DOI 10.1140/epje/i2008-10432-2

Regular Article

Eur. Phys. J. E (2009) THE EUROPEANPHYSICAL JOURNAL E

Eective potentials of dissipative hard spheres in granular matter

R.A. Bordallo-Favela1, A. Ramrez-Sato1, C.A. Pacheco-Molina1, J.A. Perera-Burgos1,2, Y. Nahmad-Molinari1, andG. Perez2,a

1 Instituto de Fsica, Universidad Autonoma de San Luis Potos, Av. Manuel Nava 6, Zona Universitaria, 78290 San Luis Potos,San Luis Potos, Mexico

2 Departamento de Fsica Aplicada, CINVESTAV del IPN, A.P. 73 Cordemex, 97310 Merida, Yucatan, Mexico

Received 5 September 2008 and Received in nal form 5 December 2008c EDP Sciences / Societa` Italiana di Fisica / Springer-Verlag 2009

Abstract. We present an experimental study of the spatial correlations of a quasi-two-dimensional dissi-pative gas kept in a non-static steady state via vertical shaking. From high temporal resolution imageswe obtain the Pair Distribution Function (PDF) for granular species with dierent restitution coecients.Eective potentials for the interparticle interaction are extracted using the Ornstein-Zernike equation withthe Percus-Yevick closure. From both the PDFs and the corresponding eective potentials, we nd a clearincrease of the spatial correlation at contact with the decreasing values of the restitution coecient.

PACS. 45.70.-n Granular systems 61.20.Ne Structure of simple liquids

1 Introduction

Granular materials under external excitation resemble andhave been used as model systems for atomic, molecular,and colloidal ensembles in thermal equilibrium [1,2]. How-ever, in granular matter we cannot talk about macroscopicirreversibility coming from microscopic reversibility sincea granular material has intrinsically dissipative interac-tions even at its minimal scale (one single grain). The dis-sipative nature of granular matter means that non-staticassemblies need a permanent injection of energy, and thisenergy ow in the system gives rise in turn to the appear-ance of some types of self-organization, allowing for somevery interesting phenomena such as oscillons and otherwide variety of beautiful patterns, as elegantly shown bySwinney and coworkers [3].

On the other hand, condensed phases resembling liq-uids or solids the latter being in crystalline or glassyform have been reported in granular materials undershaking excitation. The evolution of these out-of-equili-brium systems towards a steady state with stationarystructures have been described [4], and its jammed-likesystem relaxation dynamics has been reported [5]. In par-ticular, crystallization, in the form of a quasi-static phasewith hexagonal order (and, at least in one instance, squareorder [6]), has been obtained in two dierent ways: eitherby increasing densities or by reducing the amplitude ofthe driving. In the rst case the process is basically equalto the crystallization of hard disks in equilibrium [7,8]:

a e-mail: gperez@mda.cinvestav.mx

as the granular density is increased the aggregate displaysa transition from uid to hexatic phases, and a posteriortransition to the (hexagonal) crystalline phase. This seemsto be then an instance of the maximization-of-entropy pro-cess that originates the crystallization of hard spheres anddisks [9,10] although in the case of ref. [7] a slightly con-cave lower plate helps the condensation process. On theother hand, crystallization has been also observed for val-ues of 1 [11,12], but at very low packing fractions( = 0.42 in the experiment vs. 0.71 for equilibriumcrystallization in a gas of hard disks. Particularly illumi-nating here are Figs. 1(e) and (f) of Ref. [11] and Fig. 1of Ref. [12]). There is no clear understanding of the con-densation mechanisms underneath the appearance of thisparticular crystalline phase, since the low density of thegranulate precludes the standard entropy-driven mecha-nisms. This crystallization is usually described in terms ofthe inelastic collapse phenomenon [13], but the actual pro-cess of their formation is not yet experimentally well mea-sured. Therefore, there may be some alternative or com-plementary mechanisms that need to be considered for thiscondensation phenomenon; for instance, it has been sug-gested that the Prigogines minimum entropy productionrate principle is the origin of the self-organization shownin the epitaxial granular crystallization process [14].

The techniques and theoretical frameworks developedfor colloidal particles are being translated and applied forgranular materials since almost ten years (torsion bal-ances, light scattering, etc.) [2,15,16]. Among the theo-retical tools developed for colloids, a fundamental one hasbeen the development, starting from more than 60 years

2 The European Physical Journal E

ago, of an interparticle potential able to describe the in-teraction between colloidal particles. This interaction isactually mediated by the solvent and the possible pres-ence of other molecules at several size scales. Part ofthis eort is the Dejaurmin, Landau, Vervey and Overvek(DLVO) [1719] theory. Here, the two main contributionstaken into account are the van der Walls forces and thepure Coulombic electrostatic forces. Besides these energy-based interactions, the presence of intermediate-size ele-ments in a colloidal dispersion elements smaller thanthe colloidal particles themselves, but much larger thanthe solvent molecules gives origin to subtle entropic ef-fects, known as depletion interactions; these were mod-eled originally by Asakura and Oosawa [20]. Furthermore,during the last decade hydrodynamic eects have beentracked experimentally [21], but it is yet not clear how torepresent them as a new term in the whole picture of thecolloid-colloid interaction.

This article presents an approach closely related tothe DLVO and excluded-volume-forces theories, since itintends to give a rst step towards the construction ofa granular eective interaction potential suitable for de-scribing the appearance of dierent granular phases in asteady-state driven granular system. Given the lack of acomprehensive theoretical framework in which a systemmade of many dissipative particles kept in a non-staticsteady state via the cyclic injection of energy could betreated satisfactorily, we try to draft from a phenomeno-logical approach a way to describe the elusive mechanismby means of which a driven granular system forms con-densed phases, treating the system as if it were in thermo-dynamic equilibrium and extracting from its Pair Distri-bution Function (PDF) an eective pair interaction poten-tial. In this scenario, it then becomes natural to attributethe condensation itself to the existence of this eective pairpotential, and this could be a rst step in the process ofpredicting under which excitation conditions frequency,amplitude, and type of excitation and which geometri-cal and particle parameters lling fraction, restitutionand friction coecients, etc. a condensed phase shallstart to grow.

2 Experimental procedure

The experimental setup is similar to the one used by Olaf-sen and Urbach [11,22]. We use ve sets of uniform mo-nodisperse spheres made of dierent materials, with di-ameters ranging from 4.0 to 5.8mm, chosen in order toget dierent restitution coecients. These were obtainedletting a sphere fall on top of a plate of the same mate-rial. The normal velocities just before and just after thecollision were measured, using a high-speed camera. Thedata for diameters and restitution coecients are givenin Table 1. Spheres of one given kind are placed betweentwo horizontal hexagonal Plexiglas plates, with the sideof the hexagon equal to 10 cm. The distance between thetwo plates is set in all cases to 1.6 times the diameter ofthe spheres inside. To avoid condensation of the granulatethe lower plate of the cell was made non-at by means of

Table 1. Diameters and restitution coecients e of the gran-ular species used in the experiment.

Species (mm) e

Lead 4.3 0.25

Painted steel 4.4 0.62

Painted glass 4.0 0.69

Steel 4.4 0.75

Plastic 5.8 0.81

Fig. 1. Top view of a section of the lower plate in the exper-imental set-up. The right upper corner shows an oblique view(with a relative enlargement factor of 3) of a few bumps.

small protrusions that give a horizontal momentum com-ponent to the spheres each time they are compressed bythe shaking cycle against this plate, assuming they arewithin the radius of inuence of a protrusion. These bumpsare made touching the Plexiglas with a hot metallic tip,have roughly the form of an elliptic crater, and have ap-proximate lateral size and height of 1.5mm and 0.3mm,respectively. There are on average close to 16 protrusionsper cm2 of this type in the lower plate, and their distribu-tion is quasi-random, such that the distance between anyof them and its closest neighbor is around 1 to 3 times thetypical bump size (see Fig. 1).

Cells with non-at bottoms have been used in previ-ous experimental set-ups, and their main purpose is toinject transverse momentum to the granulate. Some veryinteresting results have emerged from these experiments.In particular, Prevost et al. [23] have shown that the cor-relation between velocity components parallel to the linebetween particle centers is strongly negative for cells withat bottoms, but becomes positive for a cell where smallerspheres are glued on its lower plate. This means that,for a at bottom, spheres that undergo a collision havean average relative velocity larger after the collision thanbefore. (This is a consequence of the fact that, on aver-age, dissipative collisions with a at plate that does notmove horizontally reduce the horizontal linear momentumof a grain. Maintaining a stationary situation requires thenthat grain-grain collisions increase horizontal momentum).The opposite phenomenon happens over a rough bottom.

R.A. Bordallo-Favela et al.: Eective potentials of dissipative hard spheres in granular matter 3

Also, as mentioned in the introduction, Reis et al. [8,24]nd that, in the presence of a rough bottom (sand-blastedglass), the density-driven transition to condensation hasessentially the same characteristics of the solidication asa hard disk system. Finally, a rough moving bottom hasbeen implemented in one study, laying a close-packed layerof heavy dimers over the lower plate, with a gas of lighterspheres on top of this rst layer, and keeping smallenough so that there is no layer mixing [25,26]. Here onecan get a gas that shows molecular chaos, with velocitydistributions that approach the Maxwell-Boltzmann form.

Returning to the present work, the cell is sinusoidallyvibrated in the vertical direction at a frequency f of 60Hzby means of a loudspeaker. The working frequency waschosen at this value to avoid nodes and resonances in thesystem that could lead to an inhomogeneous distributionof particles through the plate. In this regime, a homoge-neous stationary distribution of particles appears togetherwith uniform measurements of the cell acceleration. Theamplitude of oscillations A for the cell was of 0.5mm,and kept constant for all the experiments, getting in thisway an adimensional acceleration = (2f)2A/g = 7.25.The number of spheres is adjusted so that the 2D pack-ing fraction = N2/4S is given by = 0.35 in almostall experiments and for all tested materials. Here N isthe number of spheres in the system, their diameter,and S the area of the experimental cell. In this way weavoid shifts in the measured PDF due to packing fractioneects. Still, some extra experiments were performed at = 0.15. Care was taken as well of maintaining the ratiobetween linear dimensions of the cell and particle diame-ters always larger than 50, and of keeping the explorationregion (the area digitally recorded in pictures) far awayfrom the walls (more than 20 diameters away), and largeenough to allow a fair representation of the main featuresof the correlation function for this packing fraction regime.A high-speed camera (Red-Lake Motion-Meter) operatingat 500 frames per second was placed above the system,and the cell was obliquely illuminated with diuse light.In this way, the spheres images were recorded just fromthe bright spot shining at their centers, making it easierfor the particle center localization algorithm to processthe frames.

Once under vibration, the system of particles resem-bled a conned colloidal system driven by Brownian mo-tion [27], since the motion of the spheres in the plane isconstantly aected by their collisions with the scatterersintroduced in the lower plate. In the above experimentalconditions the granular uid neither presented condensedphases nor stable clusters for any of the tested materi-als, and there is no evidence of trapping of the spheresby the bottom plate scatterers. Ten thousand frames perexperiment are analyzed for center particle location us-ing a software build on the IDL platform and provided bythe UASLP-Complex Fluids Laboratory; further calcula-tion of the PDF was performed on the same data analysisprogram. Under these experimental conditions, granularuids for the ve dierent kinds of particles already men-tioned were analyzed and their PDFs obtained are shownin Figure 2.

Fig. 2. Pair distribution functions at a 2D packing fraction = 0.35, obtained for ve materials with dierent restitutioncoecients. The gure shows a very large correlation at contactfor lead beads. The points are experimental data, the lines arecubic splines interpolations.

3 Experimental results: pair distributionfunctions and eective potentials

In equilibrium, the PDF is determined along with tem-perature and density by the interparticle potential, butthe relationship between these two quantities is not easilystated. In principle one can use the Ornstein-Zernike (OZ)equation

h(r12) = c(r12) +

d3r3 c(r13)h(r32),

where rij = |ri rj |, is the particle density, h(r) =g(r)1 is called the total correlation function, and c(r) isthe direct correlation function, a function that is expectedto have a simpler dependence on the potential than h(r),and that is actually dened by the OZ equation itself [28].The relation between h and c can be written in a muchsimpler form using Fourier transform, where one nds

h(k) =c(k)

1 + c(k). (1)

Still, in the absence of an independent relationship, thisequation serves only as a denition for c. For very dilutedsystems [29] a simple approach is to take the direct corre-lation function equal to the Mayer function

c(r) f(r) = exp[u(r)/kBT ] 1,and, in the limit of high T , one may simply take the rstterm of the expansion

c(r) u(r)/kBT.On the other hand, for hard spheres and, in general,short-range potentials, in either dense or diluted situa-tions a more convenient closure relationship is given by

4 The European Physical Journal E

Fig. 3. Eective interaction potentials (divided by kBT ) ob-tained by inverting the PDFs shown in Figure 1 by means ofthe OZ equation, using the PY closure. The more attractivepotential corresponds to lead, which has the lowest restitutioncoecient.

the Percus-Yevick (PY) approximation [30]

c(r) = [h(r) + 1][1 exp(u(r)/kBT )]. (2)With this approximation, one can then generate an ap-proximation to the quotient u(r)/kBT , once the g(r) and are given. This requires the evaluation of h(k), which in2D involves the zeroth-order Hankel transform of g(r)1,solving (1) for c(k), doing the inverse Hankel transform toget c(r), and nally solving (2) to get u/kBT .

PDFs are often measured for granular gases ina stationary state [2,11,22,31], being these in out-of-equilibrium but stationary ensembles. So, it then becomesan interesting proposal to associate an eective potentialwith these structural characteristics. In these cases thetemperature dependence can be either ignored (more ex-actly, it gets incorporated in the eective potential), orsome convenient denition of T as proportional to the av-erage kinetic energy is implemented. Following this idea,we have extracted the product ue from the experimen-tally obtained g(r), a product that from now on we will re-fer to simply as eective potential; the results are shownin Figure 3.

Two features are striking on this gure: rst, the veryemergence of eective potentials with just one well-denedwell, and second, the clear assignment of a deeper wellto the softest material, in this case lead. In general thedepth of these wells in ue increases as the restitutioncoecient lowers its value. One can conclude then thatsmaller restitution coecients softer materials giverise to stronger (attractive) eective potentials.

One should be aware, however, of some limitations inthe experiments: to begin with, the g(r) curves have con-tinuous derivative at r/ = 1, and look as if coming froma soft potential, due to the occasional partial overlap ofthe spheres in the experiment. This is because the heightof the cell is larger than the spheres diameter. Also, dueto experimental constrains the grains used were in some

Fig. 4. Eective interaction potentials for steel and leadbeads at a packing fraction = 0.15, showing attractive andhard-sphere behaviors.

cases of dierent diameters (see Tab. 1), and this aectsthe interaction with the non-at bottom plate (the samebottom plate was used in all trials). Still, a clear eectof the changing restitution coecient is evident in thegraphs.

A clear resemblance of the potentials with those cor-responding to a simple liquid and a hard-sphere uid, ob-tained for lead and steel spheres, respectively, in a morediluted system (packing fraction = 0.15), is presented inFigure 4. Here the well depth for the steel spheres assem-bly is much shallower (less attractive) than in the case ofthe denser system depicted in Figure 3, and the eectivepotential found seems to be due exclusively to excluded-volume eects. The results for the denser system show thatthe measured eective potentials account for dissipative,depletion and, very likely, for the base plate roughnesseects.

With respect to this last point, there is a possibilitythat the observed correlations in position, which give riseto the proposed eective potentials, may emerge not somuch from the dissipative nature of the collisions andfrom entropic eects, but from temporal gravity-assistedtrapping of the spheres an the valleys that form betweenbumps in the lower plate. We believe that this trappingeect represents a very minor, maybe even null, contribu-tion to the dynamics of the system, mostly because, at thevalue of used here, the grains can almost be consideredto be moving freely between collisions. That is, the im-pulse received from the contacts with the horizontal platesis so large as to make gravity into a minor correction.The particles y vertically and hit both horizontal plateswith similar momentum, making very unlikely for them tosit in any given valley. The collision with the lower plateis for all purposes instantaneous, and the grains cannotreally explore the landscape at the bottom. Notice alsothat with a plate separation between 6.9 and 9.3 mm thebumps at their maximum height ( 0.3mm) representonly a 3-4% reduction in the vertical space, meaning thatthe grains (with diameters from 4.3 to 5.8mm) have more

R.A. Bordallo-Favela et al.: Eective potentials of dissipative hard spheres in granular matter 5

Fig. 5. Correlation at contact (square marks, left verticalaxis), or well depth (triangular marks, right vertical axis) vs.restitution coecient of the grains. The lines are provided onlyas rough indicators of the behavior. With the exception of onepoint, both quantities evolve monotonously. From left to right,the materials are: lead, painted steel, painted glass, steel, andplastic.

than enough free space between the plates to move around.Finally, it is worth restating that Figure 4 indicates clearlythat the restitution coecient is the fundamental param-eter that originates non-trivial eective potentials.

Another important feature shown in Figure 4 is a shiftto the right of the potential minimum which appears atr/ = 1.33 for lead spheres at this lower packing frac-tion of = 0.15, in comparison with its correspondingdenser case of = 0.35 in which the minimum is located atr/ = 1.076. This shift is easily explained in terms of thereduction (increment) in the area available for vagrancy asthe packing fraction is increased (reduced). As the pack-ing fraction is increased the random walks performed bythe particles decrease their mean square displacement andtemporary cages develop while particles become increas-ingly trapped by their neighbors [5], shifting the rst peakof the PDF closer to the contact position.

A noticeable increase in the height of the rst peak inthe PDF (Fig. 2) and also in the well depth (Fig. 3) is de-veloped as the restitution coecient decreases. In Figure 5the rst peak height of the PDF or the well depth of thecorresponding potential are plotted as a function of therestitution coecient, showing for both quantities withexception of a single point a monotonous dependence.These monotonous behaviors can be expected intuitivelysince dissipation of energy during each collision event isan extra factor that should be taken into account in de-termining not only the relative velocity [23] (which showsasymmetric correlations), but the position between col-liding particles as well. Moreover, dissipation slows downthe entire dynamics of the system allowing for this high-temporal-resolution testing of the granular uid structureto capture more information of correlated positions beforea new uncorrelating event occur (next shaking cycle).

4 Discussion

This paper investigates the structural changes found instationary-state granular gas changes with respect to anordinary ideal gas in terms of a possible eective inter-action potential. This is accomplished within the frame-work of integral equations, using the PY closure, which isthe most appropriate for very short-range potentials. Thisclosure relation has been widely used for extracting the ef-fective pair potential in colloids and macromolecules, butup to our knowledge, it has never been used to get an eec-tive attraction in a granular gases, where the constituentelements are dissipative hard spheres. As shown in Fig-ures 2 and 3, there is an increase in the pair distributionat contact and a consequent appearance of a potentialwell close to the grains, a well whose depth increases asthe associated restitution coecient is lowered. This cor-relation increment signals a more ecient clustering formore dissipative ensembles. The analysis of the PDFs interms of eective potentials was partially inspired by theemergence of condensation (crystallization) in a moder-ately dense granular gas. However, the data accumulatedup to now is clearly insucient for an actual predictionof the condensation transition. At the moment we haveonly shown that eective potentials with attractive wellscan be extracted from the observed PDFs. Future workinvolving the analysis of the eective potential as a func-tion of and is needed to connect to the crystallizationtransition.

The system described here is not a simply cooling one,since energy is fed cyclically via vertical shaking. So thestationary state corresponds to a situation where web-likepatterns and their constituent clusters start to form be-tween consecutive excitation impulses, but are destroyed(randomized) with each shaking event. The granular gascools down in a characteristic time (relaxation time) thatdepends on the particle restitution coecient as well ason the number of collisions per unit time, and so we willbe more likely to capture cluster features for gases withsmaller relaxation times. The resulting preference of moredissipative particles to form clusters or to stay at closerand well determined distances can be interpreted as ifthe particle were within a potential well. In fact, if onetakes a pair of particles in a bidisperse hard-sphere sys-tem, pairing and cluster formation appear as a result ofthe momentum transfer (pressure) exerted by the smallsurrounding particles, and this eect is interpreted as com-ing from a potential (the Asakura-Oosawa potential). Thispotential comes actually from the tracing out of the kine-matics of the small particles, and therefore disappears formonodisperse systems (there is nothing to be traced out.)For dissipative collisions clustering appears even in themonodisperse case, giving as a result a new type of eec-tive potential. It now comes from the time averaging ofthe injection-dissipation cycle in energy. It should be un-derstood that in an actual experimental situation say,the one described here the eective potential will in-clude not only dissipative eects, but also any inuencethe rough bottom may have. This eect, as argued before,is expected to be small.

6 The European Physical Journal E

Most of the research eort done in quasi-2D granulargases has been oriented towards determining the charac-ter of the velocity distribution in order to extend the ki-netic theory of gases to these systems. The analysis of thestructure by means of spatial correlations and its depen-dence on the restitution coecient is quite independentof these velocity distributions, as shown by van Zon andMcKintosh [32] who state that spatial correlations playa minor role, if any, in P (v) (P (v) being here the ve-locity distribution function). The independence of thesetwo characteristics implies that a more complete descrip-tion of a granular gas requires also the analysis of spatialstructures and correlations.

As a nal comment, some recent experiments in gran-ular ows are worth noticing, in particular the formationof a granular water-bell when a granular jet impinges ona at surface [33], the formation of drops in a falling jetof grains both in air and in vacuum [34] and the pres-ence of capillary waves in a thick falling ow of grains [35],seem to imply the presence of a surface tension. The ideaof an eective potential in aggregates of granular mattercan be perhaps used as a principle on which to build theseeective surface tensions.

We thank nancial support from PROMEP under grantsUASLP-PTC-084 and P/CA-23 2007-24-25. We acknowledgethe invaluable support given by Dr. Jose Luis Arauz and hisgroup at the Complex Fluids Laboratory in UASLP.

References

1. H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys.68, 1259 (1996).

2. B.V.R. Tata, P.V. Rajamani, J. Chakrabarti, A. Nikolov,D.T. Wasan, Phys. Rev. Lett. 84, 3626 (2000).

3. C. Bizon, M.D. Shattuck, J.B. Swift, W.D. McCormick,H.L. Swinney, Phys. Rev. Lett. 80, 57 (1998).

4. B. Knight, E.R. Nowak, E. Ben-Naim, H.M. Jaeger, S.R.Nagel, Phys. Rev. E 57, 1971 (1998).

5. P.M. Reis, R.A. Ingale, M.D. Shattuck, Phys. Rev. Lett.98, 188301 (2007).

6. F. Vega-Reyes, J.S. Urbach, arXiv:0803.1158v1 (2008).7. J.S. Olafsen, J.S. Urbach, Phys. Rev. Lett. 95, 098002

(2005).8. P.M. Reis, R.A. Ingale, M.D. Shattuck, Phys. Rev. Lett.

96, 258001 (2006).

9. B.J. Alder, T.E. Wainwright, J. Chem. Phys. 27, 1208(1957).

10. B.J. Alder, T.E. Wainwright, Phys. Rev. 127, 359 (1962).11. J.S. Olafsen, J.S. Urbach, Phys. Rev. Lett. 81, 4369 (1998).12. X. Nie, E. Ben-Naim, S.Y. Chen, Europhys. Lett. 51, 679

(2000).13. S. McNamara, W.R. Young, Phys. Rev. E 53, 5089 (1996).14. Y. Nahmad-Molinari, J.C. Ruiz-Surez, Phys. Rev. Lett.

89, 264302 (2002).15. P.K. Dixon, D.J. Durian, Phys. Rev. Lett. 90, 184302

(2003).16. G. DAnna, P. Mayor, V. Loreto, F. Nori, Nature 424, 909

(2003).17. B.V. Derjaguin, L. Landau, Acta Physicochim. (URSS)

14, 633 (1941).18. J.E. Verwey, J.T.G. Overbeek, Theory of the Stability of

Lyophobic Colloids (Elsevier, Amsterdam, 1948).19. P.C. Hiemenz, M. Rajagopalan, Principles of Colloid and

Surface Chemestry, 3rd edition (Marcel Dekker, New York,1997).

20. S. Asakura, F. Oosawa, J. Chem. Phys. 22, 1255 (1954).21. A.E. Larsen, D.G. Grier, Nature 385, 230 (1997).22. J.S. Olafsen, J.S. Urbach, Phys. Rev. E 60, R2468 (1999).23. A. Prevost, D.A. Egolf, J.S. Urbach, Phys. Rev. Lett. 89,

084301 (2002).24. P.M. Reis, R.A. Ingale, M.D. Shattuck, Phys. Rev. E 75,

051311 (2007).25. G.W. Baxter, J.S. Olafsen, Nature 425, 680 (2003).26. G.W. Baxter, J.S. Olafsen, Phys. Rev. Lett. 99, 028001

(2007).27. A. Ramrez-Saito, M. Chavez-Paez, J. Santana-Solano,

J.L. Arauz-Lara, Phys. Rev. E 67, 050403 (2003).28. M.E. Fisher, J. Math. Phys. 5, 944 (1964).29. L. Verlet, Phys. Rev. 165, 201 (1968).30. The PY and other closures for the OZ equation are well

covered in many textbooks, see for instance: D.A. Mc-Quarrie, Statistical Mechanics (University Science Books,Sausalito, 2000); D.L. Goodstein, States of Matter (DoverPublications, New York, 1985).

31. A. Burdeau, P. Viot, arXiv:0710.3713v1 (2007).32. J.S. van Zon, F.C. MacKintosh, Phys. Rev. Lett. 93,

038001 (2004).33. X. Cheng, G. Varas, D. Citron, H.M. Jaeger, S.R. Nagel,

Phys. Rev. Lett. 99, 188001 (2007).34. M.E. Mobius, Phys. Rev. E 74, 051304 (2006).35. Y. Amarouchenei, J.-F. Boudet, H. Kellay, Phys. Rev.

Lett. 100, 218001 (2008).

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