Jamming dynamics in grain mixtures : An extended hydrodynamic approach
Jamming dynamics in grain mixtures : An extended
hydrodynamic approach
Supurna Sinha
Raman Research Institute, Bangalore 560 080,India
(Dated: November 16, 2018)
Abstract
We study jamming in granular mixtures from the novel point of view of extended hydrodynamics.
Using a hard sphere binary mixture model we predict that a few large grains are expected to get
caged more effectively in a matrix of small grains compared to a few small grains in a matrix of
larger ones. A similar effect has been experimentally seen in the context of colloidal mixtures.
PACS numbers: PACS numbers: 87.15.-v,05.40.-a,36.20.-r
http://arxiv.org/abs/0704.1895v1
I. INTRODUCTION
In recent years granular matter has emerged as an active area of research[1]. Interest in
this field has grown as a result of observations coming from a large number of interesting and
relatively low-tech experiments [2]. In particular, a vibrated granular system[3] consisting of
a large number of macroscopic grains in motion, provides us with a “scaled up” fluid system
where we can explore the similarities and differences between such a large scale system and
a microscopic atomic fluid.
Recent studies[1, 4, 5] in dense granular systems indicate that there is a striking similar-
ity between the dynamics of granular materials and glassy dynamics in atomic fluids and
colloids[6]. In this Letter, we exploit this analogy to understand some facets of jamming
in granular matter. In particular, we study the role of compositional disorder in form-
ing jammed configurations which slow down the dynamics and eventually result in a state
of structural arrest characteristic of a glass[7, 8, 9, 10]. The observations made here are
analogous to jamming effects studied in atomic liquid mixtures and colloids [7, 11, 12].
The behavior of a fluid at large length and time scales is well described by hydrodynamics.
The hydrodynamic description has been extended to molecular length scales in an extended
or generalized hydrodynamic description [7, 13]. In a generalized hydrodynamic framework
the basic structure of the hydrodynamic equations is retained and the static susceptibili-
ties and transport coefficients are wave-vector dependent to account for the nontrivial static
correlations that come into play on molecular scales in a dense liquid. The generalized hydro-
dynamic description has proved to be very successful in describing the dramatic narrowing
of the central diffusive peak in a neutron scattering spectrum S(k, ω) of a dense liquid[14].
Such a slow decay of density fluctuations is a precursor to glassy dynamics in dense liquids.
A binary liquid mixture turns out to be more effective in forming glasses compared to a
one component liquid since compositional disorder leads to jammed configurations which
prevent the system from reaching its global equilibrium crystalline configuration[15]. Gen-
eralized hydrodynamic studies of dense binary hard sphere mixtures in the context of glass
transition suggest that at intermediate wavevectors (i.e. equivalently on length scales of the
order of the average of the molecular diameters of the two species) the density fluctuations of
the two species emerge as the slowest decaying fluctuations and therefore dominate the slow
dynamics of the system [7, 12]. In a self-consistent mode-coupling theory (MCT), nonlinear
couplings of these slowly decaying modes of density fluctuations lead to a glassy state where
structural relaxation is frozen[8, 9].
In this Letter we use, for the first time, generalized hydrodynamic techniques developed in
the realm of atomic liquids to understand the behavior of jammed configurations in granular
mixtures. Such an approach to granular matter enables us to take into consideration non-
trivial static correlations stemming from the granularity or finite sizes of the particles, which
in turn, influences the high density jamming dynamics of the system. We thus gain new
insight into granular mixture dynamics and make predictions for future experiments in such
systems.
II. EXTENDED HYDRODYNAMIC APPROACH TO A BINARY MIXTURE OF
GRAINS
Consider a binary granular mixture of hard spheres of diameters σ1 and σ2 (σ2>σ1),
masses m1 and m2, number densities n1 and n2 and of total packing fraction η =
1 + n2σ
2 ]. An analysis of the extended hydrodynamic equations of such a binary
hard sphere mixture suggests that on length scales of the order of the size of the hard sphere
particles [7] momentum and temperature fluctuations decay very fast and the slow dynamics
of the system can be well described in terms of the modes of density fluctuations of the two
species[12, 16]. This enables us to confine to a two mode description of the system on length
scales of the order of the diameters of the hard sphere granular particles at high densities.
It is convenient to describe the system in terms of the following linear combinations of the
density fluctuations of the two species- the total mass density fluctuation at wavevector ~k
ρ~k = ρ1~k + ρ2~k
and the concentration fluctuation at wavevector ~k
c~k =
where ρ
and ρ
are the mass density fluctuations of species 1 and 2, ρ1 = m1n1 and
ρ2 = m2n2 are the equilibrium mass densities of species 1 and 2 and ρ = ρ1 + ρ2 is the total
equilibrium mass density. The set of Laplace transformed coupled extended hydrodynamic
equations of this system is given by:
ρχT (k)γL(k)
ρ~k(z) +
βγL(k)
f1(k)
f2(k)
c~k(z) = ρ~k(t = 0) (1)
z + k2D(k)
f1(k) +
f2(k)
c~k(z)
n1n2 [m1f2(k)−m2f1(k)] ρ~k(z) = c~k(t = 0) (2)
where χT (k) is the generalized isothermal compressibility, defined in terms of the partial
static structure factors Sij(k) with i = 1, 2 and j = 1, 2:
χT (k) = χ
S11(k)S22(k)− S
12(k)
x2S11(k) + x1S22(k)− 2
x1x2S12(k)
Here χ0T is the compressibility of the granular gas in the dilute limit. γL(k) is the generalized
longitudinal viscosity[7].
f1(k) =
S22(k) +
S12(k)
S11(k)S22(k)− S
12(k)
f2(k) =
S11(k) +
S12(k)
S11(k)S22(k)− S
12(k)
are combinations of partial static structure factors and D(k) is the coefficient of mutual
diffusion[7, 17].
This set of equations leads to two extended hydrodynamic diffusive modes.
Since we are interested in exploring the packing aspects of jamming in a binary granular
mixture which is controlled by the sizes rather than the masses of the particles, we confine
ourselves to the case of equal masses m1 = m2 for the two species[15]. Here we analyze two
illuminating special cases to bring out the role of size difference and packing in the jamming
process which triggers the transition to a glassy state: (i) a system composed of a few large
spheres in a matrix of small spheres and (ii) a system composed of a few small spheres in a
matrix of large spheres.
In both these extreme limits (x2 = n2/n << 1 and x1 = n1/n << 1) the cross terms in
the expressions representing the eigenvalues for the extended diffusive modes are negligibly
small and the mode structure reduces to:
(k) ≃ −k2
ρχT (k)γL(k)
z+(k) ≃ −k
2D(k)[
f1(k) +
f2(k)] (4)
Thus, there are two relevant modes: z
(k), which governs the relaxation of total mass
density fluctuations and z+(k), which governs the relaxation of concentration fluctuations.
Let us analyze these modes for case (i). In this case, since the mixture consists of a large
number of small spheres, the static compressibility χT (k) which is the main determinant of
the dynamics of a dense liquid, is given by χT (k) ≃ χ
TS11(k), i.e. it is dominated by the
static structure factor of the small spheres. Thus, z
(k) ≃ −k2 1
S11(k)γL(k)
. Consequently
there is a significant slowing down of the dynamics of density fluctuations of the background
matrix at the location of the first peak of S11(k)[18]. In this case the static structure of large
(type 2) spheres is flat and is given by S22(k) ≃ 1. Thus, z+(k) ≃ −k
2 D02
S22(k)
≃ −k2D02,
where D02 is the dilute gas limit of the diffusion coefficient of large spheres[17]. In case (ii),
the large (type 2) and small (type 1) spheres switch roles. The final picture that emerges
is the following. Caging and slowing down of dynamics is more effective for x2 << 1 firstly
because the ratio D02/D01 = σ1/σ2 < 1 [7, 19] and a few large spheres diffuse slower in
the background of small spheres compared to a few small spheres in the background of
large spheres. In addition, in both cases there is also a significant slowing down of structural
relaxation due to softening of the mode of total density fluctuations z
(k) around the location
of the peak of the static structure factor of the majority particles.
In other words, caging is more efficient in a mixture dominated by small spheres compared
to one dominated by large spheres. This is the main prediction made in this Letter for future
experiments on dense granular mixtures designed to probe the efficiency of caging in such
systems.
The main point that we emphasize in this Letter is that by exploiting the analogy be-
tween glass forming atomic liquids and granular matter we can draw some definite testable
conclusions about caging dynamics in grains.
To summarize, we have, for the first time applied extended or generalized hydrodynamic
techniques to dynamics of granular matter. In particular, we consider a dense binary mixture
of hard spheres and analyze the modes of density fluctuations which dominate the slow
dynamics on length scales of the order of the sizes of the granular particles constituting the
system. Our analysis points to some differences in jamming behavior between a mixture
dominated by small spheres and one dominated by large spheres. Effects similar to the ones
predicted here have been observed in confocal microscopy studies in colloidal mixtures [11].
The predictions made here can be tested against experiments in vibrated dense granular
mixtures. We expect our analysis to be valid for a granular mixture consisting of nearly
elastic spheres of comparable masses [20]. While the present analysis captures the onset of
glassy behavior in granular mixtures, it would be worthwhile to do a mode-coupling study
for such a system using the extended hydrodynamic modes that stem out of our analysis
as an input to understand glass transition in granular mixtures. In future, one can do a
more complete analysis where the effects of momentum and temperature fluctuations and
inelastic collisions [21] are taken into consideration and check if the high density particle-
scale dynamics presented here survives the inclusion of these effects.
[1] See for instance, H. A. Maske, J. Brujic and S. F. Edwards, The physics of Granular Media,
Wiley-VCH, (2004).
[2] See for instance, K. Feitosa and N. Menon, Physical Review Letters 88 , 198301 (2002).
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(2002); X. Yang, C. Huan, D. Candela, R. W. Mair and R. L. Walsworth Physical Review E
69, 041302 (2004);
[4] A. J. Liu and S. R. Nagel, Nature 396, 21 (1998).
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granular media” in “Unifying Concepts in Granular Media and Glasses” edts. A. Coniglio, A.
Fierro, H. J. Herrmann and M. Nicodemi.
[6] W. Gotze and L. Sjorgen, Rep. Prog. Phys 55, 241 (1992).
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(2001).
[12] Supurna Sinha Physical Review E 49 3504 (1994).
[13] T. R. Kirkpatrick, Physical Review A 32, 3130 (1985).
[14] P. G. de Gennes, Physica 25, 825 (1959).
[15] J. N. Roux, J. L. Barrat and J. P. Hansen Journal Of Physics : Condensed Matter 1, 7171
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[16] A more familiar example is the phenomenon of de Gennes narrowing which happens in the
context of dense simple liquids.
[17] J. Ferziger and H. Kaper, Mathematical Theory Of Transport Processes in Gases, North-
Holland, Amsterdam, (1972).
[18] Also, we are focussing on a high viscosity (i.e. large γL(k)[[7, 13]]) regime which contributes
to the process of structural slowing down.
[19] See for instance, G. V. Vijayadamodar and B. Bagchi Journal Of Chemical Physics 93, 689
(1990).
[20] In such a domain the mixture can be described fairly well as one sharing a common temperature
T . See for instance P. Zamankhan Physical Review E 52, 4877 (1995) and V. Garzo and J.
W. Dufty Physical Review E 60, 5706 (1999).
[21] See, for instance, J. W. Dufty and J. J. Brey Physical Review E 68, 030302 (2003); J. J. Brey
and J. W. Dufty Physical Review E 72, 011303 (2005);
	Introduction
	Extended Hydrodynamic Approach To A Binary Mixture Of Grains
	References
