## April 26, 2005

### WHAT WE HAVE ACCOMPLISHED SO FAR

AREAS DISCUSSED:
—Granular gases – new scaling ideas have emerged
—Constitutive relations
Global (Pouliquen, GDR Midi)
Local
—Nearly jammed flows
Is jamming transition set by a critical value of phi (filling fraction) alone?
What are the state variables?
Is the silo problem different from other shear flows?
Microstructure vs. flow?
Particle size vs. other length scales?
Flow as a function of stress?
—Isostatic packings
Better experiments?
The nearly isostatic case
—Thermodynamics

How can we predict when it will work?
Application to chute flows
—Pattern formation (ripples, etc.)
—Materials science issues

AREAS WE MIGHT EXPLORE:
—Ways to improve experiments?
Correlations between rotation and translation
—“Gas” flows
—Geophysical flows
—Vibrationally excited flows
—Nonlocal theories
—What is the continuum scale?
—Force chains
—Minimal models?
—Formulating precisely defined open questions and ways of addressing them?
—What is the defining core of granular physics?
The energy scale is not kT
Energy is not conserved
Jamming – arrest/rigidity
Attracting distributions?
—Connections between different areas of granular physics

### David Dean: following on from Nicodemi

Spin glasses as model with blocked states of well defined measure.

H=-\sumij Jij Si Sj
Blocked states defined as having no down-energy moves (analogy of mechanical stability). In this sense the Edwards measure i sdependent on the dynamical rules.

Nmetastable = exp(N s(e)) s=S/N e=E/N

Now introduce an analogue of tapping: flip every spin with probability p, 0 Relaxation: energy lowering flips only.

• Canonical Edwards measure: pa = exp(-b Ea) /Z

l* leads to weak form of Edwards measure:
p(E) = exp(-b E) Nms(E) /Z'

Focus on infinite range S-K model where all pairs are interating; Gaussian distribution of Jij each with variance 1/N.
Find Nms =exp(0.198 N).

Histogram energies observed in a dynamics:
pdyn(E) = C e-b E Nms(E),
where DD can explicitly test the claimed exponential factor by plotting the (log of the) ratio. A first numerical study is encouraging, even when Nms is as small as 400.

• The Very Weak Edwards Measure:
test pdyn(E,T)/pdyn(E,T') =? e-(1/T-1/T')E.

For meaningful results need to have overlap of the distributions (close E or small system). Can test large systems because Nms no longer required, and this works relatively well.

• Another approximate weak test is to compute average Nms(E) in an annealed calculation. Taking the logarithm gives a bound on the Edwards entropy. Epxected to be valid above a certin energy scale. Analytic result matches enumeration rather well.

There are tests where the weak Edwards measure fails but the very weak test works. (Tetris….).

Qestion (Gollub): what about homogeneity which does not apply to real granular problems? Ans: relaxational dynamics is not observed to be sensitive to teh order of attempted flip for large enough systems.

### M Nicodemi (Napoli): Statisitical Mechanics of Granular Media

• Ideas behind Edwards apporach
• Recent experiments
• mixing/segragation transitions
• Hydrodynamics and instabilities

Granular systems dissipative, non-thermal.
Edwards 89: does some statistical mechanics apply?
Noted relatively few control parameters.
Macrostates should correspond to many microstates. Are the statistical weights Boltzmann-like?

Nagel Chicago expts: irreversible and reversible consolidation controlled by shaking amplitude. System clearly history dependent. Irreversible curve does (very slowly) approach the reversible one when shaking for longer.

Rennes expts: time to reach stationary state \propto 1/(shaking amplitude).

D'Anna expt: time = f(Y); Y=(g \Gamma … )1/2.

Edwards approach.

• restrict attention to mechanically stable states
• withiin that constraint consider Gibbs distribution …. can now define a temperature through 1/T = dS/dE

Experimental tests:
– does Thermodynamics appliy? – limited nuber of relevant parameters
– do ensemeble averages match time averages?

Schematic models and dynamics (Nicodemi, Coniglio, Herrmann 1997):
Hard spheres on lattice with Monte Carlo dynamics
– simple sedimentation as relaxation
– shake as limited time of free floating MC dynamics.

Can map out energy as function of shaking amplitude but the results are sensitive to the shaking time. HOWEVER the energy fluctuations plotted vs energy gives a unique curve: the same result (in these terms) is thus obtained by different shaking methods. These results also match the corresponding thermal distribution.

• Expts on stationary fluctuations (Swinney)
– 'tap' granular pile by pulses of upflowing gas.
– observe volume fraction fluctuations as function of volume, but it remains to be shown that the same curve is obtain by different experimental shaking protocols.

• Nicodemi et al have begun to model such 'flow tapping' and in this case superposition is duly obtained from different protocols.

• Binary mixtures as a harder problem
binary mixture in lattice model: use mean height difference between species as a probe. This does NOT masterplot against energy, violating the simplest Edwards idea, and at a minimum requires two granular temperatures. There is no theory to generally predict how many parameters are required.

• Dynamical instabilities also violate the simplest Edwards hypothesis. Demixing patterns under horizontal shaking even with matched size.

• Mean field calculation (Tarzia et al 2004) -> \phi vs Tconfig exhibiting supercooling and a thermodynamic glass transition.

Concluding remarks

• Is thermodynamic description possible?
• Is Edwards' or any other stat mech approach possible?
• Seek expt tests of deeper predictions, such as demixing, eqn of state, Fluctuation-Dissipation relations
• Understanding teh nature of 'universlities' present in jamming systems more broadly.
• Role of hydrodynamics, dynamical instabiilties …
• connections with flowing systems

## April 25, 2005

### EXPERIMENTS ON SLOW GRANULAR SHEAR FLOW – Jerry Gollub

(Written for Powders and Grains 2005. In the seminar, I also mentioned experiments on clustering mediated by hydrodynamic interactions in a fluid, and on vibrated chiral particles.)

ABSTRACT: The structure of dense granular materials can evolve during shear, and this evolution can af-fect the flow properties. We describe high precision shear flow measurements using index-matched imaging in which the internal velocity fields are mapped out quantitatively, while the total volume and shear stress are measured simultaneously. Results include the development of locally ordered structure consisting of sliding hexagonally ordered layers, the response to different shearing protocols and boundary conditions, the internal velocity profiles and rheology, the origin of shear banding, the anomalous mobility due to shear reversal, and the effects of polydispersity.

1.INTRODUCTION

Slow or quasi-static granular flows are less well studied than rapid “gas” flows. In fact, there is not even a universally agreed definition of such flows. We describe an experimental investigation of quasi-static shear flows in an annular version of a plane Couette geometry. A novel feature of this work is that the internal flow profiles are measured. Since the work has been submitted or published in refereed journals, we give only a summary of the main results here, with references to the original work (Tsai et al. 2003, 2004, and 2005).

The most important result of this study is the ob-servation that slow shearing can lead to internal or-dering over a long period of time, to a state consist-ing of hexagonally close packed planes that slide over each other during shear. The ordering substan-tially changes the rheological behavior of the mate-rial, such as the shear force and the internal velocity profiles. Generally, the effect of ordering on dense flows has been noted in many numerical studies of hard sphere systems, as explained in our papers, but has not been included in rheological models.
We are concerned with flow in the regime of per-sistent contacts, where kinetic energy is far less than the elastic energy of the packing. There are many previous experiments on sheared dense systems, and full references are given in our papers.

2.APPARATUS
Spherical glass beads of diameter d=0.6 mm fill the annular channel formed by two concentric smooth stationary cylinders. They are driven by a floating upper boundary, which rotates at a constant speed while exerting a fixed normal load. The driving sur-face is roughened by a glued layer of beads, and we consider various boundary conditions at the lower surface. The interstitial space is filled with a fluo-rescent fluid whose refractive index is matched to that of the particles. Slices illuminated by a horizon-tal or vertical laser sheet yield images that highlight the particles, which can be tracked using an imaging system. Fluid can enter freely as the volume occu-pied by the particles varies, while the shear force and volume are monitored. Hydrodynamic forces are demonstrated to be negligible, although lubrica-tion effects do occur.

3.RESULTS
3.1. Ordering transition
Long term shearing of the largely monodisperse par-ticles leads to an ordering transition. We have also found recently that in a binary packing, segregation occurs followed by ordering of the segregated parti-cles.

At the ordering transition, the material self-organizes into a stack of hexagonally packed layers that slide over each other. The transition results in a step change in the granular volume of about 3%, and a 15% reduction of the shear force. The typical par-ticle speed far below the shearing surface declines at the transition, i.e. the ordering causes the lower lay-ers to be influenced less strongly by the shearing.
The timescale of the transition depends on the layer thickness, but is also somewhat stochastic. It occurs after a much longer time for thick layers, and more time is also required when the material is dry. In general, sufficient time is required that the slow-est particles (i.e. those farthest from the shearing surface) translate by several particle diameters.

When the packing is thin (say 6–14 layers) and a monolayer bottom boundary condition is used (far from the shearing surface), the ordering is imperfect for certain filling volumes, but nearly perfect for other volumes. The sample volume as a function of filling mass M is then the sum of a linear rise plus a periodic function of M. We refer to these phenom-ena as “layer quantization effects”, as they are re-lated to the proximity of M to the value required to produce an integral number of complete layers.

The ordering depends on the surface properties of the solid surface that is farthest from the shearing surface, i.e. at the bottom of the sheared layer. If this surface is covered by an ordered array of parti-cles, this induces ordering above after a sufficient interval. On the other hand, for an irregular bottom surface, either an ordered or disordered packing can occur as the final state, depending on the history of the sample and the shearing protocol. We say that the system has multiple final states. We also find that AC shear can nucleate the ordered state even if ordering is otherwise suppressed, provided that the packing is not too dense. The ordered state is al-ways stable once it occurs.

3.2. Velocity profiles and rheology
The velocity profile, normalized by the driving speed, is found to be the same for all driving speeds over a wide range. In general, the velocity profile falls off faster than exponentially with depth. It can be fitted by the exponential of a quadratic function, and it falls off more steeply in the ordered state than in the disordered state. We find that the profile is sensitive to the channel width. It falls off faster than the profile of a Newtonian fluid in the same geome-try. Furthermore, the velocity profile does not de-pend strongly on the particle size, even though one might have expected some scaling with this dimension.

The transverse (i.e. radial) structure of the veloc-ity profile changes dramatically at the ordering tran-sition. It is Poiseuille-like (parabolic) in the disor-dered state, and nearly flat in the ordered state, where entire layers move coherently. In the disor-dered state, sidewalls induce local order in planes parallel to the sidewalls.

3.3. Origin of shear banding
What is the origin of shear-banding, i.e. the sharp decline of the velocity profile from the sheared sur-face? In our experiments, the localization of shear can be understood by considerations of torque bal-ance on a layer of granular material extending from the top down to a depth z. The torque on the top and bottom of this layer differ somewhat, because of friction at the channel walls. The difference is then proportional to the rate of change of shear stress  with height, which may in turn be decomposed into the following product:

The stress is expected to be only weakly dependent on shear rate for a granular packing. Therefore, a rapid rate of decline of shear rate with height can be expected, to generate the torque necessary to balance wall friction. The shear confinement can therefore be interpreted as a consequence of the weak depend-ence of stress on shear rate in conjunction with sidewall friction.

3.4. Particle trajectories
Particle motion in the ordered state is highly coher-ent, and particles do not diffuse vertically to an ap-preciable extent. On the other hand, in the disor-dered state, there is significant vertical diffusion, with a root-mean-square displacement which varies roughly as the square root of the elapsed time.

Studies of particle trajectories during AC shear-ing show that particles move more rapidly during a short transient following the instant of reversal, be-cause internal stresses vanish at that point. We refer to this phenomenon as anomalous mobility.

4. DISCUSSION
This work leads to many interesting questions. Why is the crystalline state always stable once reached? What is the role of the nonlinearity of the velocity gradient? How does the response to shear for a polydisperse packing differ from that of monodis-perse packings. (For binary mixtures, we have noted an interesting combination of segregation fol-lowed by local ordering.) How can one incorporate local order into rheological theories of dense granu-lar flows? What is the role of the grain size?

These experiments suggest that the internal struc-ture of natural packings probably evolves over long times, and that the internal velocity profiles and rheology must slowly change as a result. The assumption of steady state shear is unlikely to be valid because of this evolution.

ACKNOWLEDGEMENTS
We appreciate the support of the U.S. National Sci-ence Foundation, Division of Materials Research, under Grant DMR-0405187 to Haverford College and DMR-0079909 to the University of Pennsyl-vania.

REFERENCES
Full references to related work by other authors are given in these papers:

Tsai, J.-C., Voth, G.A., and Gollub, J.P. 2003. Internal granu-lar dynamics, shear-induced crystallization, and compaction steps. Phys. Rev. Lett. 91:064301.

Tsai, J.-C., & Gollub, J.P. 2004. Slowly sheared dense granu-lar flows: Crystallization and non-unique final states. Phys. Rev. E. 70: 031303.

Tsai, J.-C. & Gollub, J.P. 2005. Shear banding and transient response of granular packings in an annular channel. Submitted to Phys. Rev. E.

## April 19, 2005

### J Krug: Mass transfer models for Sand ripples

##### Vortex ripples and wind ripples

Expts in annular confinement, sand + water driven by oscillatory rotation lead to sharp ripple patterns around their circumferential direction. Ingenious conical mirror for imaging.

Time series indicate increasing wavelength and amplitude, but apparently reach steady state. Not thought to coarsen indefinitely.

Changing drive amplitude:

Selected wavelength \propto drive amplitude; \lambda/L =4/3 approx;
raising amplitude drives it up but lowering does not.

There is a limited range of drive frequency which drives rippling.

Basic viortex ripple picture known since Ayrtou 1910. Separation vortex on lee side of ripple leads to upflow in the lee and pile building. Hydrodynamic simulations (e.g. KH Andersen PhD) reproduce this. The coupling between successive ripples depends non-trivially on ripple height compared to spacing.

Mass transfer model

• models evolution of fully developed ripples;
• mass transfer mainly governed by size of recipient

dmi/dt = 2f(mi)-f(mi+1)-f(m~i-1)

and relabel when mi -> 0.

Uniform pattern stable when f'(m)<0 , so a whole band of stable wavelenghts exits.

f(m) assumed to exhibit a maximum; this is validated by measurements (Arbus sp??). Fitting with different functions for the three f's appearing above does give near agreement as expected.

PRL 88 234302 (2002)

##### Wavelength selection for vortex ripples

Model by simulation does predicted a selected wavelength with insensitivity to initial conditions. Thus there exsists a mechanism of selection within the stable band.

Model can be viewed as a downhill dynamics, dxi/dt = -dV/dxi .
This suggests selection to minimise under fixed total 'lemgth'. This would amount to a Maxwell construction but (in the approximations to f tested) this does not compare well with the data.

Selection mechanism remains a puzzle.

##### Stochastic coarsenning of wind ripples

Wind ripples propagate (e.g. rightwards) at speed v(\lamda) \propto 1/ \lamda. This maps into a particle dynamics with extinction.

One sided: d\lambdai/dt=f(\lambdai)-f(\lambdai-1)
f(l) \propto 1/l.

This can to some extent be solved exactly, \lamda \propto ln(t) emerging due to stochastic merging (to some extent confirmed by data).

Some analogy with compartmented models for granular gases (D. van der Meer et al, JSTAT (2004) P04004), but in these cases the fluxes are governed by the source boxes rather than predominantly by the destination.

Form of tail for f leads to sublogarithmic coarsenning.

All fits into wider context of condensation in zero range processes.

##### More complete sand-transport models.

Symmetries and desiderata:

• conservation of mass;
• invariance with resepect to global height
• x <> -x but not h <> -h.
• uphill current
• slope selection

Hence suggest:
ht + hxx + hxxxx + (hx2)xx - (hx3)x = 0
but this gives unbounded coarsenning. \lambda \propto ln(t).
Some support for logarithmic coarsenning from simulations.

Conjecture (Krug): conserved 1-d models exhibit either unlimited coarsenning or unlimited steepenning, not stable periodic solutions.

## April 13, 2005

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

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) = rhoinfty - Delta rhoinfty / (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:

palpha = exp(-Valpha/ 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(V1 + V2) = S(V1) + S(V2) + 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 $(\alpham / \Gamma(m)) vm-1 e- \alpha v, where v is the difference between the actual cell volume and the minimal cell volume, and \alpha is determined by the average specific volume constraint. The `regularity factor' m, which is equal to <(v - )2> / 2)-1, is used to specify the state of the system, where <> are averages over the volume distribution. For thermodynamic structures, the regularity factor increases with increasing density, and it increases sharply across the freezing transition, in response to the onset of order. The regularity factor also distinguishes between the dense thermodynamic structures and dense random or annealed structures. For thermodynamic states near regular close packing, the simulations show that the thermodynamic entropy is equal to the Voronoi free-volume entropy defined as sfv = - kB D \int f(v) \log{[f(v)]} d v , where D is the dimension. The maximum-entropy formalism, when applied to the Voronoi free-volume entropy along with the total volume constraint, shows that structures of maximum free-volume entropy have an exponential distribution of$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.

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