1 Dyamical approach to weakly dissipative graular collisios Italo Ivo Lima Dias Pito ad Alexadre Rosas Departameto de Física, CCEN, Uiversidade Federal da Paraíba, Caixa Postal 58, , João Pessoa, Brazil Katja Lideberg Departmet of Chemistry ad Biochemistry ad BioBircuits Istitute, Uiversity of Califoria Sa Diego, La Jolla, Califoria , USA (Dated: Jue 3, 25) Graular systems preset surprisigly complicated dyamics. I particular, oliear iteractios ad eergy dissipatio play importat roles i these dyamics. Usually (but admittedly ot always), costat coefficiets of restitutio are itroduced pheomeologically to accout for eergy dissipatio whe grais collide. The collisios are assumed to be istataeous ad to coserve mometum. Here, we itroduce the dissipatio through a viscous (velocity depedet) term i the equatios of motio for two collidig grais. Usig a first order approximatio, we solve the equatios of motio i the low viscosity regime. This approach allows us to calculate the collisio time, the fial velocity of each grai, ad a coefficiet of restitutio that depeds o the relative velocity of the grais. We compare our aalytic results with those obtaied by umerical itegratio of the equatios of motio ad with exact oes obtaied by other methods for some geometries. I. INTRODUCTION The characterizatio of graular matter is extremely broad, ad icludes essetially ay coglomeratio of discrete macroscopic particles. These ca be as small as grais of sad ad as large as asteroids, they ca be i a codesed or gas-like phase. The codesed phases may exhibit characteristics of solids or fluids or gases or various combiatios thereof. Graular matter is importat i more idustrial applicatios tha ca be listed here, ad exhibits a huge variety of iterestig behaviors that have provided food for thought over ceturies of time. Behaviors such as the so-called jammig trasitio ad the formatio of patters are frequet subjects of curret research, as is the propagatio of eergy i graular materials. A feature commo to graular matter is the fact that eergy is lost every time grais collide. Ideed, the grais have to be very hard ad difficult to compress for a collisio ot to lead to a loss of eergy. Yet it is usually the case that mometum is coserved i these ielastic collisios. The coservative limit, where oly elastic collisios are ivolved, is famously illustrated by Newto s cradle, cosistig of a row of very hard balls that just touch, each hagig o a strig of the same legth attached to a commo support. Whe the ball at oe ed is picked up ad released so that it collides with the ext ball, the eergy passes dow the row util the last ball flies up to the same height as the first ball before it was released (the other balls remai at rest). The last ball the flies back, the eergy is trasferred across the row agai, ad the first ball flies up to the same height [ 3]. This cotiues, although ot forever because of course some small amout of eergy must be lost at each collisio evet. The prototypical pheomeological descriptio of eergy loss ivolves the coefficiet of restitutio ε i the equatio that describes a collisio betwee two grais, v f2 v f = ε (v i v i2 ). () Here the v s represet the velocities, the subscripts i ad f stad for iitial (before collisio) ad fial (after collisio) ad the umbers label particles ad 2. This descriptio leads to a eergy loss at each collisio of ε 2 of the kietic eergy of the ceter of mass before the collisio. For a successfully built Newto s cradle, ε is exceedigly small. The coefficiet of restitutio is most ofte treated as a parameter idepedet of the velocities. Ad yet, it is broadly recogized that this ca ot be totally correct because it leads to problematic features i the asymptotic behavior such as the so-called ielastic collapse i a graular gas because there may be a ifiite umber of collisios i a fiite time [4, 5]. Ideed, whe oe cosiders realistic iteractio models, it is i fact uiversally the case that iteractios of ay two compressible grais are oliear. For istace, the iteractios betwee two spherical objects obey Hertz s law, where the repulsio is proportioal to the compressio to the power 3/2 rather tha the more familiar Hook s law where the repulsio is simply proportioal to the compressio. The cosequece of this o-liearity is that the duratio of a collisio depeds o the iitial velocities of the particles before the collisio. Therefore, whe a dissipative collisio occurs, the mechaism resposible for the dissipatio of eergy acts for differet legths of time depedig o the iitial velocities, leadig to distict eergy losses, ad cosequetly to a velocity-depedet coefficiet of restitutio. It is iterestig to ote that Hertz s law is frequetly used together with a velocity-idepedet coefficiet of restitutio. The history of the aalysis of the effect of velocitydepedet frictioal forces o the coefficiet of restitutio is log ad varied. Here we oly summarize some of
2 2 its saliet poits. Hertz s law assumes that there are o attractive surface forces i lightly loaded spherical graules. Perhaps the earliest work to recogize that such forces lead to a fiite cotact area betwee surfaces uder zero load (adhesio), ad that this i tur modifies the exteral force required to separate two bodies of give surface eergy ad geometry, is that of Johso et al. (commoly kow as JKR theory) . This i tur modifies the velocity depedece of the coefficiet of restitutio, as aalyzed i detail by Brilliatov et al. . While this correctio may be small, eve egligible, betwee graules of high elastic modulus such as metals or glass, it is cosiderable i soft graules such as rubber ad agrees well with experimets i these cases. Much of the literature o this topic is based o spherical graules, but other shapes are also discussed i this cotext. For istace, the work of Walto ad Brau o frictioal disks is oteworthy , as is the work of Herbold ad Nestereko . I  the coefficiet of restitutio for spherical graules obeyig Hertz s law for the elastic portio of the iteractios plus a velocity-depedet frictioal force was obtaied exactly as a ifiite series. The series ca ot be summed aalytically ad coverges very slowly, ad trucatio of the series leads to uphysical divergeces. A compact Padé approximatio to the series makes it possible to perform much more efficiet evet-drive molecular dyamics simulatios as well as direct Mote Carlo methods tha was possible with earlier methods, ad provides excellet results whe compared to those obtaied from umerical itegratios of the equatios of motio . Determiatio of the velocity depedece of the restitutio coefficiet has recetly become icreasigly detailed ad addresses a more varied rage of particle sizes ad compositios. For istace, experimetal determiatio  ad molecular dyamics calculatios  of the velocity depedece of the coefficiet of restitutio of argo aoparticles have show that these aoparticles are hard ad highly elastic at collisio velocities smaller tha the size-depedet yield velocity, while they progressively softe as the collisio velocity icreases beyod the yield velocity. I this cotributio, our goal is ot to arrive at a more efficiet approach for umerical simulatios, but rather to preset a perturbative but very broadly applicable methodology to aalytically aalyze the cosequeces of a geeral viscoelastic model of dissipatio o the outcome of a collisio i the low dissipatio regime. Our study provides a alterative approach to the problem. We arrive at approximate but accurate aalytic expressios for the iitial velocity depedece ot oly for the coefficiet of restitutio but also for the duratio of a collisio ad for the velocities of the graules at the ed of the collisio for large rages of parameter values. Our results have iterestig implicatios for the uderstadig of a full rage of realistic collisios, which should be useful for the study of graular gases ad for pulse propagatio i graular chais. Examples of graular systems with low itergraular frictio iclude sad, glass beads, ad steel beads. We accomplish three objectives, all i the limit of low but o-zero dissipative forces. The first, preseted i Sec. II, is to discuss the equatios of motio that describe the collisio of two ot ecessarily spherical grais. The sceario is this: the two grais are iitially just touchig. Grai has velocity v ad grai 2 has velocity v 2. How these grais came to have these velocities is ot importat. If our two grais are elemets i a graular gas, they may, for example, have arrived at this poit due to previous collisios with other grais. Or, oe might be preparig a experimet with just two grais, where oe grai is give a kick of some kid at time t = to cause it to start movig with velocity v while the other is iitially at rest (v 2 = ). The equatios of motio the determie the further evolutio of the two graules. I particular, we are able to calculate the relative velocities of the graules at the ed of a collisio as a fuctio of the iitial relative velocities. Our secod objective, detailed i Sec. III, is to use this result to calculate the duratio of a collisio as a fuctio of the iitial velocities. This makes explicit our assertio that collisios are i geeral ot istataeous ad that their duratio i fact depeds o the iitial velocities of the collidig graules. As poited out above, this i tur leads to a velocity-depedet coefficiet of restitutio. Our third objective is to calculate the coefficiet of restitutio ad the velocities of the collidig graules at the ed of the collisio. This is preseted i Sec. IV. I the absece of dissipative forces the coefficiet of restitutio is equal to uity. We are able to explicitly calculate the lowest order correctios to this ad thus to obtai a explicit form for the depedece of the coefficiet of restitutio o the iitial velocity differece of the graules. Fially, Sec. V cotais a summary of the paper ad commets o the possible geeralizatios of this study. II. THE MODEL Viscoelastic forces for the collisio of two spheres iclude two terms. The first is due to the elastic repulsio betwee the two particles ad has its origis i Hertz s theory [4, 5]. The secod term stads for the viscous dissipatio via a dashpot [6, 7]. Hece, the cotact force ca be writte as F = r (x x 2 ) γ (x x 2 ) α (ẋ ẋ 2 ). (2) Here x is the displacemet of a particle from its iitial positio at the begiig of a collisio. A dot deotes a derivative with respect to time, ad the subscripts o x label the two particles. The coefficiet r is a costat depedet o Youg s modulus ad Poisso s ratio. For istace, for collidig spheres of radii R ad R 2 this costat is give by r = 2 5D 2 2R R 2 R + R 2, (3)
3 3 where D 2 = 3 4 ( σ 2 + ) σ2 2, (4) E E 2 σ i is Poisso s ratio ad E i the Youg s modulus of sphere i. Returig to Eq. (2), γ is the coefficiet of viscosity, α is a costat that defies the specific viscoelastic model, ad depeds o the topology of the cotact betwee the particles. For spheres is equal to 5/2 [4, 5], ad α to /2 [8, 9], but we leave them as ad α for the sake of geerality. We say that particle is to the left of particle 2, ad that our system of coordiates icreases from left to right. Whe we work with two equal spheres so that R = R 2, σ = σ 2, ad E = E 2, we drop the subscript o D, that is, we set D D. The equatios of motio for two particles of mass m ad m 2 durig a collisio are m ẍ = r (x x 2 ) γ (x x 2 ) α (ẋ ẋ 2 ), m 2 ẍ 2 = r (x x 2 ) + γ (x x 2 ) α (ẋ ẋ 2 ). (5) From Eq. (5), coservatio of mometum immediately follows, m ẍ + m 2 ẍ 2 =, (6) so that m ẋ + m 2 ẋ 2 = cost. Equatio (5) also leads to a ucoupled equatio for the differece variable z = x x 2, z = r µ z γ µ zα ż, (7) where µ is the reduced mass µ = m + m 2. For the latter equatio, the iitial coditios are z() = because we deal with cofiguratios where the graules are iitially just touchig each other, ad ż() = v i v i2. Here v i ad v i2 are the iitial velocities of the two collidig graules. A aalytic solutio z(t) of Eq. (7) for arbitrary ad α seems ot to be available. However, we have bee able to obtai the velocities at the ed of the collisio as a fuctio of the iitial velocities i the low viscosity limit. We rewrite Eq. (7) as a first order differetial equatio of the velocity as a fuctio of the positio. Defiig v = ż, ad otig that we rewrite Eq. (7) as z = dv dt = dv dz dz dt = v dv dz, (8) v dv dz = r µ z γ µ zα v. (9) I the absece of dissipatio (γ = ), this equatio admits two solutios for v(z), v ± (z) = ± 2 2rz µ. () Obviously, the positive sig should be cosidered durig compressio, ad the egative oe durig decompressio. Our approximatio i the low dissipatio regime starts by writig the velocity as a perturbatio o the odissipative solutio. Cosequetly, durig compressio we have v(z) = v + (z) + γv comp (z), () where v comp (z) is a fuctio to be determied. Substitutig the trial solutio Eq. () i Eq. (9), ad collectig the terms of order γ, we have r µ z v comp + ( ) ( ) 2rz z µ 2 α µ + v comp =, (2) where a prime deotes a derivative with respect to z. Furthermore, the coditio v comp () = is ecessary to satisfy the iitial coditios. The solutio of Eq. (2) with the iitial coditios v comp () =, whe added to v + (z), the gives us the compressio velocity to first order i the dissipatio, v(z) = 2 2rz µ γ z+α [ 2( + α) + 2 F (, 2 + +α ; + +α ( + α)( α)µ ] ; 2rz µv 2 ), (3) where 2 F is a hypergeometric fuctio . Next we move o to the decompressio, which starts with v = ad z = z max. To calculate the maximum compressio z max, we agai look for a first order correctio i γ, ( ) 2 / µv z max = ( γz c ), (4) 2r where z c is a costat to be determied ad ( µv 2 /2r ) / is the maximum compressio i the absece of dissipatio, obtaied by settig v ± (z) = i Eq. (). The relative velocity must vaish for the maximum compressio. This is the poit at which the collidig grais stop movig ad begi to separate. Therefore, we evaluate Eq. (3) at z = z max as give i Eq. (4),
4 4 expad to first order i γ, ad set the left had side equal to zero. Solvig the resultig equatio for z c ad isertig the result i Eq. (4), we fid for the maximum compressio ( 2 µv z max = 2r ) γ ++α 2 ( π µ ) +α r 2 +2α 2 µγ ( α Γ ( ) +α ). (5) While it might be temptig to assume that the collisio eds whe z =, that is, whe the particles lose cotact, this would eglect the fact that the particles have ot recovered their origial shape at this poit . The effect of eglectig this cotributio leads to serious spurious attractive forces. The collisio actually eds whe the force is zero, z =, at which poit the iitial shape is recovered. Settig z =, z = z f, ad ż = v f i Eq. (7), where the subscript f stads for fial, we have ( γvf z f = r ) α. (6) I what follows we restrict to values > α +, so that z f is small for low viscosity. We write the perturbative solutio for the decompressio velocity i the low viscosity case as v(z) = v (z) + γv decomp (z). (7) Agai substitutig ito Eq. (9) ad expadig the latter to first order i γ, we fid v decomp (z) = C µv2 2rz 4rz ++α z+α ( α) µ (2rz µ2 ) [2 + 2α + 2 F (, 2 + +α ; + +α ( + α) ( α) µ )] ; 2rz µv 2. (8) Here C is a costat to be determied by the cotiuity of the solutios (8) ad (3) at z max. Rememberig that v(z max ) =, where z max is give by Eq. (5), ad expadig up to first order i γ, C is foud to be C = 2 π(µ) ( 2 + ) r v +2α+2 Γ ( ) ). Γ ( (9) To use this result to calculate the leadig cotributio to the fial relative velocity we write v f = v(z f ) = v() + v ()z f, (2) sice z f is small. A prime deotes a derivative with respect to the argumet z. Sice the force is zero at the ed of the collisio, we have z = = dż/dt = dż/dz dz/dt = v (z f )v(z f ) = [v() + v ()z f ][v () + v ()z f ] = v()v () + [v()v () + (v ()) 2 ] + O(zf 2 ). (2) This equatio has two solutios for v (): either v = v()/z f or v =. The first leads to a fial relative velocity that vaishes, which is ot physical i our perturbative approach where we expect the fial relative velocity to go to whe γ. We must thus choose v () = (or, more rigorously, v () = O(zf 2 )). Therefore, v f = v() + O(z 3 f ) = v() + O(γ 3/( α) ). (22) We choose values of ad α such that < α < 3. For spheres with = 5/2 ad α = /2 this coditio is clearly satisfied. We later also cosider = 5/2, α = (used by others), for which the coditio is also satisfied. The correctio i the fial velocity ca the be eglected i our aalysis to first order i γ. The first term, v(), is obtaied by substitutig Eq. (9) ito Eq. (8), usig this result together with Eq. (7), ad takig the limit z. We the obtai v f = + γ III. ( µv 2 2r ) +α πγ( + +α ) µγ( α ). (23) COLLISION TIME Our ext objective is to calculate the collisio time, which we do i two parts. First we calculate the com-
5 5 pressio time of the collisio as the two grais compress oe aother ad the, as the grais move apart, the attedat decompressio time. The collisio time is the the sum of the two. First we calculate the compressio time as T compressio = zmax dz (24) v(z) where, to order γ, v(z) is obtaied from Eq. (3) as v(z) = 2 2rz µ z [2( +α + α) + 2 F (, 2 + +α + γ ] ; 2rz µv 2 ) ; + +α ( + α) ( α) µ(v 2 2rz µ ). (25) Substitutig this ito Eq. (24) ad itegratig leads to a cotributio of order γ, ad a cacellatio of two terms of order γ /2. Cosequetly, the compressio time does ot show ay γ depedece up to first order: T compressio = 2 + πr µ + v 2 Γ( ) Γ( 2 + ) +O(γ 3/2 ). (26) Next we move o to decompressio which starts with v = ad z = z max (maximum compressio) ad eds whe z = (force is zero), at which poit z = z f as give i Eq. (6). Hece, the decompressio time is T decompressio = = zf z max z max dz v(z) dz v(z) + zf dz v(z). (27) We first deal with the first itegral. Sice the zeroth order term of the velocity durig decompressio, Eq. (7), is the egative of the compressio velocity i Eq. (3), ad the limits of itegratio of the compressio ad the first term i the decompressio times are switched, the latter time is the same as the compressio time up to terms of order γ 3/2. This still leaves the secod term i the decompressio time give i Eq. (27). Sice z max is small, zf dz v(z) z f v(z f ) z f = ( γv ) α. (28) r For α >, this correctio is ideed small ad our perturbative approach is valid. I particular, for spherical grais = 5/2, ad the particular values of α = /2 (commoly used i the literature) ad α = (used by others, see below), the value of the expoet of γ is 2/3 ad respectively. This i tur implies that this term is a small perturbatio. We have thus established that for the parameters used herei, the decompressio time is equal to the compressio time up to order γ 2/3 or order γ depedig o the value of α. Hece the total collisio time to this order is twice the compressio time plus a additioal cotributio to the decompressio time, T = 2 + πr µ + v 2 Γ( ) Γ( 2 + ) ( γv ) α, (29) r ad we have arrived at our secod objective. We have ot foud ay other calculatio of the velocity-depedece of the collisio time i the literature. I a recet paper, the merits ad problems of differet choices of the parameter α, ad eve a geeralizatio of the above model, were discussed . Here, for = 5/2 we cosider two choices of this parameter that have bee commoly used i the literature to test Eq. (2). The simplest case, α =, was proposed i  ad further developed i , but i our subsequet results below we see that this choice leads to problematic outcomes. The case α = /2, proposed idepedetly i [8, 23], is more widespread i the graular gas commuity. Yet aother combiatio of expoets was foud experimetally for chais of o-rigs i ad aalyzed theoretically i. I this model dissipatio was ot icluded, but the elastic force is a double power-law rather tha a sigle oe, with expoets 5/2 ad 7. I the latter work it was show that depedig o the characteristics of the o-rigs ad the experimetal setup, either the oe or the other cotributio ca be domiat. I Fig. we show the duratio of the collisio for two equal spherical grais ( = 5/2) as a fuctio of the relative velocity at the begiig of the collisio for both values of α. As ca be see i the figure, for small γ the data is very well predicted by our approximatio, idepedetly of the expoet α (the approximatio oly starts to deviate from the theoretical predictio for γ =., represeted by the plus sigs). This α-idepedece of the duratio of the collisio is oe of the strikig predictios of our theory. Aother iterestig characteristic of our solutio is the power-law depedece of T o the iitial relative velocity, as evideced i the iset of Fig.. I Fig. 2 we agai show T as a fuctio of, but this time for spheres of differet sizes. Fixig the radius of graule ad varyig the radius of graule 2 (assumig
6 6 T log T log IV. FINAL VELOCITIES AND COEFFICIENT OF RESTITUTION We ext tur our attetio to the fial velocities. From the coservatio of mometum, we kow that the total mometum at the begiig ad ed of the collisio must be the same, m v i + m 2 v i2 = m v f + m 2 v f2, (3) where the subscripts i ad f oce agai label the iitial ad fial velocities of the grais. O the other had, we also kow from the defiitio of z(t) that ż(t ) = v f = v f v f2. (3) FIG.. Collisio duratio for several values of γ (from top to bottom,.,.,. ad.) ad two differet values of α ( for lower group ad /2 for higher group), obtaied via umerical itegratio of the equatios of motio (symbols) ad from our theory (lie). The lie represet the theoretical predictio of Eq. (29). The iset shows the same data o a log-log scale. I this figure, the parameters are as follows: m = m 2 =, R = R 2 =, ad 2/(5D) =. As predicted, for these values of γ ad α the collisio duratios are essetially idepedet of these parameters. T FIG. 2. Duratio of collisios for several values of R 2 (from bottom to top, R 2 varies from. to.9 i steps of.) obtaied via umerical itegratio of the equatios of motio. The lies represet the theoretical predictio Eq. (29). I this figure the parameters are as follows: m =, R =, α = /2, γ =., ad 2/(5D 2) =. The desities of the two collidig spheres are equal. that they have the same desity), we ca see that the collisio takes loger for larger values of R 2. Solvig the set of the two equatios give above for v f ad v f2, ad usig Eq. (7), we fid v f = m m 2 m + m 2 v + 2m 2 m + m 2 v m 2 m + m 2 ( πγ ) ( r µ rγ ( v f2 = m m 2 m + m 2 v 2 + 2m m + m 2 v 2 m m + m 2 ( πγ ) ( r µ rγ ( ) ) (v i v i2 ) 2() γ, ) (32) ) (v i v i2 ) 2() γ. As expected, the result for a elastic collisio is recovered whe γ =. Further, the ifluece of the dissipatio is greater o the lighter particle, ad the ifluece of the iitial coditio o the chage i the fial velocities due to dissipatio depeds oly o the relative velocity. We coclude this sectio by usig the above results i Eq. () to calculate the coefficiet of restitutio ad thus completig our third ad pricipal objective: ε = γ 2 ( πγ ) ( r µ ) ) (v i v i2 ) +2α+2. rγ ( (33) I Fig. 3 we show the coefficiet of restitutio for several values of α. The agreemet is equally good for all of them. A importat characteristic of ε is that its qualitative depedece o the iitial relative velocity is drastically differet for α larger tha or smaller tha ( 2)/2 (i the case of spheres, this value is /4): α smaller tha this value leads to the uphysical situatio of egative coefficiets of restitutio for very small relative velocities. For larger α, the collisio approaches the elastic case for small relative velocities. Perhaps most importatly, the power ( + 2α + 2)/ of the iitial relative velocities of the two graules is equal to /5 with the physically justified values = 5/2 ad α = /2 for spheres, see e.g. Eq. (2) i . I additio to recoverig this expoet, as i Eq. (4) of that referece, we also obtai
7 ε ε FIG. 3. Coefficiet of restitutio as a fuctio of the iitial relative velocity. Each curve correspods to a differet value of α (from bottom to top o the left side, α varies from to.9 i steps of.). The other parameters are: γ =., m = m 2 =, R = R 2 =, ad 2/(5D) =. The lies correspod to the theoretical predictio of Eq. (33) (a)α = the liear depedece of the correctio of the coefficiet of restitutio o the coefficiet of viscosity γ. I Fig. 4, we show the coefficiet of restitutio as a fuctio of for particles of differet sizes (same desity) for α = ad α = /2. It is evidet from the figure that ε icreases with the radius i both cases. However, the qualitative behavior is idepedet of the sizes of the grais. V. CONCLUSIONS We have succeeded i calculatig quatities that characterize the collisio of two graules that lead to the loss of eergy (but ot mometum) to the eviromet via viscous dissipatio. We started with a equatio of motio (Newto s Law) cotaiig a kietic eergy cotributio, a force due to the elastic repulsio betwee the two graules, ad a dashpot viscous dissipatio term. I additio to parameters related to the shape ad size of the graules, the model cotais two importat parameters: a coefficiet of viscosity γ, ad a costat α that defies the specific viscoelastic model, cf. Eq (2). Our calculatios are perturbative i the coefficiet of viscosity, that is, we preset lowest order correctios to elastic (eergy-coservig) collisios. A collisio begis with the two graules just touchig head-o toward each other with a relative velocity. This velocity ad cofiguratio defie the collisio stregth. The collisio begis at this iitial momet with compressio of the graules util their relative velocity is zero (at which poit the compressio is a maximum). Decompressio the follows, util the force betwee the graules vaishes, at which poit the collisio eds. ε (b)α = /2 FIG. 4. Coefficiet of restitutio as a fuctio of the iitial relative velocity. Each curve correspods to a differet value of the radius of graule 2 (from bottom to top, R 2 varies from.5 to.4 i steps of.). The other parameters are: γ =., m =, R =, ad 2/(5D 2) =. The desities of the two collidig spheres are equal. The lies correspod to the theoretical predictio of Eq. (33). Itegratio of the equatios of motio leads to aalytic results for several importat quatities usually specified simply as pheomeological parameters. The first is the relative velocity of the graules durig compressio ad durig decompressio. We calculate the fial relative velocity as a fuctio of the separatio of the ceters of the graules ad fid the depedece o iitial relative velocity ad o the parameters γ ad α, cf. Eq. (23); if the collisio were elastic, the fial ad iitial relative velocities would of course just be the egatives of oe aother. A secod set of useful results are the fial velocities of each grai, for which we obtai explicit expressios as a fuctio of the parameters ad of the iitial velocities of each grai, cf. Eq. (32). These are importat for simu-
8 8 latios of graular gases. The third quatity we calculate is the duratio of the collisio, cf. Eq. (29). I most pheomeologies, collisios are assumed to be istataeous. Collisios are of course ot istataeous. Fially, we fid a aalytic expressio for the coefficiet of restitutio ε defied i Eq. (). This coefficiet, usually chose pheomeologically, recogizes the ielasticity of graular collisios. We have foud the depedece of this coefficiet o particle shape (via the expoet i the force that determies the topology of the cotact betwee the graules), the coefficiet depedet o Youg s modulus ad Poisso s ratio, ad, most importatly, o iitial relative velocity ad o the parameters γ (to lowest order) ad α that defie the viscoelastic model, cf. Eq. (33). If γ = the coefficiet of restitutio is uity, that is, there is o eergy loss i the collisio. Similarly, if the iitial velocities of the two graules are equal, the coefficiet is trivially uity agai. The depedeces o these quatities are otrivial ad, we submit, essetially impossible to arrive at pheomeologically. This the provides a physical basis for the usual pheomeological choice ε <. While our results are perturbative i γ ad thus ot as geeral for elastic spheres ( = 5/2, α = /2) as is the ifiite series developed i , our model allows differet values of ad α ad yields relatively simple explicit results for the iitial velocity depedece of the fial velocities of the collidig graules, of the duratio of a collisio, ad of the restitutio coefficiet. I this paper we have oly dealt with two collidig graules, takig ito accout the eergy loss due to a explicit viscoelastic force i the equatios of motio. This reders our results immediately applicable to graular gases where at low desities biary collisios are the most commo iteractios. The geeralizatio to a graular chai or to eve higher dimesioal graular arrays is ot trivial, cf. Ref. , but is ow made cosiderably easier by the fact that we have explicitly foud the pricipal igrediets of the problem. There is evertheless a great deal of work to be doe, especially toward higher dimesioal geeralizatios. Ackowledgmets I. L. D. Pito ad A. Rosas akowledge the CNPq ad Bioaotec-CAPES for fiacial support.  S. Hutzler, G. Delaey, D. Weaire ad F. Macleod, Am. J. Phys. 72, 58 (24).  P. Glediig, Phys. Rev. E 84, 672 (2).  K. Sekimoto, Phys. Rev. Lett. 4, 2432 (2).  Sea McNamara, W. R. Youg. Ielastic collapse ad clumpig i a oe-dimesioal graular medium. Phys. Fluids A 4 (3), 496 (992).  B. Beru ad R. Mazighi J. Phys. A:Math. Ge. 23, 5745 (99).  K. L. Johso, K. Kedall ad A. D. Roberts, Proc. R. Soc. Lodo A 324, 3 (97).  N. V. Brilliatov, N. Albers, F. Spah ad T. Pöschel, Phys. Rev. E 76, 532 (27).  O. R. Walto ad R. L. Brau, J. Rheol. 3, 949 (986).  E. B. Herbold ad V. F. Nestereko, Appl. Phys. Lett. 9, 2692 (27).  T. Schwager ad T. Pöschel, Phys. Rev. E 78, 534 (28).  P. Mueller ad T. Pöschel, Phys. Rev. E 84, 232 (2).  A. I. Ayesh, S. A. Brow, A. Awasthi, S. C. Hedy, P. Y. Covers, ad K. Nichol, Phys. Rev. B 8, (2).  Y. Takato, S. Se, ad J. B. Lechma, Phys. Rev. E 89, 3338 (24).  H. Hertz ad J. Reie Agew. Math. 92, 56 (88).  L. D. Ladau ad E. M. Lifshitz, Theory of Elasticity, Addiso-Wesley, MA, (959).  S. Ji ad H. H. She, J. Eg. Mech. 32, 252 (26).  E. Alizadeh, F. Bertrad ad J. Chaouki, Powder Techology 237, 22 (23).  N. Brilliatov, F. Spah, J.-M. Hertsch ad T. Pöschel, Phys. Rev. E 53,5382 (996).  W. A. M. Morgado ad I. Oppeheim, Phys. Rev. E 55, 94 (997).  George B. Arfke, Has J. Weber, Frak Harris, Mathematical Methods for Physicists (Academic Press, 5th editio, 2).  J. Lee ad H. J. Herrma, J. Phys. A: Math. Ge. 26, 373 (993).  A. Rosas, A. H. Romero, V. F. Nestereko, ad K. Lideberg, Phys. Rev. Lett. 98, 643 (27).  G. Kuwabara ad K. Koo, Japaese J. Appl. Phys. 26, 23 (987).  I. L. Dias Pito, A. Rosas, A. H. Romero, ad K. Lideberg, Phys. Rev. E 82, 338 (2).  T. Pöschel ad N. V. Brilliatov, Phys. Rev. E 63, 255 (2).