Hernán A. Makse and Jorge Kurchan
Levich Institute and Physics Department, City College of New York,
New York, NY 10031, US
P.M.M.H., Ecole Supérieure de Physique et Chimie
Industrielles, 10 rue Vauquelin 75231 Paris, France.
Years ago Edwards proposed a thermodynamic description of dense slowly flowing granular matter [1,2,3] in which the grains (the `atoms' of the system) interact with inelastic forces and enduring contacts. In Edwards' ensemble-- one of the very few generalizations of standard statistical mechanics-- thermodynamic quantities are computed as flat averages over configurations where the grains are static or jammed, leading to a natural definition of configurational temperature. The approach is intriguing but is not justified from first principles and hence, in the absence of explicit tests of its validity, has not been widely accepted. Here, we perform a numerical experiment specially conceived to be reproducible in the laboratory, using a realistic model of slowly sheared granular matter. The results strongly support the thermodynamic picture. Considering particles of different sizes in a slowly sheared dense granular system, we extract an ``effective temperature'' from an Einstein relation connecting the diffusivity and the mobility of all particles. We then perform an explicit computation to show that the effective temperature measured from Einstein's relation and Edwards' configurational temperature coincide-- thus rendering Edwards' temperature potentially accessible to experiments.
Consider a `tracer' body of arbitrary shape immersed in a liquid in thermal equilibrium. As a consequence of the irregular bombardment by the particles of the surrounding liquid, the tracer performs a diffusive, fluctuating `Brownian' motion. The motion is unbiased, and for large times the average square of the displacement goes as < |x(t)-x(0)|^2 > = 2 D t, where is the diffusivity. On the other hand, if we pull gently on the tracer with a constant force , the liquid responds with a viscous, dissipative force. The averaged displacement after a large time is <[x(t)-x(0)]> = f \chi t, where is the mobility. Clearly, the same liquid molecules are responsible both for the fluctuations and for the dissipation. Although both and strongly depend on the shape and size of the tracer, they turn out to be always related by the Einstein relation (a form of the Fluctuation-Dissipation Theorem), where is the temperature of the liquid. Conversely, if in a fluid of unknown properties we find that several tracers (of different sizes, for instance) having different diffusivities and mobilities yield the same ratio , we may take this as a strong evidence for thermalisation at temperature .
The Einstein relation is strictly valid for an equilibrium thermal system. However, recent analytic schemes for out-of-equilibrium glassy systems have shown that the Einstein relation gives rise to a well-defined ``effective temperature'' which is different from the bath temperature and governs the heat flow and the slow components of fluctuations and responses of all observables [4,5]. The existence of an effective temperature with a thermodynamic meaning suggests a hidden form of `ergodicity' for the slow modes of relaxation, which turns out to be closely related to Edwards' statistical ideas [1] for systems undergoing slow compaction. Explicit checks of this approach have been made so far within the mean-field/mode-coupling models of the glass transition [5], as well as for schematic finite-dimensional models of glassy systems [6,7,8,9,10].
On the experimental side, recently Nowak et al. [11] studied in detail the density fluctuations in a vibrated granular material, and proposed that these fluctuations should reflect an underlying thermodynamics, much as they do in an ordinary thermal system. However, as far as the actual verification of the granular thermodynamics, the evidence they found was negative: we shall discuss below some possible reasons for this. On the other hand, although Edwards' thermodynamics was originally formulated for granular matter, it has never been tested in realistic models of granular materials, characterized by the fact that energy is supplied by external driving (via, for instance, shear) and dissipated by inelastic collisions and slippage between the grains -- a very different heat exchange mechanism from that of a thermal bath in a molecular system. In order to test the existence of an effective temperature with a thermodynamic meaning for dense slowly moving granular matter, we perform, with a realistic numerical model, a diffusion-mobility experiment in conditions that can be reproduced in the laboratory. In the model, deformable spherical grains interact with one another via non-linear elastic Hertz normal forces, non-linear elastic and path-dependent Mindlin transverse forces, dissipative viscous damping force terms proportional to the relative normal and angular velocities, and via sliding friction with friction coefficient . [see Methods (i)] [12,13,14].
We perform molecular dynamics (MD) simulations (also known as discrete element methods, DEM) [13,14,15,16] for a binary system of large and small spheres in a periodic 3D cell. We apply a gentle simple shear flow at constant volume in the -direction taking care that the gradient of the velocity along is uniform [see Methods (ii)]. We focus our study on the slow shear rate, where the system is always close to jamming, but moves just barely enough to avoid stick-slip motion [17]. After a transient of the order of the inverse shear rate, we start measuring the spontaneous and force-induced displacements along the -direction for two types of particles. To measure we apply an external small constant force ( times the mean value of the contact forces) to the large and small particles separately and monitor their displacements as a function of time, averaging over all the small and large particles, respectively. The results are shown in Fig. 1. We notice that the diffusivities and the mobilities are different for the two type of particles, as expected. However, when we draw the parametric plot of versus (Fig. 2) we find parallel straight lines for large time scales, implying an extended Einstein relation:
We also repeat the numerical experiment for a system of Hertz spheres without transverse forces (: experimental realizations of elastic spheres with viscous forces but without sliding friction are foams and compressed emulsions [18,19,20]) and find that is well-defined at long time scales for this case as well (see Fig. 1). Thus, our results suggest that the validity of an effective temperature for long-scale displacements (larger than a fraction of the particle size) holds in the presence of viscous forces between grains or even of a sliding threshold (Coulomb's law).
The existence of a single temperature is an instance of the zero-th law of thermodynamics, for which we find positive evidence here, at variance with the experimental result in Nowak et al [11]. Three possible reasons for the apparent violation of the `zero-th law' in their experiments are i) the effect of an unknown height-dependent pressure (as pointed out by authors), ii) a rather high tapping amplitude (there is only a well-defined in the high compaction limit) and iii) the fact that the density fluctuations considered are integrated over all frequencies, thus including also `fast' relaxations.
Next, we treat the question whether it is possible to relate the effective temperature obtained above to the thermodynamic construction of out-of-equilibrium systems proposed by Edwards. While in the Boltzmann-Gibbs construction of equilibrium statistical mechanics one assumes that the physical quantities are obtained as average over all possible configurations, Edwards ensemble consists of only the jammed (blocked or static) configurations at the appropriate energy and volume. The strong `ergodic' hypothesis is that all jammed configurations of given volume and energy can be taken to have equal statistical weights. This formulation leads to a configurational entropy , where is the number of jammed configurations, and the corresponding configurational temperature and compactivity .
In order to calculate and compare with the obtained we need to count the number of jammed configuration at a given energy and volume (for this calculation we concentrate in the case without tangential forces and sliding friction, in order to avoid path dependency which would lead to an ambiguity in the definition of jammed configurations--see below). Counting directly all jammed configurations is impossible, except for small systems. To do it in practice, we resort here to an indirect `auxiliary model' method [7] suitably modified to the case of deformable grains. It consists of computing, using any standard method (Monte Carlo, MD, etc.) the equilibrium properties of the granular system in the periodic cell with the modified partition function , where is the elastic compressional potential energy leading to the Hertz contact force, and , with the total contact force exerted on particle by its neighbours. Thus, the dynamical system studied previously via the Einstein relation is now augmented by a temperature setting the mean elastic energy per grain, plus an auxiliary temperature, , which relates to the (artificial) term which selects, in the limit , the configurations where the grains are jammed ( ).
One can show, although this may not be immediately obvious, that the following protocol yields the correct Edwards temperature at energy . One starts by equilibrating the system at high temperatures ( and ), and anneals slowly the value to zero while tuning so as to achieve a given final elastic potential energy . At the end of the procedure ( one reads directly Edwards' temperature as . This simulation is performed with the same system parameters as in the previous dynamical simulations within the shearing cell, but without viscous internal dissipation and external shear. We note that the procedure is a calculational trick, and it does not correspond to any real experimental protocol.
In Fig. 3 we plot the elastic compressional energy as a function of for three annealing protocols using different values of and the distribution of compressional elastic energies obtained during the shearing experiments superimposed. The -intercept of each annealing curve gives, as mentioned above, the value of at . Only when we set equal to (as obtained through the Einstein relation in the shearing cell), do we find that the final coincides, within the accuracy of the simulations, with the mean elastic compressional potential energy of the system under shear (see Fig. 3). This shows the agreement between Edwards' and the effective temperature .
We conclude with some remarks: i) Since the jammed configurations are the same whatever the inter-grain dissipation coefficient, Edwards' ensemble (and hence its temperature) are insensitive to viscous dissipation, as long as we are in the slow flow regime of interest in this study.
ii) On the contrary, tangential forces and sliding friction may or may not block certain configurations, depending on how they are accessed: the ensemble of jammed configurations is then ill-defined. We have not tried to construct a suitable ensemble in this case, but content ourselves with the observation that is also in this case independent of the particle size (Fig. 2)-- our results suggest that thermodynamic concepts still apply, but the relevant ensemble for frictional systems necessarily goes beyond Edwards' construction as it stands.
iii) We have tested the validity of the thermodynamics in an ideal homogeneous system with periodic boundary conditions by explicitly avoiding structural features of dense granular flows such as inhomogeneities and shear bands (by imposing a uniform shear rate), segregation of the species or long range order [21,22]. Thus, the experimental test of our computational results may be complicated by, for instance, the formation of shear bands which tend to be present in physical systems. Even though it remains to be seen whether the thermodynamic picture may account for these highly nonlinear and dissipative effects, our ideal system may prove to be useful in deriving constitutive relations to be used in macroscopic theories of slow granular flows.
To summarize: we have first
performed numerically a diffusion-mobility experiment with a dense
slowly sheared granular systems specially conceived to be a `dress rehearsal'
for the real laboratory one.
The independence of
on the
tracer' size provides a strong test for an underlying thermodynamics.
We have then independently computed the configurational temperature
based on the entropy of jammed configurations
and verified that, remarkably, it
coincides with
-- thus supporting
Edwards' statistical mechanical ideas.
This last step cannot be performed in
the laboratory, so the numerical simulation provides the missing link
between thermodynamic ideas and diffusion-mobility checks.
METHODS
(i) Microscopic model
In the simulations two grains in contact interact via a normal Hertz force, , and a tangential Mindlin force [12]. For two spherical grains with radii and : Here , the normal overlap is , where , are the positions of the grain centers. The normal force acts only in compression, when . The variable is defined such that the relative shear displacement between the two grain centers is . The prefactors and are defined in terms of the shear modulus and the Poisson's ratio of the material from which the grains are made (typically GPa and , for glass beads). When exceeds the Coulomb threshold, , the grains slide and , where is the friction coefficient between the spheres (typically ). We assume a distribution of grain radii in which mm for half the grains and mm for the other half. The observables are measured in reduced units: length in units of , force in units of , time in units of , where is the density of the particles.
(ii) MD simulations
We perform MD simulations for a system of particles. Our calculations begin with a numerical protocol designed to mimic the experimental procedure used to prepare dense packed granular materials [15]. The simulations begin with a gas of spherical particles located at random positions in a periodically repeated cubic cell of side . The system is then compressed and extended slowly until a specified value of the pressure and volume fraction-- above the random close packing fraction-- is achieved at static equilibrium. We then apply a gentle shear in the -direction at constant volume by moving the periodic images at the top and bottom of the cell with velocities , where is the shear rate (Lees-Edwards boundary conditions) and using a suitable modified set of equations (see Chapter 8 of [16] and [23]) to generate a linear uniform shear velocity profile along the direction. Periodic boundary conditions are enforced in the direction and in the direction of the flow. We checked that shear induced segregation is absent at the times scales of our simulations.
For small time scales (fast rearrangements, not seen in Fig. 2) the fluctuation-dissipation plot of Fig. 2 yields no evidence of a well-defined temperature. We expect this result since the fast motion of the grains depend on the microscopic interactions dominated by inelastic collisions between grains. This two-relaxation scenario is analogous to the one found for glasses, although in the case of glasses a well-defined temperature is also found for the fast rearrangements of the particles.
Acknowledgments.
Acknowledgment is made to The Petroleum Research Fund for support of this research.
Correspondence should be addressed to H. A. M. (e-mail: makse@mailaps.org).