Abstract
Crystalline solids typically contain large amounts of defects such as dislocations and interstitials. How they travel across grain boundaries (GBs) under external stress is crucial to understand the mechanical properties of polycrystalline materials. Here, we experimentally and theoretically investigate with singleparticle resolution how the atomic structure of GBs affects the dynamics of interstitial defects driven across monolayer colloidal polycrystals. Owing to the complex inherent GB structure, we observe a rich dynamical behavior of defects near GBs. Below a critical driving force defects cannot cross GBs, resulting in their accumulation near these locations. Under certain conditions, defects are reflected at GBs, leading to their enrichment at specific regions within polycrystals. The channeling of defects within samples of specificallydesigned GB structures opens up the possibility to design novel materials that are able to confine the spread of damage to certain regions.
Introduction
The plastic deformation of crystalline materials typically takes place via the elementary flow of topological defects such as dislocations and interstitials^{1,2,3,4,5}. Therefore, the dynamics of such defects under external stress is of central importance for understanding the mechanical behavior of crystals. In contrast to their rapid propagation within single crystals, the motion of the defects is severely influenced by grain boundaries (GBs) in polycrystals, leading to a mechanical reinforcement of polycrystalline materials which increases with the inverse average grain size^{6,7,8,9,10}. This empirically observed Hall–Petch relation has been explained with the GBassisted accumulation of defects, which leads to an increasing yield strength^{9,10}. Evidence for this pileup mechanism is provided by electronmicroscopy experiments, where defects, which have been created by indentation of nanometersized tips, are observed to accumulate at GBs^{11,12,13,14}. While the interactions of defects with GBs have been intensively studied in atomic simulations^{15,16,17,18,19}, such detailed and particleresolved investigations of the GBdefect interactions are limited from the experimental side. In particular, how the inhomogeneous atomic GB structure locally influences the incoming defects has not been thoroughly investigated in experiments. Such knowledge, however, is mandatory to provide quantitative relationships between the structure and the mechanical properties of polycrystalline materials.
Here we report an experimental and theoretical study to unravel the properties of driven interstitial defects in polycrystalline colloid monolayers with singleparticle resolution. Experimentally, this is achieved by injecting colloidal particles into a colloid monolayer interacting with a patterned triangular substrate emulating polycrystalline grains with various GB topology. Colloids are “magnified atoms” with a length scale ~4 orders of magnitude larger than atoms and time scale ~6 orders of magnitude slower^{20}. Due to the easily accessible time and length scales and the possibility of tuning the relevant microscopic interactions in colloid experiments, colloidal systems have been established as versatile analogic models to provide detailed insight at singleparticle level, e.g., in phase transitions, nanofriction, clogging and jamming in the flow of particles hindered by obstacles^{21,22,23,24}. When injecting interstitial defects into the system by an external driving force, we observe that their motion is strongly hampered only at specific positions of the GB, leading to their distortion and splitting upon crossing the GB. Below a critical driving force the defects are not able to cross GBs, which leads to their accumulation (pileup) at these locations. A Hall–Petchlike relation is recovered by measuring the critical force as a function of the grain size. Even though our investigations pertain to twodimensional colloid monolayers where the detailed structure and dynamics of the defects are very different, the observed Hall–Petch strengthening relation and positiondepend dynamic behaviors are qualitatively similar to those observed in defectGB interactions in threedimensional real materials^{15,16,17,18,19}. Remarkably, we also find that, under certain conditions, the GB can reflect incoming defects. Their confinement to specific regions of the polycrystalline sample suggests the fabrication of polycrystalline monolayers with directiondependent mechanical properties.
Results
Experiment
Our experimental setup is illustrated in Fig. 1a. Silica particles with hardsphere interaction and diameter σ = 4.28 μm are driven by gravity F = mg sinα across a flat surface (reservoir) toward a patterned surface with hexagonal symmetry and lattice spacing b > σ. Specifically, this periodic substrate is patterned with cylindrical wells created by photolithography. Each well (with radius r ≈ 1.6 μm and depth h ≈ 0.5 μm, see Supplementary Fig. 1) can be occupied by one particle only. When reaching the patterned area, the colloids become trapped in the wells, eventually forming a hexagonal crystal with lattice spacing b. Under the gravitational driving force, extra “interstitial” colloidal particles can be injected from the reservoir into the crystal phase, which leads to the formation of zerodimensional (point like) and quasionedimensional (i.e., aggregated) interstitials (see Supplementary Movie 1). These interstitials propagate along the lattice vector being closest to the driving force. To study the interaction of interstitials with GBs, we construct patterns of substrate wells arranged as singlecrystalline domains separated by welldefined GBs (Supplementary Fig. 1). More experimental details are in the “Methods” section.
Interstitial defects in uniform lattice
Before discussing the behavior of interstitials near GBs, we summarize their properties on a uniform, i.e., single domain, lattice. Figure 1b shows an optical image of an interstitial defect, which is formed by three interstitial particles. To better visualize the corresponding strain field, we color code each particle by its distance δr to the nearest substrate well in Fig. 1c, see “Methods” for the calculation of δr. It reveals a crowdion (a specific type of interstitial) configuration where the strain caused by the interstitial is confined to an elongated area with length L ≈ 6.0b and w = 3 lines of particles wide. Given the mismatch between the particle size σ and the substrate lattice spacing b, the length L of the interstitials varies as shown in Fig. 1e. In the rest of the paper, we select b = 4.6 μm, generating interstitials of relative length L/b ≈ 6.0, which matches the size of crowdions in metallic crystals^{25}. As sketched in Fig. 1d, the features and motion of our interstitials resemble the topological solitons described by the Frenkel–Kontorova model^{26,27}. In addition to the Peierls–Nabarro (PN) potential arising due to the interaction with the underlying periodic surface, our interstitials experience a lateral friction force when moving through the lattice due to interactions with particles in neighboring motionless lines (Supplementary Fig. 2). Since the lateral friction force exists only at the boundary between the interstitial and the undistorted lattice, the interstitials become faster with increasing width, as shown in Fig. 1f, which compares the forcedependent velocity of interstitials with w = 1 and 3. Further evidence that the lateral friction of interstitials is essentially given by their boundary with the surrounding is also provided by the fact that the interstitial velocity rapidly saturates with increasing w (inset Fig. 1f). An immediate consequence of this lateral friction is their aggregation once they approach each other from different directions. This is exemplarily shown in Fig. 1g and Supplementary Movie 2 for the case of two interstitials with w = 1 merging into a single one with w = 2. Such aggregations lead to reduced collisions between interstitials and therefore enhance the average interstitial velocity (Supplementary Movie 3). In contrast, repulsion is observed (Fig. 1h, Supplementary Movie 4) when a fast (w = 2) interstitial approaches a slower one (w = 1) along the same line. This situation is similar to the interaction of running kinks in onedimensional systems, which also exhibit a repulsive interaction due to the overlap of the compressive strain fields^{28}.
Driving interstitial defects across smallangle GBs
The dynamics of interstitials becomes strongly affected by the presence of a GB. In general, GBs in a twodimensional crystal are characterized by two angles θ_{1} and θ_{2} indicating the lattice orientations on each side of the boundary (Fig. 2a). To create equilibrium lowenergy GBs, however, one also has to consider the structural relaxation of the two domains when bringing them into contact. It has been shown that equilibrium GB configurations in two dimensions with lowest energy can be constructed using a method based on centroidal Voronoi tessellation^{29}. Supplementary Fig. 1 shows some examples of a substrate containing GBs with different values of θ_{1} and θ_{2}, which have been realized according to the centroidal Voronoitessellation method. Notably, the GBs consist of an almost linear sequence of localized defects, which are characterized by pairs of fivefold and sevenfold coordinated potential wells (5–7 pairs). Between such pairs, the lattices remain almost undistorted.
When a colloidal monolayer is absorbed on such a structure, it closely matches the underlying substrate geometry. This can be seen from the small particle displacements δr which are exemplarily shown in Fig. 2a for a symmetric GB with θ_{1} = θ_{2} = 4.72°. The Voronoi tessellation of the particle positions clearly indicates the positions of the 5–7 pairs of the underlying substrate. When interstitials are inserted and driven perpendicular to the GB, their behavior strongly depends on whether they hit a 5–7 pair or cross the GB in between. When an interstitial passes the only marginally distorted GB region between 5–7 pairs, its velocity remains almost unaffected (Fig. 2b, d and Supplementary Movie 5). Opposed to that, a pronounced time delay of about 130 s in the interstitial trajectory is found when crossing the GB at a 5–7 pair (Fig. 2c, e and Supplementary Movie 5). In Fig. 2f, we show the measured time delay as a function of the driving force for three different crossing points at the GB as indicated in Fig. 2a. With increasing distance from a 5–7 pair, the measured (symbols) delay times systematically decrease in qualitative agreement with numerical simulations (lines, see “Methods”). The deviations from the experiments are possibly due to outofplane particle motions and the colloid’s polydispersity which are not considered in the simulations. To understand why the delay time strongly depends on the position where the particle passes through the GB, we have numerically calculated the potential energy U_{c} of an interstitial as a function of its centerofmass distance x_{c} to the GB. As seen in Fig. 2g, U_{c} is strongly nonmonotonic at the GB. In particular when the interstitial passes near a 5–7 pair, the spatial variation of U_{c} becomes most pronounced, which explains why the delay time is largest for these regions. The inset of Fig. 2g illustrates the correlation between the depth/height of U_{c,ex} and the Voronoiarea deviation δA = A_{GB} − A_{0}, where A_{GB} is the Voronoi area of the GB lattice point, and A_{0} is the Voronoi area of a regular lattice point. When δA_{g} < 0, i.e. the lattice is locally compressed, U_{c,ex} is positive and the GB acts as a barrier. For δA_{g} > 0, i.e. when the lattice is locally expanded, U_{c,ex} is negative and the GB acts as a trap. Far away to the left and right of the GB, U_{c} displays a tiny oscillation whose wavelength is identical to the periodicity of the substrate. The amplitude of this oscillation corresponds to the PN potential for an interstitial moving across a perfect periodic lattice. As a result of the strong variation of the potential barriers along the GB, interstitials with widths comparable with or even larger than the spacing between 5–7 pairs display a rather complex behavior crossing a GB. This is exemplarily shown in Fig. 2h and Supplementary Movie 6, displaying the dynamics of an interstitial with w = 14. Upon approaching the GB, its front becomes distorted and eventually splitted when passing the GB.
Pileup of interstitial defects at largeangle GBs
On average, the total slowing down of interstitials at GBs strongly depends on the density ρ_{p} of 5–7 pairs along the GB, which is a function of the GB misalignment angles, as illustrated in Fig. 3a. For symmetric GBs θ = θ_{1} = θ_{2}, ρ_{p} is:
Eq. (1) is plotted in Fig. 3b (top). The discontinuity at θ = 15° stems from the geometrical origin of the defects, and is discussed in the Methods section, alongside the derivation and extension of Eq. (1) to GBs with arbitrary orientations. Near θ = 0° and 30° the two grains are almost perfectly aligned, resulting in dilute 5–7 pairs at the GB. The density of 5–7 pairs becomes largest for θ = 15°, where the misalignment of the two grains is maximum. Figure 3b (bottom) reports the average delay time (“Methods”) of interstitials when they traverse a symmetric GB of angle θ. The results show that the delay time reaches a maximum near θ = 15°. Therefore, the denser the 5–7 pairs, the longer the delay time is. With θ = 19.1°, the density of 5–7 pairs is so large that a pileup of interstitials is observed, as shown in Fig. 3c and Supplementary Movie 7, by gradually lowering the driving force toward F = 23 fN. To quantitatively describe this pileup, we define a pileup ratio p (“Methods”) that describes the strength of pileup. Figure 3d reports the observed p as a function of F for the θ = 19.1° GB and for the θ = 4.72° GB. For the θ = 19.1° GB, p reaches a relatively high value when F < 30 fN. Instead, for the θ = 4.72° GB, p remains small across the entire range of experimental driving forces: generally we observe little or no pileup against smallangle GBs.
Together with the accumulation of interstitials near GBs, mechanical stress gradients are expected to increase in these regions. Accordingly, the spatial distribution of internal stress should become more homogeneous with decreasing grain size and thus leads to an increased yield stress^{9,10}. To verify this, in Fig. 3e we show v as a function of F for different grain sizes d in numerical simulations for the θ = 19.1° GB, while keeping the total size of the system and the density of interstitials fixed. d is the spacing between two successive GBs as shown in the inset of Fig. 3f. The smaller the value of d, the more numerous GBs the interstitials has to cross. Figure 3f shows the critical force F_{c} as a function of grain size d. Under such force, all interstitials are effectively stuck (pileup) at one of the GBs. Interestingly, the data in Fig. 3f can be well fitted (solid line) to the Hall–Petch relation F_{c} = F_{0} + k/d^{1/2}, which describes the strengthening of materials at small grain sizes due to the increasing yield stress with decreasing separations between GBs.
Reflection of interstitial defects by GBs
In addition to the observed slowing down and pileup of interstitials when crossing GBs, reflection of interstitials can also occur. This is shown in Fig. 4a where part of an interstitial with w = 4 is being reflected after hitting a 5–7 pair at the GB. The reflection probability R(θ_{1}, θ_{2}) for interstitials in the θ_{1} side approaching the θ_{2} side (and likewise the transmission probability 1 − R) depends on the direction of driving force as well as GB angles θ_{1} and θ_{2}. R(0°, θ_{2}) and R(θ_{2}, 0°) are plotted in Fig. 4b as a function of θ_{2}, for nonsymmetric GBs with θ_{1} = 0° and F parallel to the GB. Similar to the time delay, the reflection is also related to the potential energy barrier at the GB. This is consistent with our observation that reflection preeminently occurs at 5–7 pairs at the GB where the potential barrier is largest (see Fig. 4a).
However, contrary to a transmission process, reflection requires both the stopping and a redirection of an interstitial. Then the reflection rate is the product of the probability p_{hit}(θ_{1}, θ_{2}) for an incoming interstitial to hit a 5–7 pair at the GB and the probability p_{reflect}(θ_{1}, θ_{2}, F) for a stopped interstitial to be reflected. p_{hit}(θ_{1}, θ_{2}) is proportional to the density of 5–7 pairs divided by the cosine of the interstitial incidence angle. p_{reflect}(θ_{1}, θ_{2}, F) is obtained by solving an equilibrium twostate distribution problem with p_{reflect} ∝ exp(F_{reflect}(θ_{1}, F) b/k_{B}T) and p_{cross} ∝ exp(F_{cross}(θ_{2}, F) b/k_{B}T). Here F_{reflect}(θ_{1}, F) and F_{cross}(θ_{2}, F) are the appropriate projections of F to the lattice symmetry directions corresponding to reflection and crossing respectively. Finally, one arrives at the total reflection rate:
where h is the sigmoid function h(x) = 1/(e^{−x} + 1). The projections F_{reflect} and F_{cross} involve also the normal force exerted by the GB, and are described in the “Methods” section. Given b = 4.6 µm, k_{B}T = 4.14 zJ, θ_{1} = 0°, and a driving force parallel to the GB with magnitude F = 66 fN, the theoretical reflection rates R(0°, θ_{2}) and R(θ_{2}, 0°) as a function of θ_{2} are plotted in Fig. 4b alongside the experimental data, showing good agreement. By taking advantage of the large difference of R(0°, θ_{2}) and R(θ_{2}, 0°) at θ_{2} > 15°, defects can be strongly localized within a single grain as shown in Fig. 4c and Supplementary Movie 8. The interstitials easily cross the GB from grain 2 to grain 1 but not in the opposite direction, effectively trapping all moving interstitials inside one single region. This is also confirmed by numeric simulations, see Supplementary Movie 9. Note that Fig. 4c shows even direction changes within the grains: this is due to the nonuniform PN barriers and the existence of thermal fluctuations in experiments, and should be distinguished from the reflections that occur at the GBs.
Discussion
Various techniques have been developed to process materials in order to optimize their properties regarding specific technological applications. Many of these methods, however, are based on empirical findings rather than on a detailed microscopic understanding how defects affect the material properties. This lack of knowledge is partially due to the difficulty to observe the atomistic kinetics of defects moving across GBs with singleparticle resolution and in real time. Colloidal monolayers with interstitial defects and externally designed atomic scale GBs, as reported here, provide an ideal twodimensional emulator where much novel experimental and theoretical understanding can be obtained regarding the GBdefect interactions. While the explored GB topologies are specific to twodimensional hexagonal crystals, the GBs in threedimensional metals involve similar patterns of sites with varying coordination and atomic volume which affect the mobility of vacancies and selfinterstitials^{30,31,32}. Accordingly, the distortions and splitting of defects by GBs as well as the confinement of defects within certain regions of the polycrystal might also be observed in threedimensional polycrystals. As GBs and defects are known to assist in annealing of crystal damage^{33,34,35,36}, the observed channeling of interstitials within polycrystals suggests an intriguing possibility for controlling the damage, failure, and selfhealing of materials under mechanical stress. Our experimental approach illustrates the premises for predicting the dynamic behavior under external stress of physical systems characterized by similar topological features, such as the mechanical behavior of twodimensional materials and nanofriction^{37}.
Methods
Substrate preparation and characterization
The polycrystalline structure on the sample substrate was created by photolithography. We adopt the algorithm in ref. ^{29} to generate the GB packing geometry. This geometry is transferred to a photo mask, the lattice points on which are opaque circular disks. After exposing the negative photoresist SU8 2000.5 (~500 nm in thickness coated on a glass surface) under the mask, the unexposed disk regions on the photoresist dissolve away in photoresist developers, resulting in circular wells of depths ~500 nm on the photoresist, arranged in a polycrystalline packing. We scanned the SU8 structure under an atomic force microscope with a Bruker OTESPAR3 tip (tip curvature radius ~7 nm). The scans (Supplementary Fig. 1) show that, for the b = 4.60 μm substrate, the diameter of the wells is about 3.6 μm and the depth of the wells is about 550 nm. The values (3.6 μm and 550 nm) can vary slightly (±10%) in experiments from sample to sample.
Sample preparation
The colloidal suspension is composed of monodispersed silica spheres in deionized water. They have a diameter σ = 4.28 ± 0.12 μm, buoyant weight mg = 348 fN and a gravitational height of h_{g} = k_{B}T/mg = 11.7 nm at room temperature T = 295 K. The colloidal suspension is injected into a sample cell of about 20 mm × 30 mm × 300 μm in size, where 300 μm is the distance from the bottompatterned substrate to the top cover slide. Under gravity, the particles sediment down to the bottom of the sample cell and uniformly distribute on the substrate which contains regions of flat surface and regions of topographically patterned surface. The colloidal particles are slightly smaller than the lattice spacing of the periodic surface, therefore each well on the periodic surface can host at most one particle. The initial particle coverage (~0.3) is not enough to form a crystalline phase on the periodic surface. To facilitate the formation of crystalline phase on the periodic surface, the sample cell is tilted by 15–20 degrees so that particles can move in from the flat reservoir to the patterned surface. The driving force is such that it is too small to drive isolated particles across the periodic surface, but much larger than needed to drive interstitials across the crystals. Under the driving force, the newly arrived particles from the reservoir will move into the crystalline phase and become interstitials (Supplementary Movie 1). The interstitials will further be driven across the crystalline phase until they reach unoccupied wells. In such a way, the crystalline phase grows larger and larger in the patterned region.
Particle tracking and interstitial characterization
We recorded experimental images at a 3 Hz frame rate. Using a standard particletracking algorithm, we can track the positions of the center of the colloidal particles with a 50 nm accuracy. To calculate the distance δr of a particle relative to its equilibrium position (i.e., the nearest well), we fit the positions of all particles in the image to a perfect triangular lattice of lattice spacing b, with fitting parameters x_{0}, y_{0}, θ, where (x_{0}, y_{0}) are the coordinates of one of the lattice points and θ the lattice orientation. With (x_{0}, y_{0}) and θ, the position of all other lattice points are calculated. δr of a particle is then the distance between the particle and the nearest lattice point. An interstitial is defined as a cluster of closepacked particles whose δr > 1.2 μm. The position of an interstitial is the centerofmass position of the cluster. The velocity of individual interstitials can then be calculated from the time series of their centerofmass displacements.
Calculation of average delay time
To calculate the average delay time of interstitials crossing symmetric GBs with angle θ, as shown in Fig. 3b, we measure the average velocity v(θ) of interstitials crossing the GB and traveling for an horizontal distance Δx. As a reference, we take the average velocity for interstitials in a single crystal—i.e., v(θ = 0). The average delay time is then defined to be: Δx/v(θ) − Δx/v(0). Δx = 300 μm in both experiments and simulations. To calculate v(θ) in experiments, the average velocity v_{p}(θ) of each colloidal particle in the field of view during a time period of ~1000 s is measured. Then, v(θ) = N_{p}v_{p}(θ)/N_{int}, where N_{p} ~ 3500 is the number of particles and N_{int} = Σ_{i}w_{i} ~ 100 is the total number of interstitials (weighted by their width w_{i}). w_{i} ≥ 3 in most cases. The calculated v(θ)/v(0) is shown in Supplementary Fig. 5 for both experiments and simulations. v(θ)/v(0) reaches a minimum around θ = 15°, corresponding to the longest delay of interstitials.
Determination of pileup ratio of interstitials
We define the pileup ratio as p = (∫_{−Δx/2<x<0}δr^{2}(x)dx − ∫_{0<x<Δx/2}δr^{2}(x)dx)/∫_{−Δx/2<x<Δx/2}δr^{2}(x)dx, i.e. the difference of the shaded areas at x < 0 and at x > 0 in Supplementary Fig. 6, normalized by the entire shaded area, where δr^{2}(x) = 1/Y × ∫_{0<y<Δy} δr^{2}(x,y)dy is the yaveraged δr^{2}, Δx = 300 μm and Δy = 240 μm.
Evaluation of the grain boundary defect density
The precise local structure of the GBs depends on the alignment of the two neighboring grains relative to the direction of the GB itself. We can predict very accurately the average defect concentration and its composition in terms of topological defects. As shown in Supplementary Fig. 7, due to the lattice misalignment, the number of lattice lines joining at the GB from the left is generally different from the number of lattice lines joining from the right. To account for this discrepancy, one 5–7 pair is generated at the GB for each lattice line difference. Using this simple geometric observation, we propose simple formulas to describe the density of 5–7 pairs in any GB between triangular lattices. In general, there are only two independent ways (highlighted in red and blue in Supplementary Fig. 7) for the lattice lines in the two grains to join at the GB with minimum deflection, therefore there are two sets of 5–7 pairs at the GB. For one set, the (average) distance between 5–7 pairs is D′ = √3/2 b/(cos(θ_{1}) − cos(β)); for the other set, the distance between 5–7 pairs is D″ = √3/2 b/(cos(θ_{1} + 60°) − cos(β − 60°)). The angle β depends on the relative orientation of the two grains and is related to which directions present minimal deflection: β = θ_{2} if θ_{1} + θ_{2} < 30° and β = 60° − θ_{2} if θ_{1} + θ_{2} > 30°. The total density of 5–7 pairs is then ρ_{p} = 1/D′ + 1/D″. We find that these formulas accurately describe the structural properties of the GBs constructed according to the algorithm of ref. ^{29}. The value of ρ_{p} is 0 at θ_{1} + θ_{2} = 0° and 60° and reaches the maximum value near θ_{1} + θ_{2} = 30°. More illustrations of GBs and their 5–7 pairs with different values of θ_{1} and θ_{2} are shown in Supplementary Fig. 8.
Reflection and transmission analysis
In a first approximation a GB can be considered as a barrier that exerts a normal force onto approaching interstitials. See Supplementary Fig. 9, a driving force F parallel to the GB, and with magnitude F, has component F_{crossing} = F cos(30° + θ_{2}) in the direction of transmission, and F cos(30° + θ_{1}) in the direction of reflection. The interstitial is attracted to the GB in the direction of the incoming lattice line with a force equal to P = F cos(30° − θ_{1}). The resulting normal force, approximating the GB as a hard wall, is N = F cos(30° − θ_{1}) sin(30° − θ_{1}). When this normal force is projected back to the reflection direction, it gives a contribution F cos(30° − θ_{1}) sin(30° − θ_{1}) cos(60° − θ_{1}). Consequently, the total force in the direction of reflection is F_{reflect} = F cos(30° + θ_{1}) + F cos(30° − θ_{1}) sin(30° − θ_{1}) cos(60° − θ_{1}). This quantity is an ingredient of Eq. (2).
Modeling and molecular dynamics
The substrate corrugation V(r) explored by a single particle at position r is the sum of terms V_{dimple}(r − r_{i}), where r_{i} are the centers of the individual potential wells, as placed in a regular twodimensional lattice or, near a GB, according to the Voronoi algorithm^{29}. V_{dimple}(r) is a smooth approximation of the potentialenergy profile for a sphere of radius R located at a distance r from the center of a cylindrical well. To avoid cusps in the potential energy, we use V_{dimple}(r) = −ϵ for r < r_{m}; V_{dimple}(r) = −ϵ/2 tanh((ξ − w_{d})/(ξ(1 − ξ)) − 1.0) for r_{m} < r < r_{M}; V_{dimple}(r) = 0 for r > r_{M}. Here, ξ = (r − r_{m})/(r_{M} − r_{m}). The parameters w_{d} = 0.29, r_{m} = 0.6 µm and r_{M} = 2.0 µm have been fitted to best replicate the experimental profile experienced by the σ = 4.28 μm spheres on the b = 4.6 µm substrate. We adopt an energy corrugation depth ϵ = 170 zJ, consistent with a well depth of ~500 nm. For the investigation of the colloidalparticles dynamics, we use a sum of twobody interaction potentials of the form v_{int}(r_{i} − r_{j}) = +∞ for r_{i} − r_{j} < r_{0}; v_{int}(r_{i} − r_{j}) = v_{LJ}(r_{i} − r_{j} − r_{0}) for r_{i} − r_{j} > r_{0}. This combines a hardcore repulsion at distances smaller than r_{0} = 1.0 µm and a softer LennardJones repulsion at larger distances, with ϵ_{LJ} = 1 zJ, σ_{LJ} = 3.6 µm and a “truncated and shifted force” cutoff at 1.6 σ_{LJ}. The parameters have been chosen to fit structural properties of experimental interstitials. We perform T = 0 Langevin dynamics with a damping rate γ = 20.0 ms^{−1}, within a fourthorder Runge–Kutta integration scheme.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We would like to thank Prof. Oded Hod and Dr. Xiang Gao for discussions on the construction of lowenergy grain boundaries. X.C. acknowledges funding from Alexander von Humboldt Foundation. E.T. acknowledges support by ERC ULTRADISS Contract No. 834402. N.M. and A.V. acknowledge support by the Italian Ministry of University and Research through PRIN UTFROM N. 20178PZCB5.
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C.B. and X.C. designed the experiments; X.C. carried out the experiments; A.V., N.M., and E.P. wrote the computer code; E.P. performed the numerical simulations and constructed the theoretical formulas of grain boundary defect density and reflection probabilities; X.C. and E.P. analyzed the data; and E.T. contributed to the interpretation of the results. All authors contributed to the theoretical understanding, discussed the results, and wrote the paper.
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Cao, X., Panizon, E., Vanossi, A. et al. Pileup transmission and reflection of topological defects at grain boundaries in colloidal crystals. Nat Commun 11, 3079 (2020). https://doi.org/10.1038/s4146702016870w
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DOI: https://doi.org/10.1038/s4146702016870w
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