38 research outputs found
Strain-dependent localization, microscopic deformations, and macroscopic normal tensions in model polymer networks
We use molecular dynamics simulations to investigate the microscopic and
macroscopic response of model polymer networks to uniaxial elongations. By
studying networks with strands lengths ranging from to 200 we cover
the full crossover from cross-link to entanglement dominated behavior. Our
results support a recent version of the tube model which accounts for the
different strain dependence of chain localization due to chemical cross-links
and entanglements
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
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
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
‘‘Lozenge’’ contour plots in scattering from polymer networks
We present a consistent explanation for the appearance of “lozenge” shapes in contour plots of the two dimensional scattering intensity from stretched polymer networks. By explicitly averaging over quenched variables in a tube model, we show that lozenge patterns arise as a result of chain material that is not directly deformed by the stretch. We obtain excellent agreement with experimental data
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
Dynamical signatures of the vulcanization transition
Dynamical properties of vulcanized polymer networks are addressed via a
Rouse-type model that incorporates the effect of permanent random crosslinks.
The incoherent intermediate scattering function is computed in the sol and gel
phases, and at the vulcanization transition between them. At any nonzero
crosslink density within the sol phase Kohlrausch relaxation is found. The
critical point is signalled by divergence of the longest time-scale, and at
this point the scattering function decays algebraically, whereas within the gel
phase it acquires a time-persistent part identified with the gel fraction.Comment: 4 page
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