637 research outputs found
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
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
One-step replica symmetry breaking solution of the quadrupolar glass model
We consider the quadrupolar glass model with infinite-range random
interaction. Introducing a simple one-step replica symmetry breaking ansatz we
investigate the para-glass continuous (discontinuous) transition which occurs
below (above) a critical value of the quadrupole dimension m*. By using a
mean-field approximation we study the stability of the one-step replica
symmetry breaking solution and show that for m>m* there are two transitions.
The thermodynamic transition is discontinuous but there is no latent heat. At a
higher temperature we find the dynamical or glass transition temperature and
the corresponding discontinuous jump of the order parameter.Comment: 10 pages, 3 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
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