Directional grain growth from anisotropic kinetic
roughening of grain boundaries in sheared
colloidal crystals
Shreyas Gokhalea,1, K. Hima Nagamanasab,1, V. Santhoshc, A. K. Sooda,c, and Rajesh Ganapathyc,2
a
Department of Physics, Indian Institute of Science, Bangalore 560012, India; bChemistry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced
Scientific Research, Bangalore 560064, India; and cInternational Centre for Materials Science, Jawaharlal Nehru Centre for Advanced Scientific Research,
Bangalore 560064, India
Edited by James A. Warren, National Institute of Standards and Technology, Gaithersburg, MD, and accepted by the Editorial Board October 25, 2012
(received for review June 19, 2012)
The fabrication of functional materials via grain growth engineering implicitly relies on altering the mobilities of grain boundaries
(GBs) by applying external fields. Although computer simulations
have alluded to kinetic roughening as a potential mechanism for
modifying GB mobilities, its implications for grain growth have
remained largely unexplored owing to difficulties in bridging the
widely separated length and time scales. Here, by imaging GB
particle dynamics as well as grain network evolution under shear,
we present direct evidence for kinetic roughening of GBs and
unravel its connection to grain growth in driven colloidal polycrystals. The capillary fluctuation method allows us to quantitatively extract shear-dependent effective mobilities. Remarkably, our
experiments reveal that for sufficiently large strains, GBs with
normals parallel to shear undergo preferential kinetic roughening,
resulting in anisotropic enhancement of effective mobilities and
hence directional grain growth. Single-particle level analysis shows
that the mobility anisotropy emerges from strain-induced directional enhancement of activated particle hops normal to the GB
plane. We expect our results to influence materials fabrication
strategies for atomic and block copolymeric polycrystals as well.
colloids
| grain boundary migration | anisotropic grain growth
T
he motion and rearrangement of grain boundaries (GBs) is
central to our understanding of recrystallization (1), grain
growth and its stagnation (2), and superplasticity (3) in a broad
class of polycrystalline materials including metals (4), ceramics (5),
colloidal crystals (6), and block copolymers (7, 8). Polycrystals are
pervasive as engineering materials and elucidating mechanisms
that determine the structure and dynamics of their GBs continue to
be a central goal of materials research (9). Advances in computer
simulation methods (2, 10, 11) and experiments (12, 13) have
provided substantial insights into the microscopic origins of these
mechanisms. In conventional materials, nevertheless, establishing
a direct link between the dynamics at the single/few atom length
scale and the collective behavior of the many thousands of atoms
that constitute GBs and grains poses a serious challenge (14).
A bridging of length scales is vital for grain growth studies. The
key parameter in grain growth and its stagnation is the mobility of
GBs, which is determined by their roughness (2, 15). Although
transmission electron microscopy (TEM) is well-suited for grain
growth measurements, no technique exists that can nonintrusively
quantify the atomic scale roughness of buried GBs (14). These
experimental shortcomings are compounded in driven polycrystals,
which assume practical significance in the fabrication of functional
materials via grain growth engineering (2, 7). Driven GBs (16) are
thought to kinetically roughen (17), which may result in significantly enhanced mobilities (15) and influence grain growth. Further, in nanocrystalline materials, which often possess superior
mechanical properties compared with their coarse-grained counterparts (18), kinetic roughening is expected to have a profound
impact on mechanical behavior (16). Despite its technological
20314–20319 | PNAS | December 11, 2012 | vol. 109 | no. 50
relevance, the implications of kinetic roughening for GB mobility
and grain growth remain poorly understood. Although molecular
dynamics simulations can model atomistic phenomena realistically,
they often are limited to small system sizes (19) and short time
scales (11), which may influence measurements of GB roughness
(15) and predictions of grain growth laws (2, 20). Kinetic Monte
Carlo simulations (21) and phase field models (22) are insensitive
to microscopic details. Thus, there is a need for a complementary
multiscale approach that can probe kinetic roughening and its link
to grain growth in driven polycrystals.
In this work, we combined techniques in colloid science to investigate the dynamics of GBs in sheared 3D colloidal crystals over
length scales spanning from the single-particle level to that of the
GB network. Our single-particle resolution experiments reveal
that for large oscillatory strains, boundaries with normals oriented
parallel to shear are preferentially kinetically roughened. We find
that the equilibrium capillary fluctuation method (CFM) allows us
to extract effective mobilities, even under shear. Analysis of dynamics at the single-particle level reveals that strain-induced anisotropy in particle displacements is manifested as a preferential
enhancement of the effective mobilities. Most strikingly, by imaging the GB network under shear, we show that this shear-induced anisotropy in the mobility leads to directional grain growth.
Because colloid models are valuable analogs of atomic systems
(23–27), insights gained from the present studies should be relevant for atomic and block copolymeric polycrystals as well.
Our system consisted of thermo-responsive size-tunable poly
(N-isopropylacrylamide) (PNIPAm) colloids tagged with the fluorophore rhodamine 6G for confocal imaging (Materials and Methods). We adjusted the suspension volume fractions ϕ such that at
311 K, where the particle diameter σ = 0:60 μm, ϕ < 50%, and the
equilibrium phase is a liquid. Annealing the samples to 296 K led to
particle swelling, with σ = 0:95 μm, resulting in a ϕ ∼ 70%, and the
suspension crystallized into a polycrystalline state. A particular
advantage of PNIPAm colloids is that they allow us to control the
grain size systematically by altering the annealing rate (SI Text and
Fig. S1). To mimic the nanocrystalline regime in which GBmediated response is predominant (12, 18) and grain sizes typically are 50–200 atom diameters across, we aimed for average
grain sizes of the order of 100 colloid diameters. We restricted
Author contributions: S.G., K.H.N., A.K.S., and R.G. designed research; V.S. and R.G.
designed the confocal rheometer; S.G. and K.H.N. performed research; V.S. and R.G.
contributed new reagents/analytic tools; S.G. and K.H.N. analyzed data; and S.G., K.H.N.,
A.K.S., and R.G. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission. J.A.W. is a guest editor invited by the Editorial
Board.
1
S.G. and K.H.N. contributed equally to this work.
2
To whom correspondence should be addressed. E-mail: rajeshg@jncasr.ac.in.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1210456109/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1210456109
Results and Discussion
Using our confocal rheometer, we simultaneously imaged the
real-space particle-level microstructure of colloidal polycrystals
subjected to a controlled shear deformation (Materials and
Methods, SI Text, and Fig. S2). Samples were loaded in a customdesigned temperature-controlled parallel-plate shear cell with
a typical plate separation of g = 45 μm, corresponding to roughly
60 crystalline layers. Thermal annealing of the colloidal suspension at a rate of 0.125 K/min yielded the desired grain size with
the ½1; 1; 1" crystal planes parallel to the walls of our shear cell;
therefore, our studies are confined to pure tilt boundaries. The
plane pqrs in Fig. 1A shows a snapshot of an HAGB. We quantify
the dynamics of the GB interface (SI Text) by the height function
hðx; tÞ, which is the 1D string formed by the intersection of the
A
y
b`
x
f`
s
c`
z
g
n
r
g`
d`
e`
Θ = 19°
∇v
q
h`
B
v
h(x,t
x,t)
v
x
p
∇
a`
C
D
E
Fig. 1. Kinetic roughening of HAGBs. (A) Schematic of the shear geometry.
The planes a′b′c′d′ and e′f ′g′h′ form the top and bottom plates of our shear
cell. The imaging plane pqrs shows two crystallites with Θ = 198. The GB plane
is the shaded region denoted by normal n. This figure corresponds to the
configuration n k v. (B–E) Probability distribution of height fluctuations
PðΔhÞ for increasing γo : (B) γo = 0%, (C) γo = 2%, (D) γo = 8:3%, and (E)
γo = 18:3%. In C–E, solid curves and dashed curves are gaussian fits to histograms that correspond to GBjj and GB⊥, respectively. The circles correspond to
w, and the width of the gray-shaded region corresponds to the w for γo = 0%.
Gokhale et al.
GB plane n with the imaging plane pqrs (thick white line in Fig.
1A). The image plane pqrs typically is 8–10 layers above the
bottom plate to avoid wall effects. To investigate shear-induced
anisotropic effects, experiments consisted of studying the dynamics of hðx; tÞ at different oscillatory shear strains, γ, for two
GB configurations: n k v (denoted by GBk) and n k ∇ × v
(denoted by GB⊥), where v is the velocity (Movie S1).
γ = γo sinðωtÞ was changed in our experiments by keeping the oscillation frequency fixed at ω = 1 rad/s and changing the strain
amplitude γo . The values of γo quoted here have been corrected
to incorporate the effect of wall slip (SI Text and Fig. S3). GB
properties sensitively depend on Θ (4, 27, 28), and hence we focused on HAGBs within a narrow range of Θs, 198 < Θ < 218, to
facilitate quantitative comparison of results from experiments on
GBk and GB⊥.
Kinetic Roughening of GBs. To characterize the roughness of the
interface, we quantify a related parameter: the interface width, w,
defined as the root-mean-squared (rms) fluctuations of hðx; tÞ (29).
In Fig. 1 B–E, we plot the normalized histograms of height fluctuations PðΔhÞ for increasing γo . Here, Δhðx; tÞ = hðx; tÞ − hhðx; tÞit .
For all γo , PðΔhÞ is gaussian for GBk (solid curves) and GB⊥
(dashed curves), allowing us to estimate w (shown by circles) directly from the fits (Fig. 1 C–E). For γo ≤ 2%, wk ≈ w⊥ (Fig. 1 B and
C). Remarkably, we find that for γo ≥ 8:3%, GBk undergoes
preferential kinetic roughening with wk > w⊥ (Fig. 1 D and E).
Earlier studies addressed equilibrium roughening of HAGBs, in
the context of faceting–defaceting transition (30), and kinetic
roughening of Σ GBs (16). An important finding from the present
study is that general nonfaceted HAGBs also undergo kinetic
roughening (15). Further, the migration mechanism of GBs is
expected to change from a step-type motion to a continuous
motion across the roughening transition. In Fig. S4, we plot the
GB center-of-mass, defined as hhðx; tÞix , with t. Although it is
difficult to estimate the roughening transition temperature for our
HAGBs, we indeed find that the motion of GBs at zero-shear and
GB⊥ at γo = 8:3% is stepped and that of the kinetically roughened
GBk (γo = 8:3%) is continuous (Fig. S4).
Shear-Induced Anisotropy in GB Mobility. Having quantified the
roughness of GBs, we next focused on determining their mobility,
M, which relates the migration velocity of the boundary vGB = MF
to the driving force F (4). The net driving force acting on
a boundary is composed of two parts, F = Γκ + Pext , where the first
term is due to intrinsic boundary curvature, κ, and the second term
is from externally imposed stresses. Here, Γ is the GB stiffness
(31). To calculate M, we first determined Γ using the CFM (32–
34). In the present studies, a 1D section of the 2D GB interface is
considered, which is not unreasonable as recent experiments have
shown that Γ for an equilibrium crystal–melt interface obtained by
ignoring the third dimension completely is in good agreement with
that obtained from 3D computer simulations (35). The equilibrium roughness of an interface is a tradeoff between the interfacial
free energy, which is minimum for a smooth interface, and thermal
energy. In CFM, thermally excited capillary fluctuations of the
interface are decomposed into normal modes and the amplitude
kB T
of the modes decays as hjAðkÞj2 i = LΓk
2 . Here, kB T is the thermal
energy, k is the wavevector, and L is the interface length. Albeit
CFM is an equilibrium statistical mechanics approach, studies on
a driven colloidal liquid–gas interface have shown that shear
leaves the k dependence of hjAðkÞj2 i unchanged, which allows the
definition of an effective shear-dependent stiffness, even when the
system is out of equilibrium (36).
We perform a Fourier decomposition of Δhðx; tÞ (SI Text and
Fig. S5) and plot LhjAðkÞj2 ik2 vs. k for GBk (solid circles) and
GB⊥ (hollow circles) for increasing γo (Fig. 2 A–C). In close
parallel with the sheared colloidal liquid–gas interface (36), the
equilibrium CFM analysis is valid for driven solid–solid interfaces
PNAS | December 11, 2012 | vol. 109 | no. 50 | 20315
APPLIED PHYSICAL
SCIENCES
our attention to high-angle grain boundaries (HAGBs) formed
when adjacent grains that straddle the boundary have a misorientation angle Θ > 128. Geometrically, GBs can be described by
a planar array of misfit dislocations (28). In HAGBs, the density of
geometrically necessary dislocations is large and the dislocation
cores overlap, leading to a continuous disordered interface.
C
B
A
D
γο = 18.3%
γο = 8.3%
γο = 2%
E
F
Fig. 2. Shear-induced anisotropy in GB mobility. LhjAðkÞj2 ik2 vs. k for (A) γo = 2%, (B) γo = 8:3%, and (C) γo = 18:3% for GBjj (●) and GB⊥ (○). (D) Stiffness Γ as
a function of γo obtained from A–C by averaging LhjAðkÞj2 ik2 over the shaded region. (E) Mobility M and (F) reduced mobility M* = MΓ as a function of γo for
GBjj (●) and GB⊥ (○). In D–F, (□) corresponds to γo = 0%.
as well (Fig. S6). We find that LhjAðkÞj2 ik2 is nearly a constant kBΓT
over a decade in k, for both boundary configurations and for all
γo . Notably, we find that Γ, obtained by averaging LhjAðkÞj2 ik2
over the shaded region (Fig. 2 A–C), is preferentially lowered
for GBk (solid circles) compared with GB⊥ (hollow circles) for
γo ≥ 8:3% (Fig. 2D and Fig. S5A). An anisotropic lowering of Γ for
GBk vs. GB⊥ is a result of anisotropic enhancement in the interface
roughness and is consistent with Fig. 1 C–E. It is important to note
_ whereas for HAGBs
that in ref. 36 Γ depends on the shear rate γ,
we find that Γ depends strongly on γo (SI Text and Fig. S7).
For capillary fluctuations that decay as hAðk; 0ÞA * ðk; tÞi =
2
hjAðkÞj2 ie−MΓk t , the dynamic correlation function gd ðτÞ =
!
"
1
ðΓMtÞ2
kB Tξ
(SI Text, Fig. S8, and
hΔhðx; tÞΔhðx; t + τÞix;t = Γ erfc
ξ
Fig. S9 A–C) yields the product MΓ (37). Here, ξ is the lateral
correlation length (38) obtained from exponential fits to the height–
height correlation function gh ðδxÞ = hΔhðx; tÞΔhðx + δx; tÞix;t (Fig.
S9 D and E) and erfcðtÞ is the complementary error function. Fig. S9
A–C shows gd ðτÞ for GBk (solid circles) and GB⊥ (hollow circles) for
increasing γo . Plugging in the value of Γ (Fig. 2D), obtained earlier,
into the complementary error function fits of the data (black lines in
Fig. S9 A–C) yields an effective shear-dependent mobility M. We find
that the increase in M with γo for the kinetically roughened boundary,
GBk (Fig. 2E, solid circles), is more rapid compared with GB⊥ (Fig.
2E, hollow circles) and is almost an order of magnitude larger at
γo = 18:3% (Fig. S5B). It generally is believed that kinetic roughening
alters GB mobilities and dictates microstructure evolution in driven
polycrystals. However, even in the zero-driving force limit, a connection between roughness and mobility has been shown only in computer simulations (15) and its extension to nonzero driving forces has
not been explored. By integrating fast confocal microscopy with
rheology, we have shown that the shear-induced anisotropy in M is
a direct consequence of preferential kinetic roughening.
Will shear-induced anisotropy in mobility, Mk > M⊥ , lead to
directional grain growth? To answer this question, we take a closer
look at the mobility relation v = MΓκ + MPext (31). Focusing on the
second term, one would naively expect that an anisotropy in M
implies an anisotropy in v. Although it is difficult to characterize
Pext acting on GBs in a driven polycrystal (2), it chiefly consists of
contributions from the applied bulk stress and its coupling to the
elastic anisotropy in adjacent grains (39). In our experiments, we
expect the driving force due to bulk stress to be zero because of the
20316 | www.pnas.org/cgi/doi/10.1073/pnas.1210456109
oscillatory nature of the applied strain. Further, the driving force
due to elastic anisotropy becomes significant only for grain sizes
larger than about 1 μm (∼3,000σ) in atomic systems (39), which is
much larger than the grain sizes used in our experiments (∼100σ).
It therefore is likely that in our experiments, Pext = 0 and hence
v = MΓκ. We note that with increasing γo , M increases (Fig. 2E)
whereas Γ is found to decrease (Fig. 2D) for both GBk and GB⊥.
Thus, in the approximation Pext = 0, we expect an anisotropy in GB
motion and grain growth only if the reduced mobility, M* = MΓ, is
itself anisotropic. Remarkably enough, Fig. 2F shows that anisotropic kinetic roughening indeed results in anisotropic-reduced
mobilities with Mkp > M⊥p at large γo . Although we considered a
narrow range of Θs to allow comparison across the two GB configurations, given that HAGBs exhibit qualitatively similar dynamics over a broad range of Θs (27, 40), we expect the anisotropy
in M and M* to be a generic feature of these boundaries.
Directional Grain Growth Under Shear. Even as our single-particle
resolution measurements revealed a shear-induced anisotropy in
both M and M*, to link this to grain growth it is necessary to probe
GB dynamics under shear at the grain network length scale. To this
end, we performed Bragg diffraction microscopy (BDM), which is
the analog of TEM for colloidal crystals (41), simultaneously with
rheometry (Materials and Methods, SI Text, and Fig. S2B). Here,
the colloidal crystal sample, confined between the shear plates, is
illuminated by a white-light source and a detector is placed at the
first Bragg diffracted spot. A perfect single crystal would result in
uniform intensity at the detector. For a polycrystal, however, the
Bragg condition is met only by crystallites of a particular orientation, resulting in a diffraction contrast image. We note here that
BDM does not yield an orientation map of the polycrystal but
allows us to distinguish between low-angle GBs, which appear as
arrays of discrete spots corresponding to dislocation cores, and
HAGBs, which appear as continuous curves (42). A typical BDM
snapshot of the colloidal polycrystal is shown in Fig. 3A, with GBks
shown by solid lines and GB⊥s shown by dashed lines. Fig. 3 B and
C shows hhðx; tÞix vs. t for γo = 3:9%. In complete concord with
results thus far, anisotropy in M and M* indeed results in anisotropic grain growth with vk > v⊥ on an average (Movie S2; see also
SI Text, Fig. S10, and Movie S3). As a consistency check, we estimated the expected GB velocity by plugging in the value of M* and
assuming κ = d1, where d is the average grain radius, which is ∼100
Gokhale et al.
A
B
6
7
C
v
10
11
100 µm
5
3
4
1
2
8
9
μm. The calculated GB velocity is in close agreement with those
obtained from BDM measurements.
Microscopic Origins of Directional Grain Growth. Our observations
cannot be rationalized within the theoretical framework of shearcoupled GB migration because our HAGBs possess no identifiable structural units, the resolved shear stress has no component
in the glide plane of boundary dislocations (43–45), and the capillary fluctuation spectrum decays as k12 (46). Instead, we chose to
exploit the analogy between HAGBs and glass-forming liquids
(27, 40). For HAGBs, the rate-controlling events for migration are
single-atom hops across the boundary plane (47, 48) and the
characteristic time associated with these hops is the cage-breaking
A
D
time t*. We extracted t* from the inflection point in the mean first
passage time τðrÞ (48), defined as the average time taken by
a particle to traverse a distance r for the first time. We find that t*
for γo = 2% is greater than t* for γo = 8.3% for both GBk and GB⊥
(Fig. 4 A), implying that the activation barrier for cage breaking is
lowered at a higher γo (SI Text and Fig. S11). This lowering of
activation barriers is reminiscent of the strain-induced deformation of the potential energy landscape of sheared glasses
(49). For a migrating boundary, the self-part of the van Hove
correlation function Gs ðr; ΔtÞ for Δt ≥ t* develops a characteristic peak at an intermediate distance r with rcage < r < σ. Here,
rcage corresponds to the cage size. However, Gs ðr; ΔtÞ (Fig. 4 B and
C) do not exhibit peaks between rcage and σ for both GBk and GB⊥
B
C
E
Fig. 4. Single-particle dynamics at GBs. (A) Mean first passage time τðrÞ for GBjj [γo = 2% (■), γo = 8:3% (●)] and GB⊥ [γo = 2% (□), γo = 8:3% (○)]. The dashed
curves are Hill function fits used to extract the inflection point t*, shown as solid lines for GBjj and dashed lines for GB⊥. (B and C) Self part of the van Hove
correlation function Gs ðr; ΔtÞ for γo = 8:3% for GBjj (B) and GB⊥ (C) at Δt = 0:5t*, t*, and 2t*. (D and E) Angular distribution of displacements contributing to
the nearest-neighbor peak of Gs ðr; ΔtÞ for γo = 8:3% over Δt = t* for GBjj (D) and GB⊥ (E). In D and E, the solid arrow denotes the shear direction and the
dashed arrow denotes the GB normal. About 50% of particle hops happen within a 40° window centered on the shear direction for both GBjj and GB⊥.
Gokhale et al.
PNAS | December 11, 2012 | vol. 109 | no. 50 | 20317
APPLIED PHYSICAL
SCIENCES
Fig. 3. Directional grain growth under shear. (A) Snapshot of a colloidal polycrystal obtained using BDM. The white arrow labeled v shows the shear velocity
direction. The curves highlight GBs for which migration velocities were measured. Solid curves indicate GBjjs and the dashed curves indicate GB⊥s. The arrows
point along the direction of GB motion, and their length is proportional to the GB migration velocity vGB . (B) hhðx; tÞix vs. t for GBjjs. (C) hhðx; tÞix vs. t for GB⊥s.
The lines are least-squares fits to the data. The numbers shown adjacent to the GB center-of-mass profiles vs. time in B and C correspond to those in A.
Boundary 4 was used as a reference for the drift correction for GBjjs and GB⊥s.
even for Δt > t*, in striking resemblance to observations on stationary GBs (48). This is not entirely surprising given that over the
duration of our confocal rheology experiments (420 s), the net
displacement of hhðx; tÞix is only about 0:5σ, even for GBjj. Further, although displacements in all directions contribute to the
peak at r = σ in Gs ðr; ΔtÞ (Fig. 4 B and C), hops normal to the GB
plane are the ones that primarily influence boundary mobility. To
determine whether shear biases activated hops, we plot the angular distribution of displacements contributing to the peak in
Gs ðr = σ; Δt = t*Þ for GBk and GB⊥. For γo = 2%, the displacements are more isotropically distributed (Fig. S12) compared with
γo = 8:3% for which we find a significant enhancement along the
shear direction for both GBk and GB⊥ (Fig. 4 D and E). For GBk,
these shear-enhanced hops are normal to the boundary plane and
therefore lead to a preferential increase in hjAðkÞj2 i (Fig. S6) and
M (Fig. 2E). Collectively, our observations show that strain
enhances HAGB interface fluctuations without contributing to
a bulk driving force for migration, consistent with our earlier assumption of Pext = 0, and is therefore analogous to temperature.
As expected, similar to the variation of M and Γ with temperature
(50), the dependence of M on γo is much stronger than that of Γ
(Fig. 2). Not only does this rationalize the success of the CFM
in extracting Γ and M for sheared HAGBs, but it also accounts
for the anisotropic enhancement in M* and therefore directional
grain growth.
Conclusions
By bringing together experimental techniques in colloid science,
we have bridged vastly disparate length and time scales to unravel
the microscopic underpinnings of grain growth in sheared polycrystals. We have shown conclusively that preferential kinetic
roughening (Fig. 1 C–E) and the resulting anisotropy in effective
mobilities (Fig. 2E) ultimately lead to directional grain growth
(Fig. 3). We find that strain enhances only the amplitudes of
capillary fluctuation modes (Fig. 2 A–C) but leaves the fluctuation spectrum itself qualitatively unchanged (Fig. S6). This
strongly suggests that the paradigm of shear as an effective
temperature, routinely adopted to describe driven glasses (51), is
germane even to the HAGB interfaces investigated here (27). We
observe that the strain-induced lowering of activation barriers
and the concomitant anisotropic enhancement of particle displacements satisfactorily explain the observed anisotropy in GB
mobility (Fig. 4). Owing to the close similarity in the physics of
GBs across diverse systems (4, 8, 27, 37), we expect our results can
be extended to atomic and block copolymeric polycrystals as well.
Further, by combining the temperature-tunability of PNIPAm
1. Humphreys FJ, Hatherly M (2004) Recrystallization and Related Annealing Phenomena (Elsevier, Oxford, UK), 2nd Ed.
2. Holm EA, Foiles SM (2010) How grain growth stops: A mechanism for grain-growth
stagnation in pure materials. Science 328(5982):1138–1141.
3. Lu L, Sui ML, Lu K (2000) Superplastic extensibility of nanocrystalline copper at room
temperature. Science 287(5457):1463–1466.
4. Gottstein G, Shvindlerman LS (2010) Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications (Taylor & Francis, Boca Raton, FL), 2nd Ed.
5. Powers JD, Glaeser AM (1998) Grain boundary migration in ceramics. Interface Sci 6
(1):23–39.
6. Palberg T, Monch W, Schwarz J, Leiderer P (1995) Grain size control in polycrystalline
colloidal solids. J Chem Phys 102(12):5082–5087.
7. Torija MA, Choi SH, Lodge TP, Bates FS (2011) Large amplitude oscillatory shear of
block copolymer spheres on a body-centered cubic lattice: Are micelles like metals?
J Phys Chem B 115(19):5840–5848.
8. Ryu HJ, Fortner DB, Rohrer GS, Bockstaller MR (2012) Measuring relative grain-boundary
energies in block-copolymer microstructures. Phys Rev Lett 108(10):107801-1–5.
9. Thompson CV (2000) Structure evolution during processing of polycrystalline films.
Annu Rev Mater Sci 30(1):159–190.
10. Janssens KGF, et al. (2006) Computing the mobility of grain boundaries. Nat Mater 5
(2):124–127.
11. Trautt ZT, Upmanyu M, Karma A (2006) Interface mobility from interface random
walk. Science 314(5799):632–635.
12. Rupert TJ, Gianola DS, Gan Y, Hemker KJ (2009) Experimental observations of stressdriven grain boundary migration. Science 326(5960):1686–1690.
20318 | www.pnas.org/cgi/doi/10.1073/pnas.1210456109
colloids with template-directed growth (25, 41), in principle it
should be possible to design bicrystals and polycrystals of controlled grain crystallography and size and investigate their response to shear deformation. More importantly, our experiments
exemplify a multiscale approach that can be applied readily to
elucidate fundamental as well as technologically relevant phenomena, such as grain rotation and coalescence, GB sliding, and
texture evolution in driven polycrystals.
Materials and Methods
PNIPAm colloids of diameter 950 nm (polydispersity <5%) were synthesized
by the standard emulsion polymerization route. All the chemicals used were
purchased from Sigma-Aldrich and had a purity in excess of 98%. Particles
were purified using the method described in ref. 27. The purified samples
were concentrated to yield ϕ ∼ 70% at 296 K.
Confocal Rheology. To facilitate confocal imaging under shear, we integrated
a fast confocal microscope (Visitech VT-Eye) with a commercial rheometer
(MCR-301, Anton Paar) mounted on a homemade mechanical stage (SI Text
and Fig. S1A). Samples were imaged using a Leica objective (Plan Apochromat 100× N.A. 1.4, oil immersion) and a laser excitation centered at 514
nm. The field of view was a 54 × 54-μm slice containing ∼3,200 particles.
Images were captured at 10 frames per second (fps) for γo = 2% and 8.3%
and 26 fps for γo = 18.3%. The temperature was maintained at 296 K for all
experiments. Standard codes were used for particle tracking (52), and subsequent analysis was performed using algorithms developed independently.
The particle-tracking resolution as calculated from the micrometer to pixel
ratio is 0.07 μm.
BDM Under Shear. For grain growth measurements under shear, we integrated
BDM with rheometry (SI Text). The sample was illuminated by a white light
LED source incident at an angle θ = 56˚ (see Fig. S1B for a schematic). A lowmagnification Leica objective (Plan Apochromat 10× N.A. 0.4, dry) was used
to image the GB network. The temperature was maintained at 296 K, and an
oscillatory strain of frequency ω = 1 rad/s and amplitude γo = 3.9% was applied to the sample. The 0.915 × 0.680-mm field of view contained ∼25 grains.
Images were captured at 1 fps for 4 hours, and GB profiles were tracked
manually at periodic intervals to generate their center-of-mass time series. To
subtract drift contributions from the GB motion, a flat immobile boundary
was used as a reference.
ACKNOWLEDGMENTS. The authors thank Jack Douglas and Vikram Deshpande for useful discussions. The authors also thank the anonymous
reviewers for their valuable suggestions. S.G. thanks the Council for
Scientific and Industrial Research (CSIR) India for a Shyama Prasad
Mukherjee Fellowship, K.H.N. thanks CSIR India for a Senior Research
Fellowship, A.K.S. thanks CSIR India for a Bhatnagar Fellowship, and R.G.
thanks the International Centre for Materials Science and the Jawaharlal
Nehru Centre for Advanced Scientific Research for financial support.
13. Wang Z, et al. (2011) Atom-resolved imaging of ordered defect superstructures at
individual grain boundaries. Nature 479(7373):380–383.
14. Van Swygenhoven H (2002) Polycrystalline materials. Grain boundaries and dislocations. Science 296(5565):66–67.
15. Olmsted DL, Foiles SM, Holm EA (2007) Grain boundary interface roughening transition and its effect on grain boundary mobility for non-faceting boundaries. Scr
Mater 57(12):1161–1164.
16. Lee SB, et al. (2010) Kinetic roughening of a ZnO grain boundary. Appl Phys Lett 96
(19):1919061-3.
17. Halpin-Healy T, Zhang Y-C (1995) Kinetic roughening phenomena, stochastic growth,
directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Phys
Rep 254(4):215–414.
18. Meyers MA, Mishra A, Benson DJ (2006) Mechanical properties of nanocrystalline
materials. Prog Mater Sci 51(4):427–556.
19. Cherkaoui M, Capolungo L (2009) Atomistic and continuum modeling of nanocrystalline materials. Springer Series in Material Science, eds Hull R, Parisi J, Osgood RM, Jr.,
Warlimont H (Springer Science+Business Media, New York), Vol 112, p 81.
20. Farkas D, Mohanty S, Monk J (2007) Linear grain growth kinetics and rotation in
nanocrystalline Ni. Phys Rev Lett 98(16):165502-1–4.
21. Anderson MP, Srolovitz DJ, Grest GS, Sahni PS (1984) Computer simulation of grain
growth—I Kinetics. Acta Metall 32(5):783–791.
22. Chen L-Q (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res
32(1):113–140.
23. Suresh S (2006) Crystal deformation: Colloid model for atoms. Nat Mater 5(4):
253–254.
Gokhale et al.
39. Haslam AJ, et al. (2003) Stress-enhanced grain growth in a nanocrystalline material by
molecular-dynamics simulation. Acta Mater 51(7):2097–2112.
40. Zhang H, Srolovitz DJ, Douglas JF, Warren JA (2009) Grain boundaries exhibit the
dynamics of glass-forming liquids. Proc Natl Acad Sci USA 106(19):7735–7740.
41. Schall P, Cohen I, Weitz DA, Spaepen F (2006) Visualizing dislocation nucleation by
indenting colloidal crystals. Nature 440(7082):319–323.
42. Pieranski P (1981) Physics of Defects, eds Balian R, Kleman M, Poirier JP (NorthHolland, Amsterdam), pp 183–200.
43. Cahn JW, Mishin Y, Suzuki A (2006) Coupling grain boundary motion to shear deformation. Acta Mater 54(19):4953–4975.
44. Cahn JW, Taylor JE (2004) A unified approach to motion of grain boundaries, relative
tangential translation along grain boundaries, and grain rotation. Acta Mater 52(16):
4887–4898.
45. Trautt ZT, Adland A, Karma A, Mishin Y (2012) Coupled motion of asymmetrical tilt
grain boundaries: Molecular dynamics and phase field computer simulations. Acta
Mater 60(19):6528–6546.
46. Karma A, Trautt ZT, Mishin Y (2012) Relation between equilibrium fluctuations and
shear-coupled motion of grain boundaries. Phys Rev Lett 109(9):095501-1–5.
47. Zhang H, Srolovitz DJ, Douglas JF, Warren JA (2007) Atomic motion during the migration of general [001] tilt grain boundaries in Ni. Acta Mater 55(13):4527–4533.
48. Zhang H, Srolovitz DJ, Douglas JF, Warren JA (2006) Characterization of atomic
motion governing grain boundary migration. Phys Rev B Condens Matter 74(11):
115404-1–10.
49. Sollich P, Lequeux F, Hebraud P, Cates ME (1997) Rheology of soft glassy materials.
Phys Rev Lett 78(10):2020–2023.
50. Zhang H, Du D, Srolovitz DJ, Mendelev MI (2006) Determination of grain boundary
stiffness from molecular dynamics simulation. Appl Phys Lett 88(12):121927-1–3.
51. Haxton TK, Liu AJ (2007) Activated dynamics and effective temperature in a steady
state sheared glass. Phys Rev Lett 99(19):195701-1–4.
52. Crocker JC, Grier DG (1996) Methods of digital video microscopy for colloidal studies.
J Colloid Interface Sci 179(1):298–310.
APPLIED PHYSICAL
SCIENCES
24. Schall P, Weitz DA, Spaepen F (2007) Structural rearrangements that govern flow in
colloidal glasses. Science 318(5858):1895–1899.
25. Ganapathy R, Buckley MR, Gerbode SJ, Cohen I (2010) Direct measurements of island
growth and step-edge barriers in colloidal epitaxy. Science 327(5964):445–448.
26. Han Y, et al. (2008) Geometric frustration in buckled colloidal monolayers. Nature 456
(7224):898–903.
27. Nagamanasa KH, Gokhale S, Ganapathy R, Sood AK (2011) Confined glassy dynamics
at grain boundaries in colloidal crystals. Proc Natl Acad Sci USA 108(28):11323–11326.
28. Read WT, Shockley W (1950) Dislocation models of crystal grain boundaries. Phys Rev
78(3):275–289.
29. Barabasi A-L, Stanley HE (1995) Fractal Concepts in Surface Growth (Cambridge Univ
Press, New York).
30. Yoon DY, Cho YK (2005) Roughening transitions of grain boundaries in metals and
oxides. J Mater Sci 40(4):861–870.
31. Lobkovsky AE, Karma A, Mendelev MI, Haataja M, Srolovitz DJ (2004) Grain shape,
grain boundary mobility and the Herring relation. Acta Mater 52(2):285–292.
32. Hoyt JJ, Asta M, Karma A (2003) Atomistic and continuum modeling of dendritic
solidification. Mater Sci Eng Rep 41(6):121–163.
33. Trautt ZT, Upmanyu M (2005) Direct two-dimensional calculations of grain boundary
stiffness. Scr Mater 52(11):1175–1179.
34. Foiles SM, Hoyt JJ (2006) Computation of grain boundary stiffness and mobility from
boundary fluctuations. Acta Mater 54(12):3351–3357.
35. Nguyen VD, Hu Z, Schall P (2011) Single crystal growth and anisotropic crystal-fluid
interfacial free energy in soft colloidal systems. Phys Rev E Stat Nonlin Soft Matter
Phys 84(1):011607-1–6.
36. Derks D, Aarts DGAL, Bonn D, Lekkerkerker HN, Imhof A (2006) Suppression of
thermally excited capillary waves by shear flow. Phys Rev Lett 97(3):038301-1–4.
37. Skinner TOE, Aarts DGAL, Dullens RPA (2010) Grain-boundary fluctuations in twodimensional colloidal crystals. Phys Rev Lett 105(16):168301-1–4.
38. Werner A, Schmid F, Muller M, Binder K (1997) Anomalous size-dependence of interfacial profiles between coexisting phases of polymer mixtures in thin-film geometry: A Monte Carlo simulation. J Chem Phys 107(19):8175–8188.
Gokhale et al.
PNAS | December 11, 2012 | vol. 109 | no. 50 | 20319
Supporting Information
Gokhale et al. 10.1073/pnas.1210456109
SI Text
Controlling Grain Size. The average grain size was altered by
controlling the annealing rate. Samples were annealed from 311 K
to 296 K at three different rates, and the grain size was found to
increase with decreasing annealing rate, as shown in Fig. S1.
Shear Geometry for Confocal Rheology and Bragg Diffraction
Microscopy. Our rheometer is custom-designed to be mounted on
a confocal microscope. A parallel plate geometry consisting of
a homemade shear cell was used in all experiments. A glass plate of
roughness λ=10, thickness 3 mm, and diameter 20 mm was used as
the top plate, and a glass coverslip of thickness 170 μm was used as
the bottom plate. The sample was surrounded by oil to prevent
evaporation. The sample temperature and annealing rate were
regulated using the microscope objective heater for confocal rheology experiments, and by circulating water through metal tubes
attached to the bottom plate for Bragg diffraction microscopy
(BDM) experiments.
Interface Profile. Various regions of the sample were scanned to
locate boundaries that are nearly flat over the field of view, with
desired orientations with respect to the shear direction. The grain
boundary (GB) configurations chosen were GBk, which is nearly
perpendicular to the y-axis of the imaging plane (Fig. 1A in the
main text), and GB⊥, which is nearly perpendicular to the x-axis.
The macroscopic motion of the sample due to thermal gradients, as
well as the systematic displacement imposed by the applied shear,
was subtracted from the particle displacements to obtain driftcorrected coordinates. The image was divided into bins parallel to
the y-axis for GBk and x-axis for GB⊥, and the bin width chosen was
slightly larger than a particle diameter. To define the GB interface,
all particles belonging to the GB were first identified using the
bond-order parameter method described in ref. 1. Any given bin
might contain more than one particle because the GB region
has a finite width. Within each bin, the interface position was
determined by computing the arithmetic mean of the maximum
and minimum value of the y coordinate for GBk and x coordinate
for GB⊥. The profiles thus obtained were rotated appropriately to
study fluctuations normal to the interface. Linear interpolation was
then performed on the rotated interface positions to obtain a continuous interface profile.
Calculation of True Strain Amplitudes. The glass plates used for
rheology experiments are smooth on the particle length scale and
hence may result in wall slip. An oscillatory strain with frequency
ω ¼ 1 rad/s was applied to the sample. The gap between the rheometer plates was set to 53 μm, which corresponds to 68 crystalline
layers. It is not possible to quantify the strain profile over the entire
sample thickness from confocal rheology experiments because
spherical aberration does not allow imaging of the top plate.
However, it is possible to calculate the true strain experienced by
the imaging plane (Fig. 1A), which is located 5.9 μm from the
bottom plate. To calculate this strain, images were captured at
a distance, z = 1.3 μm, 5.9 μm, and 12.1 μm from the bottom plate,
for applied strain amplitudes of 1.9%, 7.6%, and 18.9%. At each of
these distances, the maximum displacement averaged over ∼40
strain cycles Δs was calculated from the images. By fitting straight
lines to the displacement profiles (Fig. S3), the true strains experienced by the imaging plane were computed to be 2%, 8.3%, and
18.3%. These values are very close to the applied strain amplitudes, and the extrapolated fits nearly pass through the points
denoting the no-slip displacement at the top plate. The ratio of the
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
applied strain to the true strain, γA =γo , is nearly constant for all
three applied strain amplitudes. The true strain in BDM experiments cannot be calculated directly. However, by scaling the applied strain by γA =γo , it was estimated to be 3.9%.
Effect of Baseline on GB Stiffness and Mobility. The capillary fluctuation method (CFM) requires the interface profile to be defined
relative to a baseline. In our experiments, interface fluctuations
defined relative to any fixed baseline are influenced by GB migration. To circumvent this problem, the least-squares fit straight
line to the instantaneous GB interface profile was chosen as the
baseline for the stiffness calculations shown in the main text. To test
the robustness of our results, stiffness values also were calculated
using the time-averaged GB profile as the baseline (Fig. S5A). The
change in baseline resulted in changes in the absolute values of Γ
and M, leaving the trend in M vs. γo unaltered (Fig. S5B).
Shear Rate Dependence of GB Stiffness. To ascertain whether GB
stiffness Γ depends on the shear rate γ_ or the strain amplitude γo
itself, we calculated Γ for GBk at a fixed γo ¼ 2%, for ω ¼ 1 rad/s
and ω ¼ 10 rad/s, and compared it with the results obtained in
Fig. 2D for GBk and GB⊥ (Fig. S7). We observe that Γ is essentially constant for γ_ ¼ γo ω ¼ 0:02 s−1 , irrespective of the GB
under consideration. For GBk, however, Γ is lowered dramatically with increasing γo for a fixed ω ¼ 1 rad/s, but exhibits only
a modest lowering with increasing ω for a fixed γo ¼ 2%. This
shows conclusively that Γ has a stronger dependence on γo
_
compared with γ.
Dynamic and Static Height–Height Correlation Functions. GB motion
leads to a drift of the interface center-of-mass with time (Fig. S8A).
Because CFM has been devised to probe the zero-driving force
limit, the drift affects the dynamic height–height correlation
function gd ðτÞ and needs to be subtracted. A second-order
polynomial was fit to the hhðx; tÞix time series to compensate for
the drift. The normalized dynamic height–height correlation
!
1"
2
ðτÞ
¼ erfc ðΓMtÞ
calculated using the drift-corrected
function ggdd ð0Þ
ξ
interface profile is shown in Fig. S9 A–C. Although the drift-corrected center-of-mass hh1 ðx; tÞix fluctuates about zero (Fig. S8B), it
exhibits residual oscillations at the frequency of the imposed strain
(Fig. S8B, Inset), which are reflected in gd ðτÞ (Fig. S9 A–C). These
oscillations arise because it is not possible to completely eliminate
the systematic contribution of shear to particle displacements in
dense suspensions (2).
The oscillations in Fig. S8B suppress the decay of the static
height–height correlation function gh ðδxÞ ¼ hΔhðx; tÞΔhðxþ
δx; tÞix;t ¼ kB2ΓTξ e−x=ξ (3) because their amplitude is comparable to
that of the interface fluctuations. To minimize the effect of these
oscillations, only frames for which hh1 ðx; tÞix lies within the
shaded region in Fig. S8B were considered for calculating gh ðδxÞ.
Fig. S9 D and E shows gh ðδxÞ for GBk and GB⊥ for all three
values of γo . The lateral correlation length ξ was extracted by
fitting decaying exponentials to the data.
GB Migration Under Zero Shear. To confirm that anisotropic grain
growth is indeed a consequence of the imposed shear, BDM
experiments were performed in the absence of shear at 300 K (Fig.
S10A). In contrast to grain boundary migration under shear (Fig.
3), no preferential enhancement of the GB velocities was observed
in the absence of shear as shown in Fig. S10B and Movie S3.
1 of 9
GB Diffusion. The shear-induced anisotropic enhancement of
(Fig. S11A). However, for γo ¼ 8:3%, we find that for both
GBk and GB⊥, hΔyðtÞ2 i > hΔxðtÞ2 i (Fig. S11B). Moreover, for
GBk as well as GB⊥, we observe that hΔxðtÞ2 i and hΔyðtÞ2 i are
larger for γo ¼ 8:3% compared with γo ¼ 2%. This is in accordance with the strain-induced lowering of activation barriers for
cage breaking.
1. Nagamanasa KH, Gokhale S, Ganapathy R, Sood AK (2011) Confined glassy dynamics at
grain boundaries in colloidal crystals. Proc Natl Acad Sci USA 108(28):11323–11326.
2. Skinner TOE, Aarts DGAL, Dullens RPA (2010) Grain-boundary fluctuations in twodimensional colloidal crystals. Phys Rev Lett 105(16):168301.
3. Cheng X, Xu X, Rice SA, Dinner AR, Cohen I (2012) Assembly of vorticity-aligned hardsphere colloidal strings in a simple shear flow. Proc Natl Acad Sci USA 109(1):63–67.
activated particle hops also is reflected in the mean squared
displacements hΔxðtÞ2 i and hΔyðtÞ2 i (Fig. S11). Here, Δy and
Δx denote particle displacements nearly parallel and perpendicular to the shear direction, respectively. We observe
that for both GBk and GB⊥, hΔyðtÞ2 i ∼ hΔxðtÞ2 i for γo ¼ 2%
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
2 of 9
Fig. S1.
BDM images of the polycrystal for three different annealing rates: (A) 2.6 K/min, (B) 0.125 K/min, and (C) 0.015 K/min.
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
3 of 9
Fig. S2. Schematic of the shear geometry. (A) Confocal rheology and (B) BDM. CP, colloidal polycrystal; HC, heating/cooling coils; LS, liquid seal; OH, objective heater.
Fig. S3. Calculation of true strain amplitudes at the imaging plane. Shear-induced displacement Δs vs. z for γA ¼ 1:9% (★), 7:6% (●), and 18:9% (▲). Solid
black lines are linear fits to the shear profiles, and dashed lines are extrapolations of the fits up to the top plate.
Fig. S4. GB migration mechanisms. (A) Time series of the GB center-of-mass for γo ¼ 0%, showing pronounced steps. (B) Time series of center-of-mass of GB
and GB for γo ¼ 8:3%. GB exhibits step-like features, whereas GB exhibits continuous motion.
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
4 of 9
Fig. S5. Stiffness and mobility using the time-averaged GB profile as the baseline. (A) Stiffness Γ and (B) mobility M as a function of γo for GB (●) and GB (○).
The □ corresponds to γo ¼ 0%.
Fig. S6. Power spectrum of the interface height profile. The interface height function was defined using the least-squares fit straight line to the instantaneous
GB profile as the baseline. hjAðkÞj2 i vs. k for (A) GBk and (B) GB⊥ for strain amplitudes γo ¼ 2% (stars), 8:3% (circles), and 18:3% (triangles). The solid black line in
A and B has slope = −2 and serves as a guide to the eye.
o
o
Fig. S7. GB stiffness Γ as a function of γ_ . The bottom x-axis corresponds to experiments on GBk (●) and GB⊥ (○) in which γ_ was changed by fixing ω ¼ 1 rad/s
and varying γo . The top x-axis corresponds to experiments on GBk (▲) in which γ_ was changed by fixing γo ¼ 2% and varying ω. The □ corresponds to γo ¼ 0%.
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
5 of 9
Fig. S8. Center-of-mass hhðx; tÞix drift correction. (A) Time series of hhðx; tÞix of GBk for γo ¼ 8:3%. The black curve is a second-order polynomial fit to the data.
(B) Time series of the drift-corrected center-of-mass hh1 ðx; tÞix obtained by subtracting the fit from the data shown in A. The shaded region corresponds to
jhh1 ðx; tÞix j < 0:1σh , where σh is the range of the distribution of hh1 ðx; tÞix . (Inset) Magnified portion of B, showing residual oscillations in the time series of
hh1 ðx; tÞix .
Fig. S9. Dynamic and static height–height correlation functions. (A–C) Normalized dynamic height–height correlation function gd ðτÞ for GBk (●) and GB⊥ (○)
for γo ¼ 2% (A), 8:3% (B), and 18:3% (C). The black curves are complimentary error function fits to the data. (D and E) Normalized static height–height
correlation function gh ðδxÞ for GBk (D) and GB⊥ (E) for γo ¼ 2% (stars), 8:3% (circles), and 18:3% (triangles). The curves are exponential decay fits to the data.
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
6 of 9
Fig. S10. Grain growth under zero shear and at T = 300 K. (A) A BDM snapshot of the polycrystal. The curves highlight GBs for which migration velocities were
measured. The arrows point along the direction of GB motion, and their length is proportional to the GB migration velocity. (B) hhðx; tÞix vs. t for the
boundaries identified in A. Different symbols represent different boundaries. The solid lines are the least-squares fits to the data. The numbers shown adjacent
to the GB center-of-mass profiles vs. time in B correspond to those in A.
Fig. S11. GB diffusion. Mean squared displacements hΔxðtÞ2 i (squares) and hΔyðtÞ2 i (circles) of GB particles for (A) γo ¼ 2% and (B) γo ¼ 8:3% for GBk (solid
symbols) and GB⊥ (open symbols). The black line has a diffusive slope. The oscillations in hΔxðtÞ2 i and hΔyðtÞ2 i have the same source as those observed in gd ðτÞ.
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
7 of 9
Fig. S12. Angular distribution of displacements for γo ¼ 2%. (A) GBk (●) at Δt ¼ t* ¼ 64 s. (B) GB⊥ (○) at Δt ¼ t* ¼ 68 s. The solid arrow corresponds to the
direction of shear, and the dashed arrow corresponds to the GB normal. Only about 30% of particle hops happen within a 40° window centered on the shear
direction for both GBk and GB⊥ .
Movie S1. Raw confocal microscopy video of GBs subjected to an oscillatory strain of amplitude γo = 8.3% and angular frequency ω = 1 rad/s for two boundary
configurations: GBk (Left) and GB⊥ (Right).
Movie S1
Movie S2. BDM video of directional grain growth under shear for an applied oscillatory strain of amplitude γo = 3.9% and angular frequency ω = 1 rad/s.
Movie S2
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
8 of 9
Movie S3. BDM video of isotropic grain growth at zero shear.
Movie S3
Gokhale et al. www.pnas.org/cgi/content/short/1210456109
9 of 9