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A signature of a thermodynamic phase transition in jammed granular packings: growing
correlations in force space
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J. Stat. Mech. (2011) L07002
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J
ournal of Statistical Mechanics: Theory and Experiment
LETTER
M Mailman and B Chakraborty
Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454,
USA
E-mail: mailmm@brandeis.edu and bulbul@brandeis.edu
Received 9 May 2011
Accepted 25 June 2011
Published 19 July 2011
Online at stacks.iop.org/JSTAT/2011/L07002
doi:10.1088/1742-5468/2011/07/L07002
Abstract. An outstanding question in the physics of jammed packings concerns
the nature of the correlations that arise near the unjamming transition. In this
work, we study unjamming in an assembly of frictionless grains that are hard
but not infinitely rigid. We demonstrate that a static correlation function, which
probes sensitivity to boundary conditions, exhibits a diverging correlation length
as the packing is decompressed. An analytical expression for the length scale
divergence is obtained from a scaling relation of the entropy, and shown to have
logarithmic corrections to mean-field results.
Keywords: granular matter, disordered systems (theory), structural correlations
(theory), jamming and packing
c
2011
IOP Publishing Ltd and SISSA
1742-5468/11/L07002+10$33.00
J. Stat. Mech. (2011) L07002
A signature of a thermodynamic phase
transition in jammed granular packings:
growing correlations in force space
Growing correlations in jammed granular packings
Contents
2
2. The model
3
3. A PTS measure of correlations
4
4. Sampling boundary-constrained MSS-FNE
5
5. Length scales and scaling
5
6. The origin of the length scale
6
7. The relationship of the PTS length with the isostaticity length
8
8. Conclusions
8
Acknowledgments
9
References
9
1. Introduction
In dry granular materials, resistance to shear arises only as a consequence of compression,
and such solids lose their rigidity as they are decompressed to the point of zero
pressure. Experiments in granular systems have shown that this unjamming transition is
accompanied by large fluctuations of stress [1, 2]. In equilibrium systems, large fluctuations
commonly occur in the vicinity of critical points. It has been argued that for frictionless
grains, a critical point (Point J) [3] controls unjamming [4]. Experiments in colloids,
foams and emulsions have analyzed the emergence of elasticity at the critical volume
fraction associated with Point J [5]–[7]. Real granular materials have friction, and
their properties are usually protocol dependent [8]; however, experiments on isotropically
compressed 2D granular packings show features associated with Point J [2]. In this letter,
we provide evidence for growing correlations and a phase transition which survives in
the thermodynamic limit as Point J is approached from the jammed side in packings of
frictionless grains.
There has been evidence of a growing length scale associated with unjamming of
frictionless grains at Point J from measurements of vibrational spectra [9, 10] and from the
response to a point force [11]. The existence of such a length scale has been deduced from
arguments based on balancing bulk degrees of freedom and surface constraints [10, 12].
Use of two-point correlation functions has, however, failed to detect a growing correlation
length above Point J [13]–[16], whereas a growing length scale is observed [17, 18] on
approaching Point J from below. These observations raise the question of what type of
correlations, if any, are responsible for the growing length scale that has been observed
and argued to exist. Studies of glassy systems where static two-point correlations fail to
detect a growing length scale have inspired the definition of a static, point-to-set (PTS)
correlation function that probes the growing influence of the boundary [19], and hence is a
true indicator of a second-order thermodynamic phase transition. If growing correlations
doi:10.1088/1742-5468/2011/07/L07002
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J. Stat. Mech. (2011) L07002
1. Introduction
Growing correlations in jammed granular packings
exist, they will be detected by the PTS correlation function even if they are not detected
in any finite-point correlation function [19]–[22].
We analyze a model granular system in 2D to demonstrate that a PTS correlation
of forces provides a robust signature of a diverging length at unjamming. We
connect this divergence to the vanishing entropy of force networks. Sensitivity of force
propagation in granular materials to boundary conditions has been studied and discussed
extensively [10, 12]. The PTS measure provides a quantitative description of these effects.
The degrees of freedom which define a mechanically stable state (MSS) of frictionless
grains are the positions of grains, which define a geometry, and the contact forces. A
PTS correlation function generically involves the overlap of two configurations. It is
difficult to construct an overlap function for two interacting sets of variables. In the
limit of hard but not infinitely rigid grains, however, there is an approximate separation
between the force and geometry since in this limit there are large force fluctuations
associated with infinitesimal deformations of the grains [23]. This feature has been used to
construct the force network ensemble (FNE) which is a minimal model that captures the
most robust features of contact force distributions, and mechanical properties of static
granular packings even in frictional systems [24]. The FNE is a flat distribution of all
force configurations that (a) satisfy the constraints of mechanical equilibrium on a given
geometry [23], and (b) have only compressive forces. We calculate the PTS correlation at
a given overcompression by constructing the FNE on an ensemble of amorphous packing
geometries of frictionless disks interacting via short-range, repulsive potentials [4].
Our bidisperse system of M grains is composed of 2/3 small grains with diameter d
and 1/3 with diameter 1.4d. To create an ensemble of MSS at a given compression we
start by sampling an ensemble of just-touching disk packings (zero pressure), each of which
has a particular packing fraction φc [25, 26]. These packings are then overcompressed by
increasing the packing fraction using a force law that varies linearly with grain overlap.
Starting from the φc for a particular packing, the grains are inflated uniformly until
the packing fraction has increased by a fixed amount δφ and a new MSS with overlapped
disks is reached in which forces are balanced on
MSS has a characteristic
grain.0 Each
Meach
α β
value of the full force–moment tensor σαβ = i=1 {ij} Fij rij rij /|rij |, where {ij} denotes
a sum over all contacts j of grain i, and rij is the vector associated with contact {ij},
bearing a force of magnitude Fij0 determined by the force law. The pressure of the MSS
is P = Trσ̂/M.
To construct the FNE on a given
MSS geometry, we would solve the equations of
mechanical equilibrium (ME) [23]:
rij /|rij | = 0, with the additional constraint
{ij} Fij that the force–moment tensor of the FNE has the value characteristic of the MSS. The
resulting linear system of equations can be represented as Af = b. The matrix A has z
columns and 2M + 3 rows, where z is the total number of force-bearing contacts in the
MSS. There are three additional rows corresponding to the three independent elements of
σ̂ in 2D. The vector f is a list of length z of contact force magnitudes Fij , which represents
the ‘force network’. Each MSS has a characteristic force network f0 , which is a list of the
Fij0 obtained from the compression algorithm. The vector b has 2M + 3 elements. Only
doi:10.1088/1742-5468/2011/07/L07002
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J. Stat. Mech. (2011) L07002
2. The model
Growing correlations in jammed granular packings
the last three elements are non-zero, and correspond to the σ̂ determined by f0 . Sampling
all f with only non-negative elements and with a flat measure yields the MSS-FNE. Only
non-negative force networks are sampled since grains exert only compressive forces.
3. A PTS measure of correlations
By definition, a PTS correlation function at a distance r provides a measure of the
correlation between a variable at a point and a set of variables at and beyond a boundary
a distance r away [21]. The generic protocol [21, 22, 27] for measuring a PTS correlation
function is: (i) create a reference configuration of random variables [21, 27], (ii) generate
a new configuration inside a boundary, while outside of the boundary the configuration is
kept ‘frozen’ at the reference configuration, and (iii) define an overlap function to measure
the covariance between the old and new configurations. For example, in supercooled
liquids, a length scale has been deduced by freezing an equilibrium liquid configuration
outside of a cavity of radius r, allowing the liquid to re-equilibrate inside the cavity, and
measuring the overlap of the new configuration with the original one around the center
of the cavity [27]. To study unjamming, we use the overlap of force networks to define
the PTS correlations. We choose our bounded region to be a square of linear size R (cf
figure 1), and construct a boundary-constrained MSS-FNE. All lengths are measured in
terms of the smaller grain diameter. The members of a boundary-constrained MSS-FNE,
f(R), satisfy the conditions of ME within the square, and the additional constraints that
the forces outside are fixed by the components of the MSS-specific f0 :
(1)
A(R)f(R) = b(f0 , R).
The dimension of A(R) depends on R. The last three elements of b(f0 , R) are determined
by the boundary forces f0 through σ̂. In addition, the net force on the interior grains
from grains that are outside the boundary leads to inhomogeneous constraints, which
contribute non-zero elements to b. For instance, for a grain i which has α unfrozen
rij /|rij | =
contacts and β frozen contacts, there are two ME constraints:
{ij}={α} Fij 0
0
F rij /|rij | ≡ bi , where F is an element of f0 . These inhomogeneous
−
{ij}={β}
ij
doi:10.1088/1742-5468/2011/07/L07002
ij
4
J. Stat. Mech. (2011) L07002
Figure 1. (a) The contact network at low P (left) and at higher P (right).
(b) Vectors f(R) within R are solutions to ME equations constrained by the
forces outside being fixed to f0 .
Growing correlations in jammed granular packings
4. Sampling boundary-constrained MSS-FNE
We sample the f(R)
by constructing a random walk in the null space of the matrix
A(R). The null space basis vectors, {g }, span a space of solutions for the homogeneous
= 0. All f0 (R) + cg with an arbitrary coefficient c gives a solution
equation A(R)f (R)
to (1). The random walk progresses by choosing a random step size c from a uniform
distribution, as well as a random direction in the null space g . In addition, since every
contact force in a granular packing is non-negative, the random walk is subject to reflecting
boundary conditions. Random walk type algorithms were developed earlier to sample force
networks [24], [30]–[32]. The approach based on the null space basis vectors [30] naturally
identifies the nonlocal moves required for such a sampling, creating an efficient algorithm,
especially close to unjamming where the moves can involve many contact forces.
5. Length scales and scaling
For system sizes ranging from M = 30 to 900 grains, and overcompressions ranging from
δφ = 10−3 to 10−1 with 40 packings at each δφ, we measured C(R). Only if the nullity
of A(R) is non-zero is it possible to find solutions to (1) that are different from f0 (R), and
then C(R) can be less than unity. For each MSS, singular value decomposition of A(R)
identifies a unique scaled bounding box size R0 and the measured nullity becomes greater
1
We denote all such ensemble-averaged quantities by g , while the are reserved for quantities averaged over
the FNE only, given a particular MSS.
doi:10.1088/1742-5468/2011/07/L07002
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J. Stat. Mech. (2011) L07002
constraints do not fully constrain the ME problem, since the sum over the β contacts can
be the same for two different sets of the Fij0 [28]. Taken together with the σ̂ constraints,
however, they unambiguously define the boundary ME problem.
Given an A(R), for each f(R)
we calculate the overlap, C(R) = fˆ0 · fˆ(R), where fˆ0
and fˆ include only contact forces on grains in the core of the bounded region, away from
the boundary. The core is made up of all grains within a square of side length two grain
diameters, centered on the origin. Since they share a single underlying geometry, the two
force networks have the same number of elements. Due to the positivity constraint, they
are never orthogonal to each other, so C(R) will always asymptote to a positive value.
The magnitudes of the core for each network are divided out so that the largest value
of C(R) is 1. To obtain the PTS correlation function particular to A(R), we compute
C(R), the average of C(R) over all of the sampled f(R).
To determine the PTS correlations in the MSS-FNE ensemble, we calculate C(R) for
an A(R) at a given δφ, and then compute C(R)g by sampling the ensemble of A(R)
through the compression protocol1 . The positivity constraint on the forces influences
C(R)g through the statistics of both the FNE and the geometry-specific matrices
A(R). The ensemble of A(R), characteristic of disks with purely repulsive interactions,
is different from that of elastic networks [29]. Since unjamming of repulsive grains occurs
strictly at zero pressure, we analyze our results in a fixed P g ensemble, where the
subscript g is used to signify that P is averaged over all MSS at a given δφ. As shown in
figure 1, the positions of grains, the contacts and the forces in an MSS depend sensitively
on δφ (P g ).
Growing correlations in jammed granular packings
than 0. For R > R0 the correlation function decays monotonically to its asymptotic value
C(L), where L is the linear size of the system. Figure 2 shows R0 for individual MSS
at the largest system size (M = 900). The properties of R0 g depend on the statistics
of A(R) at a given δφ, and reflect the purely repulsive character of the grains. This
with ν = 0.461 ± 0.012. Measuring R0 g
geometry-averaged length increases as P −ν
g
over the full range of M and P g , we obtain data collapse (figure 2) using the finite-size
scaling form R0 g /L = g(L1/ν P g ) for ν = 0.46. There is a visible failure of the collapse
outside of the interval ±0.02. The finite-size scaling collapse provides compelling evidence
of a length scale that dominates as P g → 0 [33].
In the inset of figure 3, we show the measured values of C(R)g . Also, shown
in the main figure is a connected correlation function, defined as 1 if C(R) is 1 and
C(R) − C(R = L) otherwise, averaged over the geometries. The connected correlation
functions exhibit a rapid decay to zero and a collapse for different pressures after scaling
Rg by R0 g . The shape of the connected correlation functions is very similar to that
predicted by the random first-order transition (RFOT) theory of glasses [19, 34].
6. The origin of the length scale
Given the nature of the PTS, the increase in R0 g indicates decreasing number of solutions
to (1). Particular properties of the solution space are dependent on the MSS-FNE; however
doi:10.1088/1742-5468/2011/07/L07002
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J. Stat. Mech. (2011) L07002
Figure 2. R0 g plotted against P g for M = 900. Inset: finite-size scaling
collapse of R0 g using ν = 0.461, and L ∝ M 1/2 (30 ≤ M ≤ 900, 0.001 ≤
δφ ≤ 0.1). Different colors (symbols) correspond to different δφ, with the open
symbols ranging from 0.01 to 0.1 (increments of 0.01), and the filled symbols
ranging from 0.001 to 0.008 (increments of 0.001). Error bars represent three
standard deviations of ln[R0 g ].
Growing correlations in jammed granular packings
approximating the solution space as a hypersphere of dimension D equal to the nullity of
A(R) is a reasonable approximation that allows us to estimate the entropy. For D > 2, a
random walk beginning at the origin is transient, and in the asymptotic limit, reaches the
surface of the hypersphere. In this regime, the normalized radius of the hypersphere can
be approximated by η = |f(R)
− f0 (R)|/[f0 ], the distance from the initial force network
to a network near the surface2 . Since each valid force network is considered to be equally
likely, and the solution space is simply connected, the volume of this hypersphere estimates
the number of solutions. The force networks are created by a random walk in a space of
dimension
D, spanned by g : f(R)
= f0 (R)+c
g. Therefore, |f(R)−
f0 (R)| can be estimated
√
√
as D[f0 ], leading to the relationship η = D. This relation defines a special class of
hyperspheres whose volume can be calculated from the knowledge of the radius, which
also dictates the dimension of the hypersphere. As shown in figure 4, η 2 g has a scaling
1/2
1/2
form: P g η 2(R, P )g = G(Rg P g ). At a given P g , the scaling relation thus
identifies a Rg at which the solution space volume takes on a given value. In particular,
at each P g there is a R0 g for which η 2g = η02 at which the volume approaches unity
1/2
1/2
(or the entropy of force networks goes to zero): P g η02 = G(R0 g P g ). At small
α
values of y the scaling function G(y) is fitted well by G0 e−(b/y) (figure 4); therefore,
1/α
1/2 P g
G0 lnP g
−1
R0 g =
−
.
(2)
ln
b
η0
2
2
[f ] is the average of single contact forces of f0 .
doi:10.1088/1742-5468/2011/07/L07002
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J. Stat. Mech. (2011) L07002
Figure 3. The connected correlation function (M = 900). Inset: the measured
PTS correlation function, with different symbols that are the same as in figure 2
denoting different values of δφ, and hence P g . The highest pressures correspond
to the curves in the inset which decay to the lowest values.
Growing correlations in jammed granular packings
As P → 0, the −lnP g /2 term dominates, and modifies the P 1/2 behavior. With the
fitted parameters, (2) yields an apparent exponent ν = 0.46, in agreement with the one
obtained from C(R)g .
7. The relationship of the PTS length with the isostaticity length
For a d-dimensional, disordered elastic network with M nodes and z springs, a minimum
condition for mechanical stability is z/2 ≥ ziso /2 ≡ dM [12]. For these networks, an
argument comparing bulk and boundary terms [10, 12] leads to a length scale that diverges
as M/(z − ziso ). Such a divergence has been detected in the response of grains to point
forces [11]. Indirect evidence for a growing length scale has also been obtained from the
frequency spectrum of vibrational modes in granular packings [9]. Relating the boundary
versus surface argument to the nullity of A(R) predicts a quadratic form for the scaling
function G(y). As seen from figure 4, the observed scaling form of G(y) is different, and this
is the origin of the logarithmic correction close to Point J (2). The difference of the scaling
forms of G(y) implies that, at low pressures, packings of grains behave differently from
elastic networks. The difference arises from a combination of the ensemble of geometries,
through A(R), and the positivity constraint on the forces.
8. Conclusions
We have identified a static correlation function that exhibits a diverging length scale as
the pressure goes to zero in frictionless granular packings. This PTS correlation function,
doi:10.1088/1742-5468/2011/07/L07002
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J. Stat. Mech. (2011) L07002
Figure 4. Scaling of η̄ 2 is shown for M = 900 (the colors and symbols match
α
figure 2). The solid line is a fit to G(y) = G0 e−(b/y) with b = 161, α = 0.86,
G0 = 3 × 104 .
Growing correlations in jammed granular packings
Acknowledgments
This work was supported by NSF-DMR0905880, and has derived benefit from the facilities
and staff of the Yale University High Performance Computing Center and NSF CNS0821132. We acknowledge useful discussions with S Franz, G Biroli, Dapeng Bi, N Menon
and Corey O’Hern, and with participants at the 2010 Les Houches Winter School. BC
acknowledges the Aspen Center for Physics, and the Kavli Institute for Theoretical Physics
where some of this work was done.
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