Hydrodynamic Model for Particle Size

Segregation in Granular Media

Leonardo Trujillo1 and Hans J. Herrmann1,2

1 Laboratoire de Physique et Mecanique des Milieux Heterogenes, UMR–CNRS7636, Ecole Superieure de Physique et de Chimie Industrielles, 10 rue Vauquelin,

75231 Paris Cedex 05, France2Institut fur Computeranwendungen 1, Universitat Stuttgart, Pfaffenwaldring, 27,

D-70569 Stuttgart, Germany

Abstract

We present a hydrodynamic theoretical model for “Brazil nut” size segregationin granular materials. We give analytical solutions for the rise velocity of a largeintruder particle immersed in a medium of monodisperse fluidized small particles.We propose a new mechanism for this particle size-segregation due to buoyant forcescaused by density variations which come from differences in the local “granulartemperature”. The mobility of the particles is modified by the energy dissipationdue to inelastic collisions and this leads to a different behavior from what onewould expect for an elastic system. Using our model we can explain the size ratiodependence of the upward velocity.

Key words: granular materials, Brazil-nut effect, segregation, buoyancyPACS: 45.70.Mg, 05.20.Dd, 05.70.Ln

1 Introduction

The physics of granular materials is a subject of current interest [1]. A granularmedium is a system of many macroscopic heterogeneous particles with dissi-pative interactions. One of the outstanding problems is the so–called “BrazilNut effect” [2]: When a large intruder particle placed at the bottom of a vi-brated bed tends to the top. This size segregation is due to the nonequilibrium,dissipative nature of granular media. Granular materials are handled in manyindustries. Many industrial machines that transport granular materials usevertical vibration to fluidize the material, and the quality of many productsis affected by segregation. Size segregation is one of the most intriguing phe-nomena found in granular physics. A deeper understanding of this effect is

Preprint submitted to Physica A Date: May 29, 2003

therefore interesting for practical applications, and also represent theoreticalchallenge.

A series of experiments [3–12] and computer simulations [2,13–23] have elu-cidated different size segregation mechanisms, including vibration frequencyand amplitude [3–8,14,19]; particle size [2,3,5,6,13–15,17,18,23] and size dis-tribution [16,19]; particle shape [21]; and other properties such as density[9–12,17,22,23] and elastic modulus [19].

Several possible mechanisms for size segregation have been proposed. One issegregation in presence of convection observed experimentally in three dimen-sions by Knight et al., [4], and by Duran et al., in two dimensions [5], underconditions of low amplitude and high acceleration vibration. In this case, bothintruder and the small particles are driven up along the middle of the cell, andwhile the smaller particles are carried down in a convection roll near the wallsthe intruder remains trapped on the top. In experiments performed by Vanelet al., in three dimensions, they observed two convective regimes separated bya critical frequency [7]. The first regime is associated with heaping and thesecond regime is similar to the one observed in Ref. [4]. Also, they reported anonconvective regime observing a size dependent rise velocity. Employing largemolecular dynamics simulations in two dimensions Poschel and Herrmann [16],and in three dimension Gallas et al., [19], have recovered several aspects thatare seen in experiments and recognize the lack of a theoretical description ofthe exact mechanism driving the segregation and the role of convection.

Other segregation mechanism is associated to the percolation of small grains.Based on a Monte Carlo computer simulation, Rosato et al., [2,13] argue thateach cycle of the applied vibration causes all the grains to detach from thebase of the container. Then, the smaller particles fall relatively freely, whilethe larger particles require larger voids to fall downward. The large graintherefore effectively rise through the bed. In the context of large–amplitude,low–frequency vertical shaking process (tapping), Jullien et al., predict a crit-ical size ratio below which segregation does not occur [14,15]. This provokedsome controversy [24–27] and this threshold may be an artifact of the simu-lation model based on the “steepest descent algorithm” [14]. Experiments inHelle Shaw cells [3,5,6] observed an intruder size dependent behavior, wherethe segregation rate increases with the size ratio between the intruder and thesurrounding particles. Duran et al., formulated a geometrical theory for segre-gation based on the arching effect [3]. They also claim experimental evidencefor a segregation size threshold [5]. In this picture the intruder contributesto the formation of an arch sustained on small grains on both sides. Betweeneach agitation the small particles tend to fill the region below the arch. So,at each cycle the small particles move downward and the intruder effectivelyrises. Using a modification of the algorithm proposed by Rosato et al., Dippeland Luding find a good qualitative agreement with the non–convective and

2

size–dependent rising [18].

In another context, Caglioti et al., considered the geometrical properties ofmixtures in the presence of compaction [21]. They established a relation forthe effective mobilities of different particles in heterogeneous situations.

The effect of the intruder density was studied experimentally by Shinbrotand Muzzio [9], and Mobius et al., [11] in three dimensions, and by Liffmanet al., [10] in two dimensions. Shinbrot and Muzzio observed an oscillatingmotion of the intruder on the top, which corresponds to the “whale effect”predicted by Poschel and Herrmann [16]. Also they claim a “reverse buoyancy”in shaken granular beds. Mobius et al., analyzed the segregation effect in thepresence of air and the interplay between vibration–induced convection andfluidization. They reported the intruder rising time dependence on density.Liffman et al. [10], analyzed the intruder ascent speed dependence on den-sity and shaking frequency. Recently, the interplay between the intruder’s sizeand material density have been considered by Hong et al. [22]. They proposea phase diagram for the upward/downward intruder’s movement. Ohtsuki etal., performed molecular dynamics simulations in two dimensions and studiedthe effect of intruder size and density on the height, and found no segregationthreshold [17]. Recently Shishoda and Wassgren performed two dimensionalsimulations to model segregation in vibrofluidized beds [23]. They reported anheight dependence with the density ratio between the intruder and the sur-rounding particles. In their model the intruder position result from a balancebetween the granular pressure (buoyant force) within the bed and the intruderweight. Their approach is in some sense similar to the model that we proposein this article.

Subject to an external force, granular materials locally perform random mo-tions as a result of collisions between grains, much like the molecules in a gas.This picture has inspired several authors to use kinetic theories to derive con-tinuum equations for the granular flow–field variables [28–34]. Some of thesetheories have been generalized to multicomponent mixtures of grains [35–38].For different size particles in the presence of a temperature gradient, Arnarsonand Willits, found that larger, denser particles tend to be more concentrated incooler regions [37]. This result was confirmed by numerical simulations [39,40].However, this mechanism of segregation is a natural consequence of the im-posed gradient of temperature and its not related to the nature of the grains[40].

In this article we address the problem of size segregation using a kinetic theoryapproach in two and three dimensions (D = 2, 3). We consider the case of anintruder particle immersed in a granular bed. We assume that the materialdensity of all particles is the same. We propose a segregation mechanism basedon the difference of densities between different regions of the system, which give

3

origin to a buoyant force that acts on the intruder. The difference of densitiesis caused by the difference between the mean kinetic energy among the regionaround the intruder and the medium without intruder. The dissipative natureof the collisions between the particles of a granular media is responsible forthis mean energy difference, and modifies the mobility of the particles.

The plan of this article is as follows. In Section 2, we derive a continuumformulation for the granular fluid, and introduce the definition of the “granulartemperature”. In Section 3, we propose an analytic method to estimate thelocal temperature in the system. In Section 4, we introduce the coefficient ofthermal expansion. In Section 5, explicit solutions of the time dependence ofheight and velocity of the large particle are calculated. We can explain the sizeratio dependence of the rise velocity and address the issue of the critical sizeratio to segregation. To validate our arguments we make comparisons withprevious experimental data.

2 Continuum formulation

We consider an intruder particle of mass mI and radius rI immersed in agranular bed. The granular bed is formed of N monodisperse particles of massmF and radius rF . The particles are modeled by inelastic hard disks (D = 2)or spheres (D = 3) in a D–dimensional volume V = LD of size L. The sizeratio is denoted φ = rI/rF . The particles interact via binary encounters. Theinelasticity is specified by a restitution coefficient e ≤ 1. We assume thisrestitution coefficient to be a constant, independent on the impact velocityand the same for the fluid particles and the intruder. The post collisionalvelocities v′ are given in terms of the pre–collisional velocities v by

v′1,2 = v1,2 ∓ mred(1 + e)

m1,2

[(v1 − v2) · n]n, (1)

where the labels 1 or 2 specify the particle, n is the unit vector normal tothe tangential contact plane pointing from 1 to 2 at the contact time, and thereduced mass mred = m1m2/(m1 +m2). To calculate the dissipated energy weconsider that energy is dissipated only by collisions between pairs of grains.In a binary collision the energy dissipated is proportional to ∆E = −mred(1−e2)v2/2, where v is the mean velocity of the particles.

In this work we use a generalized notion of temperature. In a vibrofluidizedgranular material a “granular temperature” Tg can be defined to describe therandom motion of the grains and is the responsible for the pressure, and thetransport of momentum and energy in the system [31]. The granular temper-ature Tg is defined proportional to the mean kinetic energy E associated to

4

the velocity of each particle

D

2Tg =

E

N=

1

N

N∑

i=1

(1

2miv

2i

). (2)

We expect a continuum limit to hold for N À 1, when the small particles maybe considered as forming a granular fluid [41]. In order to develop an analyticstudy, we assume that the uniformly heated granular fluid can be describedby the standard hydrodynamic equations [41] derived from kinetic theories forgranular systems [28–34]. In this study, we focus on a steady state with nomacroscopic flow.

The balance equation for the energy is

∇ · q = −γ, (3)

where q is the flux of energy and γ is the average rate of dissipated energydue to the inelastic nature of the particles collisions. The constitutive relationfor the flux of energy,

q = −κ∇Tg, (4)

defines the thermal conductivity κ. Consequently, we have

∇ · (κ∇Tg) = γ. (5)

An uniformly fluidized state can be realized when the granular material isvibrated in the vertical direction, typically as z0(t) = A0 sin(ω0t), with theamplitude A0 and the frequency ω0 = 2πf , so that one can define a typicalvelocity u0 = A0ω0. In the experiments the excitation is described by thedimensionless acceleration Γ0 = A0ω

20/g, where g is the gravitational acceler-

ation. As a first approximation the effect of the external force experienced bythe fluid particles due to the gravitational field is neglected in the descriptionof the granular flow. Experimentally this correspond to the regime Γ0 À 1.So, the momentum balance, in the steady state, implies that the pressure p isconstant throughout the system.

The hydrodynamic equations close with the state equation, the collisionaldissipation γ and the transport coefficients for a granular medium. In the limitN À 1 the constitutive relations are determined as function of the propertiesof the small grains. The transport coefficients are assumed to be given by theEnskog theory for dense gases in the limit of small inelasticity.

5

The total pressure should be essentially equal to that of the small particles,the contribution of the intruder being negligible, since N À 1. For a densesystem the pressure is related to the density by the virial equation of state,which in the case of inelastic particles is [28,34]

p = nTg [1 + (1 + e)(D − 1)G] , (6)

where n = N/V is the number density and G = νg0, where ν = ΩDnrDF /D

is the volume fraction, with ΩD = 2πD/2/Γ(D/2) as the surface area of a D–dimensional unit sphere and g0 is the pair correlation function. For disks g0 istaken to be that proposed by Verlet and Levesque [42]:

g0 =1

1− ν+

9

16

ν

(1− ν)2, (7)

with the area fraction ν = nπr2F , and that proposed by Carnahan and Starling

for spheres [43]:

g0 =1

1− ν+

ν

2(1− ν)2

[3 +

ν

2(1− ν)

], (8)

with the volume fraction ν = 4πnr3F /3.

It is important to mention that the equations (7) and (8) only work for mod-erate densities. In Ref. [44] Luding and Strauß showed that g0 should divergeat the close packing limit (ν = νmax), rather than at ν → 1. Therefore, in thepresent model we assume that ν < νmax.

The state–dependent thermal conductivity possesses the general form [28,32]

κ = κ0

√Tg, (9)

where the prefactor κ0 is a function of the fluid particle properties, and can becalculated using a Chapman–Enskog procedure through the solution of Enskogtransport equation [28,32,34,45]. The explicit expressions of these prefactorsare given in Appendix A.

To estimate the collisional dissipation rate γ we consider the loss of averagekinetic energy per collision and per unit time. In a binary collision the kineticenergy dissipated can be expressed in terms of the granular temperature as∆E = −(1 − e2)Tg/2. For the fluid particles, the average collision frequency

ωF is proportional to ωF ∼√

Tg, and we assume that it is given by the Enskog

6

collision frequency [45]

ωF =ΩD√2π

ng0(2rF )D−1(

2

mF

)1/2

T 1/2g . (10)

This form for the frequency of collisions is justified for a granular medium.This is a consequence that the average spacing between nearest neighbor sis supposed to be less than the grain diameter (s ¿ 2rF ) [41]. Multiplying∆E by the collision rate ωF and the number density n = N/V , we obtain thecollisional dissipation rate γF for the fluid particles

γF =ΩD

2√

2π(1− e2)n2g0(2rF )D−1

(2

mF

)1/2

T 3/2g . (11)

In order to simplify the mathematical notation let us express γF as

γF = ξF T 3/2g , (12)

where the dissipation factor ξF contains the prefactors which multiply T 3/2g in

Eq.(11), this is

ξF ≡ ΩD

2√

2π(1− e2)n2g0(2rF )D−1

(2

mF

)1/2

. (13)

To understand the essential features of the intruder’s presence in the granularmedium, it is adequate to adopt a simplified point of view. If the mean velocity

of the fluid particles is v ∝√

Tg/m, the flux of fluid particles which strikesthe intruder’s surface can be estimate as nv. Multiplying this flux by the areaof the intruder ΩDr

(D−1)I , we can calculate the number of fluid particles which

strike the surface of the intruder per unit time, and written in terms of thegranular temperature we have

ωI =ΩD√2π

ng0(rF + rI)D−1

(mI + mF

mImF

)1/2

T 1/2g , (14)

where g0 is the pair correlation function of the granular fluid in presence ofthe intruder.

The local density of kinetic energy dissipated in the region near the intruderis

γI =ΩD

2√

2π(1− e2)

n

Vg0(rF + rI)

D−1(

mI + mF

mImF

)1/2

T 3/2g . (15)

7

In a simplified form Eq.(15) can be expressed as

γI = ξIT3/2g , (16)

where the dissipation factor ξI is defined as

ξI ≡ ΩD

2√

2π(1− e2)

n

Vg0(rF + rI)

D−1(

mI + mF

mImF

)1/2

. (17)

Finally, in order to estimate γI an explicit expression for g0 is needed. To ourknowledge, no theory exists for the exact calculation of the pair correlationfunction of a single intruder (tracer particle) immersed in a granular fluid.Rather than going into details concerning the calculation of g0, we proposeto use the generalization of equations (7) and (8) for binary mixtures [46,47],restricted in our case to the limit where N >> 1 and rI > rF . For twodimensions we can verify that

g0 ' 1

1− ν+

9

8

ν

(1− ν)2, (18)

and for three dimensions

g0 ' 1

1− ν+

ν

(1− ν)2

[3 +

ν

1− ν

]. (19)

Let us remark that g0 and g0 are not very different and both quantities tendto 1 in the diluted limit. Here too, we assume that ν < νmax.

3 Local temperature difference

The intruder’s presence modifies the local temperature of the system due tothe collisions that happen at its surface. The number of collisions on the sur-face increases with the size of the particle, but the local density of dissipatedenergy diminishes. From Eq.(5) we can calculate within a sphere of radius r0

the value of the temperature in the granular fluid in presence of the intruderand compare it with the temperature in the granular fluid without intruder,we will denote these temperatures T1 and T2 respectively (see Fig.(1)). Thesespherical regions are considered to be placed in the reference frame of the in-truder particle. This is a simple method to estimate the temperature differencebetween a region with intruder and a region without intruder ∆Tg = T1 − T2.Let us concentrate on solutions with radial symmetry. The solutions of Eq.(5),

8

Fig. 1. Schematic representation of the regions used to calculate the granular tem-perature. (a) Region around the intruder within a sphere of radius r0 and (b) regionwithout intruder.

for an arbitrary dimension D, satisfy the equation

1

rD−1

d

dr

(rD−1κ0T

1/2g

dTg

dr

)= ξT 3/2

g . (20)

This nonlinear differential equation can be simplified by the fact that thepressure is considered constant throughout the system and remembering thatp ∼ Tg. So, linearizing Eq.(20) the resulting equation may be written in termsof w ≡ T 1/2

g ,

d2w

dr2+

D − 1

r

dw

dr= λ2w, (21)

where

λ2 ≡ ξ

2κ0

. (22)

It is useful at this point to see the implications of the requirement of constantpressure invoked to derive Eq.(21). This estimate is suggested by the fact thatthe effect of the gravitational field can be neglected in the regime of strongperturbations. Then, the momentum balance equation for fluid particles satis-fies ∇p = 0. This is a good approximation if the kinetic energy is much greaterthan the change of the gravitational potential energy experienced over the av-erage spacing between grains. However, we can not forget that, the granularfluid is a system out of “thermal” and mechanical equilibrium (See Section 4bellow). A change in the local granular temperature does change the pressureof the system. To make the model analytically tractable, we have neglectedall the gradients of the pressure. This heuristic assumption is based on the

9

fact that a dense granular fluid can be considered as an incompressible system[41]. Qualitatively, this picture is correct if the “isothermal compressibility”kp (See Sec. 4 Eq.(58)) is bigger than the coefficient of ”thermal expansion” α(See Sec. 4 Eq.(57)). To be more concrete, using the thermodynamic relation(∂p/∂T )V = α/kp, it follows that if kp >> α, then ∂p/∂T ' 0.

Therefore, in the present model, the gradients ∂p/∂T and∇p do not contributeto the description of the granular fluid. The effect of g must be included inthe description of the system at the level of individual grains. These roughapproximations, which should be sufficiently accurate for our purpose, mightnot always be valid. It is necessary to be aware that in proceeding alongthis way some part of the dynamics may be lost. For example, a completeanalysis of Eq.(20) requires much more information, such as the behavior of“compressive waves” arising from the variations of the pressure, which drivesthe system toward the mechanical equilibrium.

The collisional dissipation rate can be decomposed in two parts. We proposethis decomposition supposing that the energy dissipation around the intruderis dominated by the collisions between the small grains and the intruder, thenthe dissipation rate in this region is given by Eq.(15). In the rest of the systemthe dissipation rate is dominated by the collisions between small grains only.In this case the dissipation is given by Eq.(11).

First, let us consider the “inhomogeneous case” when the intruder is localizedin the center of the system (r = 0), see Fig.(1a). The dissipation factor ξ canbe decomposed in two parts: ξ = ξI for the region near the intruder (r = rI),and ξ = ξF for the region (rI < r ≤ r0), where r0 is the radius of the consideredregion.

For the inhomogeneous case we express Eq.(21) as

d2w

dr2+

D − 1

r

dw

dr=

λ2Iw for 0 < r ≤ rI ,

λ2F w for rI < r ≤ r0,

(23)

where, λ2I ≡ ξI/2κ0 and λ2

F ≡ ξF /2κ0. The solution of Eq.(23) is determinedby the boundary condition imposed upon the system. As boundary conditionwe suppose that the system is enclosed by an external surface of radius r0 attemperature Tg(r0) = T0 (respectively, w(r0) = w0).

Let us denote T1−(r) the granular temperature for the region (0 < r ≤ rI), andT1+(r) the granular temperature for the region (rI < r ≤ r0) (respectively,

w−(r) ≡√

T1−(r) and w+(r) ≡√

T1+(r)). The intruder’s presence imposes

10

internal boundary conditions. On the inner surface, the temperature shouldsatisfy

w−(r)|r=rI= w+(r)|r=rI

. (24)

The flux of energy also imposes another internal boundary condition. If wesuppose the flux of energy continuous on the inner surface, from Eq.(4) thegranular temperature should satisfy

dw−(r)

dr

∣∣∣∣∣r=rI

=dw+(r)

dr

∣∣∣∣∣r=rI

. (25)

3.1 Solution for 2D

The solutions to Eq.(23) for D = 2 are a linear combination of the modi-fied Bessel function of order zero w1(r) = I0(λr), K0(λr) [48]. The generalsolution is

w−(r) = A−I0(λIr) + B−K0(λIr) for 0 < r ≤ rI , (26)

and

w+(r) = A+I0(λF r) + B+K0(λF r) for rI < r ≤ r0, (27)

where A−, A+, B− and B+ are constants that must be determined from theboundary conditions.

The function K0(λr) diverges when r → 0, then

B− = 0. (28)

When r = r0 the Eq.(27) should satisfy the boundary condition

w+(r)|r=r0= w0, (29)

this is,

A+I0(λF r0) + B+K0(λF r0) = w0. (30)

On the inner surface the boundary condition (24) w−(rI) = w+(rI) leads to

11

A−I0(λIrI) = A+I0(λF rI) + B+K0(λF rI)

=⇒ A− = A+I0(λF rI)

I0(λIrI)+ B+

K0(λF rI)

I0(λIrI). (31)

The inner boundary condition (25) leads to

A−λII1(λIrI) = A+λF I1(λF rI)−B+λF K1(λF rI)

=⇒ A− =

(λF

λI

) [A+

I1(λF rI)

I1(λIrI)−B+

K1(λF rI)

I1(λF rI)

]. (32)

Equating Eqs.(31) and (32) we find

A+

B+

=λF I0(λIrI)K1(λF rI) + λII1(λIrI)K0(λF rI)

λF I0(λIrI)I1(λF rI)− λII1(λIrI)I0(λF rI),

≡ΘAB (33)

From Eqs. (30) and (33) the constant B+ should be

B+ =w0

ΘABI0(λF r0) + K0(λF r0). (34)

Substituting Eq.(34) into (33) we have

A+ =w0ΘAB

ΘABI0(λF r0) + K0(λF r0). (35)

Substituting Eqs.(34) and (35) into (31) we have

A− =ΘABI0(λF rI) + K0(λF rI)

ΘABI0(λF r0) + K0(λF r0)

(w0

I0(λIrI)

). (36)

The granular temperature in the inhomogeneous case is

T1(r) =

(A−I0(λIr))2 for 0 < r ≤ rI ,

(A+I0(λF r) + B+K0(λF r))2 for rI < r ≤ r0,

(37)

where the constant A−, A+ and B+ are given by the Eqs.(36), (35) and (34),respectively.

12

In the “homogeneous case”, see Fig.(1b), the prefactor λI = 0. Then thegranular temperature T2(r) is

T2(r) =

(I0(λF r)

I0(λF r0)

)2

T0. (38)

Now we are interested in determining the temperature difference ∆Tg betweencase 1 and 2 in the granular fluid. For this we calculate the granular temper-atures at r = rI . So, Eq.(37) and (38) lead to

T1(rI) =

(ΘABI0(λF rI) + K0(λF rI)

ΘABI0(λF r0) + K0(λF r0)

)2

T0, (39)

T2(rI) =

(I0(λF rI)

I0(λF r0)

)2

T0. (40)

Then, the temperature difference is

∆Tg =

(ΘABI0(λF rI) + K0(λF rI)

ΘABI0(λF r0) + K0(λF r0)

)2

−(

I0(λF rI)

I0(λF r0)

)2 T0 (41)

in two dimensions.

3.2 Solution for 3D

When D = 3, the solution of Eq.(23) is given in terms of the spherical mod-ified Bessel functions of zero order w1(r) = i0(λr) = sinh(λr)/λr, k0(λr) =e−λr/λr [48]. The general solution in this case is

w−(r) = A−i0(λIr) + B−k0(λIr) for 0 < r ≤ rI , (42)

and

w+(r) = A+i0(λF r) + B+k0(λF r) for rI < r ≤ r0. (43)

The function k0(λr) diverges when r → 0, then

B− = 0. (44)

The constants A−, A+ and B+, are calculated from the boundary conditionsin a similar way as before:

13

A− =ΘABi0(λF rI) + k0(λF rI)

ΘABi0(λF r0) + i0(λF r0)

(w0

i0(λIrI)

), (45)

A+ =w0ΘAB

ΘABi0(λF r0) + k0(λF r0), (46)

B+ =w0

ΘABi0(λF r0) + k0(λF r0), (47)

where in this case the factor ΘAB is

ΘAB =λF i0(λIrI)k1(λF rI) + λIi1(λIrI)k0(λF rI)

λF i0(λIrI)i1(λF rI)− λIi1(λIrI)i0(λF rI). (48)

The granular temperature in the inhomogeneous case in 3D is

T1(r) =

(A−i0(λIr))2 for 0 < r ≤ rI ,

(A+i0(λF r) + B+k0(λF r))2 for rI < r ≤ r0,

(49)

where the constant A−, A+ and B+ are given by the Eqs.(45), (46) and (47).

In the “homogeneous case” the prefactor λI = 0. Then the granular tempera-ture T2(r) is

T2(r) =

(i0(λF r)

i0(λF r0)

)2

T0. (50)

Again the temperature difference ∆Tg between case 1 and 2 is calculated atr = rI ,

T1(rI) =

(ΘABi0(λF rI) + k0(λF rI)

ΘABi0(λF r0) + k0(λF r0)

)2

T0, (51)

T2(rI) =

(i0(λF rI)

i0(λF r0)

)2

T0. (52)

Then, the temperature difference is

∆Tg =

(ΘABi0(λF rI) + k0(λF rI)

ΘABi0(λF r0) + k0(λF r0)

)2

−(

i0(λF rI)

i0(λF r0)

)2 T0 (53)

in three dimensions.

14

2 4 6 8 101

1.2

1.4

1.6

1.8

2

2.2

φ=rI/r

F

τ = T

1/T2 e = 0.90

e = 0.99

e = 0.80

Fig. 2. Ratio τ = T1/T2 of the granular temperatures, showing non–equipartition ofenergy (τ 6= 1) for different values of the coefficient of restitution e.

3.3 Energy equipartition breakdown

Let us define the temperature ratio τ ≡ T1/T2. In two dimension we have

τ =

(I0(λF r0)[ΘABI0(λF rI) + K0(λF rI)]

I0(λF rI)[ΘABI0(λF r0) + K0(λF r0)]

)2

, (54)

and for three dimensions,

τ =

(i0(λF r0)[ΘABi0(λF rI) + k0(λF rI)]

i0(λF rI)[ΘABi0(λF r0) + k0(λF r0)]

)2

(55)

since λF > λI we can verify that T1 > T2, this means τ > 1. So, the tempera-tures ratio between the region with intruder and the region without intruderare different. In our model this lack of equipartition is due to a difference be-tween the collisional dissipation rate related to the particle sizes. In the elasticlimit e → 1 the energy equipartition is restored τ → 1. In Fig. 2, we presentthe qualitative behavior of τ with the size ratio φ = rI/rF , for different valuesof the coefficient e. The granular temperature difference increases with φ anddepends on e. We can see that τ is nearly constant and very close to unitywhen e = 0.99.

Recently, this quantity was directly measured in experiments performed byWildman and Parker [49] and Feitosa and Menon [50]. They observed thatenergy equipartition does not generally hold for a binary vibrated granularsystem. They reported that the ratio of granular temperatures depends onthe ratio of particle mass densities. Also in fluidized binary granular mixturesthe breakdown of energy equipartition was observed experimentally [51] anddescribed theoretically in the framework of the kinetic theory [52]. Certainly

15

these experiments don’t correspond to the typical conditions for size segrega-tion experiments, but they support the idea of a temperature difference in thesystem due to the presence of the intruder particle. The experimental resultsreported by Wildman and Parker, show that the granular temperature of thelarger particles was higher than that of the smaller particles, this evidencesupports the new picture proposed in this work.

4 Thermal expansion

Granular materials are non-equilibrium systems and certainly they can not beconsidered ergodic in the traditional sense. The system increases its energy asa result of external driving (e.g., vibration) while its decreases its energy bydissipation. There have been different attempts to define a statistical mechan-ics for granular media [53–55]. Recent studies suggest that the thermodynamicdescription proposed by Edwards [54] opens a door towards a statistical de-scription of compact granular matter [56]. However, these findings are for weakdriving and the generalization to stronger forcing is not evident.

The thermodynamic formulation proposed by Herrmann (See Ref. [53] for adetailed discussion) starts from the energy flux balance and the analog forthe “equilibrium” is a steady state driven by the energy flux. If one allowsfor changes in the volume of the system the energy conservation will become∆I = ∆Eint + ∆D + ∆W , where ∆I is the energy that was pumped into thesystem in a given time, ∆Eint is the change of “internal energy” (e.g., kineticenergy), ∆D is the energy dissipated in a given time and ∆W is the work doneto change the volume. Theoretically we can derive the energy relaxation to thesteady state for a driven granular medium [57]. The system can be consideredto be in “equilibrium” when the excess of dissipated energy (∆D = ∆I−∆D)should be zero [53]. Under this theoretical assumption and in the frameworkof the kinetic theory we proceed to calculate the thermal expansion coefficientfor a granular fluid.

The change of mean energy of the system is basically due to a mechanicalinteraction with their external parameters (e.g., the amplitude A0 and thefrequency ω0 = 2πf of vibration, the volume of the system V , and the pressurep). The work W done to change the volume of the system from V to a certainquantity V + dV is equal to the change of its mean energy and its relatedto the mean pressure and volume by dW = pdV + V dp. From the definitionof granular temperature, the change of the granular temperature depends onthe mean kinetic energy of the particles. A volume change dV is related to atemperature change dTg by the equation of state (6).

We can express V as a function of Tg and p, V = V (Tg, p). Thus given in-

16

finitesimal changes in Tg and p, we can write

dV =(

∂V∂Tg

)pdTg +

(∂V∂p

)Tg

dp,

= αV dTg − kpV dp,(56)

where α is the thermal expansion coefficient defined as

α ≡ 1

V

(∂V

∂Tg

)

p

= − 1

n

(∂n

∂Tg

)

p,N

, (57)

and kp is the “isothermal compressibility” defined as

kp ≡ − 1

V

(∂V

∂p

)

Tg

=1

n

(∂n

∂p

)

Tg

. (58)

If in a first approximation we neglect the variations of the coefficients α andkp, we can integrate Eq.(56) and find

V (Tg, p) = V0 exp [α∆Tg − kp∆p] ,

≈ V0 [1 + α∆Tg − kp∆p] .(59)

Now, under the assumption of negligible compressibility, the density changesare caused by temperatures changes alone. From the temperature difference,Equations (41) and (53), the thermal expansion coefficient is

ρ = ρ(1− α∆Tg). (60)

Here, the constant density ρ acts as a reference density corresponding to thereference temperature T0, which can be taken to be the mean temperature inthe flow. This is valid only in some average sense, when all the particles havethe same density.

The thermal expansion coefficient can be derived from the equation of state (6)and definition (57). The general form of the coefficient of thermal expansionis

α =1

T0

C(ν), (61)

where C(ν) is a correction due to the density of the system. In the dilute limitν → 0 and C(ν) → 1, and the above expression tends to the expected value

17

for a classical gas α = 1/T0. The explicit form of C(ν) is given in AppendixB.

5 Segregation forces

Buoyancy forces arise as a result of variations of density in a fluid subject togravity. In the previous section we have introduced the change in the densityof the granular fluid through the thermal expansion produced by the differenceof granular temperatures calculated in Section 3. Now we propose that thisdensity difference leads to a buoyancy force fb, similar to the Archimedeanforce

fb = ∆ρVIg, (62)

where ∆ρ = −αρ∆Tg, VI = ΩD

DrDI is the D–dimensional volume of the intruder

and g is the gravity field. Densities variations driven by granular temperaturegradients due to inelastic collisions, were observed by Ramirez et al., [58] inmolecular dynamic simulations, which in the presence of gravity, produces abuoyancy force driving the onset of convection cells. Recently, experimentalevidence for this buoyancy–driven convection has been reported by Wildman etal., [59]. In our model the temperature gradient is obtained from the differencesof the local density of dissipated energy between the region around the intruderand the region without intruder. The region with intruder is hotter than theregion without intruder. This is a well established theoretical result showsin Section 3. It is important to note here that we are considering particleswith equal material densities, so the buoyant force due to the material densitydifferences fA = (ρF − ρ)VIg doesn’t play any role in our analysis.

The intruder also experiences a viscous drag of the granular fluid. In the gran-ular physics literature we find scarce studies of the forces on objects embeddedinside granular flows [60,61]. Experiments performed by Zik et al., reportedmeasurements of the mobility/friction coefficients of a sphere dragged hori-zontally through a vertically vibrated granular system. They observed a lineardependence of the drag force on the sphere velocity. If the resistive force fdis either linear or quadratic in the velocity, the problem admits an analyticalsolution. In this work the drag force fd is considered to be linear in the velocityof segregation u(t), analogous to the Stokes’ drag force.

fd = −6πµrIu(t), (63)

18

where µ is the coefficient of viscosity of the granular fluid. The state–dependentviscosity possesses the general form [28,32]

µ = µ0

√Tg, (64)

where the prefactor µ0 is a function of the fluid particle properties, and canbe calculated using a Chapman–Enskog procedure for the solution of Enskogtransport equation. The explicit expressions of these prefactors are given inAppendix A.

Equation (63) is assumed to be valid for the particle Reynolds numbers Re =2rF ρu/µ less than unity. Calculating the settling velocity (see below) and thecoefficient of viscosity of the granular fluid (calculated in Appendix A), wecan show that Re ∼ 0 and that the Stokes law assumption should be valid.

Equations (62) and (63) express the acting forces in the segregation process

fseg = fb + fd. (65)

Therefore, the equation of motion that governs the segregation process is

ΩD

DrDI ρ

du(t)

dt= −ΩD

DrDI αρ∆Tgg − 6πµrIu(t). (66)

Now we suppose the granular system contained between two large parallelplates perpendicular to the gravitational field. We take the reference framepositive in the upward vertical direction. Arranging terms in Eq.(66) we findthe following differential equation

du(t)

dt= α∆Tgg− 6πDµ(φrF )1−D

ΩDρu(t), (67)

where we have expressed the intruder’s radius as function of the size ratiodependence rI = φrF , and the solution of this differential equation is the risevelocity of the intruder

u(t) =α∆Tggt0

φ1−D

[1− exp

(−φ1−D t

t0

)], (68)

where the time–scale t0 is given by

t0 ≡ ΩDρ

6πDµr1−DF

. (69)

19

The force balance between the drag force fd and the buoyant force fb gives thesettling velocity us

us =α∆Tggt0

φ1−D. (70)

The time dependent intruder height z(t) is

z(t) =α∆Tggt0

φ1−D

[t− t0φ

(1− exp

(−φ1−D t

t0

))]. (71)

To estimate the granular temperature T0, we adopt the scaling relationshipbetween the granular temperature and the experimental parameter A0 and ω0

proposed by Sunthar and Kumaran for dense vibrofluidized granular systems[57]:

T0 =2√

2

π

mF L(A0ω0)2

NrF (1− e2). (72)

On a qualitative level our model satisfactorily reproduces the observed phe-nomenology: a large intruder migrates to the top of a vibrated bed, and therise velocity increases with the intruder size. The solutions (68) and (71) areplotted in Figs. 3 and 4. Our results resembles the experimental intruder heighttime evolution described in Refs. [5] and [6]. However the model can not de-scribe the intermittent ascent of the intruder since we calculate the meanvelocities. Using the following model parameters: mass particle density of 2.7gcm−3 (Aluminum), rF = 0.1 cm, e = 0.95, ν = 0.7, N = 5 × 103, g = 100cms−2, r0 = L/3 and Γ0 = 1.33 we obtain that the order of magnitude of z(t)(Fig. 4) coincides with the values reported by Cooke et al. (See Fig. 3 Ref.[6]).

From the settling velocity us (70) we show explicitly the dependence on size.This is proportional to the size ratio φ and the granular temperature differ-ences ∆Tg which also depends on the size ratio. Its agrees with the experi-mental fact that the larger the radius of the intruder, the faster is the ascent,reported by Duran et al. [5]. The plotted solution (70), describes qualitativelythe experimental results of Ref. [5], for φ > 4, show in Fig. 5. In this ex-periment, Duran et al., claim the experimental evidence of a segregation sizethreshold at φc = 3.3, below which the intruder does not exhibit any up-wards motion. Our model’s continuous aspect doesn’t allow for the existenceof this threshold. We argue that this discrepancy comes from the fact thatexperimental measures in this regime should be very difficult to carrying out.

20

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

t (s)

v(t)

(cm

/s)

φ = 4

φ = 2

φ = 6

φ = 8

φ = 10

Fig. 3. Intruder segregation velocity u(t). The parameters are: mass particle densityof 2.7 gcm−3 (Aluminum), rF = 0.1 cm, e = 0.95, ν = 0.7, N = 5 × 103, g = 100cms−2, r0 = L/3 and Γ0 = 1.33.

Fig. 4. Intruder height time dependence z(t). The parameters are: mass particledensity of 2.7 gcm−3 (Aluminum), rF = 0.1 cm, e = 0.95, ν = 0.7, N = 5 × 103,g = 100 cms−2, r0 = L/3 and Γ0 = 1.33. Inset: Measured intruder height (Fig. 3,Ref.[6]).

6 Conclusions

We derived a phenomenological continuum description for particle size segre-gation in granular media. We propose a buoyancy–driven segregation mech-

21

2 4 6 8 10 12 14 160

0.005

0.01

0.015

0.02

0.025

φ=rI/r

F

u(φ)

(cm

/s)

Fig. 5. Intruder segregation velocity dependence on φ. The parameters are: massparticle density of 2.7 gcm−3 (Aluminum), rF = 0.75 cm, e = 0.95, ν = 0.7,N = 5 × 103, g = 100 cms−2, r0 = L/3 and Γ = 1.25. The data points come fromRef.[5].

anism caused by the dissipative nature of the collisions between grains. Thecollisional dissipation rate naturally leads to a local temperature differenceamong the region around the intruder and the medium without intruder. Inthis model we proposed that the intruder’s presence develops a temperaturegradient in the system which gives origin to a difference of densities. The gran-ular temperature difference is due to the fact that the number of collisions onthe surface increases with the size of the intruder, but the local density of dis-sipated energy diminishes. So, the region around the intruder its hotter thanthe region without intruder. From this temperature difference we can con-clude that we have a change in the density of the granular fluid. This lead toa buoyancy force that is the responsible for the intruder’s upward movement.

In this work we made use of the tools of kinetic theory of gases to calculate thegranular temperature. We observed a breakdown of the energy equipartition.And this is in agreement with other reported experiments and models. Ina certain sense our theory unifies the different aspects observed in the sizesegregation phenomenon. Explicit solutions of the dependence of height andvelocity are calculated. The geometrical effect of a segregation threshold is notsupported by our model. The intruder size dependence appears naturally inour model.

It seems that in most of the segregation experiments the granular convection isunavoidable [4–7,10,11,20]. It is also important to note that, changes in the sidewalls can induce a transition to flow downward at the center of the container

22

and upward along the boundaries [62,63]. This situation was observed in theexperiments of Knight et al., [4] where the intruder particle moves downwardin the middle of a conical container. However, in spite of the convective flowappeared in experimental setups, in the size–dependent regime there is notdiscernible convective flow in the center of the bed [2,3,5–7,10,13,18], and wecan conclude that convection had no influence (in this regime) on the intrudermotion.

Very recently it has been shown experimentally [59] and by computer simu-lations [58], that the convection phenomenon in granular fluids comes fromthe effect of spontaneous granular temperature gradients, due to the dissi-pative nature of the collisions. This temperature gradient leads to a densityvariations. The convection rolls are caused by buoyancy effects initiated byenhanced dissipation at the walls and the tendency of the grains at the centerto rise. So, this segregation mechanism could be described in the hydrody-namic framework proposed in this work subject to the appropriate boundaryconditions. In the convection regime an additional drag force should appearcoupling the intruder’s movement with the bulk convection stream. Furtherinvestigation is required to deduce the forces associated with the convectiondrag on the intruder and the role of the container geometry.

In this work we only considered the case of the size–dependence on the segre-gation of a single intruder in a granular medium. The interplay between theintruder size and material density dependence will be the subject of futurework.

Acknowledgments

We thank M. Alam for helpful comments on the manuscript. One of the au-thors (L.T.) would like to thank A.R. Lima for friendly support, Prof. SidneyR. Nagel for discussions concerning the experimental aspects of the segrega-tion problem and convection, and the ICA–1 for their hospitality while partof this project was carried out.

Appendix

A Transport coefficients

In this appendix the prefactors appearing in Eqs.(9) and (64) are derived.Using a Chapman–Enskog procedure for the solution of the Enskog trans-

23

port equation, the transport coefficients for nearly elastic particles have beenderived in Refs. [28] and [32].

In 2D the thermal conductivity κ is [28]

κ = 3nrF

(π

mF

)1/2 [1 +

1

3

1

G+

3

4

(1 +

16

9π

)G

]T 1/2

g , (A.1)

where G is νg0, g0 is the 2D pair correlation function given in Eq.(7), and ν isthe area fraction ν = nπr2

F . It is convenient to express Eq.(A.1) introducingthe prefactor κ0 defined as

κ0 ≡ 3nrF

(π

mF

)1/2 [1 +

1

3

1

G+

3

4

(1 +

16

9π

)G

]. (A.2)

The result (A.1) takes the form

κ = κ0

√Tg. (A.3)

In 3D the thermal conductivity is [32]

κ =15

8nrF

(π

mF

)1/2 [1 +

5

24

1

G+

6

5

(1 +

32

9π

)G

]T 1/2

g , (A.4)

where G is νg0, g0 is the 3D pair correlation function given in Eq.(8), and νis in this case the volume fraction ν = 4πnr3

F /3. In 3D the prefactor κ0 isdefined as

κ0 ≡ 15

8nrF

(π

mF

)1/2 [1 +

5

24

1

G+

6

5

(1 +

32

9π

)G

]. (A.5)

The shear viscosity µ in 2D is [28]

µ =1

4nrF (πmF )1/2

[2 +

1

G+

(1 +

8

π

)G

]T 1/2

g . (A.6)

It is convenient to express Eq.(A.6) introducing the prefactor µ0 defined as

µ0 =1

4nrF (πmF )1/2

[2 +

1

G+

(1 +

8

π

)G

]. (A.7)

So, the result (A.6) takes the form

µ = µ0

√Tg. (A.8)

24

In 3D the shear viscosity is [32]

µ =1

3nrF (πmF )1/2

[1 +

5

16

1

G+

4

5

(1 +

12

π

)G

]T 1/2

g , (A.9)

and the prefactor µ0 in 3D is defined as

µ0 =1

3nrF (πmF )1/2

[1 +

5

16

1

G+

4

5

(1 +

12

π

)G

]. (A.10)

B Thermal expansion coefficient

We can consider the volume of the system as a function of the granular tem-perature and the pressure V = V (Tg, p). A change in the granular temperaturedTg and the pressure dp, leads to the corresponding change in the volume dV

dV =

(∂V

∂Tg

)

p

dTg +

(∂V

∂p

)

Tg

dp. (B.1)

As we have supposed that the pressure of the system is more or less constant,we can approximate dp ∼ 0. The increment of volume dV with an incrementof the granular temperature dTg is

dV =

(∂V

∂Tg

)

p

dTg. (B.2)

Thus,

dV

dTg

=

(∂V

∂Tg

)

p

, (B.3)

or

(∂V

∂Tg

)

p

=

(∂Tg

∂V

)

p

−1

, (B.4)

and in terms of the number density n, we have

(∂n

∂Tg

)

p,N

=

(∂Tg

∂n

)

p,N

−1

. (B.5)

25

From the definition of the coefficient of thermal expansion Eq.(57), and fromthe above statement, we find

α = − 1

n

(∂n

∂Tg

)

p,N

= − 1

n

(∂Tg

∂n

)

p,N

−1

. (B.6)

The partial derivative (∂Tg/∂n)p,N can be calculated from the equation ofstate (B.7). In 2D the equation of state is

p = nTg

[1 + (1 + e)

(ν

1− ν+

9

16

ν2

(1− ν)2

)]. (B.7)

where ν = nπr2F . So, an elementary calculation leads to

(∂Tg

∂n

)

p,N

= (ν − 1)p

n2

×[

716

(1 + e)− 1](ν3 − 3ν2)− (1− 2e)ν + 1

[716

(1 + e)− 1]ν2 + (1− e) ν − 1

2 (B.8)

From (B.6) one obtains:

α =n

(1− ν)p

×[

716

(1 + e)− 1]ν2 + (1− e) ν − 1

2

[716

(1 + e)− 1](ν3 − 3ν2)− (1− 2e)ν + 1

(B.9)

Using the equation of state (B.7) we can express α in function of the granulartemperature

α =1

Tg

(1− ν) [1 + (1 + e)G]−1

×[

716

(1 + e)− 1]ν2 + (1− e) ν − 1

2

[716

(1 + e)− 1](ν3 − 3ν2)− (1− 2e)ν + 1

(B.10)

this is

α =1

Tg

C(ν), (B.11)

where the correction coefficient due to the density of the system is defined as

26

C(ν)≡(1− ν) [1 + (1 + e)G]−1

×[

716

(1 + e)− 1]ν2 + (1− e) ν − 1

2

[716

(1 + e)− 1](ν3 − 3ν2)− (1− 2e)ν + 1

(B.12)

For three dimensions the equation of state is

p = nTg

[1 + (1 + e)

(ν

1− ν+

9

16

ν2

(1− ν)2

[3 +

ν

2(1− ν)

])]. (B.13)

In a similar way we find for 3D that the coefficient of thermal expansion is

α =1

Tg

(1− ν)2 [1 + 2(1 + e)G]

−1

×ν3 − (2− e)ν2 + (1− 2e)ν − 12

ν4 − 4ν3− (5− e)ν2 + 4eν + 1(B.14)

and the correction coefficient C(ν) in 3D is defined as

C(ν) =(1− ν)2 [1 + 2(1 + e)G]

−1

×ν3 − (2− e)ν2 + (1− 2e)ν − 12

ν4 − 4ν3− (5− e)ν2 + 4eν + 1(B.15)

where ν = 4nπr3F /3.

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