Previous discussions of the kinetic theory of rubber elasticity have dealt with individual long chain molecules, but the theory of the structure of bulk rubber has been almost entirely undeveloped. The present paper goes beyond earlier ones both in the more detailed treatment of the individual chains and in the development of a clear-cut model of the bulk material. From the consideration of familiar properties of rubber, it is concluded that in the lightly vulcanized state it consists of a coherent network of flexible molecular chains (this involving a considerable fraction of the total material) together with other molecules not actively involved in the network, but acting like a fluid mass through which the network extends and in which it moves with Brownian motion. The idea of an effective internal pressure is advanced and discussed. A simplified model for bulk rubber is proposed, consisting of a network of idealized flexible chains extending through the material and a fluid filling it, the bounding surfaces being in equilibrium under all the forces acting on them—internal pressure, pull of the molecular network, and any external forces. For the quantitative treatment of this model the principal problem is that of computing the forces exerted by the molecular network, because of its thermal agitation, on the bounding surfaces. Two models of the flexible molecular chains have been used—one with a linear stress-extension relation (Gaussian chain), the second with independent links of fixed length, having, like real molecular chains, a definite maximum extension. Methods for the statistical treatment of chains of independent links at all extensions are developed and applied to the computation of stress-strain relations for the second of the above models. It is shown that an irregular network of Gaussian chains is equivalent to a simple set of independent chains; the corresponding reduction in the case of non-Gaussian chains is only approximate. Making this reduction in all cases, the model is applied to the quantitative computation of stress-strain curves for unilateral stretch of rubber and for stretch in two directions. A prediction is made of the change in elastic properties of rubber in the swollen state. The thermoelastic properties of rubber are worked out, and the change in slope of the nearly linear stress-temperature curves from negative to positive values with increasing strain is explained. The location of the thermoelastic inversion point, at which this slope is zero, is shown to depend only upon the cubic expansion coefficient of unstretched rubber. The linear thermal expansion coefficient of stretched rubber in the direction of the stress is shown to be positive (and of the order of the same coefficient for unstretched rubber) below the thermoelastic inversion point. Above the thermoelastic inversion point this coefficient becomes negative and is—for appreciable extensions—of the order of the cubic expansion coefficient for a gas, independent of the composition of rubber. The linear coefficient of thermal expansion perpendicular to the direction of stress is always positive. Stress-strain curves for bulk rubber at constant temperature have been compared with the theory for extensions up to 400 percent. The agreement is particularly good as concerns that part of the total stress due to changing entropy. In some rubber compounds minor deviations, variable from sample to sample, are found for small extensions. These are attributed to van der Waals forces not taken into account by the theory. The characteristic knee in rubber stress-strain curves is shown to be due to change in internal pressure in the material. This varies inversely with the extended length, and in unstretched rubber is of the order of 5 kg/cm2. The upward curvature of the stress-strain curve for larger extensions, which appears even before the onset of crystallization, is explained as due to the approach of the molecular network to its maximum extension, which is of the order of 10 times its extension in the unstretched bulk rubber. Crystallization may in general enhance the S-shape of the stress-strain curve for natural cured rubber, Neoprene and butyl rubber. Synthetic rubber of the Buna type, while having an S-shaped stress-strain curve, does not exhibit any crystallization at all. It is estimated that, in the materials considered, roughly a fourth of the rubber chains are actively involved in the network. Internal pressure, maximum extensions, and the fraction of active material all change with the state of vulcanization. The general theory developed in this paper provides a basis for the treatment of many other physical properties of stretched rubber.

The rigidity of rubber, considered as a network of flexible molecules with Gaussian configuration functions, can be calculated for a particular sample if one is given a complete description of the molecular network or certain types of statistical description. In particular, it is sufficient to know the distribution of lengths and vector-mean extensions of the segments of the network, or only the latter distribution if it is of Gaussian form. The assumption that the vector-mean extensions have a Gaussian distribution corresponds to a similar postulate in the theory of Wall, and has been applied to a simplified network theory by Flory. In the complete theory it leads to calculation of the same proportionality between the rigidity of the material and the number Ga of segments, per unit volume, in the ``active'' part of the molecular network. However, consideration of the process of cure shows that this assumption cannot be expected to be correct, though it does lead to results of the right order of magnitude. An alternative approach to the problem is based on a study of the increase in rigidity of the material as cure proceeds. It uses a more realistic picture of the process of cure than those hitherto employed, and uses Gaussian distribution functions only where they can be logically justified. The final result is of the same form as that given by the previous theory, except that Ga is replaced by Ba, the number of bonds formed within the molecular network during cure. Since Ba is smaller than Ga by a factor of about two or more, the improved theory leads to the prediction of somewhat lower rigidities per bond formed. Available experimental results of Flory on Butyl rubber vulcanizates do not appear to give an adequate check on the quantitative features of the theory.

The method used by Hylleraas in treating the He atom has been extended to the H2 molecule. The method consists of setting up a wave function as a series in the five variables required, electronic separation being introduced explicitly as one of the variables. The coefficients are then determined so as to produce the lowest energy. The energy found is within 0.03 v.e. of the most probable experimental value, while the form and location of the potential energy curve

A Gaussian network is defined as a network of flexible chain segments, linked to each other and to a system of fixed points, in which each unbranching chain segment can take on a number of configurations which is a Gaussian function of the distance between its ends. Real molecular networks, such as those of rubber, can under certain circumstances be treated as Gaussian networks. The present paper carries out a systematic mathematical discussion of the statistical properties of Gaussian networks: the total number of possible configurations of the network as a function of the fixed-point coordinates, the probability of finding a given element of the network in a given position, or of finding two elements of the network in given relative positions, and so on. All probability-density functions appear as exponentials of quadratic forms, with constants explicity expressible in determinant form. An explicit reduction to a sum of squares is given for all quadratic forms occurring in the theory of coherent Gaussian networks, and an explicit general formula is found for integrals of the form ∫ −∞+∞dX1 ∫ −∞+∞dX2··· ∫ −∞+∞dXq exp{− ∑ in ∑ jnγijXiXj}.There is described a mechanical analog of a Gaussian network, by consideration of which the statistical properties of the Gaussian network can be determined. The method is applied to the discussion of the statistical properties of a Gaussian network with the connectivity of a regular cubic lattice.

This work compiles, reviews, and discusses the available data and information on the electrical resistivity of ten selected binary alloy systems and presents the recommended values resulting from critical evaluation, correlation, analysis, and synthesis of the available data and information. The ten binary alloy systems selected are the systems of aluminum–copper, aluminum–magnesium, copper–gold, copper–nickel, copper–palladium, copper–zinc, gold–palladium, gold–silver, iron–nickel, and silver–palladium. The recommended values for each of the ten binary alloy systems except three (aluminum–copper, aluminum–magnesium, and copper–zinc) are given for 27 compositions: 0 (pure element), 0.5, 1, 3, 5, 10(5)95, 97, 99, 99.5, and 100% (pure element). For aluminum–copper, aluminum–magnesium, and copper–zinc alloy systems, the recommended values are given for 26, 12, and 11 compositions, respectively. For most of the alloy systems the recommended values cover the temperature range from 1 K to the solidus temperature of the alloys or to about 1200 K. For most of the nine elements constituting the alloy systems, the recommended values cover the temperature range from 1 K to above the melting point into the molten state. The estimated uncertainties in most of the recommended values are about ±3% to ±5%. Key words: alloy systems; alloys; conductivity; critically evaluated data; data analysis; data compilation; data synthesis; electrical conductivity; electrical resistivity; metals; recommended values; resistivity.