Long-range strain correlations in sheared colloidal glasses
Vijayakumar Chikkadi1, Gerard Wegdam1, Daniel Bonn1, Bernard Nienhuis2, Peter Schall1.
1Van der Waals-Zeeman Institute,
University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands.
2Institute for Theoretical Physics, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands.
Glasses behave as solids on experimental time scales due to their slow relaxation. Growing
dynamic length scales due to cooperative motion of particles are believed to be central to this slow
response. For quiescent glasses, however, the size of the cooperatively rearranging regions has never
been observed to exceed a few particle diameters, and the observation of long-range correlations
that are signatures of an elastic solid has remained elusive. Here, we provide direct experimental
evidence of long-range correlations during the deformation of a dense colloidal glass. By imposing
an external stress, we force structural rearrangements that make the glass flow, and we identify
long-range correlations in the fluctuations of microscopic strain, and elucidate their scaling and
spatial symmetry. The applied shear induces a transition from homogeneous to inhomogeneous
flow at a critical shear rate, and we investigate the role of strain correlations in this transition.
PACS numbers: 82.70.Dd, 64.70.pv, 62.20.F-, 61.43.-j
arXiv:1110.2884v1 [cond-mat.soft] 13 Oct 2011
Glasses have attracted considerable attention due to their ubiquity in nature, and for
the scientific challenges they pose . Due to their long relaxation time, glasses behave as
solids on experimental time scales; this slow response is often attributed to cooperative mo-
tion of particles . Dynamic correlations that are used to characterize the length scales of
the cooperatively rearranging regions are indeed observed to grow on approach to the glass
transition [3, 5]. However, despite the rapid growth of the glass relaxation time and the
glass viscosity by many orders of magnitude, the length scale associated with the dynamic
correlations increases only weakly on approach to the glass transition. Simulations as well as
experiments have shown that in quiescent glasses, the dynamic length scales remain limited
to a few particle diameters [3, 5]. On the other hand, recent simulations on driven, athermal
amorphous materials have evidenced the existence of avalanche-like, long-range correlated
flow . Correlations in slowly driven thermal glasses, however, are poorly explored. Here,
we show that even a glass exhibits long-range correlations in the microscopic flow field when
it is driven so slowly that the rearrangements induced by the flow happen on a timescale
similar to that of the spontaneous (thermal) rearrangements. By imposing very slow shear,
we explore the transition between a regime where the spontaneous fluctuations are suffi-
ciently rapid to accommodate the flow, to a regime where this is no longer possible, and this
allows us to probe long-range correlations. Nevertheless, in strained molecular glasses, direct
observation of such correlations is prohibitively difficult, because the small size of molecules
inhibits direct observation. We therefore use direct real-space observation in suspensions of
colloidal particles; the individual particles can be imaged and their motion be tracked accu-
rately in three dimensions with confocal microscopy . Hard-sphere colloidal suspensions
have become a much-used model system for studying glasses; at high packing densities, the
motion of the colloidal particles becomes increasingly frustrated, and structural relaxations
slow down dramatically at particle volume fractions larger than φg ∼ 0.58, the colloidal
glass transition . By taking advantage of recent imaging techniques to image large sample
volumes containing as many as ∼ 2.5 × 105particles, we show that the flow of glasses is
governed by surprisingly long-range correlations .
We follow correlations in the deformation of the glass directly in three dimensions and
real time by visualizing fluctuations in the microscopic strain and non-affine displacements.
These strain fluctuations exhibit remarkably long-range correlation that extend over the full
system size, far beyond the range of dynamic correlations reported for quiescent glasses. [3,
5]. We investigate these correlations during two different modes of macroscopic deformation:
during homogeneous flow at very low applied shear rates ˙ γ much smaller than the inverse
structural relaxation time τ−1of the glass, and during inhomogeneous flow at strain rates
˙ γ > τ−1. These two modes of deformation are also observed for molecular glasses , and
the direct visualization of strain fluctuations allows us to identify their important role in
the transition between both modes. Surprisingly, we find that the scaling behavior of the
strain correlations is uniform over the investigated range of shear rates, suggesting a robust,
scale-free organization of the flow of glasses.
We used sterically stabilized fluorescent polymethylmethacrylate (PMMA) particles sus-
pended in a density and refractive index matching mixture of Cycloheptyl Bromide and
Cis-Decalin. The particles have a diameter of σ = 1.3µm, with a polydisperity of 7% to
prevent crystallization. A colloidal glass with a volume fraction of φ ∼ 0.60 is prepared by
diluting samples centrifuged to a sediment with φ ∼ 0.64. The suspension is loaded in a cell
between two parallel plates a distance of 65µm apart. Boundary-induced crystallization is
suppressed by a layer of polydisperse particles on the plates. We used a piezoelectric trans-
lation stage to move the top plate to apply shear at very slow rates between ˙ γ = 10−5s−1
and ˙ γ = 10−4s−1, of the order of the inverse structural relaxation time of the glass. We
used confocal microscopy to image individual particles in a 107µm by 107µm by 65µm vol-
ume, and determined their positions in three dimensions with an accuracy of 0.03µm in the
horizontal, and 0.05µm in the vertical direction . We tracked the motion of ∼ 2 × 105
particles during a 25 min time interval by acquiring image stacks every 60 sec. The mean
square displacement < ∆r2> of particles in the quiescent glass is shown in Fig. 1a; we
estimate the structural relaxation time of the glass τ ∼ 2 × 104sec from the requirement
< ∆r(τ)2>= (σ/2)2; this relaxation time is a factor of 5 × 104larger than the Brownian
relaxation time at infinite dilution, τB= 0.4s. Because the hard-sphere glass exhibits aging,
this structural relaxation time changes with time. To obtain reproducible results, all mea-
surements presented here were taken consistently ∼ 3 hours after mechanical rejuvenation
of the sample.
We study correlations in the deformation of the glass by decomposing the particle motion
into affine and non-affine components. To do so, we follow particle trajectories and identify
the nearest neighbors of each particle as those separated by less than r0, the first minimum
of the pair correlation function. We subsequently determine the best affine deformation
tensor Γ describing the transformation of the nearest neighbor vectors, di, over the time
of Γ is the local strain tensor. The remaining non-affine component D2
interval δt , by minimizing D2
i=1(di(t+δt)−Γdi(t))2. The symmetric part
minhas been used
as a measure of plastic deformation . We define correlations in the fluctuations of the
principal shear strain component ?xz, and the non-affine displacement D2
CA(δr) =?A(r + δr)A(r)? − ?A(r)?2
?A(r)2? − ?A(r)?2
min, and angular brackets denote ensemble averages. CAcorrelates
where A = ?xzor A = D2
values of ?xzor D2
minat locations separated by δr.
We investigate strain correlations during homogeneous flow of the glass by subjecting the
glass to a shear rate of ˙ γ ∼ 1.5 × 10−5s−1, a factor of 6 smaller than the inverse structural
relaxation time. At this shear rate, thermally activated relaxation occurs sufficiently fast,
and after an initial transient the colloidal glass flows homogeneously. We shear the glass to
γ ∼ 1 to address steady-state flow, and show the displacements ∆x of the particles during a
small shear increment as a function of height in Fig. 1b. The average particle displacement
(dashed red line) increases linearly with height, indicating uniform flow. Strong fluctuations
exist, however, at the level of the individual particles. We determine values of the local shear
strain from the motion of a particle with respect to its nearest neighbors during time intervals
of 3 min and 25 min and plot the relative frequency of occurrence of shear strain values
as a function of shear strain magnitude in Fig. 1c, inset. For short times, the probability
distribution of strain values decreases as a power-law over almost three orders of magnitude
in probability. Such power-law decay is a fingerprint of highly correlated dynamics .
This is also evident from the real-space distribution of the microscopic shear strain ?xz
(Fig. 1c). Red regions indicate zones where large shear strain is localized and irreversible
rearrangements occur, also known as shear transformation zones [8, 11, 12]. We determine
correlations between these zones by calculating the correlation functions of the shear strain,
C?xz, and of the non-affine displacements, CD2
min. At the short time intervals considered here,
the correlation functions do not depend significantly on the specific time interval chosen,
and appear robust. Correlations in the x-z plane are obtained by taking δr = (δx,0,δz); a
corresponding color coded representation of the correlation function C?xzis shown in Fig. 1d.
Remarkably, the correlation function shows a four-fold pattern at its center, which reveals
the elastic response of the glass to local shear transformation zones . This glass elasticity
leads to strong correlations between shear transformation zones as evidenced by the regular
pattern of yellow zones. This is most evident if we average C?xzwithin angular wedges
around the horizontal, vertical, and diagonal directions, and plot the corresponding angle-
specific correlation function versus r in Fig. 1e. The correlation function CD2
be isotropic. We average over all directions and plot the magnitude of CD2
minas a function
of distance in Fig. 1f. A remarkable power-law decay is observed, which is truncated at the
vertical system size, δr/σ ∼ 50; thus the correlations span the entire system. We further
confirmed that these correlations are independent of the specific measure of non-affinity by
using different definitions of non-affine motion. The correlation function remained robust.
These results provide direct evidence of the existence of long-range strain correlations in
a glass, and they highlight the scale-free character of the non-affine rearrangements that
govern plastic deformation. The scale invariance appears to be a generic feature of elasto-
plastic deformation in other materials, too: the dislocation motion in crystals , and the
aftershocks in earth quakes  display similar scale-free patterns.
We probe these strain correlations by subjecting the glass to increasing shear rates. While
the flow remains homogeneous over a range of low strain rates, at a critical strain rate of
˙ γc∼ τ−1we observe a sudden transition to inhomogeneous flow. The glass separates into two
bands that flow at different rates, as illustrated by the particle displacements as a function of
height obtained at a shear rate of ˙ γ ∼ 1×10−4s−1(Fig. 2a). Linear fits to the displacement
profile for z < 23µm and z > 27µm (dashed lines in Fig. 2b) yield strain rates of 4×10−5s−1
and 2.2×10−4s−1, respectively, that differ by a factor of five. A reconstruction of the shear
strain distribution shows that highly non-affine shear transformation zones accumulate in
the upper part (Fig. 2b). We investigate the robustness of the scaling observed in Fig. 1e
by determining CD2
min(δr) separately for the high and low shear band. The resulting angle-
averaged correlation functions are shown together with those of homogeneous flow in Fig.
2c. A remarkable collapse of the data is observed. While the magnitude of fluctuations in
the two bands differs largely, the normalized correlation function shows very similar power
law decay (Fig. 2c): the same scaling exponent applies to the low and the high shear band,
as well as to homogeneous flow. We find a scaling exponent of α = 1.3 ± 0.1 from the
best fit to the data. Athermal and quasi-static shear simulations of amorphous solids 
have shown similar long-range correlations; however, the effect of finite shear rate and finite
temperature on the statistical correlations between shear transformation zones remained
unclear . Our results conclusively show long-range correlations even at finite shear rate,
and at finite temperatures.
The difference between the two bands becomes evident when we investigate particle dy-
namics as a function of time. The mean square displacement of particles around the mean
flow, < r?2(t) >=< (∆r(t) − ?∆r(t)?z)2>, where ?∆r(t)?zis the average displacement at
height z, reveals mobilities in the two bands (Fig. 2d) that are larger than those at rest (Fig.
1a), and that differ significantly between the two bands. For particles in the high shear band,
the mean square displacement is significantly larger indicating enhanced particle diffusion
compared to particles in the low shear band. Remarkably, the strain correlation function,
C?xz, calculated separately for both bands (Figs. 2e and f) reveals a spatial symmetry
change. While for the low shear band, the central four-fold symmetry is still predominant
(Fig. 2e), for the high shear band, this symmetry is lost, and the pattern appears isotropic
(Fig. 2f). This symmetry change reflects the transition from a solid to a liquid-like response
of the glass. These results highlight the central importance of strain correlations: While the
transition from a reversible elastic to an irreversible viscous response is usually associated
with a symmetry change in time, our results show that this transition is also associated with
a spatial symmetry change in the strain correlation function, underlining its fundamental
character. A similar interpretation is given to fracture surfaces of metallic glasses that dis-
play striking evidence of such a solid to liquid transition . Our colloidal glass allows
us to directly visualize the strain correlations and identify their central importance in this
How does this transition emerge? To elucidate this, we follow correlations during the
initial stages of shear banding. We determine strain distributions during the transient stages,
before shear bands manifest, and calculate correlation functions for the entire field of view
(Figs. 3a and b). In the early stage, the correlation shows a strong bias in the horizontal
direction (arrows). This horizontal bias signals the excitation of additional elastic modes at
higher shear rates that cause strong correlations between shear transformation zones in the
horizontal direction. This bias lowers the effective resistance to flow in the direction of shear,
thereby leading to shear bands in the later stages of deformation (Fig. 3b). These results
highlight the importance of long-range correlations in the shear banding of glasses. Shear
experiments at much higher shear rates show that shear bands can originate from density
fluctuations . The model proposed by the authors suggests a linear viscous flow at very
low shear rates that is hard for them to address experimentally. Here, we have successfully
addressed this low shear rate regime; our measurements at the onset of shear banding show
strong changes in the correlations of microscopic strain, but no measurable dilatancy.
Our results establish the existence of long-range strain correlations in the flow of glasses.
The long-range elastic interactions between shear transformation zones lead to scale-free
deformation of the glass. While our results for shear banding are obtained for colloidal
glasses, they should be generic to glassy flows. The formation of shear bands has often been
linked to strain softening of the material, caused by excess dilation, that accompanies the
formation of shear transformation zones [8, 19]. The direct imaging of strain correlations
that we have reported here demonstrates that long-range elastic correlations play a central
role in the manifestation of such instabilities . Finally, the robust scaling that we observe
suggests a naturally scale-free flow and relaxation of glasses. We propose similar analysis
of shear flows in systems like granular, foams and emulsions to test the universality of
the scaling exponent, and to determine the universality class of the flow and relaxation of
We thank D. Frenkel, and F. Spaepen for helpful discussions. This work was supported
by the Innovational Research Incentives Scheme (”VIDI” grant) of the Netherlands Organi-
zation for Scientific Research (NWO).
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FIG. 1: Homogeneous deformation at a shear rate of ˙ γ = 1.5 × 10−5s−1. (a) Mean square dis-
placement of the quiescent glass without applied shear. The structural relaxation time, τ, of the
glass is estimated by extrapolating the mean square displacement in the diffusive regime with a
line of slope one (dashed black line). (b) Displacements of individual particles (+) and average
displacement (dashed line) along the shear direction during δt = 10 min of shear. (c) 7µm thick
reconstruction of the distribution of shear strain after δt = 3 min of shear. Particle color indicates
the value of ?xz. Inset in (c) shows the relative frequency of shear strain magnitudes ?xzfor time
intervals δt = 3 min (green stars) and 25 min (blue squares). (d) Angle resolved spatial correla-
tion, C?xz, of the fluctuations of shear strain, in the x-z plane. (e) C?xzas a function of distance
averaged over angular wedges of 10◦around the horizontal (red line), the vertical (green line), and
the two diagonal directions (blue line). (f) Angle-averaged correlation function CD2
minas a function
of distance in a double-logarithmic representation.
FIG. 2: Inhomogeneous deformation at a shear rate of ˙ γ = 1×10−4s−1. (a) Particle displacements
along the shear direction during δt = 4 min of shear. Dashed red lines are linear fits to the shear
profiles for z < zl= 23µm (low shear band) and z > zh= 28µm (high shear band). (b) 7µm
thick reconstruction of the distribution of incremental shear strain ?xz during the time interval
δt = 7 min. (c) Angle-averaged correlation function CD2
minas a function of distance δr, for the
low shear band (blue squares), the high shear band (yellow diamonds), and for homogeneous shear
at ˙ γ = 1.5×10−5s−1(blue dots) and 3×10−5s−1(orange triangles). A least square fit to the data
gives a slope of α ∼ −1.3±0.1 (dashed line). (d) Mean square displacement of particles in the low
shear band (black stars and line) and high shear band (red dots and line). (e and f) Angle-resolved
spatial correlations of shear strain, C?xz(δr), in the x − z plane, for the low and the high shear
bands, respectively. Correlations are computed over a time interval of δt = 3 min.
FIG. 3: Evolution of strain correlations during shear band formation. Shear strain correlation Download full-text
functions C?xzcomputed for the entire field of view before (a) and after (b) the manifestation of
shear bands. The arrows in (a) indicate the strong correlation in the direction of shear that lead
to shear banding in later stages (b).