Stephan Herminghaus

Granular matter and irreversibility

stephan.herminghaus at ds.mpg.de

(+49) 551-5176-200
(+49) 551-5176-202

2.77

 

Director

Wet granular matter and irreversibility

Introduction

Landslide in La Conchita, CA
Fig. 1: Massive landslide at La Conchita, California, which occurred in spring 1995 after heavy rain. Fortunately, no one was killed or injured (courtesy of U.S. Geological Survey [2]).

Judging from the ease at which it quietly runs through the orifice of an hourglass, or forms wandering dunes in the wind like waves on the ocean, dry sand may be legitimately called a fluid. It will not maintain any shape imposed on it, and tends to form a horizontal boundary to the air above, very much resembling regular liquids, like water. However, when some of the latter is mixed into the dry sand, a pasty material emerges, which can be sculptured into quite stable structures, such as sand castles. It is this pasty material, which may be seen as a particular type of 'soft matter', on which we focus in this project [1].

As an example, let us consider a land slide in wet soil, such as the one depicted as a bird's eye view below (Fig. 1). First of all, we see that the spontaneous liquefaction of the hill slope seems to be a threshold phenomenon. Some part of the slope has massively moved downwards, whereas in the adjacent parts, where conditions were certainly similar, nothing happened at all. Furthermore, it is instructive to study the remains of the trail, which is clearly visible as a straight line to the left and right of the landslide region. On the slide region itself, it has been strongly deformed into a shape similar to a parabola. This shows that the slope has not come down as a more or less solid piece, sliding on a fault zone deeper inside the hill, but that it rather has liquefied, and crept down like a viscous fluid. Although some individual blocks of soil are still apparent, the overall liquid-like behavior is obvious. It is tempting to suspect some kind of a discontinuous phase transition (from a solid to a liquid state) as the underlying phenomenon. These questions are far from settled presently.

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Model systems

Immersion
Fig. 2: Closeup view into a model wet granulate, consisting of spherical glass beads (300 microns in diameter) and water as the wetting liquid. The latter is clearly seen to accumulate in little rings at the points of contact of adjacent grains. The space in between the spheres has been filled by an immersion liquid (mixture of toluene and di-iodomethane) in order to minimize refraction losses.

As we are doing basic research, we are focused on model systems of wet granular matter. These should be as simple as possible in order to make theoretical descriptions accessible, but just as complex as necessary to reproduce the key features of wet granulates observed in 'real' systems. In order to develop such a model, we have to dwell shortly on the basic forces which are active in wet granular systems.

If some liquid is added to a granular pile and allowed to distribute in its interior, it will preferentially gather around the points of contact between mutually adjacent grains. This is demonstrated in Fig. 2, which shows a closeup on the interior of a model granulate, with spherical glass beads as the 'grains' and some immersion liquid replacing the interstitial air for better view. A few percent of water are added which wet the glass and form little capillary bridges where adjacent grains touch. They exert an attractive force upon the grains by virtue of the surface tension of the liquid. This is expected to alter the mechanical properties of the pile considerably. In particular, we may expect three possible mechanisms. First, if adjacent grains are pushed harder together by the capillary force, the sticking friction between them is increased, and thus the strength of the pile. Second, if one attempt to shear the pile, liquid must be redistributed, which causes dissipation due to the viscosity of the liquid.

The third mechanism is conceptually more subtle, but turns out to be dominant in many cases: In order for a capillary bridge to form, adjacent grains must come into real contact, i.e., the distance of their surfaces, s, must become equal to zero. However, in order for this bridge to rupture, this distance must exceed a critical value, sc, which depends upon the amount of liquid in the bridge, and may be of the same order of magnitude as the size of the grains [3]. Hence the attractive capillary force between adjacent grains is intrinsically hysteretic. Repeated formation and rupture of such bridges therefore results in dissipation, and thus has a noticeable impact on the mechanical properties of the granular material.

An important question in the physics of wet granular matter is thus: Is the effect of interstitial liquid upon the material properties of the granulate mainly due to increased friction between grains, to the viscosity of the liquid, or to the hysteretic nature of the capillary force? It is to be expected that the answer to this question may depend on the particular situation, on the properties of the liquid, the size and shape of the grains, and whether static or dynamic properties are considered. A promising way to study the role of viscosity effects is to investigate a model granulate which is wetted by liquid Helium, and search for changes in the behaviour of such 'quantum paste' at the superfluid transition (which occurs at 2.17 K). These experiments are currently performed in our lab.
 

Fig. 3: Schematic representation of the force hysteresis in the minimal capillary model (MCM). Positive forces are attractive, negative forces are repulsive.

We choose to start from a model which concentrates on the hysteretic nature of the interaction force mediated by the interstitial liquid. We study how much of the observed phenomena we are able to describe successfully with this model, and refine it wherever appropriate. The simplest way to describe a hysteretic interaction force (cf. Fig. 3) is to assume that the force is strictly zero when two grains approach, that it jumps instantaneously upon contact to a well defined (attractive) value f0, and that it remains constant during recess, until s reaches the critical vaule sc, at which the force jumps instantaneously to zero again. In addition, we assume a hard core repulsion which sets in at contact. This model will be called the minimal capillary model (MCM) below. It turns out to account for many of the characteristic mechanical properties of wet granulates.

As mentioned above, the hysteretic interaction force, we introduce dissipation to the system, since an energy given by f0 x sc is lost whenever a bridge ruptures. Furthermore, the hysteresis breaks the time inversion symmetry in the dynamics of the system. The latter aspect is of particular interest with respect to non-equilibrium statistical physics, since many modern concepts to describe irreversible processes, like the recently developed so-called fluctuation theorems [4-6], explicitly assume time reversibility in the dynamics.

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Wet granular matter in-silico: external shear stress

Fig. 4: Illustration of the simulated model granulate. Periodic boundary conditions extend in all three dimensions. The pile is subject to a spatially varying gravitational acceleration field which exerts a shear stress.

Let us test the MCM by means of a simple computer simulation [7]. We subject a pile of spheres (a few hundred, with periodic boundary conditions) to a shear field and observe the induced reaction of the pile. Fig. 4 shows the system, into which we have introduced a polydispersity in the sphere radii of more than 10% in order to prevent crystallization. If fluidization occurs, we expect that some kind of diffusive behaviour sets in, with the mean square displacement of individual spheres scaling linearly with the simulation time.

The mean square displacement, averaged over the full ensemble, is shown in Fig. 5 for four different values of the applied shear field, F. The latter is measured in units of the capillary bridge force. Indeed we find diffusive behavior, as it is expected for a fluid material. The only exception is found for F = 0.412, for which we find repeated quiescent periods. Furthermore, we see that the diffusion constant, D, which corresponds to the slope of the lines shown in the figure, varies in a strongly non-linear way with the applied shear field.

Fig. 5: Ensemble average of the mean square displacement of the spheres, as a function of simulation time. Straight lines correspond to diffusive behavior in this plot.
Fig. 6: The diffusivity as a function of the applied shear stress. There is clearly a sharp transition from solid to liquid-like behavior, with a critical exponent below unity. The inset shows the critical shear for varying pinch off distance. From the MCM, one would indeed expect a monotonous increase.

It is instructive to plot the diffusion constant as a function of F. This is presented in Fig. 6. Clearly, there is a threshold in the applied shear below which no fluidization takes place. Fluidization sets in at F = 0.412, where the system is found to exhibit strong avalanching as well as occasional quiescence, as might be expected for a system close to criticality.

We see that our model system exhibits at least one very prominent feature of wet granulates: It behaves like a solid for a wide range of applied shear, and abruptly fluidizes when a critical shear stress is reached. Furthermore, it turns out that the critical exponent of D(F), which is 1/2 as indicated by the solid curve in Fig. 6, can be well understood within a simple mean field picture which directly employs the MCM [1,8].

Fluidization by vertical agitation

Fig. 7: The system simulated for the vertical agitation studies. The pink rings indicate the liquid bridges.

In addition to subjecting the granulate to external shear stress, which models the gravitational stress upon a hill slope, it is also common to fluidize granular matter by means of vertically agitating its container in an oscillatory fashion. As soon as the peak acceleration of the container exceeds the acceleration of gravity, there are free-flight phases in the granular motion with successive impacts on the container bottom, which tends to fluidize the whole pile [9]. We have empolyed the MCM to simulate this situation and see whether we find similar behaviour as in experiments [8,10]. The system is shown in Fig. 7. The capillary bridges are indicated by the pink rings which occur occasionally between adjacent spheres. The polydispersity has been chosen to be about 10% (full width of radius distribution) in order to match the characteristics of typical glass bead samples used in experiments. One clearly sees that the pile still crystallizes under these conditions.

Quite generally, we found that fluidization sets in when the peak acceleration is somewhat above one g. The actual threshold increases monotonically with the liquid content, i.e., with sc, as one finds in experiments as well [8,10]. Furthermore, we find again that the motion in the fluidized state is diffusive [11]. As in experiments, we observed that the pile melts from above, and the fluidization gradually extends downwards as the acceleration is increased.

Leidenfrost
Fig. 8: Simulation of a wet model granulate at high acceleration (35 g). Left: the initial state. Center: the early stage, in which a hot granular gas fills a large fraction of the sample volume. Right: after some time, a fluid plug develops which hovers above a gaseous layer, very much like in the well known Leidenfrost phenomenon.

Interestingly, we also find a Leidenfrost phenomenon above a certain threshold acceleration, which is characterized by a compact plug hovering above a 'hot' granular gas layer. This is shown in Fig. 8, which has been obtained with f0 equal to ten times the weight of a single sphere, sc being one tenth of a sphere radius, and a peak acceleration of 35 g. The three pictures correspond to different times, increasing from left to right, starting at t=0. After some time, the plug is fully developed, as seen in the rightmost image. A Leidenfrost phenomenon has very recently been reported for dry granulates, with the hovering plug being in a solid phase. In our case, the plug is clearly fluidized, pointing to a different mechanism here.

Fig. 9: Phase diagram for vertical agitation, as derived from the MCM.

If we create a straightforward mean field model on the basis of the MCM, we can derive analytically a phase diagram for the vertically agitated system which distinguishes the various phases as solid, fluidized and gaseous. This is shown in Fig. 9, where h denotes the height within the pile, normalized with respect to its total height, and Γ the normalized external agitation. Regions I and II indicate the solid phase, III and IV the fluidized phase, and V and VI the hot granular gas. Clearly, both features reported above are reproduced: the melting from the free surface (between ΓM and Γ0) and the Leidenfrost effect (between Γ0 and ΓF), which is characterized by a fluidized plug above a gaseous layer.

Aspects of irreversibility

The physics of non-equilibrium and irreversibility has made considerable progress recently with the developments of so-called fluctuation theorems [4-6]. These concern the spread of the distribution in the total entropy production which is found when a system is carried along a certain path in parameter space many times. It has been shown that this directly relates to the probability that a system decreases (temporarily) the total entropy instead of increasing it. Many papers have been published in recent years which showed that the entropy production in (dry) granular systems is very much in accordance with what is predicted by fluctuation theorems [12].

However, this is contrary to what one would expect, since the fluctuation theorems explicitly use a time-reversible microscopic dynamics, while granular systems are not time reversible. In particular, time inversion in a dry granulate is equivalent to replacing the restitution coefficient, ε, by 1/ε. In a wet granulate, in particular as modeled by the MCM, time inversion is manifestly broken by the hysteresis in the capillary bridge formation. Each bridge rupture adds one bit of information entropy (since it is just one on-off information!), and is intrinsically non-reversible. It is one of the advantages of the MCM that it allows to follow the dissipation process very closely, since its phase space trajectory is continuous and Hamiltonian (it just crosses from one leaf of the Hamiltonian to the next). A veritable task is to lift the conditions of the fluctuation theorems such as to include naturally also the granular systems (and, in particular, the MCM), and thus to account for the observed validity of the fluctuation theorems for granular systems. This is one of our main concerns in this project [13].

A paradigm situation to study dissipative gases is to start with a finite granular temperature, T, and just let the system cool down by the dissipative impacts between the grains. In a dry granulate, where the dissipativeness of the impacts is modeled by taking a certain percentage of the impact energy (1-ε²) out of the motion of the impact partners, free cooling leads to the well known Haff's law, which predicts an algebraic decay of the granular temperature [14]

Planetestimals
Fig. 10: The sticky gas limit is taken as a model for the formation of planetesimals from interstellar dust.

This describes well how the sand in a heavily shaken contanier cools down after the agitation has terminated, as long as the density stays rather uniform. However, granular gases tend to cluster, and this finally precludes Haff's law to be valid for a reasonably extended period of time. There is a completely different context in which the cooling and clustering of dissipative gases play a role, namely the formation of planetesimals and primordial clouds (cf. Fig. 10). These systems represent the limiting case of ε = 0, i.e., grains which touch will be glued together and continue their voyage as a single body. This is the so-called sticky gas, for which there are well known asymptotic scaling laws as well [15]. This is of great importance for modeling the mass distribution in such clouds and its temporal evolution.

Fig. 11: Free cooling of the wet granulate (solid curve) and the sticky gas (dashed curve) for comparison. The ordinate represents the number of clusters in the system. The peak corresponds to a sudden re-evaporation, quite to the contrary of what one expects at first glance. The results are from one-dimensional simulations of the isochoric ensemble.

Interestingly, the MCM is somewhat intermediate between these two cases. Here, we have a certain fixed amount of energy Eloss which is taken out upon impact. As Eloss approaches infinity, we obtain again the sticky gas. The MCM does not only add new structure, such as condensation in the isobaric ensemble when the granular temperature is close to Eloss, but also leads to novel phenomena in free cooling. In Fig. 11, we show the average number of clusters in isochoric free cooling of a wet granulate as modeled by the MCM, as a function of time (solid curve). The free cooling of the sticky gas is shown as the dashed curve for comparison. Clearly, there is an unexpected increase in the number of clusters for the wet gas, which occurs when the temperature has cooled to Eloss. Thus we observe a sudden precipitation of granular droplets out of the hot granular gas in the isochoric ensemble, while there is monotonic clustering in the isobaric ensemble at the same energy. This is another example for the sometimes unexpected features of wet granular matter, providing a wide field of research which has just begun to be explored.

Literature

[1] S. Herminghaus:
     "Dynamics of wet granular matter”
      Advances in Physics 54 (2005) 221
      [Journal URL]

[2] E. C. Spiker and P. L. Gori:
     "National Landslides Mitigation Strategy"
      (U.S. Dept. of the Interior, Reston, Virginia, 2003)

[3] C. D. Willett, M. J. Adams, S. A. Johnson, and J. P. K. Seville:
     "Capillary Bridges between Two Spherical Bodies”
      Langmuir 16 (2000) 9396
      [Journal URL]

[4] C. Jarzynski:
      "Nonequilibrium Equality for Free Energy Differences"
       Phys. Rev. Lett. (78) (1997) 2690
       [Journal URL]

[5] D. J. Evans, E. G. D. Cohen, and G. P. Morriss:
      "Probability of second law violations in shearing steady states"
      Phys. Ref. Lett. 71 (1971) 2401
      [Journal URL]

[6] U. Seifert:
      "Entropy Production along a Stochastic Trajectory and an Integral Fluctuation Theorem"
      Phys. Ref. Lett. 95 (2005) 040602
      [Journal URL]

[7] M. Schulz, B. M. Schulz, and S. Herminghaus:
      "Shear-induced solid-fluid transition in a wet granular medium"
      Phys. Ref. E 67 (2003) 052301
      [Journal URL]

[8] Z. Fournier, D. Geromichalos, S. Herminghaus, M. M. Kohonen, F. Mugele, M.
      Scheel, M. Schulz, B. Schulz, Ch. Schier, R. Seemann, and A. Skudelny:
      "Mechanical properties of wet granular materials";
      J. Phys.: Condens. Matter 17 (2005) S477
      [Journal URL]

[9] J. Duran:
      "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials"
      Springer-Verlag, New York, (2000), ISBN: 9780387986562

[10] M. Scheel, D. Geromichalos, and S. Herminghaus:
       "Wet granular matter under vertical agitation"
       J. Physics.: Condensed Matter 16 (2004) S4213
       [Journal URL]

[11] K. Roeller:
       "Diploma Thesis"
       Ulm (2005)

[12] K. Feitosa and N. Menon:
       "Fluidized Granular Medium as an Instance of the Fluctuation Theorem"
       Phys. Rev. Lett. 92 (2004) 164301
       [Journal URL]

[13] A. Fingerle, S. Herminghaus, and V. Zaburdaev:
       "Kolmogorov-Sinai Entropy of the Dilute Wet Granular Gas"
       Phys. Rev. Lett. 95 (2005) 198001
       [Journal URL]

[14] N. V. Brilliantov and T. Pöschel:
       "Kinetic Theory of Granular Gases"
       Oxford University Press (2004)

[15] L. Frachebourg:
       "Exact Solution of the One-Dimensional Ballistic Aggregation"
       Phys. Rev. Lett. 82 (1999) 15502
       [Journal URL]

[16] K. Roeller, S. Herminghaus und A. Hager-Fingerle:
       „Sinusoidal shaking in event-driven simulations“
       Comp. Physics Communications 183 (2012) 251
       [Journal URL]

[17] K. Roeller und S. Herminghaus:
       „Solid-fluid transition and surface melting in wet granular matter“
       EPL 96 (2011) 26003
       [Journal URL]

[18] K. Roeller, J. P. D. Clewett, R. M. Bowley, S. Herminghaus and M. R. Swift:
       “Liquid-Gas Phase Separation in Confined Vibrated Dry Granular Matter”
       Phys Rev Lett 107 (2011) 048002
       [Journal URL]

[19] S. Khan, A. Steinberger, R. Seemann and S. Herminghaus:
       "Wet granular walkers and climbers”
       New Journal of Physics 13 (2011) 053041
       [Journal URL]

[20] S. H. Ebrahimnazhad Rahbari, J. Vollmer, S. Herminghaus and M. Brinkmann:
       "Fluidization of wet granulates under shear”
       Phys. Rev. E 82 (2010) 061305
       [Journal URL]

[21] K. Huang, K. Röller and S. Herminghaus:
       "Universal and non-universal aspects of wet granular matter under vertical vibrations"
       Eur. Phys. J. Special Topics 179 (2009) 25
       [Journal URL]

[22] K. Roeller, J. Vollmer and S. Herminghaus:
       "Unstable Kolmogorov flow in granular matter"
       Chaos 19 (2009) 041106
       [Journal URL]

[23] S. Ulrich, T. Aspelmeier, A. Zippelius, K. Roeller, A. Fingerle, and S. Herminghaus:
       "Dilute wet granular particles: Nonequilibrium dynamics and structure formation”
       Phys. Rev. E 80 (2009) 031306
       [Journal URL]

[24] S. H. Ebrahimnazhad Rahbari, J. Vollmer, S. Herminghaus, and M. Brinkmann:
       "A response function perspective on yielding of wet granular matter”
       EPL 87 (2009) 14002
       [Journal URL]

[25] S. Ulrich, T. Aspelmeier, K. Roeller, A. Fingerle, S. Herminghaus, and A. Zippelius:
       "Cooling and Aggregation in Wet Granulates"
        Phys. Rev. Lett. 102 (2009) 148002
       [Journal URL]

[26] K. Huang, M. Sohaili, M. Schröter, and S. Herminghaus:
       "Fluidization of Granular Media Wetted by Liquid"
       Phys. Rev. E 79 (2009) 010301
       [Journal URL]

[27] M. Scheel, R. Seemann, M. Brinkmann, M. Di Michiel, A. Sheppard, and S. Herminghaus:
       “Liquid distribution and cohesion in wet granular assemblies beyond the capillary bridge regime”
       J. Phys.: Condens. Matter 20 (2008) 494236
       [Journal URL]

[28] A. Fingerle, K. Roeller, K. Huang and, S. Herminghaus:
       “Phase transitions far from equilibrium in wet granular matter”
       New Journal of Physics 10 (2008) 053020
       [Journal URL]

[29] M. Scheel, R. Seemann, M. Brinkmann, M. Di Michiel, A. Sheppard, B. Breidenbach, and S. Herminghaus:
        "Morphological Clues to Wet Granular Pile Stability”
       Nature Materials 7 (2008) 189
       [Journal URL]

[30] A. Fingerle and S. Herminghaus:
        "Equation of State of Wet Granular Matter”
        Phys. Rev. E 77 (2008) 011306
       [Journal URL]