119 research outputs found
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
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
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
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
Conformations of Randomly Linked Polymers
We consider polymers in which M randomly selected pairs of monomers are
restricted to be in contact. Analytical arguments and numerical simulations
show that an ideal (Gaussian) chain of N monomers remains expanded as long as
M<<N; its mean squared end to end distance growing as r^2 ~ M/N. A possible
collapse transition (to a region of order unity) is related to percolation in a
one dimensional model with long--ranged connections. A directed version of the
model is also solved exactly. Based on these results, we conjecture that the
typical size of a self-avoiding polymer is reduced by the links to R >
(N/M)^(nu). The number of links needed to collapse a polymer in three
dimensions thus scales as N^(phi), with (phi) > 0.43.Comment: 6 pages, 3 Postscript figures, LaTe
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