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 (\phi/\phistar)^\alpha \propto \phi^\alpha \exp(\delta E) with exponents $\alpha_d=\delta_d=1/(1+\gamma) \approx 0.46$ and $\alpha_s = [1+(\gamma-1)/(\nu d -1)]/2 \approx 0.60, \delta_s=1/2$ 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 $\gamma = 1.165 \pm 0.01$.Comment: 6 pages, 4 figures, LATE

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 ...Comment: 38 pages, 24 Postscript figures, LATEX, minor misspellings corrected, Journal de Physique I, Ref. Nr. 6.1125, accepte

Active Brownian Particles and Run-and-Tumble Particles: a Comparative Study

Active Brownian particles (ABPs) and Run-and-Tumble particles (RTPs) both self-propel at fixed speed $v$ along a body-axis ${\bf u}$ that reorients either through slow angular diffusion (ABPs) or sudden complete randomisation (RTPs). We compare the physics of these two model systems both at microscopic and macroscopic scales. Using exact results for their steady-state distribution in the presence of external potentials, we show that they both admit the same effective equilibrium regime perturbatively that breaks down for stronger external potentials, in a model-dependent way. In the presence of collisional repulsions such particles slow down at high density: their propulsive effort is unchanged, but their average speed along ${\bf u}$ becomes $v(\rho) < v$. A fruitful avenue is then to construct a mean-field description in which particles are ghost-like and have no collisions, but swim at a variable speed $v$ that is an explicit function or functional of the density $\rho$. We give numerical evidence that the recently shown equivalence of the fluctuating hydrodynamics of ABPs and RTPs in this case, which we detail here, extends to microscopic models of ABPs and RTPs interacting with repulsive forces.Comment: 32 pages, 6 figure

Diffusion and rheology in a model of glassy materials

We study self-diffusion within a simple hopping model for glassy materials. (The model is Bouchaud's model of glasses [J.-P. Bouchaud, J. Physique I 2, 1705 (1992)], as extended to describe rheological properties [P. Sollich, F. Lequeux, P. Hebraud and M.E. Cates, Phys. Rev. Lett. 78, 2020 (1997)].) We investigate the breakdown, near the glass transition, of the (generalized) Stokes-Einstein relation between self-diffusion of a tracer particle and the (frequency-dependent) viscosity of the system as a whole. This stems from the presence of a broad distribution of relaxation times of which different moments control diffusion and rheology. We also investigate the effect of flow (oscillatory shear) on self-diffusion and show that this causes a finite diffusivity in the temperature regime below the glass transition (where this was previously zero). At higher temperatures the diffusivity is enhanced by a power law frequency dependence that also characterises the rheological response. The relevance of these findings to soft glassy materials (foams, emulsions etc.) as well as to conventional glass-forming liquids is discussed.Comment: 39 page (double spaced), 2 figure

Dilatancy, Jamming, and the Physics of Granulation

Granulation is a process whereby a dense colloidal suspension is converted into pasty granules (surrounded by air) by application of shear. Central to the stability of the granules is the capillary force arising from the interfacial tension between solvent and air. This force appears capable of maintaining a solvent granule in a jammed solid state, under conditions where the same amount of solvent and colloid could also exist as a flowable droplet. We argue that in the early stages of granulation the physics of dilatancy, which requires that a powder expand on shearing, is converted by capillary forces into the physics of arrest. Using a schematic model of colloidal arrest under stress, we speculate upon various jamming and granulation scenarios. Some preliminary experimental results on aspects of granulation in hard-sphere colloidal suspensions are also reported.Comment: Original article intended for J Phys Cond Mat special issue on Granular Materials (M Nicodemi, Ed.

Osmotic Pressure of Solutions Containing Flexible Polymers Subject to an Annealed Molecular Weight Distribution

The osmotic pressure $P$ in equilibrium polymers (EP) in good solvent is investigated by means of a three dimensional off-lattice Monte Carlo simulation. Our results compare well with real space renormalisation group theory and the osmotic compressibility K \propto \phi \upd \phi/\upd P from recent light scattering study of systems of long worm-like micelles. We confirm the scaling predictions for EP based on traditional physics of quenched monodisperse polymers in the dilute and semidilute limit. Specifically, we find $P\propto \phi^{2.3}$ and, hence, $K \propto \phi^{-0.3}$ in the semidilute regime --- in agreement with both theory and experiment. At higher concentrations where the semidilute blobs become too small and hard-core interactions and packing effects become dominant, a much stronger increase % \log(P/\phi)\approx \log(\Nav^2/\phi) \propto \phi is evidenced and, consequently, the compressibility decreases much more rapidly with $\phi$ than predicted from semidilute polymer theory, but again in agreement with experiment.Comment: 7 pages, 4 figures, LATE

Glass transitions and shear thickening suspension rheology

We introduce a class of simple models for shear thickening and/ or `jamming' in colloidal suspensions. These are based on schematic mode coupling theory (MCT) of the glass transition, having a memory term that depends on a density variable, and on both the shear stress and the shear rate. (Tensorial aspects of the rheology, such as normal stresses, are ignored for simplicity.) We calculate steady-state flow curves and correlation functions. Depending on model parameters, we find a range of rheological behaviours, including `S-shaped' flow curves, indicating discontinuous shear thickening, and stress-induced transitions from a fluid to a nonergodic (jammed) state, showing zero flow rate in an interval of applied stress. The shear thickening and jamming scenarios that we explore appear broadly consistent with experiments on dense colloids close to the glass transition, despite the fact that we ignore hydrodynamic interactions. In particular, the jamming transition we propose is conceptually quite different from various hydrodynamic mechanisms of shear thickening in the literature, although the latter might remain pertinent at lower colloid densities. Our jammed state is a stress-induced glass, but its nonergodicity transitions have an analytical structure distinct from that of the conventional MCT glass transition.Comment: 33 pages; 19 figure