The theory of rubber elasticity

1976 ◽  
Vol 280 (1296) ◽  
pp. 317-353 ◽  
Keyword(s):  
Free Energy ◽  
Statistical Mechanics ◽  
General Expression ◽  
Degrees Of Freedom ◽  
Amorphous Materials ◽  
Sum Of Squares ◽  
Rubber Elasticity ◽  
Cross Link ◽  
Polymer Molecules ◽  
Gibbs Formalism

This paper attempts to improve several weaknesses in the classical theories of rubber elasticity. It develops a formulation of the statistical thermodynamics of amorphous materials analogous to the Gibbs formalism for conventional statistical mechanics. This then permits the replacement of ‘phantom chains’, i.e. long polymer molecules with the fictitious property that they experience no forces except at cross link points and are transparent to one another, by realistic molecules which do experience forces and which can become entangled. The crosslinked points are no longer assumed to deform affinely with the gross behaviour of the solid. Under the simplest conditions forms like the classical are recovered but with a different coefficient, and the term representing the degrees of freedom lost by crosslinking, over which the classical theories are in dispute, is found to lie between the previous values in a formula which can reproduce the classical results by making different assumptions. The entanglements give rise to more complicated forms than the classical sum of squares of strain ratios, which under certain circumstances can reproduce the Mooney-Rivlin term which when added empirically to the free energy usually improves the fit with experiment. The general expression is complicated, but is nevertheless an explicit function of the density of crosslinks, the density of the rubber and the interchain forces.

scholarly journals Evidence for a 5d F-theorem

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Martin Fluder ◽  
Christoph F. Uhlemann
Keyword(s):  
Free Energy ◽  
Renormalization Group ◽  
General Expression ◽  
Geometric Parameters ◽  
Regularity Conditions ◽  
Large Sample ◽  
Large Numbers ◽  
Renormalization Group Flows

Abstract Renormalization group flows are studied between 5d SCFTs engineered by (p, q) 5-brane webs with large numbers of external 5-branes. A general expression for the free energy on S5 in terms of single-valued trilogarithm functions is derived from their supergravity duals, which are characterized by the 5-brane charges and additional geometric parameters. The additional geometric parameters are fixed by regularity conditions, and we show that the solutions to the regularity conditions extremize a trial free energy. These results are used to survey a large sample of $$ \mathcal{O} $$ O (105) renormalization group flows between different 5d SCFTs, including Higgs branch flows and flows that preserve the SU(2) R- symmetry. In all cases the free energy changes monotonically towards the infrared, in line with a 5d F -theorem.

The theory of rubber elasticity

1976 ◽  
Vol 351 (1666) ◽  
pp. 397-406 ◽  
Keyword(s):  
Equation Of State ◽  
Statistical Mechanics ◽  
Equations Of State ◽  
Rubber Elasticity ◽  
Elastic Behaviour ◽  
Excluded Volume ◽  
Simple Pattern ◽  
Dynamical Approach ◽  
Equilibrium Approach ◽  
The Future

It is argued that since statistical mechanics has developed in two ways, the dynamical approach of Boltzmann and the equilibrium approach of Gibbs, both should be valuable in rubber elasticity. It is shown that this is indeed the case, and the generality of these approaches allows one to study the problem in greater depth than hitherto. In particular, damping terms in the elastic behaviour of rubber can be calculated, and also the effect of entanglements and excluded volume on the equation of state. It is noticeable that although the calculated equations of state are quite complex, they do not fit into a simple pattern of invariants. The future for these developments is briefly discussed.

scholarly journals Random boundary geometry and gravity dual of $$ T\overline{T} $$ deformation

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Shinji Hirano ◽  
Masaki Shigemori
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Energy Spectrum ◽  
Correlation Functions ◽  
Degrees Of Freedom ◽  
Dual Language ◽  
Renormalization Group Flow ◽  
Conformal Field ◽  
Logarithmic Corrections ◽  
Group Flow

Abstract We study the random geometry approach to the $$ T\overline{T} $$ T T ¯ deformation of 2d conformal field theory developed by Cardy and discuss its realization in a gravity dual. In this representation, the gravity dual of the $$ T\overline{T} $$ T T ¯ deformation becomes a straightforward translation of the field theory language. Namely, the dual geometry is an ensemble of AdS3 spaces or BTZ black holes, without a finite cutoff, but instead with randomly fluctuating boundary diffeomorphisms. This reflects an increase in degrees of freedom in the renormalization group flow to the UV by the irrelevant $$ T\overline{T} $$ T T ¯ operator. We streamline the method of computation and calculate the energy spectrum and the thermal free energy in a manner that can be directly translated into the gravity dual language. We further generalize this approach to correlation functions and reproduce the all-order result with universal logarithmic corrections computed by Cardy in a different method. In contrast to earlier proposals, this version of the gravity dual of the $$ T\overline{T} $$ T T ¯ deformation works not only for the energy spectrum and the thermal free energy but also for correlation functions.

scholarly journals A new entropy model for RNA: part II. Persistence-related entropic contributions to RNA secondary structure free energy calculations

2013 ◽  
Vol 4 (1) ◽  
pp. 2 ◽  
Author(s):  
Wayne Dawson ◽  
Kenji Yamamoto ◽  
Kentaro Shimizu ◽  
Gota Kawai
Keyword(s):  
Free Energy ◽  
Secondary Structure ◽  
Degrees Of Freedom ◽  
Free Energy Calculations ◽  
Local Entropy ◽  
Entropy Model ◽  
Stem Loop ◽  
Entropy Correction ◽  
The Cross ◽  
Folded Rna

In previous work, we have shown that the entropy of a folded RNA molecule can be divided into local and global contributions using the cross-linking entropy (CLE) model, where, in the case of RNA, the cross- links are the base-pair stacking interactions. The local contribution to the CLE is revealed in the Kuhn length (a measure of the stiffness of the RNA). The Kuhn length acts as a scaling parameter. When the size of the system is rescaled, the relationship between local and global free energy must be renormalized to reflect this rescaling. In this renormalization process, the Kuhn length increases, the local entropy also increases due to freezing out of the local conformational degrees of freedom. At the same time, as the number of degrees of freedom decrease, there is a significant reduction in the global entropy. Here we present a method, based on the concepts of renormalization theory, to quantitatively estimate the size of the contribution from the local entropy as a function of the Kuhn length. The local entropy correction is used to predict the current empirically derived constant in the Jacobson-Stockmayer equation. The variation in the Kuhn length is shown to be largely influenced by the length of the double-stranded RNA stems formed in the secondary structure of folded RNA. This result is used to test the resulting entropy under a variable Kuhn length in stem-loop structures. Comparisons between a variable Kuhn length and a static Kuhn length on a short stem-loop of RNA are also examined. The model is quite general and is also directly applicable to protein structure and folding problems.

scholarly journals Aspects of emergent symmetries

2016 ◽  
Vol 31 (10) ◽  
pp. 1630009 ◽  
Author(s):  
Pedro R. S. Gomes
Keyword(s):  
Field Theory ◽  
Statistical Mechanics ◽  
Degrees Of Freedom ◽  
Condensed Matter ◽  
Lorentz Invariance ◽  
Low Energy ◽  
Background Material

These are intended to be review notes on emergent symmetries, i.e. symmetries which manifest themselves in specific sectors of energy in many systems. The emphasis is on the physical aspects rather than computation methods. We include some background material and go through more recent problems in field theory, statistical mechanics and condensed matter. These problems illustrate how some important symmetries, such as Lorentz invariance and supersymmetry, usually believed to be fundamental, can arise naturally in low-energy regimes of systems involving a large number of degrees of freedom. The aim is to discuss how these examples could help us to face other complex and fundamental problems.

ENTROPY-ENTHALPY COMPENSATION BEHAVIOR REVISITED

2004 ◽  
Vol 03 (04) ◽  
pp. 511-520 ◽  
Author(s):  
WILLIAM R. KIRK
Keyword(s):  
Free Energy ◽  
Statistical Mechanics ◽  
Solvent Effect ◽  
Liquid State ◽  
Compensation Behavior ◽  
Solute Interaction ◽  
Entropically Driven ◽  
Thermodynamic Perturbation

The origin of genuine (not statistically spurious) entropy-enthalpy compensation is investigated within standard liquid state statistical mechanics formalism. We treat two extreme cases of solvent-solute interaction: (1) those in which the changes "internal" to the solute dominate, and (2) those in which the solvent effect is dominant. They are evaluated in different ways, and lead to different predictions for compensation behavior. The first involves a conventional "thermodynamic perturbation" formalism.1,2 The second case leads to an Ornstein–Zernicke-like relationship that predicts predominantly entropically driven changes in free energy of reaction.

A Constraint Dynamics Approach to Rubber Elasticity

1993 ◽  
Vol 66 (5) ◽  
pp. 817-826 ◽  
Author(s):  
K. L. Ngai ◽  
C. M. Roland ◽  
A. F. Yee
Keyword(s):  
Nmr Spectroscopy ◽  
Statistical Mechanics ◽  
Elasticity Theory ◽  
31P Nmr ◽  
Coupling Model ◽  
Rubber Elasticity ◽  
Mechanical Relaxation ◽  
31P Nmr Spectroscopy ◽  
Dynamical Models ◽  
Relaxation Data

Abstract The coupling model of relaxation is applied to crosslinked rubber, with a connection drawn between the dynamics of network junctions and statistical mechanics derived rubber elasticity theory. The suggestion that unifying concepts must underlie dynamical models and thermodynamic theories is shown to be supported by analyses of recent 31P NMR spectroscopy measurements by Shi, Dickinson, MacKnight and Chien on polytetrahydrofuran networks and of our mechanical relaxation data on stretched and unstretched polycarbonate.

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