Elastic and Thermoelastic Properties of Rubberlike Materials. A Statistical Theory

1941 ◽  
Vol 14 (3) ◽  
pp. 596-605 ◽  
Author(s):  
Eugene Guth ◽  
Hubert M. James
Keyword(s):  
Statistical Theory ◽  
Second Law Of Thermodynamics ◽  
Statistical Interpretation ◽  
Free Rotation ◽  
Extended Form ◽  
Thermoelastic Properties ◽  
Vulcanized Rubber ◽  
Carbon Carbon Bond ◽  
Kinetic Effects ◽  
Molecular Compounds

Abstract A statistical theory of the elasticity of rubber and similar high-molecular compounds was proposed by one of the authors several years ago, and an extended form of the theory was presented recently. The theory is applied here to vulcanizates, and shows the dependence of stress on both temperature and deformation for both extension and compression. The foundations of the theory will be stated precisely, inasmuch as some recent articles dealing with it are likely to lead to misunderstandings concerning the assumption of free rotation and structure of long-chain compounds, such as vulcanized rubber. The possible ro^le of kinetic effects for the rubber type of elasticity has been discussed qualitatively by Wöhlisch, Busse, Karrer, Meyer, Susich and Valkó, and Shacklock. The first quantitative theory was developed by one of the authors in 1934 and published later, partly in collaboration with Mark. In the statistical theory, the main assumption is that of quasi-free rotation in a rubber molecule around a single carbon-carbon bond. An important consequence of this free rotation is that the molecules are coiled in the unstretched state. The contraction of stretched rubber is due mainly to the tendency of the stretched chain to change from a less probable stretched form back to the most probable coiled form. This happens in accordance with the statistical interpretation of the second law of thermodynamics, and is not caused by forces. Our theory, then, is that the kinetic motion of freely rotating groups in the rubber molecule is the main cause of contraction.

Planck, the Second Law of Thermodynamics, and Black‐body Radiation

2019 ◽  
pp. 45-83
Author(s):  
Anthony Duncan ◽  
Michel Janssen
Keyword(s):  
Second Law Of Thermodynamics ◽  
Black Body ◽  
Black Body Radiation ◽  
Second Law ◽  
Statistical Interpretation ◽  
Low Frequencies ◽  
High Frequencies ◽  
Proportionality Constant ◽  
Entropy Formula ◽  
Planck Law

Planck’s work on black‐body radiation grew out of a failed attempt to use electrodynamics to show that entropy can never decrease, i.e., that the second law of thermodynamics is not just a statistical but a strict law of nature. This original interest is reflected in his approach to the problem of black‐body radiation. Planck derived the formula for the spectral distribution of black‐body radiation from the formula for the entropy of a resonator interacting with the radiation at its resonance frequency. He initially chose an entropy formula that gave him a black‐body radiation formula proposed by Wien. Deviations from this Wien law at low frequencies led him to adopt a new entropy formula, which gives a law that reduces to the Wien law at high frequencies and to (what is now known as) the Rayleigh‐Jeans law at low frequencies. This new Planck law agreed remarkably well with all experimental data. Planck thus set out to find a derivation of the entropy formula leading to it. Although he continued to resist Boltzmann’s statistical interpretation of the second law for another decade, Planck borrowed some of Boltzmann’s techniques for this derivation. The derivation critically depends on energy elements with sizes proportional to the frequency of the radiation and Planck’s constant as the proportionality constant. Planck’s papers of 1900–01, however, leave open the question of how these energy elements are to be interpreted.

scholarly journals Particle absorption by black holes and the generalized second law of thermodynamics

2009 ◽  
Vol 466 (2116) ◽  
pp. 1155-1166 ◽  
Author(s):  
Scott Funkhouser
Keyword(s):  
Black Hole ◽  
Cross Section ◽  
Probability Function ◽  
Second Law Of Thermodynamics ◽  
Quantum Mechanical ◽  
Statistical Interpretation ◽  
Schwarzschild Black Hole ◽  
Particle Absorption ◽  
Generalized Second Law ◽  
Fundamental Physical

The change in entropy, Δ S , associated with the quasi-static absorption of a particle of energy ε by a Schwarzschild black hole (ScBH) is approximately ( ε / T )− s , where T is the Hawking temperature of the black hole and s is the entropy of the particle. Motivated by the statistical interpretation of entropy, it is proposed here that the absorption should be suppressed, but not forbidden, when Δ S <0, which requires the absorption cross section to be sensitive to Δ S . A purely thermodynamic formulation of the probability for the absorption is obtained from the standard relationship between microstates and entropy. If Δ S ≫1 and s ≪ ε / T , then the probability for the particle not to be absorbed is approximately exp[− ε / T ], which is identical to the probability for quantum mechanical reflection by the horizon of an ScBH. The manifestation of quantum behaviours in the new probability function may intimate a fundamental physical unity between thermodynamics and quantum mechanics.

Interaction between Rubber and Liquids. IV. Factors Governing the Absorption of Oil by Rubber

1943 ◽  
Vol 16 (4) ◽  
pp. 818-833 ◽  
Author(s):  
G. Gee
Keyword(s):  
Kinetic Energy ◽  
Alternative Explanation ◽  
Second Law Of Thermodynamics ◽  
Quantitative Interpretation ◽  
Thermal Agitation ◽  
Quantitative Expression ◽  
Strong Force ◽  
Vulcanized Rubber ◽  
Gas Molecules ◽  
The Relationship

Abstract A piece of vulcanized rubber, dropped into benzene, swells to several times its size but retains its shape. With raw rubber, a further stage ensues, in which the rubber flows and ultimately disperses. Inevitably, one tends to form a picture of the rubber attracting and holding the liquid with some strong force. It is the purpose of this paper to explain why this picture is believed to be entirely false, and to give an alternative explanation of the phenomena of swelling and solution. It will be necessary first to consider briefly the way in which simpler materials mix with one another. The simplest possible system is that of two gases which do not react with each other. In a gas the molecules spend most of their time a long way from one another, and the total energy of the system is therefore made up largely of the kinetic energy of thermal motion. As a consequence of this kinetic energy the gas molecules tend, on the average, to distribute themselves uniformly, so that any pair of gases mix completely. A quantitative interpretation can be given to this mixing tendency, in terms of the concept of entropy. Of the various ways in which entropy may be regarded, the most useful for the present purpose is in terms of probability. Qualitatively it is evident that gas molecules in violent thermal agitation are extremely unlikely to arrange themselves so all molecules of one type are confined to one part of the vessel. This is expressed in thermodynamic language by saying that the entropy of such an arrangement would be small. The second law of thermodynamics states that if, as in a gas, there is no change of energy, a system tends to take up the state of maximum entropy, or maximum randomness. A quantitative expression to the relationship between the entropy S and probability W, takes the form:

Thermodynamics of Stressed Vulcanized Rubber

1930 ◽  
Vol 3 (2) ◽  
pp. 304-314 ◽  
Author(s):  
Roscoe H. Gerke
Keyword(s):  
Modulus Of Elasticity ◽  
Strain Curve ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Equilibrium Stress ◽  
Permanent Set ◽  
Vulcanized Rubber ◽  
Laws Of Thermodynamics

Abstract The first and second laws of thermodynamics are applied to the stretching of vulcanized gum rubber stocks. Equilibrium stress-strain curves without appreciable hysteresis are described. The modulus of elasticity of vulcanized rubber for higher elongations obtained from the equilibrium stress-strain curve is capable of giving agreement with predictions of the second law of thermodynamics and the Joule heat effect. The modulus of elasticity from the equilibrium stress-strain curve is practically independent of the time of cure for a range of cures for elongations less than 600 per cent. The customary stress-strain curves show the rubber to be stiffer with increased cure. These facts are additional evidence that the important effect caused by vulcanization is a greater resistance to plastic flow or permanent set.

Is the Theory of Natural Selection a Statistical Theory?

1988 ◽  
Vol 14 ◽  
pp. 187-207 ◽  
Author(s):  
Alexander Rosenberg
Keyword(s):  
Statistical Theory ◽  
Natural Selection ◽  
Boundary Conditions ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Local Boundary ◽  
Long Run ◽  
Equilibrium Level ◽  
Statistical Version ◽  
Average Fitness

In The Structure of Biological Science (Rosenberg [1985]) I argued that the theory of natural selection is a statistical theory for reasons much like those which makes thermodynamics a statistical theory. In particular, the theory claims that fitness differences are large enough and the life span of species long enough for increases in average fitness always to appear in the long run; and this claim, I held, is of the same form as the statistical version of the second law of thermodynamics.For the latter law also makes a claim about the long run, and its statistical character is due to this claim: thermodynamic systems must in the long run approach an equilibrium level of organization that maximizes entropy. Over finite times, given local boundary conditions, an isolated mechanical system, like the molecules in a container of gas, may sometimes interact so as to move the entropy of the system further from, instead of closer to the equilbrium level. But given enough interacting bodies, and enough time, the system will always eventually move in the direction prescribed by the law. Thus, we can attach much higher probabilities to the prediction that non-equilibrium systems will reflect greater entropy in future periods than we can to predictions that they will move in the opposite direction. And as we increase the amount of time and the number of bodies interacting, the strength of the probability we can attach to the prediction becomes greater and greater.

Dependence of Elastic Properties of Vulcanized Rubber on the Degree of Cross-Linking

1950 ◽  
Vol 23 (1) ◽  
pp. 9-26
Author(s):  
Paul J. Flory ◽  
Norman Rabjohn ◽  
Marcia C. Shaffer
Keyword(s):  
Statistical Theory ◽  
Network Structure ◽  
Short Circuit ◽  
Elastic Equilibrium ◽  
Butyl Rubber ◽  
Rubber Elasticity ◽  
Cross Linking ◽  
Substantial Part ◽  
Vulcanized Rubber ◽  
Low Degrees

Abstract The results reported above demonstrate a progressive increase in the force of retraction τ at fixed elongation with increase in the fraction p of the structural units which are cross-linked from ρ=0.10×10−2 to 3.0×10−2. Over this range, τ at 100 per cent elongation increases about thirteenfold. Swelling measurements indicate that the increase in τ with ρ continues over an additional tenfold range in ρ. Previous assertions that the modulus of elasticity of soft gum rubber vulcanizates depends largely on chain interaction and entanglements other than those imposed by the cross-linkages, and that the modulus is, therefore, not directly related to the degree of cross-linking, are without foundation. The statistical theory of rubber elasticity expresses the force of retraction as a function of the temperature, vulcanizate structure and elongation; no arbitrary constants are involved. The magnitudes of τ for α=2 are in remarkably close agreement with the predictions of the theory over most of the range in ρ. This fact is of the utmost significance in confirmation of the statistical theory of rubber elasticity and of the analysis of the network structure of vulcanized rubber. On the other hand, τ increases less rapidly with ρ than the direct proportionality prescribed by theory. Forces of retraction are higher than the theory predicts at low degrees of cross-linking, and an opposite deviation is observed for values of ρ greater than about 1×10−2. Previous observations on Butyl rubber, vulcanized to p values from about 0.16×10−2 to 0.28×10−2 indicated forces of retraction (for infinite molecular weight M) which exceed by about threefold those predicted from the theory. This deviation is decidedly larger than has been observed here in the same range for ρ. A substantial part of the discrepancy observed for Butyl rubber may have arisen from failure to secure elastic equilibrium, however. Deviations in the values of τ from theory probably originate largely from oversimplifications in the treatment of the network structure. Entanglements of the sort previously discussed tend to enhance the restraints imposed on the chains when the rubber is elongated. Their percentage effect should be greatest for low degrees of cross-linking, hence the observed τ values are higher than theory at low degrees of cross-linking. “Intramolecular” cross-linkages, yielding short-circuit structures contributing nothing to the elastic reaction of the network, should become increasingly important at higher degrees of cross linking. Such wastage of cross-linkages may account for the low values of τ obtained for higher ρ values.

Intrinsic kinetic effects during martensitic transformations

2003 ◽  
Vol 112 ◽  
pp. 133-137 ◽  
Author(s):  
A. Fraile-Rodriguez ◽  
P. P. Rodriguez ◽  
R. B. Pérez-Saez ◽  
A. Lopez-Echarri ◽  
J. San Juan
Keyword(s):  
Martensitic Transformations ◽  
Kinetic Effects ◽  
Intrinsic Kinetic
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