Statistical mechanics of permanent random atomic and molecular networks: Structure and heterogeneity of the amorphous solid state

Under sufficient permanent random covalent bonding, a fluid of atoms or small molecules is transformed into an amorphous solid network. Being amorphous, local structural properties in such networks vary across the sample. A natural order parameter, resulting from a statistical-mechanical approach, captures information concerning this heterogeneity via a certain joint probability distribution. This joint probability distribution describes the variations in the positional and orientational localization of the particles, reflecting the random environments experienced by them, as well as further information characterizing the thermal motion of particles. A complete solution, valid in the vicinity of the amorphous solidification transition, is constructed essentially analytically for the amorphous solid order parameter, in the context of the random network model and approach introduced by Goldbart and Zippelius [Europhys. Lett. 27, 599 (1994)]. Knowledge of this order parameter allows us to draw certain conclusions about the stucture and heterogeneity of randomly covalently bonded atomic or molecular network solids in the vicinity of the amorphous solidification transition. Inter alia, the positional aspects of particle localization are established to have precisely the structure obtained perviously in the context of vulcanized media, and results are found for the analogue of the spin glass order parameter describing the orientational freezing of the bonds between particles.Comment: 31 pages, 5 figure

Goldstone fluctuations in the amorphous solid state

Goldstone modes in the amorphous solid state, resulting from the spontaneous breaking of translational symmetry due to random localisation of particles, are discussed. Starting from a microscopic model with quenched disorder, the broken symmetry is identified to be that of relative translations of the replicas. Goldstone excitations, corresponding to pure shear deformations, are constructed from long wavelength distortions of the order parameter. The elastic free energy is computed, and it is shown that Goldstone fluctuations destroy localisation in two spatial dimensions, yielding a two-dimensional amorphous solid state characterised by power-law correlations.Comment: 7 pages, 2 figure

Scaling of Entropic Shear Rigidity

The scaling of the shear modulus near the gelation/vulcanization transition is explored heuristically and analytically. It is found that in a dense melt the effective chains of the infinite cluster have sizes that scale sub-linearly with their contour length. Consequently, each contributes k_B T to the rigidity, which leads to a shear modulus exponent d\nu. In contrast, in phantom elastic networks the scaling is linear in the contour length, yielding an exponent identical to that of the random resistor network conductivity, as predicted by de Gennes'. For non-dense systems, the exponent should cross over to d\nu when the percolation length becomes much larger than the density-fluctuation length.Comment: 4 pages, 2 eps figure

Universality and its Origins at the Amorphous Solidification Transition

Systems undergoing an equilibrium phase transition from a liquid state to an amorphous solid state exhibit certain universal characteristics. Chief among these are the fraction of particles that are randomly localized and the scaling functions that describe the order parameter and (equivalently) the statistical distribution of localization lengths for these localized particles. The purpose of this Paper is to discuss the origins and consequences of this universality, and in doing so, three themes are explored. First, a replica-Landau-type approach is formulated for the universality class of systems that are composed of extended objects connected by permanent random constraints and undergo amorphous solidification at a critical density of constraints. This formulation generalizes the cases of randomly cross-linked and end-linked macromolecular systems, discussed previously. The universal replica free energy is constructed, in terms of the replica order parameter appropriate to amorphous solidification, the value of the order parameter is obtained in the liquid and amorphous solid states, and the chief universal characteristics are determined. Second, the theory is reformulated in terms of the distribution of local static density fluctuations rather than the replica order parameter. It is shown that a suitable free energy can be constructed, depending on the distribution of static density fluctuations, and that this formulation yields precisely the same conclusions as the replica approach. Third, the universal predictions of the theory are compared with the results of extensive numerical simulations of randomly cross-linked macromolecular systems, due to Barsky and Plischke, and excellent agreement is found.Comment: 10 pages, including 3 figures (REVTEX

Random solids and random solidification: What can be learned by exploring systems obeying permanent random constraints?

In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state. In this random solid state, particles are permanently but randomly localized in space, and a rigidity to shear deformations emerges. Owing to the permanence of the random constraints, this phase transition is an equilibrium transition, which confers on it a simplicity (at least relative to the conventional glass transition) in the sense that it is amenable to established techniques of equilibrium statistical mechanics. In this Paper I shall review recent developments in the theory of random solidification for systems obeying permanent random constraints, with the aim of bringing to the fore the similarities and differences between such systems and those exhibiting the conventional glass transition. I shall also report new results, obtained in collaboration with Weiqun Peng, on equilibrium correlations and susceptibilities that signal the approach of the random solidification transition, discussing the physical interpretation and values of these quantities both at the Gaussian level of approximation and, via a renormalization-group approach, beyond.Comment: Paper presented at the "Unifying Concepts in Glass Physics" workshop, International Centre for Theoretical Physics, Trieste, Italy (September 15-18, 1999

Elasticity near the vulcanization transition

Signatures of the vulcanization transition--amorphous solidification induced by the random crosslinking of macromolecules--include the random localization of a fraction of the particles and the emergence of a nonzero static shear modulus. A semi-microscopic statistical-mechanical theory is presented of the latter signature that accounts for both thermal fluctuations and quenched disorder. It is found (i) that the shear modulus grows continuously from zero at the transition, and does so with the classical exponent, i.e., with the third power of the excess cross-link density and, quite surprisingly, (ii) that near the transition the external stresses do not spoil the spherical symmetry of the localization clouds of the particles.Comment: REVTEX, 5 pages. Minor change

On the relevance of percolation theory to the vulcanization transition

The relationship between vulcanization and percolation is explored from the perspective of renormalized local field theory. We show rigorously that the vulcanization and percolation correlation functions are governed by the same Gell--Mann-Low renormalization group equation. Hence, all scaling aspects of the vulcanization transition are reigned by the critical exponents of the percolation universality class.Comment: 9 pages, 2 figure

Thermal Fluctuations and Rubber Elasticity

The effects of thermal elastic fluctuations in rubber materials are examined. It is shown that, due to an interplay with the incompressibility constraint, these fluctuations qualitatively modify the large-deformation stress-strain relation, compared to that of classical rubber elasticity. To leading order, this mechanism provides a simple and generic explanation for the peak structure of Mooney-Rivlin stress-strain relation, and shows a good agreement with experiments. It also leads to the prediction of a phonon correlation function that depends on the external deformation.Comment: 4 RevTeX pages, 1 figure, submitted to PR

Elasticity of highly cross-linked random networks

Starting from a microscopic model of randomly cross-linked particles with quenched disorder, we calculate the Laudau-Wilson free energy S for arbitrary cross-link densities. Considering pure shear deformations, S takes the form of the elastic energy of an isotropic amorphous solid state, from which the shear modulus can be identified. It is found to be an universal quantity, not depending on any microscopic length-scales of the model.Comment: 6 pages, 5 figure

Phase-slip avalanches in the superflow of 4^4He through arrays of nanopores

Recent experiments by Sato et al. [1] have explored the dynamics of $^4$He superflow through an array of nanopores. These experiments have found that, as the temperature is lowered, phase-slippage in the pores changes its character, from synchronous to asynchronous. Inspired by these experiments, we construct a model to address the characteristics of phase-slippage in superflow through nanopore arrays. We focus on the low-temperature regime, in which the current-phase relation for a single pore is linear, and thermal fluctuations may be neglected. Our model incorporates two basic ingredients: (1) each pore has its own random value of critical velocity (due, e.g., to atomic-scale imperfections), and (2) an effective inter-pore coupling, mediated through the bulk superfluid. The inter-pore coupling tends to cause neighbours of a pore that has already phase-slipped also to phase-slip; this process may cascade, creating an avalanche of synchronously slipping phases. As the temperature is lowered, the distribution of critical velocities is expected to effectively broaden, owing to the reduction in the superfluid healing length, leading to a loss of synchronicity in phase-slippage. Furthermore, we find that competition between the strength of the disorder in the critical velocities and the strength of the inter-pore interaction leads to a phase transition between non-avalanching and avalanching regimes of phase-slippage. [1] Sato, Y., Hoskinson, E. Packard, R. E. cond-mat/0605660.Comment: 8 pages, 5 figure