently realized. On this basis equations are derived for predicting θ, the second-order transition temperature, of binary copolymers from the two second-order transition temperatures of the pure polymers and their coefficients of expansion in the glassy and rubbery states. Previous mechanistic theories of the second-order transition temperature of such copolymers are thus superseded by a general reduction of the problem to the mechanism of thermal expansion. Practical applications to the choice of monomers in producing synthetic rubbers are outlined, and attention is drawn to the importance of second-order transitions in kinetic measurements on the reactions of polymers.

The gel point is a well-understood critical point in polymer science (Flory, Stockmayer). A summary is given of the resolution of those aspects which may seem paradoxical at first sight. The relevant equations for the basic paradigm of f-functional polycondensation are very simple. Special attention is paid to the critically branched state of materials not far from the gel point. The quasiinvariance principle is explained according to which all solution properties appear ultimately to level off as the critical conversion is approached from below. The 'Malthusian' packing paradox is resolved by a proper treatment of the ring-chain competition situation, which also disposes of the spurious divergence of the rate of cyclization predicted by a more naive theory.Network theories not based on extinction probability (Charlesby) are not worth considering. The proper definition of an elastically active network chain (EANC) was based on this concept by Scanlan and by Case in 1960, and it greatly simplifies the graph-like-state theories of network structure (usually called 'network topology'). It allows classical rubber elasticity theory to be applied near the gel point. The point is characterised by a fifth-order Ehrenfest transition due to the contribution of long-range correlations to the configurational free energy. Though directly relevant data on reversible gelation are not available, data on isothermal crosslinking of very diverse systems support this analysis. Except possibly for some highly crosslinked systems, the parameter M (mean chain length between 'crosslinks') is an undesirable ingredient of elasticity or swelling theories. Chain-end corrections are quite undesirable (and usually done incorrectly). Everything is correctly and more simply formulated in terms of Scanlan-Case EANCs.The proper understanding of the gel point as a critical point allows the construction of reduced plots, illustrated with temperature superposition of experimental modulus-conversion plots (by M. Judd) for aqueous gelatin jellies. These fit reasonably to the basic model involving triple-helix junction zones, essentially without adjusting arbitrary parameters. Such reducedvariable treatments for critically branched materials eliminate the otherwise inescapable difficulties of characterization, purity, electrolyte content, etc., from gelatin research.

Recent data of STEPTO and WAYWELL for ring-chain competition in the kinetically controlled polymerizing system ( i e . backward rate = 0) of (polyethylene glycol)/(hexamethylene diisocyanate) have been fitted by exact computer calculations, assuming GAUSSian chain statistics. A good one-parameter fit is obtained. The optimum value of the parameter is compared with theory. The results suggest that the solvent used (benzene at 7OoC) is thermodynamically poor for this polymer system. ZUSAMMENFASSUNG:Neuere Daten von STEPTO und WAYWELL zur Ring-Ketten-Konkurrenz bei der kinetisch kontrollierten Polyaddition (d. h. mit der Riickreaktionsgeschwindigkeit = 0) im System Polyathylenglykol/Hexamethylendiisocyanat wurden durch exakte Computerberechnnngen untersucht, und zwar unter Voraussetzung einer Gauss-Statistik. Es konnte eine gute Ein-Parameter-Anpassung erzielt werden. Die optimalen Werte der Parameter wurden mit der Theorie verglichen. Die Ergebnisse deuten darauf hin, daR das verwendete Losungsmittel (Benzol bei 70 "C) fiir dieses Polymersystem ,,thermodynamisch schwach" ist.