Site-specific colloidal crystal nucleation by template-enhanced particle transport

Enhancing Particles’ Mean-Free Path.

A plausible route to help alleviate the restrictions on L, imposed by D/F, is to use surfaces with energy gradients to transport particles to desired locations. Using the facile replica imprinting technique (21), we present a design principle based on templates with spatially varying feature sizes which in the presence of short-range depletion attraction induced activation energy gradients for the diffusing colloids. Our substrates comprised linear and square moiré patterns on polymethyl-methacrylate (PMMA) layers spin-coated on glass coverslips. We fabricated these patterns, by first transferring a linear (square) array of trenches (holes), with periodicity λ, from a master grating to the PMMA substrate (8). This was followed by a second imprint at an angle θ, relative to the first, which resulted in a long wavelength modulation of the trench (hole) depths (Materials and Methods). Whereas λ decides the symmetry of the growing crystallites, the wavelength of the moiré pattern sets Lp, and can be independently controlled by varying θ as seen in Fig. 1 A and B. Long wavelength modulations in the feature size are also clearly evident from atomic force microscope (AFM) snapshot (Fig. 1C). Next, we sedimented colloidal particles [silica, diameter σ=940 nm (22)] in the presence of a depletant polymer (sodium carboxylmethyl cellulose, Rg=60 nm) on these substrates. Particle diffusion on templated surfaces has an activated form, D=D0e−Ea/kBT, where Ea is the barrier height, D0 is the attempt frequency, and kBT is the thermal energy (4). Apart from the polymer concentration c, the strength of the depletion attraction between the colloids and the substrate is also proportional to the excluded overlap volume that is freed up when they come in contact (23). Thus, on moiré patterns, gradients in trench (hole) depths induced gradients in Ea for the diffusing colloids and results in particle migration to the nearest energy minima (Fig. 1D and Movie S1).

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Fig. 1.

Particle locomotion to traps induced by surface energy gradients. (A and B) Representative optical micrographs of square moiré patterns (Bottom) and the corresponding computer-generated topography maps (Top) for two different θs. (The scale bar in A represents 15 μm.) (C) AFM image showing modulation in trench depth for the square moiré template corresponding to A. (D) Schematic representing particle–substrate interactions on a linear moiré pattern. The overlap volume increases in the direction of the arrows and results in a net migration of particles to regions of high overlap (high Ea). (E) Particle trajectories on a linear moiré pattern. The gray region corresponds to regions of the pattern with high Ea. (F) Representative image of particle localization on moiré templates.

We confirmed the enhancement in particle mean-free paths by analyzing the motion of individual particles on a linear moiré substrate, with Lp=42σ (Fig. 1E and SI Appendix, Fig. S1). The gray-shaded region in Fig. 1E represents regions of the template with a high Ea. Whereas the particles that land within this region stay localized, those that land outside (regions of low Ea) migrate nearly 25σ to the high Ea regions and are subsequently trapped. On a linear array of trenches (no surface energy gradients) and for the same c and λ as the moiré templates, we found L∼5−7σ (SI Appendix, Fig. S2 and Movie S2). Fig. 1F shows a snapshot of particles localized in trap sites on moiré templates. By quantifying the variation in D across Lp, we estimated ΔEa∼0.5kBT (SI Appendix, Fig. S3).

Nucleation Control on Linear Moiré Templates.

The accumulation of particles in traps also increased the likelihood of crystal nucleation events at these sites. For site-specific nucleation to occur with a high fidelity, however, an optimal balance between F and the particle surface mean-free path, albeit enhanced here, is needed (4, 12). Fig. 2 A and B shows representative snapshots of crystals grown on linear moiré templates with Lp=42σ at identical c and surface coverage, Θ, and for two different Fs, respectively. At the smaller F, particles were able to migrate to traps before encountering arriving monomer(s) and the crystals formed are compact and confined to trap sites (Fig. 2A and Movie S3). For the higher F, however, this is not the case and the crystals within the traps are more ramified; colloidal islands with sixfold symmetry (highlighted by circles in Fig. 2B and Movie S3) nucleate outside the traps as well. We parametrized the fidelity of nucleation events at trap sites by the nucleation control efficiency (NCE), xNCE=Nm/NTot. Here, Nm is the subset of particles that migrated to traps in a time interval Δt and then crystallized and NTot is the total number of particles that landed outside traps over the same time interval (24, 25). We note that xNCE in ref. 24 is defined as the ratio of number of traps to the number of islands. This definition, however, poses difficulties in estimating xNCE for spatially extended traps that can support multiple stable nuclei in the early stages of film growth, as is the case here. xNCE, defined here, not only circumvents this limitation but also allows for direct comparison with previous experiments.

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Fig. 2.

Nucleation control on moiré patterns.(A and B) Representative images of crystals growth on linear moiré patterns at constant c and for two different Fs. Because λ=σ, the templates promoted the growth of crystals with square symmetry. (B) Loss of nucleation control results in the nucleation of hexagonally ordered crystallites outside of the traps and is highlighted by circles. (C) Comparison of xNCE versus p* for conventional site-specific nucleation studies and on moiré patterns. The red and blue squares correspond to experiments and simulation results for vapor deposition of organic molecules (24). Here, nucleation control is lost (xNCE<1) beyond the striped region. The green circles correspond to colloid experiments on linear moiré patterns for various Lp, c, and F values. Owing to the enhancement in particle mean-free path on these substrates, xNCE<1 only for p*>6.

Next, we carried out thin-film growth experiments on linear moiré templates for different c, F, and Lp values (Movies S2 and S3). To succinctly capture the dependence of xNCE on these parameters and to isolate the contribution arising from the enhancement in mean-free paths, we plotted xNCE versus p*=Lp/L. Here, L is the mean-free path of particles under identical experimental conditions but in the absence of energy gradients. L was obtained by measuring the density of critical clusters, nc, at the onset of coalescence (SI Appendix, Fig. S2). An island with i particles is termed a critical cluster if the addition of another particle makes it stable against disintegration. We found i=2 with and without surface energy gradients (SI Appendix, Fig. S4 and Table S1). In Fig. 2C, we compare xNCE versus p* from our experiments with simulation and experimental results for site-specific nucleation and growth of organic molecules (24). The data points denoted by 1 and 2 correspond to Fig. 2 A and B, respectively. Strikingly, as opposed to previous studies where complete nucleation control (xNCE=1) (24) was possible only for p*≤1, on moiré patterns xNCE≈1 until p*=6 and beyond which it decreases. Further, the lateral flux of particles to traps on moiré templates also effectively reduced the time taken to complete the nucleation process (SI Appendix, Fig. S5).

Nucleation Kinetics and Island Growth on Square Moiré Templates.

The above findings clearly exemplify the efficacy of our approach in achieving site-specific nucleation for colloidal particles over a range of Lp values. To shed light on the nucleation kinetics, we mimicked atomic heteroepitaxy experiments (26⇓–28) and measured nc as a function of c, at constant F, on square moiré patterns with Lp=16σ (Movie S4). Here, c plays the role of an inverse temperature. At the largest cs studied (c≥0.27 mg/mL), particles are unable to overcome the energy barriers for surface diffusion even in the low Ea regions and the moiré template thus acts like a homogeneous surface. On such surfaces, for small D, atomic and colloid experiments follow rate equation predictions and find that nc depends only on Θ and saturates to a constant, nc≈0.03 for i=1 (4, 27). This was indeed found to hold in our experiments as well (triangles in Fig. 3A). For 0.2<c<0.27 mg/mL, although particles diffuse, their mean-free path remains smaller than Lp and nucleation events continue to occur at random locations (panel labeled “3” in Fig. 3A). Akin to the behavior on a homogeneous surface nc versus c exhibits an Arrhenius-like dependence. We found a second plateau in nc for c lying between 0.1−0.2 mg/mL (green squares and red circle in Fig. 3A). In this regime, particles perceive the heterogeneous nature of the surface energy landscape and thus their diffusivities are preferentially smaller at trap sites, which subsequently resulted in site-specific nucleation (panel 2 in Fig. 3A). Finally, for the lowest cs studied (c≤0.1 mg/mL), particles are oblivious to the underlying surface topography and nc decreases owing to an increase in the critical cluster size (i≥4) (panel 1 in Fig. 3A).

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Fig. 3.

Nucleation and island growth on square moiré patterns. (A) nc versus c at fixed F. The legends represent the size of the critical cluster. The red line is a guide to the eye. The blue-shaded region corresponds to the regime of organized growth. The magenta circles within the blue-shaded region represent the expected nc from homogeneous nucleation for D/F s in the plateau region (4). (B) Schematic of nucleation curves for homogeneous nucleation (black curve) and heterogeneous nucleation with and without energy gradients shown by red and green curves, respectively. Mean-field scaling predictions for L and nc for D/F>104 are also shown. (C) Representative snapshot of island growth on square moiré patterns with Lp=32σ. (Inset) Particles clustered based on their bond-order parameter. Green represents particles with Ψ4>0.7, and red represents particles with Ψ6>0.7. (D and E) Island size distributions and fractal dimensions on square moiré patterns (green circles) and on hexagonal array of holes (10) (black circles) for Θ=50%, respectively. (F) Colloidal crystal heterostructures grown on moiré templates. The image corresponds to the third layer of particles from the template. The size of the squares and hexagons represents the width of the strip.

Although the overall shape of the nucleation curve (red line in Fig. 3A) for colloids is strikingly similar to that seen for atoms (27, 28), there are fundamental differences. The black line in the schematic in Fig. 3B shows nc versus D/F, or the scaled 1/T, for homogeneous surface growth. When dimers are stable and also immobile, the rate equation predicts and experiments find that L∝(D/F)1/6 for D/F>104 (4). However, for D/F≤104, the regime accessible in colloid experiments, L saturates with decreasing D/F (4, 10). For site-specific nucleation studies, which are typically performed at a fixed F, a plateau in nc begins when L∼Lp (black solid circle in Fig. 3B) and continues until particles fail to feel the underlying substrate. More importantly, the plateau is restricted only to the yellow-shaded region, corresponding to L≥Lp, of Fig. 3B. On moiré templates, however, the enhancement in particle mean-free path due to surface energy gradients corresponds to a larger effective D/F and the plateau in nc should therefore lie below the homogeneous nucleation curve (blue-shaded region of Fig. 3B). Fig. 3A also shows the nc expected for homogeneous nucleation (magenta circles in Fig. 3A) for D/F values corresponding to the plateau region in the nucleation curve (blue-shaded region in Fig. 3A) (4). The spatially averaged D on square moiré templates has been used while estimating D/F (SI Appendix, Fig. S6). In line with expectations, nc on moiré template lies below the homogeneous nucleation curve (Fig. 3A).

In the plateau regime of nc, because crystalline islands are periodically spaced and have nearly identical monomer capture rates, the island size distribution is expected to be narrower than on a homogeneous surface (4, 27). We measured the island size distribution on square moiré patterns with Lp=32σ by clustering particles based on their local bond-order parameter (Fig. 3C, Inset) and compared it with homogeneous surface growth experiments for the same Θ (SI Appendix, Fig. S7) (10). Whereas we found a broad range of island sizes on homogeneous surfaces (black circles in Fig. 3D) (10), on moiré templates the distribution was peaked with a maximum that roughly coincided with the trap size (green circles in Fig. 3D). Further, at the same Θ∼50%, we found islands on moiré templates to be more compact with a fractal dimension df∼2 whereas on the homogeneous surface we found df∼1.7 in agreement with theoretical predictions (green and black circles in Fig. 3E, respectively) (12). With continued particle deposition, crystals with hexagonal order nucleated and grew outside the traps and subsequent layers were found to be in registry with the underlying symmetry (SI Appendix, Fig. S8). We found this to be true even on templates with complex moiré periodicities. Fig. 3F shows a representative image of the third layer of crystals grown on a template that was fabricated by making multiple imprints at different θs. The symmetry and the width of the crystals reflect the underlying substrate periodicity with a motif: 8 σ□→ 8 σ → 8 σ□→ 4 σ.