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Contributed by Joanna Aizenberg, June 26, 2021 (submitted for review on April 25, 2021; reviewed by Paul Chaikin, Eric Dufresne, and Boaz Pokroy)

Self-assembly is one of the central themes of biocontrol synthesis, and it also plays a pivotal role in the manufacture of various advanced engineering materials. In particular, the evaporation-induced self-assembly of colloidal particles can realize the multifunctional manufacturing of highly ordered two-dimensional or three-dimensional nanostructures for optical, sensing, catalysis and other applications. Although it is well known that this process leads to the formation of face-centered cubic (fcc) lattices in which the close-packed {111} planes are parallel to the substrate, the development of the crystal structure of colloidal crystals is little known. In this study, we show that the preferred <110> growth in fcc colloidal crystals synthesized by evaporation-induced assembly is achieved by gradual crystal rotation promoted by geometrically necessary dislocations induced by mechanical stress.

Unlike crystal atoms and ionic solids, there is less research on texture development due to crystallographic preferential growth in colloidal crystals. Here, we use experiments, modeling and theoretical analysis to study the potential mechanism of texture evolution in the process of colloidal assembly induced by evaporation. In this widely used method of obtaining large-area colloidal crystals, colloidal particles are driven to the meniscus by evaporation of a solvent or matrix precursor solution, where they are tightly packed to form face-centered cubic colloidal components. Through the two-dimensional large-area crystal mapping, we show that the initial crystal orientation is dominated by the interaction between the particles and the meniscus, resulting in the close packing direction co-aligned with the expected local meniscus geometry. By combining the crystal structure analysis at the single particle level, we further reveal that in the later stage of self-assembly, colloidal crystals undergo gradual rotation promoted by geometrically necessary dislocations (GNDs), and achieve a large-area uniform crystal orientation, with the close-packed direction being vertical At the meniscus and parallel to the growth direction. Classical slip analysis, finite element-based mechanical simulation, computational colloidal component modeling, and continuum theory clearly show that due to the limited shrinkage of colloidal crystals during the drying process, these GNDs are caused by the tensile stress field along the direction of the meniscus. produced. A GND with a specific slip system is generated in a single crystal grain, which causes the crystal to rotate to adapt to the mechanical stress. The mechanical understanding reported here can be used to control the crystallographic characteristics of colloidal components and may provide further insights into the preferred growth of crystallography in synthetic, biological, and geological crystals.

Similar to atomic crystals, colloidal crystals are highly ordered structures formed by colloidal particles ranging in size from 100 nm to several microns (1⇓ ⇓ ⇓ ⇓ –6). In addition to engineering applications such as photonics, sensing and catalysis (4, 5, 7, 8), colloidal crystals are also used as model systems to study some basic processes of statistical mechanics and mechanical behavior of crystalline solids (9⇓ ⇓ ⇓ ⇓ – 14). According to the nature of the interaction between particles, many equilibrium and non-equilibrium colloidal self-assembly processes have been explored and developed (1, 4). Among them, evaporation-induced colloidal self-assembly has many advantages, such as large-scale manufacturing, versatility, cost and time efficiency (3⇓ –5, 15⇓ ⇓ –18). In a typical synthesis, the substrate is immersed in a colloidal suspension vertically or at an angle, and the colloidal particles are driven to the meniscus by the fluid flow induced by evaporation, and then self-assemble to form a face-centered cubic colloidal crystal (fcc) lattice structure And the close-packed {111} plane is parallel to the substrate (2, 3, 19⇓ ⇓ ⇓ –23) (the schematic diagram of the synthesis device is shown in Figure 1A).

The evaporation of colloidal crystals induces co-assembly. (A) Schematic diagram of the evaporation-induced colloid co-assembly process. "G", "M" and "N" refer to the directions of "growth", "meniscus" and "normal" respectively. In addition to the colloid, the reaction solution also contains a silica matrix precursor (tetraethyl orthosilicate, TEOS). (B) Schematic diagram of the crystallographic system and direction used in this work. (C and D) Optical images (top left) and scanning electron micrographs (SEM) (bottom left) of a typical large-area colloidal crystal film after calcining in (C) and (D). (Right) SEM images of selected areas (yellow rectangles) at different magnifications. The corresponding fast Fourier transform (see the inset in the middle of C) shows the single crystal nature of the assembled structure. (E) Based on FIB tomography data and (right) 3D reconstruction of colloidal crystals (left) after particle detection. (F) Top view SEM image of colloidal crystals, which indicate crystal orientation.

Although previous studies focused on the use of assembled colloidal structures for different applications (4, 5, 7, 8), there is much less effort to understand the self-assembly mechanism itself in this process (17, 24). In particular, although the term "colloid crystal" is used to emphasize the long-range order of the microstructure, similar to atomic crystals, little is known about the crystallographic evolution of colloidal crystals and the self-assembly process (3, 22, 25). The underlying mechanism of the puzzling (but commonly observed) phenomenon that evaporation induces the preferential growth of colloidal crystals along the <110> direction is not clear (3, 25⇓ ⇓ ⇓ –29). <110> The growth direction has been observed in many processes with various particle chemistry, evaporation rates and matrix materials (3, 25⇓ ⇓ –28, 30), suggesting a general underlying mechanism. This behavior is particularly interesting because the colloidal particles are expected to pack closely parallel to the meniscus, which will lead to growth along the <112> direction and perpendicular to the <110> direction (16, 26, 31)*.

The preferred growth along a specific crystal direction, also known as texture development, is usually observed in crystalline atomic solids in synthetic systems, biominerals, and geological crystals. Although current knowledge recognizes mechanisms such as directional nucleation, which define the future crystallographic orientation of growing crystals and the competitive growth of atomic crystals (32⇓ -34), the basic principles of texture development in colloidal crystals are still difficult fathom. The previous assumptions based on orientation-dependent growth rate and solvent flow resistance are insufficient to provide a general explanation for different evaporation-induced colloidal self-assembly processes (3, 25⇓ ⇓ ⇓ –29). A better understanding of crystallographically preferred growth during colloidal self-assembly may provide new clues for crystal growth in atomic, ion, and molecular systems (35⇓ –37). In addition, a mechanistic understanding of the self-assembly process will allow more precise control of lattice types, crystallography, and defects to improve the performance and function of colloidal assembly structures (38⇓ –40).

In order to study the crystallographic characteristics related to the assembly process, we first used the evaporation-induced colloidal co-assembly route as a model system (3, 15). The fluid flow drives the colloid to the meniscus, and as the solution evaporates slowly in the vertical direction, colloidal crystals are formed on the substrate (Figure 1A). In the case of co-assembly, in addition to colloidal particles, the reaction solution also contains a so-called matrix precursor, which can react and fill the space of colloidal crystals. For example, in the typical co-assembly process used in this work, polystyrene (PS) particles (diameter 0.38 ± 0.01 μm; SI appendix, Figure S1) pass through the silicate sol-gel precursor tetraethylorthosilicate The condensation and silica co-assembly (TEOS), in the assembly process. Figure 1B shows a coordinate system used to describe the crystallography of assembled particles. In this coordinate system, 1) the close-packed (111) plane is parallel to the substrate, so the [111] direction is parallel to the substrate normal N; 2) the growth direction G is parallel to the close-packed direction [101¯], belonging to <110> Family, the horizontal meniscus M is parallel to the [12¯1] direction and belongs to the <110> family 112> family. * As will be discussed extensively in the following sections, this self-assembly process first nucleates the polycrystalline colloidal monolayer along the horizontal meniscus direction M, which has various crystal grain orientations, where the close-packed direction is usually Parallel to the direction of the local meniscus; then the number of layers increases depending on the concentration of colloidal particles; finally, the crystal grains in the multilayer colloidal crystals gradually rotate within hundreds of microns until the close packing direction is consistent with the growth direction G, and then the assembled The cm size film maintains a uniform <110> orientation.

Figure 1C shows a representative large area of ​​a crack-free colloidal crystal with uniform crystallographic orientation grown using the evaporation-induced co-assembly method. In addition, this method allows easy formation of silica inverse opal after removing the PS particles by chemical etching or calcination (Figure 1D). As the silica matrix changes from periodic polymer spheres to air spheres, the inverse opal shows a strong blue-shifted photon coloration (Figure 1D). Through three-dimensional (3D) reconstruction based on a focused ion beam/scanning electron microscope (FIB/SEM) tomography of a representative area, the resulting colloidal crystals can be clearly resolved at the single-particle level, which confirms the fcc lattice structure (Fig. 1E) ; See SI appendix, Figure S2). The crystal orientation of the colloidal crystals grown in the close-packed [101¯] direction (hereinafter referred to as the <110> preferred growth direction) is shown in Figure 1F.

In order to understand how to achieve the preferred growth along the <110> direction, we first performed a structural analysis in the starting area, where colloidal crystals nucleated along the horizontal meniscus (Figure 2). By combining large-area (1.6 mm × 0.2 mm) high-resolution imaging and quantitative image analysis based on 2D simultaneous compression transformation and variational regularization (41), we determined the top lattice type, crystal orientation, defects, and deformation. Most layers of colloidal crystals (see materials and methods). As shown in the representative starting area (Figure 2A, i; total length, 1.6 mm), although there is some local distortion, the starting line is approximately horizontal (parallel to the M direction). The number of layers in the colloidal crystal gradually increases from one layer to six layers within a vertical distance of about 50 μm (as shown by the numbers in Figure 2A and ii). The calculated lattice type indicates that the assembly starts with a hexagonal stacking and changes to a square stacking when additional layers are added (Figure 2A, ii, and B). This pattern can also be seen in the SEM images of the original grown and subsequently calcined structure (Figure 2F). This phenomenon has been observed in direct opal assembly before, possibly due to the limitation of the inclined meniscus during assembly (20, 42, 43). As shown by the vertical thickness distribution along the growth direction (Figure 2D), a gradual increase in thickness was observed in the transition zone, resulting in a total wedge angle of approximately 1.1° (Figure 2E). As the number of layers becomes five or more, the transition can be carried out without square accumulation, although local accumulation defects may appear (Figure 2B and G).

Morphological and crystallographic analysis of the initiation zone. (A) (i) the original SEM image, (ii) the corresponding lattice type map, and (iii) the pattern of the typical starting area. The number of layers in the colloidal crystal gradually increases from one layer to five layers, as shown in ii. (B and C) High magnification of (B) lattice type and (C) orientation of the representative area indicated in A. (D) SEM image of FIB milling cross-section, showing areas with different number of layers: 0, sub-monolayer; 1, one layer; 2, two layers; (E) the thickness distribution of colloidal crystals in the initial zone. (F) The transition zone from the three-layer to the four-layer region, indicating that the lattice type has changed from hexagon to square to hexagonal stacking. (G) The transition zone from the four-layer to the five-layer region indicates that the crystal orientation is maintained. The illustrations in F and G show the corresponding structure after calcination. As shown in the figure, the areas of F and G are taken from B and C. (H) The color scheme used in the pattern. The crystal angle θ is defined as the angle between the upper left-lower right-oriented [110] direction (ie [01¯1] direction) and the global meniscus direction (M). (I) (Top) The local crystal orientation angle θ'is defined as the angle between the local meniscus orientation (M') and the nearest close-packed orientation, and is assigned a sign according to the schematic diagram. (Bottom) Based on the one shown in A The statistical analysis of the orientation map is the distribution of grain size and local crystal misorientation (θ'). (J) The superposition of the original SEM image and the orientation map shows an extreme example of the defect area. The highly curved meniscus direction is always aligned with the local close-packed direction (ie, θ'≈0°). All directional patterns are based on the same color scheme, as shown by H. All images except D and E have the same orientation as the sample in A.

In addition to the lattice type, we also mapped the distribution of crystal orientations in the starting area (Figure 2A, iii, and C). As shown in Figure 2H, the crystal orientation θ is defined as the angle between the global meniscus orientation (M) and the first close-packed direction in the clockwise direction (ie [01¯1]). Due to the six-fold symmetry of the {111} plane, the range of θ is from 0° to 60°, where θ = 30° corresponds to the case where [101¯] is parallel to G. We also defined the local mismatch angle θ'as the angle between the local meniscus direction M'and the nearest <110> direction (Figure 2I; see the SI appendix, Figure S3 for more examples). θ'provides the intrinsic relationship between the local meniscus and the crystal orientation, because severe distortion of the meniscus may occur at the edges or defects of the substrate. Figure 2I summarizes the grain size distribution of the first layer of grains in Figure 2A and the size of θ'(average grain size 15.6 ± 12.1 μm, n = 106). The results clearly show that when colloidal assembly is first initiated, the close-packed orientation tends to align with the local meniscus (size weighted average 9.2 ± 8.2°), leading to [12¯1] growth (<112> family) initially, As expected from the directional nucleation caused by particle pinning at the local meniscus. In addition, the grains initially grown along the <112> direction are usually larger in size (Figure 2I). Figure 2J shows an extreme example in which the meniscus is locally severely distorted; however, the direction of close packing is always parallel to the direction of the local meniscus, resulting in different θ values ​​but similar θ'values.

The above analysis shows that the <110> direction is actually initially parallel to the direction of the meniscus, which contradicts the general observation that the <110> direction is consistent with the growth direction in the assembled bulk crystals. In order to further explore this interesting contradiction, we then carried out a similar analysis along the growth direction to study crystal evolution (Figure 3). As shown in Figure 3A, after reaching six or seven layers, the number of layers in the colloidal crystal becomes relatively constant. At this time, a large gradient can usually be seen in the crystal orientation within a single "crystal grain" (Figure 3A, ii) ; For large-area maps, see SI appendix, Figure S4). A single crystal grain, regardless of its initial crystal orientation, will gradually rotate in the <110> direction (Figure 3A, ii). The gradient of crystal rotation is not uniform, and the θ close to 15° and 45° is close to the highest value of 0.3°/μm (Figure 3A; see SI appendix, Figure S5).

The evolution of crystal orientation during growth. (A) (i) Lattice type, (ii) Corresponding orientation and (iii) Large area diagram of defects, indicating that the crystal orientation gradually rotates along the growth direction. The number of layers is shown, and up to seven layers are observed in the lower part of the area. (B and C) The magnified orientation map of the local crystal orientation in the range of (B) 0° <θ <30° and (C) 30° <θ <60°, obtained in the area shown in A. The circles highlight individual defects. (D and E) High magnification of the orientation map and (F and G) corresponding original SEM images showing a single defect, these images were taken in the areas shown in B and C. In F and G, the dislocation core is represented by the following formula ⊥; the white circle represents the Hamburg circuit. (H) The pattern obtained in the area where the preferred <110> growth has been achieved, showing self-correcting behavior. After the defect caused the crystal orientation error (arrow), the crystal orientation was rotated back to θ = 30°. (I) The top view SEM image of the area with dislocations (indicated by ⊥) selected for tomographic analysis. (J) Reconstruction of the top view of the same area. The red particles represent the stacking faults in the fcc lattice. In order to facilitate observation, a smaller particle radius is used to indicate it. (K) Reconstruction of the lens in the perspective view, with the front particles removed. All directions, phases, defect maps and SEM images are the same as the direction and color scheme in A.

A close inspection of the crystallographic rotation area revealed the presence of local co-parallel defects with a pitch of approximately 10 μm (Figure 3 B and C). These defects are aligned with the [11¯0] and [01¯1] directions in the regions of 0° <θ <30° and 30° <θ <60°, respectively. It is worth noting that a slight rotation towards θ = 30° occurs after each defect (note the yellow to orange transition and cyan to blue transition in Figure 3 D and E, respectively). The original SEM images show that these defects exhibit a classic dislocation structure similar to crystalline metals (Figure 3F and G and SI appendix, Figure S6). The Burgers circuit is represented by a white line, showing that the closure around the dislocation core has failed, otherwise it would close with a perfect lattice. For the regions of 0° <θ <30° and 30° <θ, the Burgers vector b connecting the start and end of the Burgers circuit is a/2[11¯0] and a/2[01¯1] respectively <60°, where a is the lattice parameter of the fcc lattice. This rotation behavior is so powerful that even if a local defect destroys the continuity of the crystal and produces a different crystal orientation, the crystal will rotate back to the preferred growth orientation (3) (as shown by the arrow in Figure 3H).

In order to further study the 3D structural characteristics of dislocations, FIB/SEM tomography was used to detect these crystal defects at the single particle level. Figure 3I shows a representative dislocation, in which tomographic data (θ = 45.2°) was acquired, from which the particle position was determined (Figure 3J). The subsurface defect structure can be visualized by representing particles in a normal fcc lattice with a smaller radius (Figure 3J). The defects here are detected by comparing the number of nearest relative neighbor pairs, which changes from six in the fcc lattice to three in the hcp lattice. Therefore, the defective particles shown in red form a stacking fault plane in which the stacking sequence changes from ABCABC to ABCBCABC. By removing some fcc particles, we confirm that the stacking fault is located on the (1¯11) plane, and its area gradually decreases from the top layer to the bottom layer, from ~10 to ~3 particles (Figure 3K; see also the SI appendix, Figure S7). Stacking faults are bounded by a pair of Shockley partial dislocations, and the Burgers vectors are a/6[11¯2] and a/6[1¯2¯1], which are also the most observed in fcc metals. Protruding dislocations (44). The relationship between this analysis and the full-edge dislocation observed from the top view SEM image is a/2[01¯1]=a/6[11¯2]+a/6[1¯2¯1 ] Consistent, for the case of 30° <θ <60°. FIB / SEM tomography measurement further confirmed that the dislocations belong to the same sliding system, and there are stacking faults of similar size in areas with multiple dislocations within a short distance (SI appendix, Figure S8).

As we all know, the evaporation-induced drying process usually produces a tensile stress field in the dried film parallel to the substrate (45), which may be high enough to cause the formation of cracks (for example, cracks in dry muddy ground). In the induced colloidal crystals, the tensile stress is mainly caused by the shrinkage rate of the colloidal crystal film (including the shrinkage of colloidal particles, the local readjustment caused by capillary force, and the shrinkage of the silica matrix in the case of co-assembly here). Caused by a mismatch. And the constraints of the rigid substrate during the drying process (46⇓⇓–49). The quantification of crack openings on the single-layer PS particles after assembly and subsequent drying allowed us to estimate the effective shrinkage rate to approx. 5% (SI appendix, Figure S9). We further noticed that during the evaporation-induced assembly process, there is a partially dry area [also known as the “gel” area (48)] before the assembly, where the assembled particles can still undergo The structural rearrangement is finally fixed at the completely dry stage (46⇓ –48) (Figure 4A).

The mechanism of crystallographic preferential growth. (A) Schematic diagram of evaporation and related self-assembly process. Horizontal (ie, along the direction of the meniscus) tensile stress is generated during the drying process of the colloidal component. (B) (Left) Determine the Schmid factor m as a function of the colloidal crystal orientation θ. Only the three close-packed planes ((111¯), (11¯1), and (1¯11)) have the highest m-values ​​in the three sliding directions shown here. (Right) A (111) stereoscopic projection, showing that the crystal gradually rotates towards the [101¯] growth direction. (CH) Results of colloidal self-assembly modeling method: (C) (Left) Schematic diagram of evaporation-induced colloidal assembly modeling. (Right) The particle size distribution used in the simulation; tstart and tend specify the time range for the particle size to decrease from dmax to dmin; tfix is ​​the time when the particle is fixed. (D) Representative simulation results showing gradual crystal rotation. (E) Stacking fault detection due to the generation of dislocations at the growth front. The location of the volume is highlighted in D. (F) The influence of initial crystal orientation, θinitial = 0°, 5°, 10°, 15°, 20°, 25° and 30°. (G) A graph of the average crystal angle distribution along the growth direction, G. (H) The effect of particle shrinkage, s = 1 – dmin/dmax = 10%, 5%, 3%, 2% and 0%.

In atomic crystals, dislocations can be generated by applying mechanical stress exceeding a critical level, which is one of the key mechanisms of plastic deformation of many materials (44). Therefore, we hypothesized that, similar to the stress-induced dislocations in atomic crystals, the drying-induced tensile stress field leads to the formation of dislocations in the partially dried colloidal crystals before the structure is completely dried and fixed. To verify our hypothesis, we determined the active slip system as a function of crystal orientation θ under tensile stress by evaluating the Schmid factor (m) of 12 equivalent slip systems in the fcc crystal. -Crystal deformation through slippage (SI appendix, Figure S10) (50). (Please note that in the current directional drying setting, the tensile stress is mainly along the M direction, because the fluid at the bottom releases the stress in the vertical [growth] direction, which is different from the equiaxial tensile field in a uniformly dried film.) Summarized in the figure In 4B, for 0° <θ <30° and 30° <θ <60°, the active slip system is [11¯0](111¯) and [01¯1](1¯11) respectively, and The experimental observations are consistent (Figure 3 BG).

Similar results have been obtained through finite element-based continuum mechanics modeling, indicating that the composite nature of co-assembled colloidal crystals does not affect the active slip system (SI appendix, Figures S11-S14). The high density of dislocations observed in the regions where θ is close to 15° and 45° is only because of their large m value, which results in greater analytical shear stress and is more prone to dislocations. In addition, the large width of the stacking fault area near the crystal surface is also consistent with the fact that the tensile stress caused by drying is greater on the film surface than near the substrate (51). Since the (111) plane is always parallel to the substrate in the colloidal crystal formed by this self-assembly process, this preferred sliding activation causes the in-plane rotation to grow towards <110> (Figure 4B, right). During this process, the slip direction gradually rotates toward the stretching axis, which is a well-known behavior in crystalline atomic materials (50, 52). When θ reaches 30°, m becomes the same for the [11¯0](111¯) and [01¯1](1¯11) systems. At this time, any perturbation of the crystal orientation will cause the rotation to return, leading to the stable growth of <110>. It should be noted that the generation of dislocations is not caused by other defect types, such as vacancies, which are evenly distributed throughout the growth area (SI Appendix, Figure S15). FIB/SEM tomography analysis also shows that there are no local accumulation defects near the dislocations, such as large or small particles, gaps, etc. These observations further confirm that these dislocation defects are caused by mechanical stresses that are strongly related to the local crystal orientation during the assembly process.

Crystal rotation is observed in some atomic crystal films obtained by direct electrodeposition (35) or thermally induced phase change (36). Although this phenomenon has not been fully understood, the concept of geometrically necessary dislocations (GNDs) has been hypothesized as a possible mechanism (36), because an array of GNDs with the same Burgers vector can theoretically lead to a crystal orientation gradient within a grain ( 53, 54). However, the relationship between crystal rotation and GND characteristics is not well established in atomic crystals, mainly because it requires structural analysis across the atomic and grain levels. The colloidal crystals formed through the evaporation-induced self-assembly process discussed here provide convincing experimental evidence that, in fact, dislocations with the same Burgers vector in the crystal grains due to drying-induced tensile stress can cause continuous Reorientation of crystallography. Following the 2D model developed for atomic crystals (53), by using the relationship ρ = θ/bd, the GND density ρ can be estimated to be ~1.4 × 1010/m2, where θ/d and b indicate that the angle has a position (0.3°/μm ) And Burgers vector (0.38 μm for a complete a/2 <110> dislocation). This corresponds to a GND spacing of ~9 μm (~20 lattice parameters), which is consistent with experimental observations (Figure 3 BG). Although there are significant differences in the length scales of their lattice parameters (36), it is estimated that the GND in atomic crystals has a similar order of magnitude, indicating that the influence of mechanical forces affects crystallization on more length scales than previously realized.

We can use a one-dimensional model to form misfit dislocations in the strained epitaxial film (9, 55, 56) to estimate the critical thickness hc, and form the first dislocation during the drying process. In this model, the colloidal crystal is regarded as an isotropic linear elastic continuum (Young's modulus E, shear modulus μ and Poisson's ratio υ), and hc=μbln(R/rc)/[4πEε0(1 −υ)cos⁡ α], where R and rc are the outer radius and core radius of the strain field related to the dislocation, ε0 is the tensile strain caused by drying, and α is the difference between the Burgers vector and its projection in the loading direction Angle (see analysis in the detailed SI appendix). Here, by taking ε0 = 0.05, rc = b/4 (9, 57, 58), υ = 1/3, and E/μ = 2(1+υ) (57), we find that hc = 1.24 µm, which corresponds to 4.2 Layered colloidal crystals. This prediction is consistent with our observation that GND-induced crystal rotation usually occurs in areas above five layers, while one or two layers of colloidal crystals rarely show any rotation.

We further developed a colloidal assembly modeling method to study the influence of some important experimental parameters and the particle-level dislocation dynamics during this self-assembly process. According to our hypothesis, the drying-induced tensile stress field controls the crystallographic rotation during the assembly process. This modeling method takes into account the "shrinking zone" after the particles are driven to the assembly front, where they undergo self-assembly and re-assembly. Row and shrink (s = 1-dmin/dmax), and then finally determine their position (Figure 4C; see SI appendix, Figure S16 and S17 and detailed description of the modeling method). In order to simplify this simulation scheme, we use the shrinkage rate of colloidal particles to represent the effective shrinkage rate of colloidal crystals. By using experimental conditions (particle size = 0.38 µm, polydispersity = 2.8%, shrinkage rate s = 0.05, initial crystal orientation θ = 5°, number of layers = 6), we successfully replicated the gradual crystal rotation to the first choice <110 >The growth of silicon self-assembled colloidal crystals, as shown in a series of snapshots at different stages of modeling (Figure 4D and Movie S1). It is worth noting that the length scale of the rotating area (200 μm to 300 μm in the G direction, θ rotating from ~5° to 30°) is similar to the experimental observation results (Figure 3A and 4G). In addition, the crystal structure analysis showed that perfect face-centered cubic crystals were formed first at the assembly front, and dislocations were subsequently generated in the shrinking zone. These dislocations exhibited the same slip system and similar spacing and size as those observed in the experiment (Figure 4E).

We confirm that consistent crystallographic rotation occurs regardless of the initial crystal orientation (Figure 4F; see SI appendix, Figures S18 and S19, and movie S2). In particular, the crystal angle profile is folded onto a single master curve, which also indicates that the rotation gradient reaches the highest value of 0.3°/µm around 15°, which matches the experimental observation (Figure 4G; see SI appendix, Figure S5). Please also note that for the initial crystal orientation θ = 0°, θ may take time to fluctuate to a high enough value (~5°) to cause rotation, because at θ = 0°, the system does not prefer the growth direction The specific sliding system operates (see the θ = 0° case in the SI appendix, Figure S18). In addition, by systematically changing the shrinkage rate of the particles, we found that when the shrinkage rate is less than 3%, no dislocations are observed in the current system, so no crystal rotation is observed, and a high shrinkage rate, such as 10%, leads to crystal orientation The occasional sudden change (Figure 4H; see SI appendix, Figures S20 and S21 and movie S3). Finally, previous work has shown that large polydispersities can completely inhibit colloidal crystallization (59, 60). Here, when the polydispersity is greater than 10%, we observe a similar close packing behavior loss, and the use of particles with a polydispersity less than 5% can achieve consistent crystal rotation behavior, including single-sized particles (SI Appendix, Figure S22 And S23 and movie S4).

We concluded that through the combination of comprehensive experiment, theory and modeling analysis, we revealed the basic mechanism of the unique characteristics of the colloidal component synthesized by the evaporation induction method, and the colloidal component grows into a large area of ​​fcc through the evaporation induction method. Crystal. 110> Direction. Our results clearly show that although the initial nucleation produces a polycrystalline region in which the domains are closely packed parallel to the meniscus line, due to the presence of geometrically necessary dislocations (GND), gradual intra-grain crystal rotation occurs. This is the mechanical stress field caused by drying. Although the continuum theory of atomic crystals can describe the GND characteristics well, our research results also show that, unlike atomic crystals, the two main mechanisms of texture development, namely epitaxial nucleation and orientation-dependent growth, are not induced by evaporation. <110> growth in colloidal crystals. Since mechanical stress is often present in the evaporation system, we expect that the dislocations and the corresponding crystal grain rotation caused by the mechanical stress disclosed here are the general principles for the preferential growth of crystals in the evaporation-driven colloidal components (3, 25⇓ ⇓ – 28, 30). In fact, we have observed similar crystal rotation behavior of direct PS and inorganic silica colloidal components (without matrix material) (SI appendix, Figures S24-S28). It is worth mentioning that due to the high thickness of the colloidal crystals (the total assembly of colloidal crystals is about 20 layers), in addition to dislocations, another form of strain release mechanism, namely cracking (3, 4, 61) ). By using the preferred <110> growth and orientation cracks, one can precisely control the formation of crack patterns in the evaporation-induced colloidal crystals (49).

Since the film growth of atomic and ionic crystals also generates mechanical stress due to mechanisms such as substrate/film mismatch and phase transition, the preferred growth induced by GND mechanical stress may provide further insights into the texture development in these systems. Insights, except for colloidal crystals. In addition, we further note that when the amorphous biomineral precursor is transformed into an oriented crystal form, it is generally believed that there is mechanical stress, which may contribute to the crystallographically preferred growth in the biological crystal structure, following the similar mechanism discussed here ( 62).

PS colloidal particles with diameters of 0.38 ± 0.1 μm and 0.54 ± 0.16 μm were prepared according to a surfactant-free emulsion polymerization process (63). A typical synthesis procedure is described below. Purify styrene (20 mL, Aldrich) by column chromatography (neutral alumina, 90-230 mesh, Aldrich) to remove any inhibitors. A three-necked 300 mL round bottom flask containing 170 mL of deionized (DI) water was deoxygenated with nitrogen for 30 minutes and heated to 70 °C. In a separate beaker, add 0.0870 g of potassium persulfate initiator (Sigma-Aldrich) to 10 mL of water. Add 10 mL of initiator solution and 20 mL of styrene into the round bottom flask. The solution was stirred at 300 rpm for 36 hours at 70±2°C. The resulting latex balls were filtered through glass wool to remove any large agglomerates. 134 ± 5 ​​nm (PC02006) and 277 ± 14 nm (PC02009) PS particles were purchased from Bangs Laboratories, Inc.

Add 400 ml of 95% ethanol, 20 ml of tetraethyl orthosilicate (TEOS, 98%, Sigma-Aldrich), 40 ml of 28% NH4OH, and 40 ml of deionized water into a 1,000 ml round bottom flask. The solution was stirred at 250 rpm for 24 hours at room temperature. The obtained SiO2 beads were collected by centrifugation, washed with ethanol 3 times, and finally dispersed in ethanol.

The method of evaporation-induced co-assembly of colloidal crystals has been described in detail in previous works (3, 15). A brief overview of the process is given here. First, the monodisperse PS colloid was diluted with deionized water (Milli-Q system) to a solid content of 0.1% for subsequent co-assembly. Prepare a fresh stock solution of pre-hydrolyzed TEOS (Sigma-Aldrich), which consists of TEOS (98%, Sigma-Aldrich), HCl (0.1 M, Sigma-Aldrich) and ethanol (100%, popular science) in a weight ratio of 1:1:1.5 Tektronix). Stir the TEOS stock solution at room temperature for 1 hour, and then add 140 µL to 20 mL of the colloidal solution with a solid content of 0.1% in the scintillation vial. The mixture was sonicated briefly. Wash the silicon substrates (~1 × 4 cm) in piranha solution, rinse them with ethanol and dry them completely, and suspend them vertically in a scintillation vial containing colloid/TEOS solution. In an oven (Memmert UF110 oven) at a temperature of 65°C, the solvent content is allowed to evaporate slowly within a period of 1 to 2 days. This process resulted in the deposition of a composite PS colloid-silica matrix colloidal crystal film (5 to 20 layers) on a silicon substrate (Wafer World, Inc.). In order to obtain inverse opal, the prepared composite colloidal crystals were calcified at the maximum temperature of 500°C for 2 hours, and the heating and cooling rate was 2°C/min (Thermo Scientific). For the direct synthesis of PS opal, the procedure remains the same, except that the TEOS and HCl mixture is not added to the final solution. To synthesize direct silica opal, add 45 µL of 10 vol% "stock" SiO2 colloidal suspension and 2 mL of absolute ethanol into a glass bottle. The Si substrate (~1 × 4 cm) pre-cleaned in piranha solution is suspended vertically in the vial. The solvent evaporates slowly on a pneumatic non-vibrating table in an oven at 40 °C to deposit the film on the suspended substrate.

Before imaging with Helios Nanolab 660 dual beam (FEI) with an acceleration voltage of 2 kV and a working distance of about 4 mm, the colloidal crystal film was coated with Pt/Pd (~5 nm) to minimize the charging effect. The images used for quantitative crystallographic analysis were obtained in the immersion mode with secondary electron mode, the beam current was 0.05 nA to 0.1 nA, and the dwell time was 10 μs to 30 μs. The imaging conditions of each data set remain unchanged to ensure the consistency of image quality. Use a standard SEM calibration sample to calibrate the magnification. The SEM image of the colloidal particles was imported into Fiji (64), and then converted into an 8-bit image for further processing. The preprocessing includes background flattening and Gaussian blur steps to smooth the intensity distribution within a single particle. Next, the image is binarized and further watershed processing is performed to separate any connected particles. Then identify and record individual particles, from which the particle area, radius and circularity are determined for statistical analysis. The bin width in the histogram is determined according to the method proposed by Scott (65), where the optimal bin width hn is given by 3.49sn-1/3. Here s is SD and n is the sample size.

The cross-sectional imaging and tomography of the colloidal crystal were obtained by FIB grinding on a Helios Nanolab 660 dual beam (FEI) instrument and subsequent SEM imaging. The typical protocol of FIB tomography based on the slice and view method is described below. First raise the sample to a working distance of ~4 mm and tilt it to 52°. Then align the region of interest at the concentric point where the electron beam and ion beam are focused. Under 30 keV and 0.28 nA ion beam conditions, a platinum protective layer (~10 × 10 × 1 µm) was first deposited in situ on top of the rectangular area of ​​interest. Three grooves on the side and front of a rectangular area (such as a U-shape) are milled by FIB (30 kV, 2.8 nA). The cross section analyzed using the FIB Slice and View method was further polished with a lower ion beam current (30 keV, 93 pA). Evaluate the electronic imaging conditions first before starting sectioning and viewing. Typically, imaging is in "immersion" mode at 2 keV, 0.1 nA to 0.2 nA, 1,535 × 1,024 or 2,058 × 1,768 pixels per frame, dwell time per pixel from 10 μs to 60 μs, and pixel sizes from 5 to 10-nm. The imaging is performed in the backscattered electron imaging mode. The automatic FIB tomography package (Slice & View G2, FEI) is used to perform data collection. Typical experimental parameters include a milling current of 80 pA and a slice distance of 5 nm to 20 nm. First use Fiji/ImageJ (NIH) to align the collected 3D tomographic image stack. Use Avizo 7.1.0 (VSG) to perform volume rendering of tomographic data.

The preprocessing of the original slices is performed on Fiji/ImageJ, including alignment, binary segmentation, inversion and erosion. The preprocessed image is saved as an image stack, which is used as an input file for the following program to identify the location of individual particles. The particle detection method used here was originally implemented in Matlab by John Crocker and David Grier (66). Use Ovito (open visualization tool, https://www.ovito.org/) to export the calculated particle position as an input file for defect analysis. The integrated algorithm Dislocation Extraction Algorithm is used to identify lattice types, dislocations and Burgers vectors.

Synchronous compression transformation technology is used for quantitative two-dimensional imaging analysis to obtain information on lattice type, defects, grain orientation and deformation field (41, 67⇓ -69). Compared with the more commonly used windowed Fourier analysis, the synchronous compression transform technique provides several advantages, especially high resolution, which is essential for current work. In short, the Bravais lattice of the crystal image at position x is encoded as the corresponding distribution of energy bumps in the frequency domain associated with x (SI Appendix, Figure S29 A and B). Crystal deformation moves atoms away from their reference positions, which is equivalent to moving energy bumps away from their reference positions. The resolution of windowed Fourier analysis is not sufficient to reflect this distortion. The synchronous compression transform provides sharp energy peaks in the frequency domain (see the SI appendix, Figure S29C for examples), making it more feasible to calculate the changes of these peaks away from their reference positions. By tracking the position and energy concentration of these peaks, an effective algorithm for crystal image analysis at the single particle or atom scale has been proposed, which can be used as a MATLAB package called SynCrystal (41) (https://github.com/SynCrystal) supply. For example, the peak position in the angular coordinate indicates the orientation of the crystal grain (SI Appendix, Figure S29C); inside the crystal grain, the synchronous compression transformation only shows a few highly concentrated energy peaks, depending on the crystal lattice (SI Appendix, Figure S29C) . On grain boundaries or isolated defects, the synchronous compression transform provides irregular energy peaks that propagate in the entire frequency domain (SI Appendix, Figure S29D). These characteristics can be used to identify defects and grain boundaries.

By using the SI appendix, the coordinate system shown in Figure S10, the sliding planes that may be activated under tensile load along the M direction include (111¯), (11¯1), and (1¯11). The separate operating sliding system is 1) [11¯0](111¯), 2) [01¯1¯](111¯), 3) [1¯01¯](111¯), 4) [101¯] (11¯1), 5) [1¯1¯0](11¯1), 6) [01¯1¯](11¯1), 7) [01¯1](1¯11), 8) [1¯01¯](1¯11) and 9) [1¯1¯0](1¯11). The three slip systems on the (111) plane are not considered because the plane normal is perpendicular to the loading axis. For a colloidal crystal with a given value of θ, the corresponding global meniscus direction, so the loading direction, can be expressed as [A, -1, 1-A], where A ∈ [0,1] represents the length SI in the appendix The segment shown, Figure S10. In addition, the relationship between crystal orientation θ and A is tan⁡θ=3A/(2-A). The Schmid factor m can be calculated by m=cos⁡λ⁡cos⁡ϕ, where λ is the angle between the loading direction and the slip direction, and ϕ is the angle between the loading direction and the normal direction of the slip surface. In a cubic crystal, the cosine of the angle between two directions is given by the dot product of the unit vectors in these directions. Therefore, for each slip system, we can calculate the angles of λ and ϕ as a function of A, which is therefore θ.

The finite element simulation is performed on a representative volume element (RVE) with 4 × 4 spherical particles in the GM plane and three layers of particles in the N direction. By applying macroscopic stretching in the GM plane, periodic boundary conditions in the G and M directions are used to simulate the microscopic deformation of colloidal crystals under evaporation-induced tension. Specifically, the influence of the stretching direction θ (relative to the [01¯1] direction) on the crystal deformation is studied by rotating the deformation gradient tensor F'=RTFR while keeping the RVE constant. Note that R is a rotation tensor completely defined by θ. In the finite element implementation (commercial finite element package ABAQUS), the macroscopic deformation gradient F is designated as the displacement components of the three virtual nodes. The tension caused by evaporation is modeled as stretching F11 = λ. Then, the principle of virtual work is used to capture the macroscopic mechanical response of RVE. The PS particles use an elastic material model with a Young's modulus of 2.9 GPa and a Poisson's ratio of 0.4 because their deformation is relatively small. The hyperelastic Neo Hooke model with G/2 = 0.1667 MPa and K = 1 GPa is used to capture the highly incompressible behavior of the matrix material, where G and K are the shear modulus and bulk modulus, respectively. The mesh convergence was verified, and the results were simulated from a finite element model with 3,457,387 C3D4 elements. After obtaining the stress and strain fields from the simulation (SI Appendix, Figure S12-S14), the stress is further projected onto four close-packed planes, highlighting the shear component (SI Appendix, Figure S11).

All data can be found in the text and SI appendix. The synchronous compression transformation technique for quantitative 2D lattice analysis is available as a MATLAB package called SynCrystal (41) (https://github.com/SynCrystal). Computational colloid assembly code implemented as MATLAB code can be provided upon request.

This research was mainly supported by the NSF Materials Research Science and Engineering Center and won the DMR-1420570 and DMR-2011754 awards. LL thanks 3M for its support through the Non-Tenured Faculty Award and the Department of Mechanical Engineering at Virginia Tech and State University. Electron microscopy and FIB/SEM tomography were performed at the Harvard Nanosystems Center. We thank Professor Mathias Kolle and Professor Benjamin Hatton for a fruitful general discussion of the manuscript. We would also like to thank Emily Redston and Joseph Zsolt Terdik for helpful discussions on particle detection analysis.

Author's contribution: LL and JA design research; LL, CG, HY, KRP, ZJ, HC, JZ, AL and JA conducted research; LL, CG, HY, KRP, ZJ, HC, LW, JZ, AL, JL , JS, MPB, FS and JA analysis data; LL and JA wrote this paper.

Reviewers: PC, New York University; ED, Eidgenossische Technische Hochschule Zurich; and BP, Technion-Israel Institute of Technology.

The author declares no competing interests.

↵*In this article, we use the standard crystal symbol of the fcc system. The conventional coordinate axis is parallel to the four-fold rotation axis, the length unit a=σ2, where σ is the nearest neighbor distance; [hkl] represents the direction defined by the vector rhkl = a(h,k,l) ​​in the crystal; h¯ represents -h ;(Hkl) represents an infinite set of parallel planes perpendicular to rhkl, uniformly spaced at a2/rhkl; {hkl} and <hkl> respectively refer to the set of all (hkl) planes or [hkl] directions with equivalent crystal symmetry. For example, a general approach can be found in Buerger (23).

This article contains online support information at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2107588118/-/DCSupplemental.

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