175 research outputs found
Partitioning of energy in highly polydisperse granular gases
A highly polydisperse granular gas is modeled by a continuous distribution of
particle sizes, a, giving rise to a corresponding continuous temperature
profile, T(a), which we compute approximately, generalizing previous results
for binary or multicomponent mixtures. If the system is driven, it evolves
towards a stationary temperature profile, which is discussed for several
driving mechanisms in dependence on the variance of the size distribution. For
a uniform distribution of sizes, the stationary temperature profile is
nonuniform with either hot small particles (constant force driving) or hot
large particles (constant velocity or constant energy driving). Polydispersity
always gives rise to non-Gaussian velocity distributions. Depending on the
driving mechanism the tails can be either overpopulated or underpopulated as
compared to the molecular gas. The deviations are mainly due to small
particles. In the case of free cooling the decay rate depends continuously on
particle size, while all partial temperatures decay according to Haff's law.
The analytical results are supported by event driven simulations for a large,
but discrete number of species.Comment: 10 pages; 5 figure
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
Orientational order and glassy states in networks of semiflexible polymers
Motivated by the structure of networks of cross-linked cytoskeletal
biopolymers, we study the orientationally ordered phases in two-dimensional
networks of randomly cross-linked semiflexible polymers. We consider permanent
cross-links which prescribe a finite angle and treat them as quenched disorder
in a semi-microscopic replica field theory. Starting from a fluid of
un-cross-linked polymers and small polymer clusters (sol) and increasing the
cross-link density, a continuous gelation transition occurs. In the resulting
gel, the semiflexible chains either display long range orientational order or
are frozen in random directions depending on the value of the crossing angle,
the crosslink concentration and the stiffness of the polymers. A crossing angle
leads to long range -fold orientational order, e.g.,
"hexatic" or "tetratic" for or , respectively.
The transition is discontinuous and the critical cross-link density depends on
the bending stiffness of the polymers and the cross-link geometry: the higher
the stiffness and the lower , the lower the critical number of cross-links.
In between the sol and the long range ordered state, we always observe a gel
which is a statistically isotropic amorphous solid (SIAS) with random
positional and random orientational localization of the participating polymers.Comment: 20 pages, added references, minor changes, final version as published
in PR
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
Goldstone-type fluctuations and their implications for the amorphous solid state
In sufficiently high spatial dimensions, the formation of the amorphous (i.e.
random) solid state of matter, e.g., upon sufficent crosslinking of a
macromolecular fluid, involves particle localization and, concommitantly, the
spontaneous breakdown of the (global, continuous) symmetry of translations.
Correspondingly, the state supports Goldstone-type low energy, long wave-length
fluctuations, the structure and implications of which are identified and
explored from the perspective of an appropriate replica field theory. In terms
of this replica perspective, the lost symmetry is that of relative translations
of the replicas; common translations remain as intact symmetries, reflecting
the statistical homogeneity of the amorphous solid state. What emerges is a
picture of the Goldstone-type fluctuations of the amorphous solid state as
shear deformations of an elastic medium, along with a derivation of the shear
modulus and the elastic free energy of the state. The consequences of these
fluctuations -- which dominate deep inside the amorphous solid state -- for the
order parameter of the amorphous solid state are ascertained and interpreted in
terms of their impact on the statistical distribution of localization lengths,
a central diagnostic of the the state. The correlations of these order
parameter fluctuations are also determined, and are shown to contain
information concerning further diagnostics of the amorphous solid state, such
as spatial correlations in the statistics of the localization characteristics.
Special attention is paid to the properties of the amorphous solid state in two
spatial dimensions, for which it is shown that Goldstone-type fluctuations
destroy particle localization, the order parameter is driven to zero, and
power-law order-parameter correlations hold.Comment: 20 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
Glassy states and microphase separation in cross-linked homopolymer blends
The physical properties of blends of distinct homopolymers, cross-linked
beyond the gelation point, are addressed via a Landau approach involving a pair
of coupled order-parameter fields: one describing vulcanisation, the other
describing local phase separation. Thermal concentration fluctuations, present
at the time of cross-linking, are frozen in by cross-linking, and the structure
of the resulting glassy fluctuations is analysed at the Gaussian level in
various regimes, determined by the relative values of certain physical
length-scales. The enhancement, due to gelation, of the stability of the blend
with respect to demixing is also analysed. Beyond the corresponding stability
limit, gelation prevents complete demixing, replacing it by microphase
separation, which occurs up to a length-scale set by the rigidity of the
network, as a simple variational scheme reveals.Comment: 7 pages, 6 figure
Variational bounds for the shear viscosity of gelling melts
We study shear stress relaxation for a gelling melt of randomly crosslinked,
interacting monomers. We derive a lower bound for the static shear viscosity
, which implies that it diverges algebraically with a critical exponent
. Here, and are the critical exponents of
percolation theory for the correlation length and the gel fraction. In
particular, the divergence is stronger than in the Rouse model, proving the
relevance of excluded-volume interactions for the dynamic critical behaviour at
the gel transition. Precisely at the critical point, our exact results imply a
Mark-Houwink relation for the shear viscosity of isolated clusters of fixed
size.Comment: 5 pages; CHANGES: typos corrected, some references added; version as
publishe
Randomly Crosslinked Macromolecular Systems: Vulcanisation Transition to and Properties of the Amorphous Solid State
As Charles Goodyear discovered in 1839, when he first vulcanised rubber, a
macromolecular liquid is transformed into a solid when a sufficient density of
permanent crosslinks is introduced at random. At this continuous equi- librium
phase transition, the liquid state, in which all macromolecules are
delocalised, is transformed into a solid state, in which a nonzero fraction of
macromolecules have spontaneously become localised. This solid state is a most
unusual one: localisation occurs about mean positions that are distributed
homogeneously and randomly, and to an extent that varies randomly from monomer
to monomer. Thus, the solid state emerging at the vulcanisation transition is
an equilibrium amorphous solid state: it is properly viewed as a solid state
that bears the same relationship to the liquid and crystalline states as the
spin glass state of certain magnetic systems bears to the paramagnetic and
ferromagnetic states, in the sense that, like the spin glass state, it is
diagnosed by a subtle order parameter.
In this review we give a detailed exposition of a theoretical approach to the
physical properties of systems of randomly, permanently crosslinked
macromolecules. Our primary focus is on the equilibrium properties of such
systems, especially in the regime of Goodyear's vulcanisation transition.Comment: Review Article, REVTEX, 58 pages, 3 PostScript figure
Dynamics of gelling liquids: a short survey
The dynamics of randomly crosslinked liquids is addressed via a Rouse- and a
Zimm-type model with crosslink statistics taken either from bond percolation or
Erdoes-Renyi random graphs. While the Rouse-type model isolates the effects of
the random connectivity on the dynamics of molecular clusters, the Zimm-type
model also accounts for hydrodynamic interactions on a preaveraged level. The
incoherent intermediate scattering function is computed in thermal equilibrium,
its critical behaviour near the sol-gel transition is analysed and related to
the scaling of cluster diffusion constants at the critical point. Second,
non-equilibrium dynamics is studied by looking at stress relaxation in a simple
shear flow. Anomalous stress relaxation and critical rheological properties are
derived. Some of the results contradict long-standing scaling arguments, which
are shown to be flawed by inconsistencies.Comment: 21 pages, 3 figures; Dedicated to Lothar Schaefer on the occasion of
his 60th birthday; Changes: added comments on the gel phase and some
reference
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