*Week 3*

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## April 13, 2005

### David Head: The onset of rigidity in particulate systems

Writing about web page http://www.kitp.ucsb.edu/online/granular05/head/

The above link is to a pdf file of the talk slides, which should be available shortly on the KITP server.

The purpose of the talk was twofold; mostly to summarise and elucidate some recent evidence, mostly from numerical simulation, on the onset or rigidity in dissipative particulate systems. The phenomonelogy with constant volume, zero gravity simulations is of a critical volume (or volume fraction) at which the elastic moduli first become non-zero. The transition appears to be continuous – certainly the elastic moduli increase continuously from zero, and there is some recent evidence of a diverging length scale, but how far the analogy with continuous transitions in thermal systems can be taken is not yet clear. Further complications include friction and gravity, but this talk deliberately focussed on the simplest case – isotropic systems of frictionless elastic spheres.

A common term used to describe the transition is that of isostaticity, or marginal rigidity (or one of any number of other terms). This is usually classified by the mean coordination number z – simple constraint counting schemes predict dimension-dependent values at which rigidity should first occur. For frictionless spheres, this is simply 2d with d the dimension. This prediction is not exact – corrections such as the removal of rattlers should be considered – but appears to give a very good first approximation of the real value.

I then briefly (to avoid boring non-theoreticians) describe a simple mean field calculation to predict the scaling of various properties near the transition. The key ingredients are a focus on mechanical stability (rather than mechanical equilbrium, which is always trivially obeyed) and the dynamics of formation of the static state, which is treated in a very basic, energy minimisation framework. The overal picture is of systems relaxing to (or very near) the boundary of a stable region and become arrested there, allowing for exponents to be derived that agree well with experiments. Calculations of Schwarz et al. derive the same exponents in infinite dimension, suggesting these are the mean field exponents for this problem.

A manuscript relating to this work should be on cond-mat within a few weeks.

## April 12, 2005

### David Dean: Thermodynamic approach to driven dense granular media

Writing about web page http://qcd.th.u-psud.fr/page_perso/Barrat/dean.pdf

The web link above is to some notes on a course I gave at the IHP in Paris is February 2005 on analogies between granular media and spin systems. The notes below are a summary of the Introductory part of this course which I outined here in at the KITP on April 11th.

Certain experiments on mechanically perturbed dense granular media show a steady state regime which is history independent. Typically one shakes granular media, e.g. glass spheres, rigid rods in a tube by accelerating the system through a sine cycle. The strength of a tap is characterized by Gamma = a/g, where a is the peak acceleration in cycle. Generally the lighter you tap the more dense the system you obtain. The dynamics can be very slow and the compaction

seems to occur extremely slowly, a popular fit is

rho(t) = rho_{infty} - Delta rho_{infty} / (1 + B ln(1 + {t/tau})})

In these dry random media experiments rho varies between 0.55 (random loose packed) and 0.64 (random close packed).

1/ln(t) decay has various explanations/models- free volume arguments, car parking model, spin models with kinetic constraints – but is it really the case in these experiments, more recents

results favor stretched exponentials

In experiments one can measure density fluctuations about the mean steady state density and the power spectrum of these fluctuations. Also seen are memory effects and strong out of equilibrium behavior e.g the Kovacs effect on varying the tapping strength Gamma (c.f.temperature change experiments on glassy polymer systems).

Recent experiments tap by fluidizing a granular bed, tapping strength is controlled by flow rate Q. The system gets more dense a Q is reduced, this experiment explores less dense regimes than the dry experiments ! Why ? Density fluctuations are Gaussian but their width is non monotone with tapping strength Q.

In systems of dry hard rods a phase transition/crossover is seen to a nematically ordered state.

Question is there a thermodynamic approach for these systems in the steady state. Edwards hypothesis in the steady state the system will have a fixed number of macroscopic observables fixed on average i.e. volume and energy, in the steady state the distribution is that which maximizes the entropy while fixing these averages. If volume is the only relevant

quantity fixed on average:

p_{alpha} = exp(-V_{alpha}/ X) /Z

X = compactivity – Lagrange multiplier fixing the average volume per particle.

Edwards entropy

S(V) = ln(Number of blocked states of volume V)

Blocked state – configuration of stable mechanical equilibrium. As mechanical equilibrium is a local concept we expect that

S(V_{1} + V_{2}) = S(V_{1}) + S(V_{2}) + surface terms

it can possibly make sense to develop a thermodynamics along these lines.

We can test these ideas on spin glasses where the relaxational dynamics is single spin flip and the entropy of metastable or blocked states is easy to find numerically and in some cases analytically. Here the volume corresponds to the energy, tapping means reversing each spin with probability p (extensive manipulation). The energy gets lower the gentler you tap and many steady state and slow relaxation phenomena seen in granular systems are seen in these systems.

### E. Ben–Naim: Kinetic Theory of Granular Gases

Writing about web page http://cnls.lanl.gov/~ebn/talks/kitp.pdf

Granular flows conserve mass and momentum but do not conserve energy. Effective continuum theories of granular flows, typically Navier-Stokes equation with an energy sink, reflect that. Consequences of energy dissipation are manifest for granular gases where the velocity distributions are nonequilibrium, in contrast with elastic gases. Experimental observations widely report these deviations but the details vary depending on how granular matter is forced.

This talk presented basic consepts from kinetic theory of granular gases and the stretched exponential tails for forced and unforced gases where derived using extreme statistics analysis and the WKB approximation. It was also described how the energy of fast particles cascades to small scales and the consequences of this cascade process, velocity distributions with power-law tails where explained.

Questions raised and answered clarified sevaral points. For example:

— Statistics of interacting particle systems do not follow immediately from the law of large numbers.

— Thermal driving is a phenomenological assumption. It is very effective in modeling experiments, though.

— Maxwell's original derivation of equilibrium distribution relies on a particular potential but the result is generic: it follows from energy conservation.

A (hopefully) misprint free pdf version of the talk is available on my web site.

Eli Ben-naim

### V. Kumaran: Entropy, Voronoi free volume distributions and the disorder parameter.

The Voronoi free-volume distributions for hard disk and hard sphere fluids are well described by a two-parameter gamma distribution $(\alpha^{m}/ \Gamma(m)) v

^{m-1}e

^{- \alpha v}, where v is the difference between the actual cell volume and the minimal cell volume, and \alpha is determined by the average specific volume constraint. The `regularity factor' m, which is equal

to <(v -

^{2}> /

^{2})

^{-1}, is used to specify the state of the system, where <> are averages over the volume distribution. For thermodynamic structures, the regularity factor increases with increasing density, and it increases sharply across the freezing transition, in response to the onset of order. The regularity factor also distinguishes between the dense thermodynamic structures and dense

random or annealed structures. For thermodynamic states near regular close packing, the simulations show that the thermodynamic entropy is equal to the Voronoi free-volume entropy defined as s

_{fv}= - k

_{B}D \int f(v) \log{[f(v)]} d v , where D is the dimension. The maximum-entropy formalism, when applied to the Voronoi free-volume entropy along with the total volume constraint, shows that structures of maximum free-volume entropy have an exponential distribution of $v^\ast$. Simulations carried out using a swelling algorithm indicate that the dense random packed states approach the distribution predicted by the maximum entropy principle, indicating that the free-volume entropy is a possible disorder parameter for disordered states. Thus, for the hard disk and hard sphere systems, the present analysis suggests a relationship between the Voronoi free-volume (cellular) entropy and the thermodynamic entropy, and indicates that maximally random close-packed states maximise the Voronoi free-volume entropy.

## April 05, 2005

### Scheduled Events 11–15 April

Monday 11 April 10am: Discussion, KITP Small Seminar Room

*Entropy of volume distributions*

(A. Zippelius, D. Dean, V. Kumaran)

Tuesday 12 April 10am: Two talks + discussion, KITP Small Seminar Room

*Kinetic theory of Granular Gases*

E Ben-Naim

A Zippelius

Tuesday 12 April 2pm Founders Room

*Simulations of Granular Materials*

Informal Discussion (A Zippelius and students of J Carlson, J Langer)

Wed 13 April 10.30am: Talk + discussion, Small Seminar Room

*The onset of rigidity in simple particulate systems*

David Head

Thurs 14 April 10am: Programme Seminar, KITP Auditorium

JF Brady (Caltech)

*From Proteins to Peas: Diffusion Across Scales*

Thurs 14 April 2pm: Founders Room

Discussion on Correlated structures in Granular Gases

Led by T. Poeschel.

Fir 15 April 10am: Two talks + discussion, Small Seminar Room

*Free surface and confined shear flows: continuum modelling*

Daniel Lhuillier

*Constitutive equations from kinetic theory*

V. Kumaran

### John Brady: Seminar 14 April

#### From Proteins to Peas: Diffusion Across Scales

John F. Brady

Division of Chemistry and Chemical Engineering

California Institute of Technology

Pasadena, CA 91125

###### Abstract

Diffusion is one of the most basic and elemental transport processes and is responsible for the molecular mixing of different chemical species. For a protein molecule, the diffusivity is given by the familiar Stokes-Einstein formula relating the Brownian diffusivity to the thermal energy times the hydrodynamic mobility of the protein: D =kT/6 \pi eta a, where \eta is the viscosity of the solvent and a is the protein size. The Brownian self-diffusivity decreases as the concentration of protein molecules increases owing to the crowding effect of near neighbors. As the diffusing species increases in size from a protein to a several micron-sized colloidal particle, the stirring of the background fluid can give rise to another mechanism of transport – ‘shear-induced’ diffusion. Here, hydrodynamic interactions among particles promote mixing and the self-diffusivity now scales as \gamma a^{2} , where \gamma is the shear rate. In this regime, the self-diffusivity is an increasing function of concentration since particle-particle ‘collisions’ are responsible for the diffusion motion. At still large particle size (millimeter or larger), the inertia of the particles becomes important, direct particle-particle collisions dominate the transport as opposed to the stirring of the background fluid, and the self-diffusivity now behaves like that in a dense gas: D ~ a Tg^{1/2} , where Tg is the ‘granular temperature’, which is set by the stirring motion and the energy dissipated upon particle-particle collision. As in a dense gas, the self-diffusivity now decreases with increasing particle concentration. The physical origin of these various behaviors and their implications for mixing and concentration distributions in flows will be discussed.