We study the crystallization of colloidal dispersions under capillary-action-induced shear as the dispersion is drawn into flat walled capillaries. Using confocal microscopy and small angle x-ray scattering, we find that the shear near the capillary walls influences the crystallization to result in large random hexagonal close-packed (RHCP) crystals with long-range orientational order over tens of thousands of colloidal particles. We investigate the crystallization mechanism and find partial crystallization under shear, initiating with hexagonal planes at the capillary walls, where shear is highest, followed by epitaxial crystal growth from these hexagonal layers after the shear is stopped. We then characterize the three-dimensional crystal structure finding that the shear-induced crystallization leads to larger particle separations parallel to the shear and vorticity directions as compared to the equilibrium RHCP structure. Confocal microscopy reveals that competing shear directions, where the capillary walls meet at a corner, create differently oriented hexagonal planes of particles. The single-orientation RHCP colloidal crystals remain stable after formation and are produced without the need of complex shear cell arrangements.
Shear processes arise whenever a fluid travels across a surface or inside a pipe. As such, these processes have great significance anywhere fluid is transported, spanning a wide range of fields from food manufacture1 to the geological study of tectonic plates.2 Under the influence of shear flow, rich rheological phenomena can occur, such as shear thickening3 and thinning,4 where the viscosity increases or decreases, respectively, with shear rate, and shear banding,5–8 where spatial discontinuities in viscosity arise. Such rheological phenomena often occur in complex fluids, where one material is dispersed inside another, for example, shear thickening dispersions of corn starch in water.9
A change in rheological properties is often accompanied by a change in microscopic structure; therefore, much attention has been directed toward using shear processes to control the micro-structure of complex fluids, with examples including the formation of nano-emulsions10 and ordering of anisotropic particles11–13 with ramifications for ceramic manufacture.11 This extends to the control of crystallinity in colloidal dispersions, which may also be manipulated under shear. Studies have shown that dispersions of colloidal spheres with volume fractions near or above the freezing transition feature aligned layers of hexagonally arranged particles within the velocity vorticity plane.7,14–19 The hexagonal layers can stack on top of one another to create close-packed structures with the spheres in the layer above sitting in the recesses created by the layer below, resulting in three lateral positions for the subsequent layers, labeled A, B, and C. Often, colloidal crystals induced by shear exhibit a random sequence of A, B, and C layers to form random hexagonal close-packed (RHCP) crystals that contain a combination of face centered cubic (FCC) and hexagonal close-packed (HCP) crystal structures. With carefully controlled oscillatory shear, the crystal structure may be tuned to select for FCC over HCP14,20,21 and to create large domains of coherent orientation and low defect density.22 The orientation of the hexagonal layers may even be manipulated so that close-packed directions align perpendicular or parallel to the direction of flow according to the magnitude of the shear strain.14,16,19 Such control over crystal structure is desirable for advanced material applications and especially for photonic applications.23
While colloidal suspensions are often sheared and monitored by counter translating7,18,24 or rotating17 parallel plates, herein, we instead initiate shear by exploiting capillary action to draw a colloidal suspension through a flat walled capillary. The suspension is subject to shear as the colloidal dispersion travels more slowly closer to the walls, and the flow contains a plane of zero shear midway between the capillary walls, unlike within common shear cell and rheometer geometries. Many studies use scattering or microscopy4,7,14,18,21,25,26 to observe structural changes during shear. Here, instead, we focus on monitoring the crystallization during and after shear has stopped, with the view to create large crystals with controlled orientation27 and structure beyond just a few crystalline layers.7,22 To characterize the structure of the colloidal crystals aligned through capillary action, we employ a combination of confocal laser scanning microscopy (CLSM) and small angle x-ray scattering (SAXS).
This paper is organized as follows: In Sec. II, we detail the preparation of the colloidal dispersions and the experimental CLSM and SAXS setups. Next, in Sec. III, we discuss the broad features of the RHCP crystals before investigating the mechanism of crystallization under capillary action induced shear using SAXS. Then, we combine SAXS and CLSM to examine the details of the RHCP crystal structure. Finally, in Sec. IV, we offer some concluding remarks and outlook.
A. Colloidal crystal preparation
Colloidal 3-(trimethoxysilyl)propyl methacrylate (TPM) dispersions were synthesized following the work by Liu et al.28 Three slightly different dispersions of TPM spheres are considered, batches 1–3; the different samples are summarized in Table I. Batch 1 consists of 1.5 µm diameter spheres suspended in tetralin (TTL) with 2% by weight OLOA 1200 (Chevron Chemical Co) charge control agent. OLOA is a polyisobutylene succinic anhydride mixed with a proprietary mineral oil, which stabilizes ionic moieties in the solvent while also adsorbing to particle surfaces to achieve a steric stabilization effect.29,30 The more there is, the shorter the repulsive range of the interactions become. Batch 2 comprises of slightly smaller TPM spheres of 1.4 µm diameter, which are suspended in a refractive index matched dispersion of decalin (DCL) and tetralin in an ∼2:3 volume ratio. OLOA 11000, a slightly different OLOA formulation, is added to 5% by weight to achieve similar charge screening to the first suspension. The particles in batches 1 and 2 interact approximately as hard spheres as the Debye length is ≪0.1 µm, significantly smaller than the particle diameter.32 Batch 3 consists of 2.1 µm diameter TPM spheres fluorescently labeled with rhodamine B isothiocyanate28 and is suspended in a DCL-TTL mixture like batch 2, however, with a lower concentration of OLOA of 0.5% by weight, so that the negatively charged33 particles display a longer Debye length of ∼0.12 µm.34 These changes allow for the individual particles in batch 3 to be resolved using CLSM.
|Batch .||D (μm) .||p .||Use .||Solvent .||[OLOA] (%) .|
|2||1.4||0.049||SAXS||2 DCL: 3 TTL||5|
|3||2.1||0.060||CLSM||2 DCL: 3 TTL||0.5|
|Batch .||D (μm) .||p .||Use .||Solvent .||[OLOA] (%) .|
|2||1.4||0.049||SAXS||2 DCL: 3 TTL||5|
|3||2.1||0.060||CLSM||2 DCL: 3 TTL||0.5|
High volume fraction, ϕ ≈ 0.4, colloidal dispersions are produced by spinning down the colloidal dispersion in a centrifuge to a random packing volume fraction ϕ = 0.64 and then adding solvent to reach the target volume fraction.35 Crystals are produced by bringing the solution into contact with the end of a hollow rectangle quartz glass capillary (CM Scientific VitroCom) with internal dimensions of 0.1 × 2 mm2 by , where capillary action draws the dispersion inside at a velocity of 1.2 mm s−1. For experiments hours in duration, the capillary ends may be sealed with two-part epoxy glue to prevent solvent leakage and evaporation. This ensures that the sample has a long lifetime and allows the sample to be transported to a confocal microscope or to be placed in an x-ray beam. A photograph of a crystal inside a capillary is shown in Fig. 1(a). Here, the crystal is illuminated from behind with white light, and the Bragg reflection is clearly visible owing to the presence of crystalline planes. The capillary is oriented in the laboratory frame, as shown in Fig. 1(b). The combination of the small length in z and the xy walls being much larger than the xz walls results in the flow (), vorticity (), and shear () directions, respectively, being parallel to the x, y, and z axes throughout the majority of the capillary volume.
B. Confocal microscopy
The CLSM setup consists of a Thorlabs 8 kHz resonant point scanner mounted on an Olympus IX73 microscope. The microscope is equipped with an Olympus Plan Apochromat oil immersion objective with 60× magnification and a numerical aperture of 1.42, identical to Ref. 34. Single particle resolution was achieved in a field of view of 100 × 100 × 100 µm3 with a pixel size of 0.2 µm. We use a laser with wavelength 532 nm to excite the rhodamine B dye in the particles. The particle coordinates were obtained using the Crocker–Grier method.36
C. Small angle x-ray scattering
The experiments were carried out during two experimental sessions at the B16 Test beamline at the Diamond Light Source, Didcot, UK. Our setup consists of the following: A Ru/B4C multilayer monochromator selected x-rays with photon energy of 8.5 keV. The beam was focused using three consecutive beryllium compound refractive lenses (radius of parabola of 200 µm and effective aperture of about 800 µm) installed in front of the sample. Lenses produced a converging beam focused on the detector position plane. The Photonic Science ImageStar9000 large-area CCD detector (3056 × 3056 pixels and 31 µm pixel size) and a Medipix4 detector (Merlin) were used to capture diffraction images. Colloidal crystals in the flat capillaries were mounted horizontally on a motorized sample stage, allowing for computer-controlled 3D translations and goniometer rotations of the sample. The sample-to-detector distance was set to . Reciprocal space is calibrated using the diffraction pattern of dry rat tail collagen.37
To initiate crystallization in a horizontally mounted glass capillary, a small amount of batch 2 dispersion was loaded into a Teflon tube coupled to a syringe on a syringe pump; the tube was mounted so its exit was positioned at the entry to the capillary. The pump was activated so that the dispersion was pushed out of the tube into contact with the entrance of the capillary where it is pulled inside by capillary action. For the crystallization experiments, the capillary ends were not sealed with epoxy glue, but left open to allow the dispersion to enter. Normal incidence diffraction patterns were taken in the center of the capillary at distances L between 1 and 6 mm from the capillary entrance. Unlike in CLSM, SAXS patterns remain unaffected by the rapid velocity of the particles entering the capillary. The Teflon tubing and the capillary were not sealed together to prevent the pump from pushing the solution out the other end of the capillary. Consequently, the flow stops when the capillary is full of the solution, after roughly 25 s. A large reservoir of air prevented the hazardous solvent vapor escaping but provided minimal resistance to fluid entering the capillary. Post crystallization, the RHCP stacking parameter and lattice constants were measured from 3D reciprocal space using goniometer scans, as illustrated in Fig. 1(c).
Post processing of the diffraction images starts with the subtraction of a blank image of a capillary containing solvent, scaled to account for sample adsorption. Next, we divide by a calculated form factor (see Sec. III of the supplementary material) to reveal the structure factor from which the Crocker–Grier method36 identifies the peak locations. Although not common practice, we took advantage of the ability of the Crocker–Grier method to identify the location and intensity of Gaussian blobs on a dark background, which is essentially a 2D fitting algorithm to find peak position with subpixel resolution and accurate integrated intensity. Then, the intensity of a structure factor peak is calculated as the sum of the pixel values within a radius of the peak center. Note this radius is set slightly larger than the radial extent of the structure factor peak.
III. RESULTS AND DISCUSSION
In Sec. III A, we study the crystallization mechanism under shear as the colloidal dispersion enters the capillary. We then go on to monitor the crystallization after the cessation of shear to create the final crystal. Following this, in Sec. III B, we use SAXS and CLSM to characterize the RHCP crystal structure by measuring the ratio of FCC to HCP and by measuring the lattice constants. An example of such a shear aligned crystal is shown in Fig. 1(a). The color of the Bragg reflections varies smoothly with viewing angle due to the single orientation of hexagonal planes. The single orientation is confirmed by observation of the same structure factor orientation throughout the capillary [like those shown in Figs. 2(c) and 6(a)].
A. Crystallization mechanism
We use SAXS to reveal that the crystallization is initiated by the formation of hexagonal layers as the dispersion flows into the capillary. Although we cannot directly observe where in the sample the hexagonal layers form, confocal microscopy on similar systems4,7,25 revealed that the layers form near the walls. It is reasonable to suggest that this is the case for the experiments herein as the shear is highest near the xy walls. At first, the layers slide over each other before settling into A, B, and C positions. However, by the time that shear stops once the dispersion has filled the capillary, it is not fully crystalline. We suggest that the central portion of the dispersion, far away from the capillary walls where shear is lowest, remains amorphous. Next, we see crystallization and suggest that the final RHCP structure with the same orientation is a result of epitaxial growth using the shear aligned hexagonal layers as templates.
As a dispersion from batch 2 flows through a capillary, we capture the structure factor changes, revealing information on the crystallization mechanism. The structure factor at L = 6 mm from the capillary entrance, shortly after the solution entered the capillary at time, t = 5.1 s, is shown in Fig. 2(a). Note that the solution takes t ≈ 5 s to travel 6 mm inside the capillary. The concentric rings around the center of reciprocal space indicate an amorphous arrangement of particles. Subsequently, peaks appear quickly within several seconds as the particles begin to crystallize under shear. At t ≈ 25 s, flow in the capillary stops as the dispersion reaches the other open end of the capillary. A snapshot of the structure factor, close to when shear stops at t = 23 s, is shown in Fig. 2(b). The hexagonal arrangement of structure factor peaks show that crystallization with hexagonal symmetry has occurred. The innermost primary ring is still visible at low q in the structure factor showing that both crystalline- and liquid-like arrangements of particles co-exist. The hexagonal layers form at the xy walls of the capillary under shear but exhibit an amorphous arrangement in the center where shear is lower. The spatial heterogeneity of crystalline and amorphous order is similar to the crystallization by shear banding of colloidal dispersions observed in CLSM studies.7,25 Note that, at this point, the sample in the irradiation volume has still only been exposed to shear for 5 s, like the snapshot in Fig. 2(a). This suggests that it takes several seconds for the shear flow profile to be established within the capillary. Gradually, the ring disappears and the existing peaks become clearer and more numerous at high q as the dispersion crystallizes throughout the capillary, as shown in the late time structure factor in Fig. 2(c).
Next, we consider the evolution of the inter- and intra-layer ordering of the hexagonal layers during crystallization. We do so through examination of the structure factor over time, where we quantify the crystallization through the structural correlations within and between hexagonal planes. To this end, we modify Wilson theory by splitting the structure factor into the contributions,
where and are the inter-layer and intra-layer structure factors, respectively. Intra-layer order increases as particles in hexagonal layers align more closely with a 2D hexagonal lattice and as the number of parallel hexagonal layers increases. The hexagonal planes in Figs. 3(a-i) and 3(a-ii) have high and low intra-layer order, respectively. This increase in order results in an increase in for both Bragg rods and spots (Miller indices h − k ≠ 3n and h − k = 3n, respectively, for integer n) at integer values of h and k. For more details on the geometry of the RHCP structure in real and reciprocal space, refer to Sec. I of the supplementary material. Inter-layer order increases when the correlations between successive A, B, or C hexagonal layers increase. This is illustrated in Figs. 3(a-i) and 3(a-iii), which have high and low inter-layer order, respectively. For Bragg rods and spots in an RHCP crystal with equal proportions of FCC and HCP and at Miller index l = 0, this increase in inter-layer order has opposing effects on the inter-layer structure factor. For Bragg rods, the inter-layer structure factor at l = 0 is
and for Bragg spots,
where the parameter σ quantifies the inter-layer disorder and is the standard deviation in lateral displacement of a hexagonal layer in a random direction within the layer [see the dashed vectors in Fig. 3(a-iii)]. A decrease in σ results in an increase in and a decrease in , as shown in Fig. 3(b). For Bragg rods, decays to for an RHCP crystal containing a random sequence of hexagonal layers perfectly arranged in A, B, or C lateral positions. A derivation for the above is found in Sec. II of the supplementary material.
Figure 4(a) shows the time dependence of Bragg spots with t for several distances from the capillary entrance, L. Here, ⟨Sspot⟩ is the mean structure factor peak intensity for the first order Bragg spots, as marked in Fig. 2(c). Figure 4(a) shows that as time evolves, the Bragg spot structure factor increases and does so similarly at all distances from the capillary entrance. Note that the duration of the ⟨Sspot⟩ plots is shorter for lower L positions because glue from the sellotape fixing between the Teflon tube and the capillary dissolved and diffused into the capillary. As the glue reached higher L over time, it caused a depletion interaction disrupting the crystalline order. Any affected positions were discarded from the analysis. An increase in the structure factor results from either an increase in inter-layer ordering, where the translational positions of A, B, and C layers become more established, or an increase in intra-layer ordering, where the particles within a hexagonal layer align more closely with a 2D hexagonal lattice; however, the contributions from each remain unclear. Recall that an increase in inter- or intra-layer order increases the value of the inter- or intra-layer structure factors or , respectively, and therefore increases the Bragg spot structure factor (at l = 0), [see Eq. (1)]. Although we see evidence for intra-layer ordering from the formation of hexagonally arranged structure factor peaks as time progress, as shown in Figs. 2(a)–2(c), we learn little about the inter-layer ordering.
To examine the inter-layer ordering, we measure the structure factor for Bragg rods, where the increase in inter-layer ordering has the opposite effect on the structure factor as does the intra-layer ordering. Figure 4(b) shows, ⟨Srod⟩, the mean structure factor peak intensity for the second order Bragg rods, as marked in Fig. 2(c). In addition, the form of the structure factor is similar at all measured distances from the capillary entrance. Instead of a monotonic increase in structure factor intensity as for the Bragg spots [see Fig. 4(a)], three distinct stages are observed in the Bragg rod structure factor [see Fig. 4(b)]. First, between t = 0 s and t ≈ 25 s, the structure factor increases sharply. Second, between t ≈ 25 s and t = 5 min, it decreases. Third, from t = 5 min onward, it reaches a plateau. These stages are a consequence of the competition between the opposing effects of increasing inter- and intra-layer order on the Bragg rod structure factor, [see Eq. (1)]. An increase in inter-layer order decreases the inter-layer structure factor for Bragg rods, as shown in Eq. (2) and Fig. 3(b). This opposes the increase in the structure factor afforded by the increase in intra-layer order. Therefore, during the first stage where the Bragg rod structure factor, ⟨Srod⟩, increases, the gains in intra-layer order out-compete any gains in inter-layer order, consistent with the idea of hexagonal planes sliding over each other under shear. This reveals that the hexagonal planes that are initially formed have poor inter-layer order. Next, during the second stage where the Bragg rod structure factor decreases, increasing inter-layer order dominates as the hexagonal layers settle into A, B, and C positions in the absence of shear. In the third stage, the increases in inter- and intra-layer order have equal effects on the structure factor. In this stage, the crystal grows epitaxially using the hexagonal layers at the capillary walls as templates. The particles find lattice positions on the surface of the crystal and, in the absence of flow, establish inter- and intra-layer order simultaneously.
More information on the mechanism is uncovered by examining the effect of increasing the thickness of the capillary on crystallization. Diffraction patterns were measured as the same colloidal dispersion (batch 2) entered a capillary with an internal dimension of 0.2 mm in the z direction in which epitaxial growth is expected. This is twice as long as before. The structure factor evolution with time is similar to that shown in Figs. 2(a)–2(c). However, the primary ring persists for a much longer time as shown by the pattern at t = 15 min in the inset of Fig. 4(d). Next, we measure the structure factor change with time in order to monitor the inter- and intra-layer ordering during crystallization. The evolution of the structure factor with time for Bragg spots for the wider capillary follows a similar trend to the narrower one, as shown in Fig. 4(c). However, the Bragg rod structure factor evolution with time is different [see Fig. 4(d)]. Here, we see a two stage process. First, in stage one, we see that the Bragg rod structure factor, ⟨Srod⟩, increases between t = 0 and t ≈ 4.5 min. In the second stage, after t ≈ 4.5 min, the structure factor reaches a plateau. Note that ⟨Srod⟩ is again measured from the second order Bragg rods. The absence of the decay tells us that the order increase (both intra- and inter-layer) resulting from epitaxial crystal growth from the hexagonal layers occurs simultaneously with the establishment of inter-layer order between the shear-aligned hexagonal layers, where the A, B, and C positions become more established. It also suggests that the proportion of colloidal particles aligned in hexagonal layers by the shear is lower in the thicker capillary. As such, a lower proportion of the colloidal crystals exhibits crystallization dominated by inter-layer ordering, thus suppressing the Bragg rod structure factor decay. The behavior of the Bragg rods and spots in the wider capillary makes it clear that inter- and intra-layer order increases simultaneously after the flow stops, which is not so apparent from the results from the thinner capillary.
Like for the narrower capillary case, the structure factor for Bragg rods and spots has a similar form regardless of distance from the capillary edge. This suggests that any effects due to solvent flow caused by evaporation from the open ends of the capillary had little effect on the crystallization. We expect any effect to be more pronounced near the capillary edge and do not expect such a contribution 6 mm deep within the capillary.
The effect of sedimentation during this experiment is minimal. During the filling, the layering phenomenon is dominated by shear. We can calculate the mean shear Peclet number using viscosity, η, for tetralin = 1.9 × 10−3 kg m−1 s−1; the particle radius, a = 0.75 × 10−6 m; temperature, T = 298 K; and mean shear rate, . This yields Pe ∼ 90. In contrast, the sedimentation Peclet number (calculated as , where Δρ = 320 kg m−3) is roughly equal to 1. The effect of shear is clearly dominant during capillary filling. Once the shear has stopped, we further note that the volume fraction profile does not change over time. Over the 15 min it takes for the sample to crystallize, the volume fraction at different points in the y-direction remains equal to each other and virtually unchanged (see Sec. IV of the supplementary material).
In summary, results for both widths of capillaries are consistent with crystallization initiating near the walls and then propagating epitaxially toward the bulk of the dispersion. Initially, the lateral A, B, and C hexagonal positions are not correlated throughout the hexagonal stacking but become more correlated as time progress. At the same time, epitaxial crystallization from the hexagonal layers toward the center of the capillary occurs. In fact, over many hours, we observe that the volume fraction increases as the crystals sediment inside the capillary. Furthermore, the melting transition is likely suppressed below ϕχ = 0.49 due to the finite range repulsions on the TPM particles.28
B. Crystal structure and orientation
In Sec. III B, we examine the crystal structure once the dispersions have fully crystallized using CLSM and SAXS in Secs. III B 1 and III B 2, respectively, and characterize the the structural distortions as a result of shear. For extensive sampling, separate capillaries for CLSM and SAXS were made, allowed to form for 20–30 min, longer than the crystallization dynamics discussed in Sec. III A, and are sealed with epoxy before observation. Note that crystals for CLSM measurements were left with the xy face parallel to the benchtop and z parallel to g during formation for ease of sample handling. Likewise, samples for SAXS measurements were held with y parallel to g. Samples were then inverted for CLSM observation but retained their vertical orientation (like in Fig. 1) for SAXS.
1. Confocal microscopy measurements
The local structure of these crystals is probed by CLSM, where several cubic fields of view (0.001 mm3) were imaged. In this case, flow-aligned colloidal crystals from batch 3 are studied. Figure 5(a) shows the typical appearance of an xy, plane in the bulk of the crystal. This image shows that the hexagonal planes of the crystal are parallel with the plane as expected. Furthermore, we observe that the close-packing direction is parallel with the flow direction, , supporting the results of several studies where high shear, γ > 1, aligns hexagonal layers with close-packed direction parallel to .15,18 However, shear decreases to zero in the middle of the capillary. Although lower shear, γ < 1, is known to produce orientations of hexagonal layers with the close packed direction perpendicular to the velocity direction,18 this is only under oscillatory shear, not under steady shear like is the case here. Figure 5(b) shows a reconstruction of the particle coordinates, typical for a field of view in the centre of the capillary spanning the entire internal 0.1 mm length in z. The FCC or HCP crystal structures are identified using the crystal structure identification package of BLoSSOM,38 which compares the similarity of nearest neighbour shells to perfect FCC and HCP nearest neighbour shells. Locally FCC and HCP particles are coloured in blue or red, respectively, and create stripes of alternating colour along the yz, and xz, faces showing changes in stacking sequence parallel to . The stacking parameter, , where NFCC and NHCP are the numbers of colloidal particles in locally HCP and FCC environments, respectively, quantifies the proportion of both crystal structures. The value varies between 0 ≤ α ≤ 1 from exclusively HCP stacking to exclusively (twinned) FCC stacking. We measure a value of α = 0.54, revealing a RHCP structure with slight preference for FCC. The slight preference for FCC may be due to mechanical perturbations during transport of the sample and mounting on the microscope as observed by Martelozzo et al. in similar colloidal suspensions.39
The high spatial resolution of CLSM enables detailed imaging at a field of view close to the xz walls of the capillary. A perspective looking along the filling direction, [see Fig. 1(b)], is shown in Fig. 5(c). Here, the orientation of the hexagonal layers changes to be parallel to its closest wall; the dotted lines mark the change in hexagonal plane orientation and are at 45° to the walls. Close to the xz walls, the directions of and rotate by 90° about the x axis because the shear direction becomes dominated by the closest wall, aligning hexagonal layers in different orientations. However, the much larger xy walls result in the alignment parallel to the xy planes dominating the hexagonal plane orientation over the capillary volume. Another hexagonal plane orientation exists at the top middle of Fig. 5(c), possibly due to competing epitaxial growth orientations from the two wall orientations.
We observe an effect of shear on the crystal structure on the orthogonal components of the inter-particle distances, parallel to each of the bulk , , and directions. The spacing parallel to along the close-packing direction is μm, parallel to is μm, and parallel to , the distance between hexagonal planes, is μm, where the error is the standard error (for more details, refer to Sec. V of the supplementary material). For an RHCP crystal with isotropic nearest neighbour separation, the orthogonal components should have the following ratios: and . We observe distorted hexagonal layers, where stretching parallel to , which suggests that close-packed lines of particles within the hexagonal planes slide over each other during crystal formation. It is notable that a hexagonal layer distorted in this manner may, in fact, accommodate an inter-layer spacing shorter than for a perfectly hexagonal layers due to the increased cross-sectional area of the inter-particle voids. However, we observe that showing that the crystal structure is stretched parallel to consistent with the notion of crystallization via shear where hexagonal planes slide over one another. Using a particle diameter of 2.1 µm and the particle spacings above, ϕ = 0.37. The persistence of the crystals suggests that the charge of the colloidal particles has lowered the volume fraction at which the system is expected to crystallize.
2. SAXS measurements
Next, we characterize the crystal structure of a shear induced colloidal crystal from batch 1 using SAXS. Figure 6(a) shows the 2D structure factor at ω = 0, i.e., the x-ray beam is normal to the xy walls of the capillary. The diffraction pattern reveals that the hexagonal planes of particles are parallel to the xy walls [see Fig. 1(b)] and, therefore, the plane. In addition, the orientation of the close-packing direction is parallel to the direction of flow.
The full 3D structure of the crystals is revealed using many 2D slices through reciprocal space from SAXS goniometer scans [see Fig. 1(c)]. Example 2D slices of the structure factor at normal incidence, ω = 0°, and at ω = −32° are shown in Figs. 6(a) and 6(b), respectively. Several Bragg spots with the same h and k at different l indices are visible in these two images; the 110 and 111 spots are marked as examples. Figure 6(c) shows the variation of against qz for 11l Bragg spots from a goniometer scan. The local maxima define the qz coordinates of the Bragg spots, which are plotted in 3D in Fig. 6(e). The orientation of the Bragg spots shows that the RHCP stacking is in the same orientation as seen in the CLSM image in Fig. 5(b).
To measure the stacking parameter, α, we compare the variation in the structure factor as a function of l, S(l), to Wilson’s theory.40 An example S(l) variation for a Bragg rod is shown in Fig. 6(d). The solid line represents the measured S(l), and the dotted lines show the fit with Wilson theory with α and M that satisfy
where Sm(lω) is the structure factor at angle ω of the goniometer scan and M is a scaling factor accounting for experimental conditions such as x-ray source intensity, illuminated volume, and detector distance. For more details, refer to Secs. III and IV of the supplementary material. S(l) takes the following form:40
Here, as before, 0 ≤ α ≤ 1 from entirely HCP stacking to entirely (twinned) FCC stacking. The mean stacking parameter across all Bragg rods is measured as α = 0.55, revealing a RHCP structure with a slight preference for FCC, in agreement with the measurements from CLSM. Similarly prepared crystals exhibit α values, slightly greater than 0.5. The slight preference for FCC may, again, be due to mechanical perturbations39 during sample transport and mounting.
From the goniometer scans, we also determine the inter-particle spacings as before: μm, μm, and μm. The errors are the standard error calculated as described in Sec. V of the supplementary material. Once more, the ratios are larger than the values for perfect RHCP crystals, and , due to the stretching effects of shear, supporting the observation from CLSM. Using a particle diameter of 1.50 µm and the particle separations measured from the SAXS goniometer scans, we calculate the volume fraction as ϕ = 0.42. In addition, due to the small charge and the persistence of the crystals, this is likely above the fluid–solid co-existence volume fraction. Finally, given the different orientations of the samples during measurement, we note that the lack of any difference in the structural distortion between CLSM and SAXS highlights the lack of a significant sedimentation effect on these time scales.
We have shown that large and singly oriented domains of random hexagonal close-packed crystals can be produced by capillary action on a high volume fraction colloidal dispersion (ϕ ≈ 0.4) within flat and narrow capillaries. We measured the evolution of crystalline order between and within the hexagonal planes during crystallization by comparing the structure factor time evolutions of Bragg rods and spots. At the walls at early times, crystallization is dominated by the action of shear, aligning particles into hexagonal planes that slide over each other. After flow stops and shear ceases, we see then that the shear aligned hexagonal planes establish hexagonal stacking A, B, and C positions, in tandem with crystallization via epitaxy from the hexagonal layers. Details of the local RHCP crystal structure were afforded by CLSM measurements, in particular, allowing the visualization of competing shear profiles on the crystal orientation. Global structure was uncovered by SAXS measurements, where goniometer scans revealed the three dimensional reciprocal structure, supporting the measurements of the direct space structure from CLSM. We find that the shear aligned crystal structures agree well with those from other studies. Finally, we provide detailed measurements of the distortions to the crystal structure as a consequence of shear. We suggest that aligning colloidal crystals using shear generated by capillary action may prove useful for the manufacture of photonic bandgap crystals.
See the supplementary material for a description of the structure of RHCP crystals in real and reciprocal space, which is given in Sec. I. We derive the Wilson theory for lateral displacement correlations in Sec. II. We detail the x-ray image processing schemes used to find the structure factor and reciprocal lattice coordinates and to discriminate between Bragg rods and spots in Sec. III. In Sec. IV, we discuss the role of sedimentation in crystal structure, and in Sec. V, we present the calculations for the orthogonal components of particle separations.
We thank Daniel Bell for preliminary work on resolving colloidal particles within these RHCP crystals using confocal microscopy. The ERC (ERC Consolidator Grant No. 724834–OMCIDC) is acknowledged for financial support. The Diamond Light Source is acknowledged for providing the beamtime on B16 beamline under Experiment Nos. MT18697-1 and MM21934-1.
Conflict of Interest
The authors have no conflicts to disclose.
Nicholas H. P. Orr: Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Taiki Yanagishima: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Igor P. Dolbnya: Investigation (equal); Methodology (equal); Writing – review & editing (equal). Andrei V. Petukhov: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). Roel P. A. Dullens: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.