E L S E V I E R Powder Technology 92 (1997) 75-80

POWD CHNOWh'Y

Effects of geometry on the characteristics of the motion of a particle rolling down a rough surface

Maria Alejandra Aguirre a Irene Ippolito b, Adriana Calvo a Christian Henrique a ,b

Daniel Bideau b,. a Grupo de Medios Porosos, Departamento de Fisica, Facultad de ingenier(a, Universidad de Buenos Aires, Paseo Coldn 850,

1063 Buenos Aires, Argentina b Groupe Mati~re Condensge et Matdriaux, Universitd de Rennes L Bdtiment IIA, Campus de Beaulieu, 35042 Rennes Cedex, France

Received 4 March 1996; revised 4 November 1996

Abstract

The different dynamic regimes of a sphere rolling on a rough inclined surface have been studied experimentally, with special attention to the effects of energy dissipation and geometry. A controlled roughness of the surface is obtained by gluing glass beads or sand onto a plane. Three different types of rolling spheres (steel, glass and plastic) have been used on two kinds of rough surface (glass beads and sand). The different regimes can be presented in a phase diagram whose control parameters are 0, the inclination angle of the plane, and R, the radius of the rolling sphere (in fact, we use @, the surface smoothness, which is the ratio of R and r, the radius of the glued beads). A comparative analysis is done, showing that the behaviour is qualitatively not dependent on the system, and a 'universal' dynamical threshold line is obtained for all the particle-surface systems studied. We also confirm that the motion is characterized by a viscous friction force, strongly dependent on the smoothness.

Keywords: Granular flow; Surface flow

1. Introduction

Granular flows are of great importance due to their natural and industrial applications in chumical engineering, construc- tion, grain storage, pharmaceuticals, etc. Surface flows are particularly important and fascinating because of their com- plexity, and have been the topic of many experimental, numerical and theoretical studies during the last ten years [ 1,2]. Even if one forgets the very important work done on avalanches and surface equilibrium following the paper by Baket al. on self-organized criticality (SOC) [ 3 ], important contributions have been proposed on segregation by surface flow [4,5 ]: the trajectories of grains rolling on arough surface are very dependent on their size, because the roughness 'seen by a grain' depends on its own size compared with the size of the bumps beneath it. A large grain has a greater chance to reach the bottom of an inclined surface than a small grain, and that explains the segregation generally observed on the slope of a mountain or at the surface of a heap.

* Corresponding author. Tel.: + 33 299-286 205; fax: + 33 299-286 717; e-mail: pmhc @univ-rennesl.fr.

0032-5910/97/$!7.00 © 1997 Elsevier Science S.A. All rights reserved Pli S0032-5910 (97) 03231-2

Some of us have recently reported [ 6,7 ] experimental and theoretical results on the motion of a sphere on an inclined rough surface (the roughness is well controlled because it is obtained by gluing glass beads of known variable size distri- bution onto a plane). Indeed, this model is not able to account for all reality. Plastic deformation of the sand at the surface of a pile under the impact of a rolling ball is absent here and the viscosity of the glue is added. However, it is for us a good one; it is as simple as possible and the geometry is preserved. In the particular case of steel spheres rolling down a rough surface made of glass beads, Riguidel et al. [6] found three regimes of motion, depending on the inclination angle 0 of the plane with the horizontal, and the surface smoothness 4' of the system, defined as the ratio between the radius R of the rolling sphere and the radius r of the glued glass beads: ~= R/r. For small angles 0 and for all the studied values of O, the motion is decelerated (regime A) and, in all cases, the ball stops suddenly a few centimetres from the point of its release. For intermediate angles, the ball, after moving down some centimetres, reaches a constant mean velocity and tra- verses the whole plane: this is called regime B. In these two regimes, some balls stop suddenly when trapped in a 'large cavity' [7]. One can say that the motion in these cases is

76 M,A. Aguirre et al. / Powder Technology 92 (1997) 75-80

characterized by two noises which originate from the disor- dered nature of the roughness: a 'small' one corresponding to the microscopic shocks of the rolling ball with the glued grains and a 'large' one which corresponds to the trapping. For large angles 0, a third regime, C, is observed. It is char- acterized by jumps on the surface of the plane: the scale of the motion is completely changed and it cannot be studied t, xperimentally even with a 2 metre long system. However, this last regime is generally transitory, eventually leading to a constant velocity, as observed in numerical simulations [ 8 ]. The dependence of this constant velocity on the experimental parameters ( 0, 4,) is not known and is possibly quite different from that of region B. The central question in all these exper- iments is how energy dissipation (shocks and friction) depends on the roughness: for example, in the constant veloc- ity regime the dissipation is essentially due to a macroscopic viscous force whose origin is not yet well understood [7]. More precisely, the question is to know the dependence on the geometry of such a viscous force (observed by Riguidel et al. [6] in a particular case) and if it is generally observed for different types of roughness.

More generally, the dynamic friction in granular flow remains unknown: between the static ( V - 0 ) friction described, for example, by Coulomb [9], and the quadratic dependence of the friction force on the velocity - - or on the shearing rate - - described by Bagnold [ 10], nothing is well understood from a general point of view [ I l ].

In this context, an analysis as complete as possible of a simple model is important, in order to know if the motion depends or not on the experimental system characteristics (roughness, material of the rolling ball, etc.), and to try to quantify its eventual effects. Such a study is the aim of this paper, in which we present a comparative experimental anal- ysis made in a geometry similar to that just described, but for three different kinds of particles - - steel, glass and plastic-- and two different surface materials-- sand or glass beads. In this work, the size of the glued grains (glass beads or sand) is kept constant.

2. Experimental set-up

For this study, it is necessary to have a rough surface with a controlled roughness and geometry. So, we use a plywood fiat plane, 1 m long, 70 cm wide, and 3 cm thick, supported by a rigid metallic frame fixed at one of the edges to a table. The other edge may be lifted by sliding it between two vertical rails, as shown in Fig. 1. In such a way, we can vary and control the inclination angle 0 with a precision of + 0.2 °. We have verified that the defects in the flatness of our system are negligible compared with the surface roughness. An analysis of the statistics of the ball trajectory - - very sensitive to the defects - - shows immediately if the plane is not flat enough. In order to do that, we analysed the dispersion of the particle trajectories in the direction transverse to the driven motion

Fig. 1. Schematic view of the inclined rough surface.

and it was found to be Gaussian, showing that there are no significant defects on the rough surface.

Our rough surfaces are obtained by gluing glass beads or sand onto contact paper. The glass beads have a mean radius r equal to 0.25 mm with a dispersion of 0.07 ram. We used sieved river rolled (not angular) sand with a mean grain size between 0.2 and 0.3 ram. The surfaces are built by spreading the glass beads (or the sand grains) on contact paper with a technique that allows us to obtain a disordered monolayer of grains. The surface packing fraction of each surface was determined by an image analysis system. For glass beads, we found a surface packing fraction of 0.75 +0.03. This is smaller than the usual packing fraction obtained in two- dimensional dense disordered packings of disks. For sand, we found a surface packing fraction of the order of 0.85 +0.05. This value is clearly overestimated due to the distribution of shape and size of the grains, which leads to a projected surface larger than the one really occupied. We can consider, then, that the packing fraction values are approxi- mately the same for both rough surfaces. The packings are nevertheless very homogeneous, for spheres as well as for sand, as can be seen in Fig. 2 which shows two typical acquired images (after processing) of the rough sandy (a) or glassy (b) planes. In consequence, we have a well-char- acterized surface smoothness 4". We note that, in this partic- ular case, the surface roughness is defined as r/R. Here r is kept constant, and then R is the true variable, but we use 4" as a dimensionless variable. This 'rough sheet' is they placed on the inclined plane.

We used rolling spheres of three different materials: steel, glass and plastic, all of them with diameters varying from 1 to 8 mm. In order to have a quantitative idea of the difference between the plastic properties of each material, we deter- mined e, the coefficient of restitution of each type of sphere, by measuring (using a video system) the height h at which a ball of a given material rebounds on a heavy plate of the same material when dropped from a constant height he = 30 cm. In fact, the restitution coefficient depends on the impact velocity, but here it is only used to compare the different materials. In this case, the restitution coefficient e is defined as e:'= h/he. We found es = 0.90 +0.02 for steel, eg-0.85 +_ 0.02 for glass and ep = 0.83 _+ 0.02 for plastic. The measured densities of the materials are #s = 7.9 g/cm 3 for steel, pg = 2.5 g/cm 3 for glass and pp = 1.4 g/cm 3 for plastic.

M.A. Aguirre et al. / Powder Technology 92 (1997) 75--80 77

of the beads is observed, and their time to traverse the surface is measured every 20 cm in the constant velocity regime. This procedure is repeated for several values of • at each 0, with 0 ranging between 0.5 ° and 18 °. Depending on the ~ and 0 values, the beads have a decelerated motion and stop suddenly on the plane before reaching the first mark (regime A) [ 12 ], descend with constant velocity, and eventually suddenly stop (regime B) or, for large 0, start to jump (regime C).

3. Experimental results

3.1. Critical static angle

Fig. 2. Photographs showing a partial view of (a) the sandy and (b) the

glassy rough surfaces.

As a characterization of the (relative) roughness of the surface, we determined the 'static' angle thresholds 0s for steel balls as follows. For a given value of ~, a row of balls is placed in equilibrium on the rough plane, initially horizon- tal; then the plane is smoothly lifted and the value of 0 for which 50% of the balls start to move is taken as the static threshold angle. The comparison between the 0s values obtained for sand and glass will be useful for the geometrical characterization of the roughness.

For each rolling particle-surface system, the motion of the beads depends only on R and 0. By varying these two control parameters, we can easily study the different regimes of the particle motion, obtaining the respective phase diagrams described in the next section. The experimental procedure to obtain the phase diagrams is as follows. For a given surface and bead size, that is, for a given smoothness ~, and for a fixed inclination angle 0, 50 balls are successively released with roughly the same small initial kinetic energy, and their velocities are then measured. Concerning the influence of the initial conditions of the motion, we have shown experimen- tally [ 12] that the ball quickly loses the memory of its initial energy both in the A and B regimes: the critical line does not depend on the initial conditions. The driven stochastic motion

Fig. 3 shows the va~'iation of the static threshold angle 0s with q~, using steel balls ot different diameters both on the sandy and glassy surfaces, on a logarithmic scale. In this representation, the two curves are parallel straight lines, which corresponds to a power law with an e~.ponent of - 0 . 6 + 0.1. The values of 0 for which the balls ie¢e their equilibrium have a Gaussian distribution, whose width, otter- mined from experiments with 200 balls, is roughly the same for both surfaces. Experiments show that the static threshold is strongly dependent on the surface. For the same smoothness • , the values of the static angle are larger for the sand surface than for the glass one, showing the influence of the grain geometry. The irregular sand grain shape (even if rolled sand is used) gives rise to more stable configurations. These stable configurations are defined by at least three contact points of the ball with the grains constituting the plane.

On the other hand, in our experiments, both surfaces are homogeneous and their packing fractions almost equal. Then, it can be expected that the cavity size distribution - - as seen by the bali m is roughly the same for the sand and glass surfaces, leading to the same power law, 0s = 0o/i~ for tile static threshold (0o~ is the value of 0 when • = 1, obtained from the linear regression in Fig. 3). For the sand surface, 0os was found equal to 43°+ 1 ° and for glass 0og = 34°+ 1 °. In addition, the experimental value found for the exponent a, - 0 . 6 + 0.1, is almost equal to that reported previously [ 6] for steel balls rolling on a glass bead surface.

a:~ 1.8 ~ 1.6 ---, !.4

1.2 !

0.8 0.6 0.4 0.2

0 i i i i ! i i ~.

0 0.2 0.4 0.6 0.8 t 1.2 1.4 t.6 log

Fig. 3. Logarithmic variation of the static threshold 0s vs. surface smoothness • for steel bails on the sandy ( I ) and glassy (I-I) surfaces.

78 M.A. Aguirre e! aL / Powder Technology 92 (1997) 75-80

3.2. Phase diagrams

Phase diagrams for each rough surface and for each kind of bead material are shown in Fig. 4. In ~I cases, zone A corresponds to the decelerated regime, zone B to the constant velocity regime and zone C to the regime in which the bead jumps .

The first thing to say when observing these six diagrams is that the behaviour is qualitatively the same in all cases: only relatively small displacements of the limit between zones are observed.

Now comparing these phase diagrams more closely, it can be seen that:

(i) the width of zone A is larger for the sand surface than for the glass one, for all kinds of beads, which is an indication

conforming with intuition ~ that the amount of energy dissipated in collisions by the rolling ball is larger in the former case due, in particular, to the different nature of the contacts;

(ii) plastic beads, less dense and with a coefficient of restitution smaller than the glass or steel ones, have a larger (in terms of 0) constant velocity regime for both surfaces.

The apparent 'universality' in the phase diagram led us to a more accurate determination of the 'critical lines'. Fig. 5 is a log-log plot giving the variation with 4 of 0^n, values of the inclination angle corresponding to the 'critical line' A- B. It can be seen that the data corresponding ~o both surfaces fall on two parallel straight lines of slope - 1.3 + 0, I, which suggests a power law variation:

18 ~' 16

14 12 ~ 10 8

18

' i ~ glass-steel 14 l \ ¢

' T ' ~ " " ' "' ' " " ' :o

" ~ ~ - e 2 ~ " ' , ~ ' ' " " ' 4

0 18 16 ~ t ~ l , , 4 q t iP o ~ ~ t i t , t 9

n ' ~ ' " \ . . . . . . . . . . . . . tO ° ~ " ~ " , , , , , , 0 , , *

s ~ , , ~ : , " , = , : ! . . . . . . . .

0 2 4 6 8 10 12 14 16 18

o

\ glass.plastic 1 1 ~ 9 , l . t , t t ~0 6 ~ g ~ , t ' t 't t t t t

a k

dV *e " e * e-~& t t t "'@~ , t 2'

O 2 4 6 8 10 12 14 IG 18

0

O

18 16 14 12 10

8

: | \ ..d.,,ool

.... , ¢ , . , ~ . . . . . . , .

; ; ; l I l ; : ."

0 2 4 6 8 10 12 14 16 18

0

. . . . . . . . . .

, o t ' " t ' " ~ ' " " ' ' st'" ~ ' '~,.~ . . . . . . ~ t . . . . . xe2 , ~ v , : . . . . 4 I . . . . . . . - N . . ~ , .

" l ' " " " : ' ' . ' " " - ' - " " 0 . . . . . .t ! ! l I

0 2 4 6 8 10 12 14 16 18

o

] \ sand-plastic o o ~ , , ~ . . . , . , , ,

4 me w m • w • m w rr..~..d~

2

0 2 4 6 8 ,10 12 14 16 18

o Fig, 4, Phase diagrams corresponding to the six studied surface-particle systems.

100

I0

I

0.1 1 10 100

O Fig. 5. Log-log plot of the plane inclination angle 0Ae corresponding to the critical line AB vs. the surface smoothness O. Open symbols correspond to the glass surface and filled ones to the sand surface.

0 A B (X 4 - ! ' 3 ( 1 )

We have also made a comparative analysis of the critical lines B-C, but in this case no universal law was found: the equation of the line depends strongly on the rolling ball- surface system.

The most important result of previous work [7 ] was to show that, in regime B, dissipation was due to a viscous force of the form fox (I/') where (V) is the mean velocity of the ball. In this regime, we have found that

(V)cz 4 ~ sin 0 (2)

The aim is to prove the general validity of this result obtained only for steel balls rolling on a glass bead surface. particularly its dependence on sin 0 and the power law on 4 (the value of/3 seer,,s to be dependent - - even if weakly - - on the system, particularly on r). In order to do that, we have perform~:d careful experiments on our six systems. In each one, we measured the constant mean velocity (V') reached by a rolling bali in regime B as a function of 0 and 4 ((V') for each triangle in Fig. 4). We found, in all cases, a linear dependence of (V~ on sin 0 for all values of 4.

In Fig 6 we show these results for the system glass-glass and for :wo values of 4 (4 and 6). It can be seen that the slopes of the straight lines increase with 4. In each case we adjusted the ~et of straight lines corresponding to one system with a relation of the type

~ 1 0

~ .

5

4 3

2 !

0 ' I I . . . . . I I

o 0.05 oJ o J5 0.2 sin 0

Fig. 6. Experimental values of <10 as a function of sin 0 for the glassy surface and glass rolling beads: r'l, O = 4; II , q~-- 6.

M.A. Aguirre et al. / Powder Technology 92 (1997) 75-80 79

~ 2.5

~v! , Oca 1.5

I

0.5

0 I I I I I I I

o 0.2 0.4 0.6 0.8 I 1.2 i.4 log

Fig. 7. Variation of log[ ( ( IO - VoD/sin 0l vs. log @ for glass (open sym- bols) and sand (filled symbols) surfaces.

The experimental relation Eq. (1) for the 0 values on the critical line A-B can be used in Eqs. (5) and (6), and one finds that all the points on this line, for each surface, have almost the same mean velocity (VAB)~. Indeed, this result is in good agreement with measured mean velocities corre- sponding to the points in zone B (triangles in Fig 4) nearest to line A-B. These (VAB) values are (VAB)s = 6.7 +0.7 cm/s for the sand surface and (VAB)g = 5.2 5:0.4 cm/s for the glass one.

4 . D i s c u s s i o n

(Vi)=Voi+Ci(@) sin 0 (3)

where the index i = 1 . . . . . 6 indicates the system. We per- formed linear regressions ever all curves, that is for each surface-ball system and for each @ value. We found that Vog = 2.4 -t- 0.4 cm/s for the glass surface and Vos = 1.6-1- 0.4 cm/s for the sand one. Note that we are measuring mean velocity values over an 80 cm length of rough surface. So, the data dispersion does not correspond to actual experimen- tal errors in time, distance or angle measurements, but essen- tially to longitudinal and transversal dispersion of the rolling ball path. Taking this fact into account, we can say that the Vo~ value is the same for both surfaces and equal to 2 + 0.4 cm/s. This value does not correspond to the velocity for which sin 0 = 0; note that the constant velocity regime is not valid for very small values of 0.

In order to study the function Ci(@), we plotted the loga- rithmic variation of C~(@) = ( ( V ) - Vo~)/sin 0 with log q~ (Fig. 7). On this scale we observe two straight lines corre- sponding to both surfaces for all the experiments, which suggests a power law of the type

C:(@) =Coi@ # (4)

where Co~ is a constant which depends on the system. In order to determine Coi and/3 we performed linear regressions on each surface-rolling ball system. The following values of/3 were found: 1.3, 1.4, 1.4, 1.4, 1.5, 1.4 for the sand surface (glass, steel and plastic balls) and the glass surface (glass, steel and plastic balls), respectively. The Co~ were found to be equal to 4.6 + 0.5 cm/s for the three sand surface systems and equal to 5.7 5:0.3 cm/s for all the glass surface systems.

Finally, a general expression can be established for the mean velocity, expressed in cm/s, on each type of surface:

(V~) =2+4 .6@ TM sin 0 sand surface (5)

(Vg) = 2 + 5.7~ TM sin 0 glass surface (6)

We can conclude that the general power law (V) cx @ ~ sin 0, with/3 = 1.4 5: 0.1 can be stated for the mean velocity in regime B for all studied systems characterized by the same r [ 13].

We are dealing here with small values of 0 (2 ° < 0< 18 °) and for values of • (2 < ~ < 16) which are not too large.

The phase diagrams described above are very similar even though they describe systems with very different geometrical (the sand surface is a priori different from the glass one) and physical properties (different restitution coefficients), with (roughly) the same value of r. These results confirm the general existence of three different regimes (results obtained by Riguidel et al. [ 6,7 ] just for a steel-glass system) for the dynamics of a ball on a rough surface independent of the nature of the particle-rough surface system. We also remark that the constant ve/ocity regime is larger than the one intui- tively expected according to the balance of energy in which the distribution of impact angles is crucial [ 14].

The base of this 'universality' is essentially of geometrical origin: using the fact that curves giving the static threshold angle (Fig. 3) are parallel, we can normalize the q~ values for glass to obtain c,nly one curve for both surfaces. If we use the same normalization for the critical line A-B obtained for a glass surface, all the data (glass and sand) almost coincide and 0^, can be represented by the same curve, shown in Fig. 8. Note that this normalization is obtained only from geometrical results: q~by itself is the only pertinent parameter for describing the static threshold. So, in this geometrical case, the normalization of R by r keeps all its sense.

As said before, the dynamics of a ball rolling on a rough surface can be phenomenologically explained if one assumes that the motion of the ball depends on two 'noises', at least in the A and B regimes: a small one which leads to dissipation in shocks through a viscous friction force and a large one which acts as a trap for the ball [7,14]. Trapping by 'large' noise can be due to (at least) two origins: the first comes

I00

m io

0.1 I •

I0 I~

Fig. 8. Same as Fig. 5 but here the values of @ for glass are normalized as indicated in the text. The two curves coincide.

80 M.A. Aguirre et al./ Powder Technology 92 (1997) 75-80

from the fact that the ball falls suddenly into a 'cavity' which is large compared to others; the second is related to the angle at which the rolling ball collides with the grain: sometimes this angle is such that it induces a back-reflection of the ball. In both cases we can consider that the ball is trapped in a well, and we have shown that these wells are uniformly dis- tfibuted on the plane: the distribution oftbe distance from the origin at which the ball stops has been found to be exponential in the B regime [7].

Small noise leads to fluctuating losses of momentum in the direction of the motion. These fluctuations can be regarded as uncorrelated. They have a non-zero mean value ~ 7)- The fact that we experimentally find a viscous dissipation leads to (~) cx ~ - ~ [7], independent of the bali velocity.

Large noise seems to give a different result. According to experimental observations, one can consider that the ball is stopped in only one step (i.e. at a distance of the order of r), during which it completely loses its momentum in the direc- tion of motion.

In the case of regime A, the velocity decreases. The amount of momentum the ball has to lose before it stops is not nec- essarily larger than (el), because it can be very small in this decelerated regime.

In regime B, the ball stops when the momentum fluctua- tions are large, with a maximum value ~m~ > (el>. This can be a criterion for distinguishing between the two regimes, depending a priori on the geometry of the rough surface, on the velocity of the ball, and eventually on the preceding collision.

With this description, the problem is clearly dominated by geometry, responsible for both kinds of noise. But if it is relatively easy to analyse experimentally the consequences of the small noise (viscous force), it is difficult to make a quantitative prediction about large fluctuations responsible for trapping, which is very important for studying size segregation.

All these experiments show that new experimental tech- niques are necessary to obtain complementary information, in particular at the microscopic level. At present, experiments

to analyse the noise of the motion of a sphere rolling on the rough surface by using a microphone are in preparation [ 12 ]. This kind of study allows us to obtain information about the number of shocks, instantaneous velocities and fluctuations, type of friction force, etc.

Acknowledgements

We would like to thank G.G. Batrouni and L. Samson from GMCM and R. Chertcoff from FIUBA for very interesting discussions. D.B. and C.H. wish to acknowledge the hospi- tality of the Grupo de Medios Porosos. This work was sup- ported by the agreement PICS 145 CNRS-CONICET and a grant from SECyT of the Universidad de Buenos Aires.

References

[ 1 ] D. Bideau and A. Hansen, Disorder and granular media, in Random Materials and Processes, Elsevier, Amsterdam, 1993.

[2] A. Mehta, Granular Matter: an Interdisciplinary Approach, Springer, New York, 1994.

[3l P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett., 59 (1987) 381. [4] S. Savage, lnterparticle percolation and segregation in granular

materials, in A.P.S. Selvadurai (ed.), Developments in Engineering Mechanics, Elsevier, Amsterdam, 1987.

[5] R. Jullien and P. Me.akin, Nature, :144 (1990) 425. [6] F.X. Riguidel, R. Jullien, G. Ristow, A. Hansen and D. Bideau, J. Phys.

/, 4 (1994) 261. [7] F.X. Riguidel, A. Hansen and D. Bideau, Europhys. Lett., 28 (1994)

13. [8] G.H. Ristow, F.X. Riguidel and D. Bideau, J. Phys. !, 4 (1994) ! 161. [9] C.A. Coulomb (de), Mdmoires de Math~matiques et de Physique

prc~sentds d I'Acaddmie Royale des Sciences, 7 (1773) 343. [ 10] R.A. Bagnold, Proc. R. Soc. London, Ser. A, 225 (1954) 49. [ 11 ] H.M. Jaeger, C.H. Liu, S.R. Nagel ar ~l T.A. Witten, Europhys. Lett.,

i i (1990) 619. [ 121 C. Henrique, M.A. Aguirre, A. Cairo, l. Ippolito and D. Bideau,

Powder Technol., submitted. [ 13l L. Samson, personal communication. [ 14] G.G. Batrouni, S. Dippel and L. Samsoit, Phys. Rev. E, 56 (1996)

6496.