Directional grain growth from anisotropic kinetic roughening of grain boundaries in sheared colloidal crystals

Abstract

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.

The 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 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 μm, , and the equilibrium phase is a liquid. Annealing the samples to 296 K led to particle swelling, with μm, resulting in a , 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 GB-mediated 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 our attention to high-angle grain boundaries (HAGBs) formed when adjacent grains that straddle the boundary have a misorientation angle . 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.

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 custom-designed temperature-controlled parallel-plate shear cell with a typical plate separation of 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 crystal planes parallel to the walls of our shear cell; therefore, our studies are confined to pure tilt boundaries. The plane in Fig. 1A shows a snapshot of an HAGB. We quantify the dynamics of the GB interface (SI Text) by the height function , which is the 1D string formed by the intersection of the GB plane n with the imaging plane (thick white line in Fig. 1A). The image plane 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 at different oscillatory shear strains, γ, for two GB configurations: (denoted by GB_∥) and (denoted by GB_⊥), where v is the velocity (Movie S1). was changed in our experiments by keeping the oscillation frequency fixed at rad/s and changing the strain amplitude . The values of quoted here have been corrected to incorporate the effect of wall slip (SI Text and Fig. S3). GB properties sensitively depend on (4, 27, 28), and hence we focused on HAGBs within a narrow range of s, , to facilitate quantitative comparison of results from experiments on GB_∥ and GB_⊥.

Fig. 1.

Kinetic roughening of HAGBs. (A) Schematic of the shear geometry. The planes and form the top and bottom plates of our shear cell. The imaging plane shows two crystallites with . The GB plane is the shaded region denoted by normal n. This figure corresponds to the configuration . (B–E) Probability distribution of height fluctuations for increasing : (B) , (C) , (D) , and (E) . In C–E, solid curves and dashed curves are gaussian fits to histograms that correspond to GB_|| and GB_⊥, respectively. The circles correspond to w, and the width of the gray-shaded region corresponds to the w for .

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 (29). In Fig. 1 B–E, we plot the normalized histograms of height fluctuations for increasing . Here, . For all , is gaussian for GB_∥ (solid curves) and GB_⊥ (dashed curves), allowing us to estimate w (shown by circles) directly from the fits (Fig. 1 C–E). For , (Fig. 1 B and C). Remarkably, we find that for , GB_∥ undergoes preferential kinetic roughening with (Fig. 1 D and E). Earlier studies addressed equilibrium roughening of HAGBs, in the context of faceting–defaceting transition (30), and kinetic roughening of 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 , 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 is stepped and that of the kinetically roughened GB_∥ () 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 to the driving force F (4). The net driving force acting on a boundary is composed of two parts, , where the first term is due to intrinsic boundary curvature, κ, and the second term is from externally imposed stresses. Here, is the GB stiffness (31). To calculate M, we first determined 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 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 of the modes decays as . Here, 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 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 (SI Text and Fig. S5) and plot vs. k for GB_∥ (solid circles) and GB_⊥ (hollow circles) for increasing (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 as well (Fig. S6). We find that is nearly a constant over a decade in k, for both boundary configurations and for all . Notably, we find that , obtained by averaging over the shaded region (Fig. 2 A–C), is preferentially lowered for GB_∥ (solid circles) compared with GB_⊥ (hollow circles) for (Fig. 2D and Fig. S5A). An anisotropic lowering of for GB_∥ 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 that in ref. 36 depends on the shear rate , whereas for HAGBs we find that depends strongly on (SI Text and Fig. S7).

Fig. 2.

Shear-induced anisotropy in GB mobility. vs. k for (A) , (B) , and (C) for GB_|| (●) and GB_⊥ (○). (D) Stiffness as a function of obtained from A–C by averaging over the shaded region. (E) Mobility M and (F) reduced mobility as a function of for GB_|| (●) and GB_⊥ (○). In D–F, () corresponds to .

For capillary fluctuations that decay as , the dynamic correlation function (SI Text, Fig. S8, and Fig. S9 A–C) yields the product (37). Here, ξ is the lateral correlation length (38) obtained from exponential fits to the height–height correlation function (Fig. S9 D and E) and is the complementary error function. Fig. S9 A–C shows for GB_∥ (solid circles) and GB_⊥ (hollow circles) for increasing . Plugging in the value of (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 for the kinetically roughened boundary, GB_∥ (Fig. 2E, solid circles), is more rapid compared with GB_⊥ (Fig. 2E, hollow circles) and is almost an order of magnitude larger at (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, , lead to directional grain growth? To answer this question, we take a closer look at the mobility relation (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 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 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, and hence . We note that with increasing , M increases (Fig. 2E) whereas is found to decrease (Fig. 2D) for both GB_∥ and GB_⊥. Thus, in the approximation , we expect an anisotropy in GB motion and grain growth only if the reduced mobility, , is itself anisotropic. Remarkably enough, Fig. 2F shows that anisotropic kinetic roughening indeed results in anisotropic-reduced mobilities with at large . Although we considered a narrow range of s to allow comparison across the two GB configurations, given that HAGBs exhibit qualitatively similar dynamics over a broad range of s (27, 40), we expect the anisotropy in M and 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 , 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 GB_‖s shown by solid lines and GB_⊥s shown by dashed lines. Fig. 3 B and C shows vs. t for . In complete concord with results thus far, anisotropy in M and indeed results in anisotropic grain growth with 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 and assuming , where d is the average grain radius, which is ∼100 μm. The calculated GB velocity is in close agreement with those obtained from BDM measurements.

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 GB_||s 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 . (B) vs. t for GB_||s. (C) 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 GB_||s and GB_⊥s.

Microscopic Origins of Directional Grain Growth.

Our observations cannot be rationalized within the theoretical framework of shear-coupled 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 (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 time . We extracted from the inflection point in the mean first passage time (48), defined as the average time taken by a particle to traverse a distance r for the first time. We find that t* for γ_o = 2% is greater than t* for γ_o = 8.3% for both GB_‖ and GB_⊥ (Fig. 4 A), implying that the activation barrier for cage breaking is lowered at a higher (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 for develops a characteristic peak at an intermediate distance r with . Here, corresponds to the cage size. However, (Fig. 4 B and C) do not exhibit peaks between and σ for both GB_∥ and GB_⊥ even for , 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 is only about , even for GB_||. Further, although displacements in all directions contribute to the peak at in (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 for GB_∥ and GB_⊥. For , the displacements are more isotropically distributed (Fig. S12) compared with for which we find a significant enhancement along the shear direction for both GB_∥ and GB_⊥ (Fig. 4 D and E). For GB_∥, these shear-enhanced hops are normal to the boundary plane and therefore lead to a preferential increase in (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 , and is therefore analogous to temperature. As expected, similar to the variation of M and with temperature (50), the dependence of M on is much stronger than that of (Fig. 2). Not only does this rationalize the success of the CFM in extracting and M for sheared HAGBs, but it also accounts for the anisotropic enhancement in and therefore directional grain growth.

Fig. 4.

Single-particle dynamics at GBs. (A) Mean first passage time for GB_|| [ (■), (●)] and GB_⊥ [ (), (○)]. The dashed curves are Hill function fits used to extract the inflection point , shown as solid lines for GB_|| and dashed lines for GB_⊥. (B and C) Self part of the van Hove correlation function for for GB_|| (B) and GB_⊥ (C) at , , and . (D and E) Angular distribution of displacements contributing to the nearest-neighbor peak of for over for GB_|| (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 GB_|| and GB_⊥.

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 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 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 = 2% and 8.3% and 26 fps for = 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 low-magnification 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 = 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.