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Shear banding of colloidal glasses: Observation of a dynamic first order
transition
Chikkadi, V.; Miedema, D.M.; Dang, M.T.; Nienhuis, B.; Schall, P.
DOI
10.1103/PhysRevLett.113.208301
Publication date
2014
Document Version
Final published version
Published in
Physical Review Letters
Link to publication
Citation for published version (APA):
Chikkadi, V., Miedema, D. M., Dang, M. T., Nienhuis, B., & Schall, P. (2014). Shear banding
of colloidal glasses: Observation of a dynamic first order transition.
Physical Review Letters
,
113
(20), 208301. https://doi.org/10.1103/PhysRevLett.113.208301
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Download date:09 Aug 2022
Shear Banding of Colloidal Glasses: Observation of a Dynamic First-Order Transition
V. Chikkadi, D. M. Miedema, M. T. Dang, B. Nienhuis, and P. Schall
van der Waals–Zeeman Institute, Univ ersity of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
(Received 9 January 2014; revised manuscript received 22 June 2014; published 12 November 2014)
We demonstrate that application of an increasing shear field on a glass leads to an intriguing dynamic
first-order transition in analogy with equilibrium transitions. By following the particle dynamics as a
function of the driving field in a colloidal glass, we identify a critical shear rate upon which the diffusion
time scale of the glass exhibits a sudden discontinuity. Using a new dynamic order parameter, we show that
this discontinuity is analogous to a first-order transition, in which the applied stress acts as the conjugate
field on the system’s dynamic evolution. These results offer new perspectives to comprehend the generic
shear-banding instability of a wide range of amorphous materials.
DOI: 10.1103/PhysRevLett.113.208301 PACS numbers: 82.70.Dd, 61.43.Fs, 62.20.F-, 64.70.kj
A central unresolved question in the physics of glasses
concerns the behavior of a glass under applied stress. While
at the glass transition, microscopic observables change
rather smoothly, yet rapidly [1,2], as a function of density
or temperature; an important question is whether a similarly
smooth variation occurs upon application of stress. Recent
experiments and simulations show that, unlike quiescent
glasses, slowly sheared glasses exhibit high dynamic
susceptibilities with long-range, directed strain correlations
[3–6]. Such long-range correlations also indicate a high
susceptibility to the applied shear. The question is then how
the highly susceptible glass responds to an increasing
applied shear field.
It is well known that application of shear on amorphous
materials can lead to intriguing shear inhomogeneity known
as shear banding [7–12], where the shear localizes in bands
that flow at a much increased rate. This phenomenon has
long been recognized in metallic glasses [8], for which
intriguing liquid vein patterns have been observed along the
shear bands [12]. Despite its importance to a wide range of
amorphous materials including metallic and soft glasses, a
fundamental understanding of shear banding is lacking.
Phenomenologically, shear banding is associated with
nonmonotonic flow curves [9,13]: the stress to maintain a
steady-state flow of the material varies nonmonotonically
with applied strain rate. This leads to two (or more) flow
rates that coexist at the same applied stress, analogous to
the van der Waals description of coexisting gas and liquid.
While such nonmonotonic flow curves have been recently
measured in colloidal glasses [14], the microscopic origin
of shear banding remains unclear; in particular, it is unclear
whether and how shear banding is related to structural and
dynamic properties of the glassy state. Structural
differences in glasses are small, often below the resolution
limit, and direct observation of the atomic dynamics in
molecular glasses is prohibitively difficult.
Colloidal glasses allow direct observation of single
particle dynamics, offering particle trajectories to be
followed at long time and large length scales [15,16].
The constituent particles exhibit dynamic arrest due to
crowding at volume fractions larger than ϕ
g
∼ 0.58, the
colloidal glass transition [17,18]. These systems exhibit
glasslike properties such as nonergodicity and aging [19],
and they show long-range strain correlations when sheared
slowly [5], demonstrating the high susceptibility of the
material under applied shear. Recent combined rheology
and structure measurements [14] have revealed nonmono-
tonic flow curves and steady-state shear banding in these
systems. The onset of shear banding occurred at shear rates
of around the inverse structural relaxation time of the glass,
suggesting a deep connection between the shear-banding
phenomenon and dynamic properties of the glass.
However, the crucial relation between shear banding and
glassy dynamics remains unclear: Does the high dynamic
susceptibility of the glass eventually lead to a dynamic
analog of a first-order transition?
In this Letter, we use direct observation of single particle
dynamics in a colloidal glass to show that the application of
shear on a glass leads to an intriguing dynamic first-order
transition. We demonstrate the existence of a critical shear
rate, at which the glass separates into two dynamic states
characterized by distinct diffusion time scales. We measure
a new dynamic order parameter [20] to demonstrate the
coexistence of two dynamic phases. We show that this
dynamic transition is accompanied by a weak, but char-
acteristic, structural modification of the glass that relates
shear-induced structural distortion to mechanical proper-
ties. These results offer a new framework to understand the
genuine shear-banding instability observed in a wide range
of colloidal and metallic glasses [9,10].
The colloidal glass consists of sterically stabilized
fluorescent polymethylmethacrylate particles with a diam-
eter of σ ¼ 1.3 μm, and a polydisperity of 7%, suspended
in a density and refractive index matching mixture of
cycloheptyl bromide and cis-decalin. A dense suspension
with particle volume fraction ϕ ∼ 0.60 well inside the
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glassy state is prepared by diluting suspensions centrifuged
to a sediment. The suspension is loaded in a cell between
two parallel plates 65 μm apart, and a piezoelectric trans-
lation stage is used to move the top boundary to apply
shear at constant rates between _γ ¼ 1.5 × 10
−5
and
2.2 × 10
−4
s
−1
, with a maximum strain of 140%.
Confocal microscopy is used to image the individual
particles and determine their positions in three dimensions
with an accuracy of 0.03 μm in the horizontal and 0.05 μm
in the vertical direction [16]. All measurements presented
here are recorded in the steady-state regime, after the
sample has been sheared to γ ∼ 1. We use the structural
relaxation time τ ¼ 2 × 10
4
s [5] of the quiescent glass to
define the dimensionless shear rate _γ
¼ _γτ; the applied
shear rates then correspond to _γ
between 0.3 and 2, smaller
and larger than 1, reflecting the transition from the thermal
to the shear-dominated regime. We note that this normal-
ized shear rate is significantly lower than in other studies of
colloidal flows [11,21].
The shear-rate-dependent flow behavior is summarized
in Fig. 1. At shear rates _γ
< 1, the glass flows homo-
geneously as shown by the particle displacements as a
function of height in Fig. 1(a).At_γ
> 1, the glass
separates spontaneously into bands that flow at different
rates as shown for _γ
¼ 2 in Fig. 1(b). Linear fits to the
displacement profiles yield flow rates of _γ
high
¼
2.2 × 10
−4
s
−1
and _γ
low
¼ 4 × 10
−5
s
−1
that differ by a
factor of 5. We specifically checked for steady state in our
measurements, as reaching steady state may require some
larger amount of strain, especially for the shear-banded
case. To do so, we first confirm that, after an initial
transient, the slopes in Fig. 1(b) remain unchanged over
the entire observation time (see red symbols). We then
carefully checked both structure and dynamics of the glass
as a function of the applied strain. We find that both reach a
plateau, conclusively indicating the emergence of steady
state for strains larger than γ ≳ 0.3, as shown in Fig. 1(b),
inset. We thus observe the spontaneous transition from
steady-state homogeneous to steady-state inhomogeneous
flow at _γ
∼ 1. This is also in agreement with recent
rheology and x-ray scattering measurements [14], revealing
shear banding starting at _γ ∼ τ
−1
. This transition from
homogeneous to inhomogeneous flow is analogous to
the shear banding in metallic glasses [8,12].
To elucidate it, we use the full trajectories of the particles
to investigate their dynamic evolution as a function of the
applied shear. For each particle i with trajectory Δr
i
ðtÞ,we
subtract the mean flow to compute displacement fluctua-
tions Δr
0
i
ðtÞ¼Δr
i
ðtÞ − hΔrðtÞi
z
, where hΔrðtÞi
z
is the
average particle displacement at height z. Typical examples
of the resulting mean-square displacements hΔr
02
ðtÞi in the
high- and low-shear bands are shown in Fig. 2 (inset). The
low-shear band (stars) reveals reminiscence of a plateau,
while the high-shear band (circles) exhibits a closely linear
increase of hΔr
02
ðtÞi, similar to the mean-square displace-
ment of particles in a liquid. This interpretation is supported
by the strain correlations: strain correlations computed
separately for the two bands reveal coexistence of an
isotropic liquidlike and an anisotropic solidlike response
[5]; similar behavior is observed for all other applied
shear rates with _γ
> 1. Interestingly, we can collapse all
mean-square displacements by rescaling the time axis by _γ
as shown in Fig. 2, main panel. The figure compiles
FIG. 1 (color online). Deformation map of colloidal glasses at
volume fraction ϕ ¼ 0.60. The flow is homogeneous at low shear
rates (a) and inhomogeneous beyond the critical shear rate _γ
c
∼
6 × 10
−5
s
−1
(b). The figures show height-dependent particle
displacements at shear rates _γ ¼ 3 × 10
−5
s
−1
(a) and _γ ¼ 1 ×
10
−4
s
−1
(b). Each cross represents a particle. Symbols in
(b) indicate average flow profile after γ ∼ 0 .3 (circles), 0.8
(squares), and 0.95 (stars), demonstrating stable shear bands.
Dashed horizontal lines (b) delineate the shear bands. Inset:
Average nonaffine displacement (during Δt ¼ 2 min, left axis)
and structural distortion (eigenvalue ratio of Minkowski tensor
W
20
1
of Voronoi volumes, see Ref. [22], right axis) versus strain
demonstrate steady state after γ ∼ 0.3.
FIG. 2 (color online). Mean-square displacements of the
particles. Upper left inset: Mean-square displacement in the
upper (circles) and lower shear bands (dots) at _γ
¼ 2. Main
panel: Mean-square displacements as a function of rescaled time
for the applied shear rates _γ
¼ 0.3 (blue), 0.6 (cyan), 1.2 (green),
2 (magenta), and 5.6 (red). Lower right inset: Strain correlations
of low shear band for t ¼ 70 (left) and 350 s (right).
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measurements both in the homogeneous and the shear-
banding regime. This collapse suggests that the different
dynamics of the bands is solely due to different underlying
diffusion time scales. Indeed, this is supported by the strain
correlation function that indicates disappearance of the
solidlike quadrupolar symmetry when correlations are
computed on the rescaled time scale (longer by a factor
of _γ
high
=_γ
low
), as illustrated in Fig. 2 (lower right-hand
insets). We thus conclude that the change of diffusion time
scale causes the symmetry change of correlations; such
discontinuous change reminds one of first-order transitions,
with the discontinuity occurring in the underlying diffusion
time scale.
To quantify this dynamic discontinuity, we search for an
order parameter that is a good measure of the dynamic
evolution. An appropriate measure of the underlying
dynamic evolution is [20]
K ¼ Δt
X
N
i¼1
X
t
obs
t¼0
jΔr
i
0
ðt þ ΔtÞ − Δr
i
0
ðtÞj
2
; ð1Þ
the time-integrated mean-square displacement, where Δt is
a short microscopic time scale. This parameter increases
linearly with observation time t
obs
[Fig. 3(a)]; hence, K=t
obs
measures the rate of the system’s dynamic evolution, and
we choose this as the dynamic order parameter. To address
the transition, we determine values of K=t
obs
in 2 μm thick
horizontal subsections and plot probability distributions for
three different observation times in Fig. 3(b). With increas-
ing observation time, two peaks appear and sharpen,
demonstrating the coexistence of two dynamic states.
The positions of the peaks demarcate the order parameter
values of the coexisting shear bands. We can now construct
the corresponding dynamic phase diagram from the peak
positions of K for all steady-state shear rates, as shown in
Fig. 3(c).At_γ
< 1, only one single peak of K exists,
indicating the homogeneous regime. At _γ
> 1, two values
coexist, indicating the coexisting shear bands. The diagram
has the characteristic topology of a phase diagram, in which
the two-phase region is entered close to a critical point. We
note that similar dynamic phase coexistence has been
recently observed by us in traffic models with interacting
cars [23]. With increasing density and in the limit of strong
braking, traffic jams exhibited long-range correlations, after
which macroscopic phase separation into jammed and free-
moving traffic occurred.
We confirmed the first-order nature of the transition in
x-ray scattering measurements on oscillatory shear [24].At
increasing strain amplitude and concomitant increasing
strain rate, the structure factor exhibited an abrupt sym-
metry change from anisotropic solid to isotropic liquid, just
like the strain correlations from confocal microscopy
(cf. Fig. 2). These measurements demonstrated the sharp-
ness of the transition: using order parameters to quantify
the structural symmetry and its fluctuations, we consis-
tently demonstrated the sharp appearance of a liquidlike
state via an abrupt change of the order parameter. The
confocal microscope measurements presented here allow us
to reveal the microscopic nature of this transition. To
elucidate the sharpness as a function of the increasing
shear field, we continuously ramped the shear rate
_
γ
from
below to above 1, crossing the transition with a continu-
ously increasing shear rate. The resulting values of K as a
function of strain rate [Fig. 3(c), inset] suggest that, indeed,
the transition occurs rapidly, in agreement with our x-ray
measurements [24]. Because of the limited system size both
spatially and along the time dimension, there are significant
fluctuations; nevertheless, the data indicate a sudden jump
of the order parameter at
_
γ
∼ 1. The position of this jump is
consistent with the steady-state measurements (main
panel), plus an eventual small delay due to metastabil-
ity [25].
This dynamic transition is surprising and suggests highly
collective dynamic behavior of the system. Dynamic first-
order transitions have been recently observed in simula-
tions on facilitated glassy dynamics [20]: under an applied
artificial field s that couples to the dynamic order parameter
distribution via PðsÞ¼P
0
expð−Ks=kTÞ, where P
0
is the
unperturbed distribution, all hallmarks of a true first-order
FIG. 3 (color online). Dynamic order parameter and phase
diagram. (a) Dynam ic order parameter as a function of obser-
vation time. K is a linear measure of the system’s dynamic
evolution. (b) Histogram of order parameter values for increasing
observation times. The emerging bimodal distribution indicates
dynamic phase coexistence. (c) Corresponding dynamic phase
diagram: Mean order parameter as a function of applied strain
rate. The dashed lines delineate boundaries of the shear-banding
regime. Inset shows the dynamic order parameter at continuously
increasing applied shear rate, for particles in the upper (red
squares) and lower regions (blue dots). Arrow demarcates sudden
change of order parameter.
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transition were observed. In the present case, the applied
shear stress σ can take the role of the conjugate field: it is
well known that the applied stress σ couples to local
rearrangements via their induced strain ϵ according to
PðϵÞ¼P
0
exp½−σΩðϵÞ=kT, where the activation volume
ΩðϵÞ¼
R
ϵdV measures the integrated local strain [8,26].
Because of the high dynamic susceptibility of the colloidal
glass under applied shear [5], likewise, the coupling
between the applied stress and the particle dynamics can
introduce a first-order transition in its dynamic evolution.
While the transition occurs in the dynamics, it is
interesting to elucidate changes in the glass structure.
Constitutive models of the flow of amorphous materials
suggest a coupling between flow and structure, often
related to small density changes [8,11,27]; we therefore
investigated structural differences in the two bands. We
show radial distribution functions in Fig. 4(a). No obvious
structural difference between the low- and high-shear bands
is observed, in agreement with earlier observations [11].
However, when we resolve gðrÞ along the compression and
dilation directions, a clear structural difference shows up
[Fig. 4(a), inset]. The high-shear band (red symbols and
line) exhibits pronounced anisotropy, while the low-shear
band (blue symbols and line) is more isotropic. The
decrease of gðrÞ in the dilation direction indicates a
depletion of nearest neighbors in the extensional sector,
while the (slight) increase of gðrÞ in the compression
direction indicates a (small) enhancement of nearest
neighbors in the compressional sector. Because of the
nature of the hard-core potential, the increase in the
compressional sector is small, smaller than the particle
loss in the extensional sector, and this results in a net
depletion of particles in the nearest-neighbor cage.
According to Ref. [28], this depletion of nearest neighbors
leads to lower shear moduli. This is precisely what is
observed in the figure: the decrease along the extension
direction is more pronounced than the increase in the
compression direction, leading to a depletion of particles in
the cage. These results reveal the structural origin of the
different mechanical behavior of the two bands. This
structural distortion should play an important role in the
coupling of structure and dynamics in the shear-banding
transition. Indeed, we can measure the resulting net dilation
in the shear band from the mean-square difference Δ
2
between the two angle-averaged gðrÞ curves as a function
of a linear stretching α that transforms r to r
0
¼ r × α.We
show Δ
2
as a function of α in Fig. 4(b), inset; the minima of
Δ
2
at α > 1 indicate a small amount of dilation. We
evaluate the minima α
min
for all shear rates and plot α
min
as a function of shear rate in the main panel. These values
indicate a dilation of ∼0.4% in the high-shear band after
shear banding. While the detected changes are small and
affected by large uncertainty, they demonstrate a character-
istic structural change accompanying the shear-banding
transition.
The direct observation of particle dynamics during shear
banding of a colloidal glass reveals the coexistence of two
dynamic steady states analogous to the coexistence of
equilibrium phases in first-order transitions. The applied
shear plays the role of a conjugate field that couples to the
dynamic evolution: sufficiently high values of the applied
shear rate cause coexistence of two dynamic states with
different time scales for diffusion. This mechanism points
out new perspectives to comprehend flow instabilities in
amorphous materials: the large dynamic susceptibility, on
the one hand (evidenced by long-range strain correlations),
and the coupling to the applied shear, on the other hand,
lead to a dynamic transition that is akin to first-order
transitions. This is supported by the different structure of
the two bands, distinct by the degree of distortion of the
nearest-neighbor cage. This distortion leads to net depletion
of particles in the cage, and to lower shear moduli. We
believe that the presented dynamic transition should be a
general feature of dynamically driven systems, and recent
traffic simulations reveal the formation of dynamic con-
densates in one-dimensional dynamically facilitated sys-
tems, consistent with this idea [23]. The observed coupling
between applied stress and diffusion time could play a role
in crowded biological systems; as the diffusion time scale is
an important underlying time scale, any direct coupling of
diffusion to external (shear) fields would greatly affect the
diffusive behavior upon mechanical perturbation.
This work was supported by the Foundation for
Fundamental Research on Matter (FOM), which is sub-
sidized by the Netherlands Organisation for Scientific
Research (NWO). We thank D. Denisov for discussion.
P. S. acknowledges Vidi and Vici fellowships from NWO.
[1] M. D. Ediger, C. A. Angel, and S. R. Nagel, J. Phys. Chem.
100, 13200 (1996).
FIG. 4 (color online). Glass structure and density. (a) Pair
correlation function of particles in the low- (blue dots) and high-
shear bands (red squares). Inset: Angle-resolved gðrÞ along the
shear-shear gradient plane, extension (π=4, diamonds) and
compression direction (3π=4, circles), for the low- (blue) and
high-shear bands (red). Arrows indicate decrease of gðrÞ in the
extension direction. (b) Dilation parameter α
min
versus shear rate
determined from the minimum of the mean-square difference of
gðrÞ (inset). Dashed lines are guides to the eye.
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![FIG. 1 (color online). Deformation map of colloidal glasses at volume fraction ϕ ¼ 0.60. The flow is homogeneous at low shear rates (a) and inhomogeneous beyond the critical shear rate _γc ∼ 6 × 10−5 s−1 (b). The figures show height-dependent particle displacements at shear rates _γ ¼ 3 × 10−5 s−1 (a) and _γ ¼ 1 × 10−4 s−1 (b). Each cross represents a particle. Symbols in (b) indicate average flow profile after γ ∼ 0.3 (circles), 0.8 (squares), and 0.95 (stars), demonstrating stable shear bands. Dashed horizontal lines (b) delineate the shear bands. Inset: Average nonaffine displacement (during Δt ¼ 2 min, left axis) and structural distortion (eigenvalue ratio of Minkowski tensor W201 of Voronoi volumes, see Ref. [22], right axis) versus strain demonstrate steady state after γ ∼ 0.3.](/figures/fig-1-color-online-deformation-map-of-colloidal-glasses-at-351etulp.webp)
