arXiv subjects

M. E. Cates

Publications and source records attributed to M. E. Cates.

140 records · Page 8Linked to original sources

arXiv1998

Topological effects in ring polymers: A computer simulation study

Unconcatenated, unknotted polymer rings in the melt are subject to strong interactions with neighboring chains due to the presence of topological constraints. We study this by computer simulation using the bond-fluctuation algorithm for chains with up to N=512 statistical segments at a volume fraction Φ=0.5 and show that rings in the melt are more compact than gaussian chains. A careful finite size analysis of the average ring size R \propto N^ν yields an exponent ν\approx 0.39 \pm 0.03 in agreement with a Flory-like argument for the topologica interactions. We show (using the same algorithm) that the dynamics of molten rings is similar to that of linear chains of the same mass, confirming recent experimental findings. The diffusion constant varies effectively as D_{N} \propto N^{-1.22(3) and is slightly higher than that of corresponding linear chains. For the ring sizes considered (up to 256 statistical segments) we find only one characteristic time scale τ_{ee} \propto N^{2.0(2); this is shown by the collapse of several mean-square displacements and correlation functions onto corresponding master curves. Because of the shrunken state of the chain, this scaling is not compatible with simple Rouse motion. It applies for all sizes of ring studied and no sign of a crossover to any entangled regime is found.

cond-mat.stat-mech↗

arXiv1998

Diffusive growth of a polymer layer by in sity polymerization

We consider the growth of a polymer layer on a flat surface in a good solvent by in-situ polymerization. This is viewed as a modified form of diffusion-limited aggregation without branching. We predict theoretically the formation of a pseudo-brush with density rhog(z) \propto z^{-2/3} and characteristic height \propto t^{3}. These results are found by combining a mean-field treatment of the diffusive growth (marginally valid in three dimensions) with a scaling theory (Flory exponent nu =3/5) of the growing polymers. We confirm their validity by Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv1998

Computational confirmation of scaling predictions for equilibrium polymers

We report the results of extensive Dynamic Monte Carlo simulations of systems of self-assembled Equilibrium Polymers without rings in good solvent. Confirming recent theoretical predictions, the mean-chain length is found to scale as $\Lav = \Lstar (ϕ/\phistar)^α\propto ϕ^α\exp(δE)$ with exponents $α_d=δ_d=1/(1+γ) \approx 0.46$ and $α_s = [1+(γ-1)/(νd -1)]/2 \approx 0.60, δ_s=1/2$ in the dilute and semi-dilute limits respectively. The average size of the micelles, as measured by the end-to-end distance and the radius of gyration, follows a very similar crossover scaling to that of conventional quenched polymer chains. In the semi-dilute regime, the chain size distribution is found to be exponential, crossing over to a Schultz-Zimm type distribution in the dilute limit. The very large size of our simulations (which involve mean chain lengths up to 5000, even at high polymer densities) allows also an accurate determination of the self-avoiding walk susceptibility exponent $γ= 1.165 \pm 0.01$.

cond-mat.soft↗

arXiv1998

Stabilisation of Emulsions by Trapped Species

We consider an emulsion whose droplets contain a trapped species (insoluble in the continuous phase), and study the emulsion's stability against coarsening via Lifshitz-Slyozov dynamics (Ostwald Ripening). Extending an earlier treatment by Kabalnov et al(Colloids and Surfaces, 24,(1987), 19-32), we derive a general condition on the mean initial droplet volume which ensures stability, even when arbitrary polydispersity is present in both size and composition of the initial droplets. We distinguish "nucleated" coarsening, which requires either fluctuations about the mean field equations or a tail in the initial droplet size distribution, from "spinodal" coarsening in which a typical droplet is locally unstable. A weaker condition for stability, previously suggested by Kabalnov et al., is sufficient only to prevent "spinodal" coarsening and is best viewed as a condition for metastability. The coarsening of unstable emulsions is considered, and shown at long times to resemble that of ordinary emulsions (with no trapped species), but with a reduced value of the initial volume fraction of dispersed phase. We discuss the physical principles relevant to the stability of emulsions with trapped species, describing how these may be exploited to restabilise partially coarsened emulsions and to "shrink" previously formed emulsion droplets to form "miniemulsions".

cond-mat.soft↗

arXiv1998

Development of Stresses in Cohesionless Poured Sand

The pressure distribution beneath a conical sandpile, created by pouring sand from a point source onto a rough rigid support, shows a pronounced minimum below the apex (`the dip'). Recent work of the authors has attempted to explain this phenomenon by invoking local rules for stress propagation that depend on the local geometry, and hence on the construction history, of the medium. We discuss the fundamental difference between such approaches, which lead to hyperbolic differential equations, and elastoplastic models, for which the equations are elliptic within any elastic zones present .... This displacement field appears to be either ill-defined, or defined relative to a reference state whose physical existence is in doubt. Insofar as their predictions depend on physical factors unknown and outside experimental control, such elastoplastic models predict that the observations should be intrinsically irreproducible .... Our hyperbolic models are based instead on a physical picture of the material, in which (a) the load is supported by a skeletal network of force chains ("stress paths") whose geometry depends on construction history; (b) this network is `fragile' or marginally stable, in a sense that we define. .... We point out that our hyperbolic models can nonetheless be reconciled with elastoplastic ideas by taking the limit of an extremely anisotropic yield condition.

cond-mat.stat-mech↗

arXiv1998

Jamming, Force Chains and Fragile Matter

We consider materials whose mechanical integrity is the result of a jamming process. We argue that such media are generically "fragile": unable to support certain types of incremental loading without plastic rearrangement. Fragility is linked to the marginal stability of force chain networks within the material. Such ideas may be relevant to jammed colloids and poured sand. The crossover from fragile (when particles are rigid) to elastoplastic behavior is explored.

cond-mat↗

arXiv1998

Shear-Induced Clustering in a Simple Driven Diffusive Model

We study a simple lattice model of shear-induced clustering in two dimensions in which clusters of particles aggregate under an imposed shear flow and fragment stochastically. Two non-equilibrium steady states are identified: an unjammed state and a jammed state characterised by a system-spanning cluster. A discontinuous jamming transition with strong hysteresis occurs as the shear rate is increased or fragmentation rate decreased. We study the kinetics of jamming and measure power law cluster size distributions. We also consider some general simulation issues including the role of Galilean invariance.

cond-mat.stat-mech↗

arXiv1997

Jamming transition in a homogeneous one-dimensional system: the Bus Route Model

We present a driven diffusive model which we call the Bus Route Model. The model is defined on a one-dimensional lattice, with each lattice site having two binary variables, one of which is conserved (``buses'') and one of which is non-conserved (``passengers''). The buses are driven in a preferred direction and are slowed down by the presence of passengers who arrive with rate lambda. We study the model by simulation, heuristic argument and a mean-field theory. All these approaches provide strong evidence of a transition between an inhomogeneous ``jammed'' phase (where the buses bunch together) and a homogeneous phase as the bus density is increased. However, we argue that a strict phase transition is present only in the limit lambda -> 0. For small lambda, we argue that the transition is replaced by an abrupt crossover which is exponentially sharp in 1/lambda. We also study the coarsening of gaps between buses in the jammed regime. An alternative interpretation of the model is given in which the spaces between ``buses'' and the buses themselves are interchanged. This describes a system of particles whose mobility decreases the longer they have been stationary and could provide a model for, say, the flow of a gelling or sticky material along a pipe.

cond-mat.stat-mech↗

arXiv1997

Spontaneous Jamming in One-Dimensional Systems

We study the phenomenon of jamming in driven diffusive systems. We introduce a simple microscopic model in which jamming of a conserved driven species is mediated by the presence of a non-conserved quantity, causing an effective long range interaction of the driven species. We study the model analytically and numerically, providing strong evidence that jamming occurs; however, this proceeds via a strict phase transition (with spontaneous symmetry breaking) only in a prescribed limit. Outside this limit, the nearby transition (characterised by an essential singularity) induces sharp crossovers and transient coarsening phenomena. We discuss the relevance of the model to two physical situations: the clustering of buses, and the clogging of a suspension forced along a pipe.

cond-mat.stat-mech↗

arXiv1997

Models of stress propagation in granular media

Stress patterns in static granular media exhibit unusual features when compared to either liquids or elastic solids. Qualitatively, we attribute these features to the presence of `stress paths', whose geometry depends on the construction history and controls the propagation of stresses. Stress paths can cause random focussing of stresses (large fluctuations) as well as systematic deflections (arching). We describe simple physical models that capture some of these effects. In these models, the `stress paths' become identified with the characteristic `light rays' of wavelike (hyperbolic) equations for force propagation. Such models account for the `pressure dip' below conical sandpiles built by pouring from a point source, and explain qualitatively the large stress fluctuations observed experimentally in granular matter. The differences between this approach and more conventional modelling strategies (based on elastoplastic or rigid-plastic models) are highlighted, focusing on the role of boundary conditions. Our models provide a continuum picture in which granular materials are viewed as fragile matter, able to support without rearranging only a subset of the static external loadings admissible for a normal elastic solid.

cond-mat↗

arXiv1997

Models of stress fluctuations in granular media

We investigate in detail two models describing how stresses propagate and fluctuate in granular media. The first one is a scalar model where only the vertical component of the stress tensor is considered. In the continuum limit, this model is equivalent to a diffusion equation (where the rôle of time is played by the vertical coordinate) plus a randomly varying convection term. We calculate the response and correlation function of this model, and discuss several properties, in particular related to the stress distribution function. We then turn to the tensorial model, where the basic starting point is a wave equation which, in the absence of disorder, leads to a ray-like propagation of stress. In the presence of disorder, the rays acquire a diffusive width and the angle of propagation is shifted. A striking feature is that the response function becomes negative, which suggests that the contact network is mechanically unstable to very weak perturbations. The stress correlation function reveals characteristic features related to the ray-like propagation, which are absent in the scalar description. Our analytical calculations are confirmed and extended by a numerical analysis of the stochastic wave equation.

cond-mat↗

arXiv1997

Diffusive Evolution of Stable and Metastable Phases I: Local Dynamics of Interfaces

We find analytical solutions to the Cahn-Hilliard equation for the dynamics of an interface in a system with a conserved order parameter (Model B). We show that, although steady-state solutions of Model B are unphysical in the far-field, they shed light on the local dynamics of an interface. Exact solutions are given for a particular class of order-parameter potentials, and an expandable integral equation is derived for the general case. As well as revealing some generic properties of interfaces moving under condensation or evaporation, the formalism is used to investigate two distinct modes of interface propagation in systems with a metastable potential well. Given a sufficient transient increase in the flux of material onto a condensation nucleus, the normal motion of the interface can be disrupted by interfacial unbinding, leading to growth of a macroscopic amount of a metastable phase.

cond-mat.soft↗

arXiv1997

Role of Metastable States in Phase Ordering Dynamics

We show that the rate of separation of two phases of different densities (e.g. gas and solid) can be radically altered by the presence of a metastable intermediate phase (e.g. liquid). Within a Cahn-Hilliard theory we study the growth in one dimension of a solid droplet from a supersaturated gas. A moving interface between solid and gas phases (say) can, for sufficient (transient) supersaturation, unbind into two interfaces separated by a slab of metastable liquid phase. We investigate the criteria for unbinding, and show that it may strongly impede the growth of the solid phase.

cond-mat.soft↗

arXiv1996

Stress Propagation and Arching in Static Sandpiles

We present a new approach to the modelling of stress propagation in static granular media, focussing on the conical sandpile constructed from a point source. We view the medium as consisting of cohesionless hard particles held up by static frictional forces; these are subject to microscopic indeterminacy which corresponds macroscopically to the fact that the equations of stress continuity are incomplete -- no strain variable can be defined. We propose that in general the continuity equations should be closed by means of a constitutive relation (or relations) between different components of the (mesoscopically averaged) stress tensor. The primary constitutive relation relates radial and vertical shear and normal stresses (in two dimensions, this is all one needs). We argue that the constitutive relation(s) should be local, and should encode the construction history of the pile: this history determines the organization of the grains at a mesoscopic scale, and thereby the local relationship between stresses. To the accuracy of published experiments, the pattern of stresses beneath a pile shows a scaling between piles of different heights (RSF scaling) which severely limits the form the constitutive relation can take ...

cond-mat↗