# Criticality and marginal stability of the shear jamming transition of
frictionless soft spheres

Varghese Babu<EMAIL_ADDRESS>Srikanth Sastry<EMAIL_ADDRESS>Jawaharlal Nehru Center for Advanced Scientific Research, Jakkur Campus,
Bengaluru 560064, India.

###### Abstract

We study numerically the critical behavior and marginal stability of the shear
jamming transition for frictionless soft spheres, observed to occur over a
finite range of densities, associated with isotropic jamming for densities
above the minimum jamming (J-point) density. Several quantities are shown to
scale near the shear jamming point in the same way as the isotropic jamming
point. We compute the exponents associated with the small force distribution
and the interparticle gap distribution, and show that the corresponding
exponents are consistent with the marginal stability condition observed for
isotropic jamming, and with predictions of the mean field theory of jamming in
hard spheres.

††preprint: APS/123-QED

Jamming is a ubiquitous phenomenon, observed in a variety of physical systems
classified as granular matter (sand, grains, powders), foams, colloids etc,
wherein thermal motion does not play a significant role. The jamming
transition is typically observed when disordered materials are compressed,
resulting in a transition to a state which can resist deformation [1].
Athermal frictionless soft-spheres have served as an idealized model for
studying the jamming transition [2, 3, 4], along with packings of hard
spheres.

The jamming density $\phi_{J}$ has many properties of a critical point. The
configurations at $\phi_{J}$ are isostatic with average co-ordination number
$z=2d$, where $d$ is the spatial dimension [2, 3, 4]. As the system is
compressed above the jamming density (for soft-spheres) the excess co-
ordination number scales as $\delta z\sim(\phi-\phi_{J})^{\frac{1}{2}}$,
independent of the interaction potential and the configurations are
mechanically stable [3, 2, 4]. The pressure vanishes linearly with
$(\phi-\phi_{J})$, with a pre-factor depending on the interaction potential
[2, 5, 4]. Close to, and above, the jamming density, the vibrational density
of states (VDOS) $D(\omega)$, where $\omega$ is the frequency, displays
anomalous behavior, with the presence of excess low-frequency modes compared
to the Debye solid which describe a normal elastic medium, with a
characteristic frequency $\omega^{*}$ vanishing as the jamming point is
approached, with a power law dependence on $(\phi-\phi_{J})$ [6, 7, 8]. Such
aspects of critical-like behaviour near the jamming point has been widely
investigated and established [4, 9].

The requirement of mechanical stability of jammed packings has been shown to
imply an inequality, or bound, between exponents that characterise the
distribution of inter-particle forces $f$, which exhibits a power law form
$P_{e}(f)\sim f^{\theta}_{e}$ at small forces, and the distribution of
distances, or gaps $h$, between particles that are nearly in contact, which
exhibits a well known power-law singularity, $g(h)\sim h^{-\gamma}$ [10, 11].
The bound, $\gamma\geq 1/(2+\theta_{e})$, was argued to be saturated at
jamming [10] and that such a marginal stability condition provided a mechanism
to explain the avalanches of rearrangement observed [12, 13].

The mean field theory of glass transition in hard spheres in the limit of
infinite dimensions [14, 15], interestingly, leads to predictions concerning
the behavior at jamming, and in particular a prediction for the exponents
$\theta$ and $\gamma$ to be $\theta=0.42311..$ and $\gamma=0.41269..$. While
the predicted value of $\gamma$ is close to those observed in two (2D) and
three dimensional (3D) packings, as well as higher dimensions [16], leading to
the possibility that $d=2$ constitutes the upper critical dimensions for the
jamming transition, the reported values of exponent $\theta$ [17, 11, 16]
exhibits a wide range. However, as noted in [11], the presence of localised
excitations leads to a modified distribution $P_{l}(f)\sim f^{\theta_{l}}$ and
marginal stability condition $\gamma\geq(1-\theta_{l})/2$. The localised
excitations were associated with sphere arrangements prone to buckling, or
bucklers, in [18], and separating out the distributions of forces
corresponding to bucklers leads to the verification of marginal stability
condition, with the exponents predicted by mean field theory.

In this letter, we examine the validity of these aspects of criticality and
marginal stability for the shear jamming transition for frictionless soft
spheres in two and three dimensions, and show that they are indeed valid. The
jamming of granular matter under shear has been observed experimentally [19,
20] and numerically [21, 22, 23] for frictional systems. However friction is
not necessary for the shear jamming transition [24, 25, 26, 27, 28, 29]. In
[26], critical behavior near the shear jamming transition for frictionless
soft spheres were considered in 3D. Although the density range over which
shear jamming occurs was seen to vanish in the thermodynamic limit, the
behavior of the pressure, contact number $z$ and the bulk modulus were shown
to exhibit the same behavior as a function of shear stress (equivalently,
shear strain above the jamming strain) as at the isotropic jamming point. The
key difference is that both the bulk modulus $B$ and the shear modulus $G$
remain finite at the shear jamming point, unlike the isotropic case where only
the bulk modulus is finite. Nevertheless, only a single eigenvalue of the
elastic modulus tensor becomes finite, and the finite values of $B$ and $G$
can be understood in terms of a rotation of the eigenvectors, leading to the
conclusion that shear jamming and isotropic have the same symmetry and
critical behavior. Similar conclusions have been arrived at in other
investigations [30, 28]

However, shear jamming can occur over a finite range of densities even in the
absence of friction [27, 24, 25, 28], a possibility related to the presence of
a range of densities (J-line) over which isotropic jamming can take place,
above the minimum jamming density $\phi_{J}$ (of $\approx 0.648$ for $d=3$ and
$\approx 0.84$ for $d=2$). When the preparation history dependent jamming
density of a packing $\phi_{j}$ is $>\phi_{J}$, decompression of the packing
to densities $\phi$, $\phi_{J}<\phi<\phi_{j}$ leads to unjamming, but such
unjammed packings can jam under the application of shear [27, 28] (The jamming
density $\phi_{j}\approx\phi_{J}$ in [26], and hence the finite shear jamming
range is not observed).

It is of interest therefore investigate the critical behavior of shear jamming
over such density intervals, which we do, building on previous work [26, 29,
23, 28]. With the distance from the shear jamming strain $\gamma-\gamma_{j}$
playing the role of the excess density above jamming, $\phi-\phi_{j}$ for
isotropic jamming, we find that the scaling of pressure, excess contact number
and shear stress and the behavior of the VDOS $D(\omega)$ is the same as at
isotropic jamming. In addition, we explore in detail the marginal stability
condition, employing the approach of [18] to distinguish localised
excitations, or bucklers, and demonstrate that the behavior for shear jamminng
is consistent with that for isotropic jamming and mean field predictions.
Other than a preliminary investigation in [22], the applicability of the
marginal stability condition for shear jamming has not been investigated. Our
results thus clearly demonstrate that properties related to criticality and
marginal stability for shear jamming are the same as for isotropic jamming.

Model and Methods: The systems we study are bi-disperse soft sphere mixtures
($50:50$) in 2D and 3D with a harmonic repulsive inter-particle potential. The
interaction potential is given by
$v(|\vec{r}_{ij}|)=\epsilon(1-\frac{|\vec{r}_{ij}|}{\sigma_{ij}})^{2}$ for
$|\vec{r}_{ij}|\leq\sigma_{ij}$ where $\vec{r}_{ij}$ is the distance between
particles $i$ and $j$ and $\sigma_{ij}=\frac{\sigma_{i}+\sigma_{j}}{2}$ with
$\sigma_{k}$ being the diameter of particle type $k$
$(\sigma_{2}/\sigma_{1}=1.4)$. The first step in our study involves generating
configurations with jamming density $\phi_{j}>\phi_{J}$. For this we follow
the protocol similar to the one used in [5] which we describe for $d=3$
briefly; other procedures that could be employed are outlined in [31, 27]. At
the packing fraction $\phi=0.5935$ we generate configurations by initializing
particle centers randomly and performing an energy minimization to generate a
configurations with no overlaps (A configuration with no overlap is considered
unjammed). This configuration is treated as a configuration of hard-spheres
and equilibrated by hard-sphere Monte Carlo simulation using HOOMD [32, 33].
We then compress the system in steps of $\delta\phi=10^{-4}$, performing an
energy minimization after each compression. When the jamming density
$\phi_{j}$ is crossed, the energy after minimization $e/N$ will be greater
than $10^{-24}$. When the energy crosses a threshold (here
$\frac{e}{N}>10^{-7}$) we stop the compression and start decompressing the
system with smaller steps of $\delta\phi=10^{-5}$. During the decompression
when we are able to minimize the energy to $\frac{e}{N}<10^{-24}$ we stop the
process and identify the jamming density. The jamming densities obtained
through the procedure is distributed around $\phi\approx 0.661$, which depends
on the density of the initial equilibrated fluid [5]. From the configurations
at $\phi_{j}$ we generate unjammed configurations at $\phi_{J}<\phi<\phi_{j}$
by scaling the volume.

These configurations are sheared uniformly using Athermal Quasi Static (AQS)
shear to observe shear jamming at a strain $\gamma_{j}$, employing LAMMPS
[34]. AQS shear for a strain step $\delta\gamma$ is carried out by performing
an affine transformation $x_{i}\rightarrow x_{i}+\delta\gamma\times
y_{i};y_{i}\rightarrow y_{i};z_{i}\rightarrow z_{i}$ of coordinates followed
by energy minimization. We generate configurations close to the jamming strain
$\gamma_{j}$ and identify $\gamma_{j}$ as follows: We increment strain in
steps of $\delta\gamma=10^{-3}$ until $\frac{e}{N}>e_{thresh}=10^{-7}$, at
which point we redefine the strain step and threshold energy as
$\delta\gamma\rightarrow-1\times\delta\gamma/10$ and $e_{thresh}\rightarrow
e_{thresh}/10$. The system is strained in the reverse direction until
$\frac{e}{N}<10^{-20}$ where updates to $e_{thresh}$ and $\delta\gamma$ are
implemented again. This procedure is stopped when $\delta\gamma<10^{-6}$ and
$\frac{e}{N}<10^{-20}$ and the system is being reverse strained.

Using this procedure we are able to obtain configurations close to the jamming
strain, but to study the marginal stability of the shear jamming transition we
need to generate configurations that are just shear-jammed. Quantitatively
this means the configurations has a single self-stress state, or the contact
network has one unique force-balance solution [18]. For a given jammed
configuration with $N_{c}$ contacts and $N$ particles which are not rattlers,
the number of self-stressed states is given by $N_{ss}=N_{c}-(N-1)d$ with
periodic boundary conditions [18]. We observe that for large system sizes
configurations obtained using the SJ procedure to obtain shear jammed
configurations are not close enough to jamming and have multiple self-stressed
states.

To obtain configurations with single self-stressed state we adapt the
procedure described for isotropic jamming in [18] for shear jamming. Starting
with a configuration at $d\gamma\equiv(\gamma-\gamma_{j})=10^{-5}$ as
determined through the SJ procedure, we iteratively reduce the strain by
exploiting the scaling of the potential energy $U\sim(\gamma-\gamma_{j})^{2}$
(which is indeed observed, as shown in the Supplemental Material (SM) 111See
Supplemental Material at https://xxx for the additional information on: i)
Details of the iterative procedure (IP), ii) Derivation of the expression for
elastic constants, iii) Fabric anisotropy of the shear jammed configuration,
iv) Results for pressure and contact number for 2D and 3D v) Size dependence
of the shear jamming line vs. density and finite size scaling of the gap
distribution, which includes additional references [37, 38], in addition to
further details of this procedure.) Using this procedure we generate shear-
jammed configurations with a single self-stressed state, whose structure and
forces we analyze to study the marginal stability condition. We follow similar
procedures for the data regarding the isotropic case, with density instead of
the strain as the control variable.

The components of the stress tensor are is calculated using
$\sigma_{\alpha\beta}=\frac{1}{V}\sum_{i<j}f^{\alpha}_{ij}r^{\beta}_{ij}$,
where $f_{ij}$ are the interparticle forces, and the pressure as $P={1\over
3}\operatorname{tr}({\bf\sigma})$. We calculate the density of states
$D(\omega)$ which is the distribution of $\omega=\sqrt{\lambda}$ where
$\lambda$’s are the eigenvalues of the Hessian, for configurations over a
range of strains above the jamming strain. As in the case of isotropic jamming
we observe a plateau in $D(\omega)$ for small $\omega$, before $D(\omega)$
decreases to zero as $\omega\rightarrow 0$. The frequency $\omega^{*}$ at
which $D(\omega)$ decreases to half the plateau value is identified as the
crossover frequency.

We note that the jammed configurations analysed contain rattlers, particles
with less than $d+1$ contacts. We remove rattlers recursively, by identifying
them in each iteration from the configurations till no rattlers remain. The
percentage of rattler particles is approximately $0.05\%$ for the cases
considered. The average contact number, as well as distribution of gaps and
forces reported, are obtained after the rattlers are removed. However, the
packing fractions we report are calculated with the total number of particles.

Figure 1:  a) Stress $\sigma_{xy}$ vs $d\gamma=\gamma-\gamma_{j}$ obtained by
IP with fit line $\sigma_{xy}\sim d\gamma$. b) VDOS $D(\omega)$ of
configurations at various $d\gamma$. The green, orange, blue and red symbols
represent $d\gamma=10^{-5},10^{-4},10^{-3},10^{-2}$ respectively. The square
and circle symbols represent $\phi=0.658,0.656$ respectively. The violet curve
is $D(\omega)$ calculated for isotropically jammed configurations with
$N_{SS}=1$. Inset: The cross-over frequency $\omega^{*}$ is calculated by
choosing the frequency at which $D(\omega)$ becomes approximately half of the
plateau value. These values are marked in b). The spaced line shows the
scaling $\omega^{*}\sim d\gamma^{\frac{1}{2}}$. c) Eigenvalues of the
stiffness matrix $\lambda_{i}$ as a function of $d\gamma$ (for $N=8192$). One
eigenvalue is significantly larger than the others and this corresponds to the
bulk and shear modulus $C_{xyxy}$. The dashed line denotes an exponent of
$1\over 2$. $\lambda_{i}>\lambda_{i+1}$ d) Inner product of the eigenvector
corresponding to the largest eigenvalue $\vec{e}_{1}$ of the stiffness matrix
(for $d\gamma=10^{-5}$) with compressive (blue) and shear (orange) strain
directions.

Results: In Fig. 1(a), we show the variation of the shear stress $\sigma_{xy}$
vs $d\gamma$ for 3D, demonstrating linear behavior $\sigma_{xy}\sim d\gamma$
above the jamming strain. The pressure exhibits the same linear behavior and
the excess contact number $\delta z=z-z_{c}$ varies with the distance from the
jamming strain as $\delta z\sim\sqrt{d\gamma}$, with $z_{c}=2d$, as observed
for isotropic jamming. These results are shown in the SM, along with the
corresponding results for 2D. In Fig. 1(b), we show the VDOS $D(\omega)$,
which exhibits a plateau at low frequencies corresponding to excess modes,
which extend towards zero frequency as the jamming strain is approached from
above. The frequency at which the crossover to the plateau occurs,
$\omega^{*}\sim\delta\phi^{\frac{1}{2}}$ for isotropic jamming [6, 8], and we
observe the same scaling near the shear jamming transition, as shown in Fig.
1(b)(inset). In Fig. 1(c), we show the eigenvalues of the stiffness matrix
(details of whose calculation are provided in the SM), investigated in [26]
for shear jamming in fricionless packings. As in [26], we find a nearly
constant largest eigenvalue that is finite at the shear jamming point, and
five nearly degenerate (but less so than in [26]) eigenvalues which are zero
at shear jamming, and whose magnitude grows roughly as $d\gamma^{1/2}$ for
larger strains. In 1d) we show the overlap of the eigenvector corresponding to
the largest eigenvalue with the bulk strain direction and the shear strain
direction, as a function of density $\phi$. Interestingly, the overlap of
shear strain with the stiffest eigenvector shows a non-monotonic behavior.
This is apparently related to the anisotropy of the configurations at shear
jamming, quantified by the fabric anisotropy, which also shows a similar non-
monotonic behavior with changing $\phi$, as shown in SM, and also observed in
[28]. These results taken together show that the nature of criticality near
the shear jamming point is the same as that near the isotropic jamming point.

Figure 2: Inter-particle forces and gaps in shear jammed configurations and
comparison with iso-tropically jammed configurations. All configurations
analyzed has a single self-stressed state. Symbols
$\Diamond,\Circle,\Box,\triangle$ represent isotropic
compression,$\phi=0.660,0.658,0.656$ respectively in $3D$ and symbols
$\Circle,\Box$ represent $\phi=0.8499,0.8485$ in $2D$. a)-b) The cumulative
distribution of forces $G(h)$. The blue symbols represent localized forces and
the orange symbols represent the extended forces. Comparison with exponents
obtained from mean-field theory is shown. c)-d) The cumulative distribution of
gaps $h$. The red line shows the exponent from mean field theory.

We now describe the results regarding the forces and the structure of the
shear jammed configurations. It is convenient to consider the cumulative
probability of forces, $G(f)=\int_{0}^{f}P(f^{\prime})df^{\prime}$. With the
gap defined as $h=\frac{r-\sigma}{\sigma}$, the cumulative probability of gaps
is $G(h)=\int_{0}^{h}P(h^{\prime})dh^{\prime}$. For isotropic jamming, the
cumulative probability for forces (normalised to the mean value) is described
by a power law $G(f/\langle f\rangle)\sim f^{1+\theta}$ and for gaps,
$G(h)\sim h^{1-\gamma}$. As shown in [11] small forces in the system can be
either mechanically isolated, i. e. opening a contact will result only in
local rearrangement of the contact network, or localized modes, or they can
correspond to extended modes. Small forces which correspond to localised modes
have a distribution characterized by exponent $\theta_{l}$ and the forces
corresponding to extended modes are characterized by exponent $\theta_{e}$.
The inequalities discussed in [11] in the two cases are,

$\gamma\geq\frac{1}{2+\theta_{e}}~{}~{};~{}~{}\gamma\geq\frac{1-\theta_{l}}{2}$
(1)

If the jammed system is marginally stable and the above inequalities Eq. (1)
are saturated (if they become equalities) then the following relation holds:

$\theta_{e}\gamma=\theta_{l}$ (2)

The force-distribution calculated by including all the forces in the system is
characterized by exponent $\theta=\min(\theta_{l},\theta_{e}$). In order to
extract the $\theta_{l}$ and $\theta_{e}$, we have to identify contacts
associated with localized and extended modes correctly. Although the mean-
field theory of hard-sphere glasses does not contain a prediction for
$\theta_{l}$, based on the predicted values $\theta_{e}=0.42311..$ and
$\gamma=0.41268..$ and Eq. 2, one has $\theta_{l}=0.17462$. Charbonneau et al.
[18] explored how to identify the contacts which carry small mechanically
isolated forces. The mechanically isolated contacts are associated with
“buckler” particles which are particles with $d+1$ contacts. As shown in [18]
the force distribution calculated by including only the bucklers, $P_{l}(f)$,
exhibits an exponent of $\theta_{l}=0.17462$. The force exponent calculated by
using the distribution of the remaining forces is $P_{e}(f)$ shows an exponent
of $\theta_{e}=0.42311$. We follow the same procedure to analyse
configurations with a single self-stressed state identified by
$N_{c}=(N-1)d+1$. As opposed to isotropic jamming, for the shear jamming
transition we need take into account the effect of shear while classifying
bucklers. However, we observe that for configurations at small-strains,
classification of bucklers as particles with $d+1$ contacts is sufficient to
obtain meaningful results. The cumulative probabilities of forces, separately
for bucklers (localised modes) and the rest (extended modes), shown in Fig. 2
(a) and (b) for 3D and 2D, show that indeed, the predicted values of
$\theta_{e}$ and $\theta_{l}$ describe the data extremely well.

In Fig. 2 (c) and (d) we show the distribution of gaps for 3D and 2D. For 3D,
while we find the mean field prediction of $\gamma=0.41268..$ closely
describes the data, a value of $\gamma=0.38$ is a better description of the
data. Indeed, results in several works [11, 16, 28, 36], both for isotropic
and shear jamming, are consistent with such a smaller exponent, which would
correspond to a weak violation of the stability condition. However, the role
of finite size effects in the observed departures at very small gaps has
recently been investigated [36], emphasizing that finite size effects are much
more pronounced for gaps rather than forces. A scaling collapse over several
orders of magnitude are described supporting the accuracy of the mean field
exponent for three dimensional packings. Our analysis of finite size effects
for shear jamming, shown in the SM, clearly support the same conclusion. On
the other hand, the results for 2D, shown in 2 (d) agree very well with the
mean field predictions. Thus, we conclude that the marginal stability
conditions (Eq. 1) is indeed valid for shear jamming, as well as they do for
isotropic jamming, with the same universal exponents.

In summary, we have numerically analysed configurations of soft spheres in two
and three dimensions accurately generated at the shear jamming point, and
above, for densities below the density $\phi_{j}$ at which they exhibit
isotropic jamming, but above the minimum isotropic jamming density $\phi_{J}$.
We show that several quantities, such as the pressure $P$, the excess contact
number, and a crossover frequency $\omega^{*}$ in the VDOS $D(\omega)$ exhibit
critical scaling that is identical to that at the isotropic jamming point,
with the shear stress in addition displaying the same scaling as the pressure.
We confirm the behavior of the eigenvalues of the stiffness matrix which have
been investigated [26] to argue that shear jamming has the same symmetry as
isotropic jamming, and show that the rotation of the eigenvector of the
largest eigenvalue in the shear strain direction is correlated with the
anisotropy of the shear jammed structures. We show that the marginal stability
condition is met for shear jamming to the same degree as for isotropic
jamming, with exponents predicted by the mean field theory of the glass
transition and jamming in hard spheres (although our results indicate that
better finite size analysis is warranted for the gap distribution). Our
results thus strongly support the idea that shear jamming displays the same
critical behavior and marginal stability as isotropic jamming.

We acknowledge support from the Thematic Unit of Excellence on Computational
Materials Science (TUE-CMS) and the National Supercomputing Mission facility
(Param Yukti) at the Jawaharlal Nehru Centre for Advanced Scientific Research
(JNCASR), for computational resources. SS acknowledges support through the JC
Bose Fellowship (JBR/2020/000015) from the Science and Engineering Research
Board, Department of Science and Technology, India.

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