We implement molecular dynamics simulations in canonical ensemble to study the effect of confinement on a $2d$ crystal of point particles interacting with an inverse power law potential proportional to $r^{-12}$ in a narrow channel. This system can describe colloidal particles at the air-water interface. It is shown that the system characteristics depend sensitively on the boundary conditions at the two {\it walls} providing the confinement. The walls exert perpendicular forces on their adjacent particles. The potential between walls and particles varies as the inverse power of ten. Structural quantities such as density profile, structure factor and orientational order parameter are computed. It is shown that orientational order persists near the walls even at temperatures where the system in the bulk is in fluid state. The dependence of elastic constants, stress tensor elements, shear and bulk modulii on density as well as the channel width is discussed. Moreover, the effect of channel incommensurability with the triangular lattice structure is discussed. It is shown that incommensurability notably affects the system properties. We compare our findings to those obtained by Monte Carlo simulations and also to the case with the periodic boundary condition along the channel width. .
We study confined colloidal suspensions as a model system which approximates the behavior of confined small molecule glass-formers. Dense colloidal suspensions become glassier when confined between parallel glass plates. We use confocal microscopy to study the motion of confined colloidal particles. In particular, we examine the influence particles stuck to the glass plates have on nearby free particles. Confinement appears to be the primary influence slowing free particle motion, and proximity to stuck particles causes a secondary reduction in the mobility of free particles. Overall, particle…
A confined colloidal glass, under the imposition of a uniform shear stress, is investigated using numerical simulations. Both at macro- and microscales, the consequent dynamics during the onset of flow is studied. When the imposed stress is gradually decreased, the time scale for the onset of steady flow diverges, associated with long-lived spatial heterogeneities. Near this yield-stress regime, persistent creep in the form of shear-banded structures is observed.
We apply a hybrid Molecular Dynamics and mesoscopic simulation technique to study the dynamics of two dimensional colloidal discs in confined geometries. We calculate the velocity autocorrelation functions, and observe the predicted $t^{-1}$ long time hydrodynamic tail that characterizes unconfined fluids, as well as more complex oscillating behavior and negative tails for strongly confined geometries. Because the $t^{-1}$ tail of the velocity autocorrelation function is cut off for longer times in finite systems, the related diffusion coefficient does not diverge, but instead depends logarith…
We use colloidal suspensions encapsulated in emulsion droplets to model confined glass-forming liquids with tunable boundary mobility. We show dynamics in these idealized systems are governed by physical interactions with the boundary. Gradients in dynamics are present for more mobile boundaries, whereas for less mobile boundaries gradients are almost entirely suppressed. Motions in a system are not isotropic, but have a strong directional dependence with respect to the boundary. These findings bring into question the ability of conventional quantities to adequately describe confined glasses.
Colloidal interactions in nematic liquid crystals can be described as interactions between elastic multipoles that depend on particle shape, topology, chirality, boundary conditions and induced topological defects. Here, we describe a nematic colloidal system consisting of mesostructures of gold capable of inducing elastic multipoles of different order. Elastic monopoles are formed by relatively large asymmetric mesoflower particles, for which gravity and elastic torque balancing yields monopole-type interactions. High-order multipoles are instead formed by smaller mesoflowers with a myriad of…
The physical mechanism of elasticity of liquid surfaces coated with colloidal particles is proposed. It is suggested that particles are separated by water clearings and the capillary interaction between them is negligible. The case is treated when the colloidal layer is deformed normally to its surface. The elasticity arises as an interfacial effect. The effective Young modulus of a surface depends on the interfacial tension, equilibrium contact angle, radius of colloidal particles and their surface density. For the nanometrically scaled particles the line tension becomes essential and has an …
Recent studies on confined crystals of charged colloidal particles are reviewed, both in equilibrium and out of equilibrium. We focus in particular on direct comparisons of experiments (light scattering and microscopy) with lattice sum calculations and computer simulations. In equilibrium we address buckling and crystalline multilayering of charged systems in hard and soft slit confinement. We discuss also recent crystalline structures obtained for charged mixtures. Moreover, we put forward possibilities to apply external perturbations, in order to drive the system out of equilibrium. These in…
We study the pressure-driven flow of concentrated colloids confined in glass micro-channels at the single particle level using fast confocal microscopy. For channel to particle size ratios $a/\bar{D} \lesssim 30$, the flow rate of the suspended particles shows fluctuations. These turn into regular oscillations for higher confinements ($a/\bar{D} \simeq 20$). We present evidence to link these oscillations with the relative flow of solvent and particles (permeation) and the effect of confinement on shear thickening.
We propose a simple theoretical model which explains a formation of dipolar 2D and 3D colloidal structures in nematic liquid crystal. Colloidal particles are treated as effective hard spheres interacting via their elastic dipole, quadrupole and octopole moments. It is shown that octopole moment plays an important role in the formation of 2D and 3D nematic colloidal crystals. We generalize this assumption on the case of the external electric field and theoretically explain a giant electrostriction effect in 3D crystals observed recently [A. Nych et al., Nature Communications \textbf{4}, 1489 (2…
Using laser tweezers and fluorescence confocal polarizing microscopy, we study colloidal interactions of solid microspheres in the nematic bulk caused by elastic distortions around the particles with strong tangential surface anchoring. The particles aggregate into chains directed at about 30 degrees to the far field director and, at higher concentrations, form complex kinetically trapped structures. We characterize the distance and angular dependencies of the colloidal interaction forces.
We show by computer simulation of a two-dimensional crystal confined by corrugated walls that confinement can be used to impose a controllable mesoscopic superstructure of predominantly mechanical elastic character. Due to an interplay of the particle density of the system and the width D of the confining channel, "soliton staircases" can be created along both parallel confining boundaries, that give rise to standing strain waves in the entire crystal. The periodicity of these waves is of the same order as D. This mechanism should be useful for structure formation in the self-assembly of vario…
We demonstrate a novel route of spatially organizing the colloid arrangements on the polymer-grafted substrate by use of self-consistent field and density functional theories. We find that grafting of polymers onto a substrate can effectively control spatial dispersions of deposited colloids as a result of the balance between colloidal settling force and entropically elastic force of brushes, and colloids can form unexpected ordered structures on a grafting substrate. The depositing process of colloidal particles onto the elastic "soft" substrate includes two steps: brush-mediated one-dimensio…
The progress of realizing colloidal structures mimicking natural forms of organization in condensed matter is inherently limited by the availability of suitable colloidal building blocks. To enable new forms of crystalline and quasicrystalline self-organization of colloids, we develop truncated pyramidal particles that form nematic elastic dipoles with long-range electrostaticlike and geometry-guided low-symmetry short-range interactions. Using a combination of nonlinear optical imaging, laser tweezers, and video microscopy, we characterize colloidal pair interactions and demonstrate unusual f…
Self-assembly of nanoparticles can enable composites with pre-designed properties but remains challenged by reproducing structural diversity of atomic and molecular crystals. We combine anisotropic elastic and weakly screened electrostatic interactions to guide both orientational and triclinic positional self-ordering of inorganic nanocrystals in a nematic fluid host. The lattice periodicity of these low-symmetry colloidal crystals is more than an order of magnitude larger than the nanoparticle size. Orientations of nanocrystals, as well as crystallographic axes of ensuing triclinic colloidal …
We investigate the tagged-particle motion in a strongly interacting quasi-confined liquid using periodic boundary conditions along the confining direction. Within a mode-coupling theory of the glass transition (MCT) we calculate the self-nonergodicity parameters and the self-intermediate scattering function and compare them with event-driven molecular dynamics simulations. We observe non-monotonic behavior for the in-plane mean-square displacement and further correlation functions which refer to higher mode indices encoding information about the perpendicular motion. The in-plane velocity-auto…
Using a novel bi-axial confocal rheoscope, we investigate the structure and rheology of sheared colloidal suspensions under confinement. Consistent with previous work [X. Cheng \textit{et al., Proc. Natl. Acad. Sci. U. S. A.}, 2011, 109, 63], we observe a vorticity-aligned string phase in moderate concentrated colloidal suspensions under uniaxial shear. Using bi-axial shear protocols, we directly manipulate the orientation and morphology of the string structures. Simultaneously, we measure the suspension rheology along both the flow and vorticity directions with a bi-axial force measurement de…
The flow of colloidal suspensions is ubiquitous in nature and industry. Colloidal suspensions exhibit a wide range of rheological behavior, which should be closely related to the microscopic structure of the systems. With in-situ small-angle neutron scattering complemented by rheological measurements, we investigated the deformation behavior of a charge-stabilized colloidal glass at particle level undergoing steady shear. A short-lived, localized elastic response at particle level, termed as transient elasticity zone (TEZ), was identified from the neutron spectra. The existence of the TEZ is a…
Using self-consistent field and density-functional theories, we first investigate colloidal self-assembling of colloid/polymer films confined between two soft surfaces grafted by polymers. With the increase of colloidal concentrations, the film undergoes a series of transitions from disordered liquid $\to$ sparse square $\to$ hexagonal (or mixed square-hexagonal) $\to$ dense square $\to$ cylindric structures in plane, which results from the competition between the entropic elasticity of polymer brushes and the steric packing effect of colloidal particles. A phase diagram displays the stable re…
Self-assembly of colloidal particles is poised to become a powerful composite material fabrication technique, but remains challenged by a limited control over the ensuing structures. We develop a new breed of nematic colloids that are physical analogs of a mathematical surface with boundary, interacting with the molecular alignment field without inducing defects when flat. However, made from a thin nanofoil, they can be shaped to prompt formation of self-compensating defects that drive pre-programmed elastic interactions mediated by the nematic host. To show this, we wrap the nanofoil on all t…
Theoretical description of the elastic interaction between colloidal particles in NLC with incorporation of the higher order elastic terms beyond the limit of dipole and qudrupole interactions is proposed. The expression for the elastic interaction potential between axially symmetric colloidal particles, taking into account of the high order elastic terms, is obtained. The general paradigm of the elastic interaction between colloidal particles in NLC is proposed so that every particle with strong anchoring and radius $a$ has three zones surrounding itself. The first zone for $a<r\lessapprox 1.…
A binary mixture of super-paramagnetic colloidal particles is confined between glass plates such that the large particles become fixed and provide a two-dimensional disordered matrix for the still mobile small particles, which form a fluid. By varying fluid and matrix area fractions and tuning the interactions between the super-paramagnetic particles via an external magnetic field, different regions of the state diagram are explored. The mobile particles exhibit delocalized dynamics at small matrix area fractions and localised motion at high matrix area fractions, and the localization transiti…
Recent theories predict that when a supercooled liquid approaches the glass transition, particle clusters with a special "amorphous order" nucleate within the liquid, which lead to static correlations dictating the dramatic slowdown of liquid relaxation. The prediction, however, has yet to be verified in 3D experiments. Here, we design a colloidal system, where particles are confined inside spherical cavities with an amorphous layer of particles pinned at the boundary. Using this novel system, we capture the amorphous-order particle clusters and demonstrate the development of a static correlat…
We calculate the configurational entropy of colloidal particles in a confined geometry interacting as hard disks using Monte Carlo integration method. In particular, we consider systems with three kinds of boundary conditions: hard, periodic and spherical. For small to moderate packing fraction $ϕ$ values, we find the entropies per particle for systems with the periodic and spherical boundary conditions tend to reach a same value with the increase of the particle number $N$, while that for the system with the hard boundary conditions still has obvious differences compared to them within the st…
Based on Brownian dynamics simulations, we investigate the thermodynamic signatures of non-equilibrium steady states in a confined colloidal suspensions under shear flow. Specifically, we consider a thin film consisting of charged particles in narrow slit-pore confinement, forming two layers with quadratic in-plane structure in equilibrium. This many-body system displays three distinct steady states, characterized by unique dynamical and rheological response to the applied shear flow. Calculating the work and heat, we find that both quantities indicate the different states by their mean and by…
The motion of an optically trapped sphere constrained by the vicinity of a wall is investigated at times where hydrodynamic memory is significant. First, we quantify, in bulk, the influence of confinement arising from the trapping potential on the sphere's velocity autocorrelation function $C(t)$. Next, we study the splitting of $C(t)$ into $C_\parallel(t)$ and $C_\perp(t)$, when the sphere is approached towards a surface. Thereby, we monitor the crossover from a slow $t^{-3/2}$ long-time tail, away from the wall, to a faster $t^{-5/2}$ decay, due to the subtle interplay between hydrodynamic b…
Particle tracking and displacement covariance matrix techniques are employed to investigate the phonon dispersion relations of two-dimensional colloidal glasses composed of soft, thermoresponsive microgel particles whose temperature-sensitive size permits \textit{in situ} variation of particle packing fraction. Bulk, $B$, and shear, $G$, moduli of the colloidal glasses are extracted from the dispersion relations as a function of packing fraction, and variation of the ratio $G/B$ with packing fraction is found to agree quantitatively with predictions for jammed packings of frictional soft parti…
Using numerical and analytical calculations we study the structure of vacancies and interstitials in two-dimensional colloidal crystals. In particular, we compare the displacement fields of the defect obtained numerically with the predictions of continuum elasticity theory for a simple defect model. In such a comparison it is of crucial importance to employ corresponding boundary conditions both in the particle and in the continuum calculations. Here, we formulate the continuum problem in a way that makes it analogous to the electrostatics problem of finding the potential of a point charge in …
Charge-stabilized colloidal spheres dispersed in weak 1:1 electrolytes are supposed to repel each other. Consequently, experimental evidence for anomalous long-ranged like-charged attractions induced by geometric confinement inspired a burst of activity. This has largely subsided because of nagging doubts regarding the experiments' reliability and interpretation. We describe a new class of thermodynamically self-consistent colloidal interaction measurements that confirm the appearance of pairwise attractions among colloidal spheres confined by one or two bounding walls. In addition to supporti…
Using Brownian dynamics (BD) simulations we investigate a dense system of charged colloids exposed to shear flow in a confined (slit-pore) geometry. The equilibrium system at zero flow consists of three, well-pronounced layers with square-like crystalline in-plane structure. We demonstrate that, for sufficiently large shear rates, the middle layer separates into two sublayers where the particles organize into moving lanes with opposite velocities. The formation of this micro-laned state results in a destruction of the applied shear profile. It has a strong impact not only on the structure of t…
We performed a thorough investigation of the drying dynamics of a charged colloidal dispersion drop in a confined geometry. We developed an original methodology based on Raman micro-spectroscopy to measure spatially-resolved colloids concentration profiles during the drying of the drop. These measurements lead, for the first time, to estimates of the collective diffusion coefficient of the dispersion over a wide range of concentration. The collective diffusion coefficient is one order of magnitude higher than the Stokes-Einstein estimate showing the importance of the electrostatic interactions…
We show that geometric confinement dramatically affects the shear-induced configurations of dense mono-disperse colloidal suspensions; a new structure emerges, where layers of particles buckle to stack in a more efficient packing. The volume fraction in the shear zone is controlled by a balance between the viscous stresses and the osmotic pressure of a contacting reservoir of unsheared particles. We present a model that accounts for our observations and helps elucidate the complex interplay between particle packing and shear stress for confined suspensions.
We study bidisperse colloidal suspensions confined within glass microcapillary tubes to model the glass transition in confined cylindrical geometries. We use high speed three-dimensional confocal microscopy to observe particle motions for a wide range of volume fractions and tube radii. Holding volume fraction constant, we find that particles move slower in thinner tubes. The tube walls induce a gradient in particle mobility: particles move substantially slower near the walls. This suggests that the confinement-induced glassiness may be due to an interfacial effect.
Strongly confined colloidal dispersions under shear can exhibit a variety of dynamical phenomena, including depinning transitions and complex structural changes. Here, we investigate the behaviour of such systems under pure oscillatory shearing with shear rate $\dotγ(t) = \dotγ_0 \cos(ωt)$, as it is a common scenario in rheological experiments. The colloids' depinning behaviour is assessed from a particle level based on trajectories, obtained from overdamped Brownian Dynamics simulations. The numerical approach is complemented by an analytic one based on an effective single-particle model in t…
The concept of a local linear elastic strain field is commonly used in the metallurgical research community to approximate the collective effect of atomic displacements around crystalline defects. Here we show that the elastic strain field approximation is a useful tool in colloidal systems. For colloidal crystals with repulsive particle interaction potentials, given similar mechanical properties, sharper potentials lead to: 1) free energies of deformation dominated by entropy, 2) lower variance in strain field fluctuations, 3) increased tension-compression asymmetry near dislocation core regi…
The new free energy functional that describes general elastic interaction between colloidal particles and nematic liquid crystal has been proposed. It generalizes results of the paper \cite{lupe} on the case of arbitrary orientation of colloidal particles and is valid for arbitrary surface anchoring strength. Formal analogies and differences between electric particles and colloidal particles in LC are found. It is first time shown that spur of the quadrupole moment tensor is different from zero $Sp\hat{Q}_μ\neq 0$ in the case of axial symmetry breaking of the director field around colloidal pa…
The theory of elastic interaction of micron size axially symmetric colloidal particles immersed into confined nematic liquid crystal has been proposed. General formulas are obtained for the self energy of one colloidal particle and interaction energy between two particles in arbitrary confined NLC with strong anchoring condition on the bounding surface. Particular cases of dipole-dipole interaction in the homeotropic and planar nematic cell with thickness $L$ are considered and found to be exponentially screened on far distances with decay length $λ_{dd}=\frac{L}π$. It is predicted that boundi…
Mixing colloid shapes with distinctly different anisotropy generates composite nematics in which the order of the individual components can be fundamentally different. In colloidal rod-disk mixtures or hybrid nematics composed of anisotropic colloids immersed in a thermotropic liquid crystal, one of the components may adopt so-called anti-nematic order while the other exhibits conventional nematic alignment. Focussing on simple models for hard rods and disks, we employ Onsager-Straley's second-virial theory to derive scaling expressions for the elastic moduli of rods and disks in both nematic …
We present a framework to segregate the roles of elastic and non-elastic deformations in the examination of real-space experiments of solid-solid Martensitic transitions. The Martensitic transformation of a body-centred-tetragonal(BCT) to a body-centred-orthorhombic(BCO) crystal structure has been studied in a model system of micron-scale ionic microgel colloids. Non-affine fluctuations, i.e., displacement fluctuations that do not arise from purely elastic(affine) deformations, are detected in particle configurations acquired from the experiment. Tracking these fluctuations serves as a highly …
The collective diffusion effects in system of a colloidal particles in a liquid crystal has been proposed. In this article described peculiarity of collective diffusion colloidal particles in a liquid crystal, which can be observe experimentally. The diffusion coefficient should be crucial dependence from temperature and concentration of particles. In system colloidal particles arise from the elastic distortion of the elastic director field the inter-particle interaction. These interaction can cause nontrivial collective behavior, as the results the non-monotonic dependence collective diffusio…
Achieving and exceeding the diversity of colloidal analogs of chemical elements and molecules as building blocks of matter has been the central goal and challenge of colloidal science ever since Einstein introduced the colloidal atom paradigm. Recent significant advances have been achieved by exploiting the powerful machinery of DNA hybridization to assemble colloidal particles, but robust physical means of defining colloidal atoms and molecules remain limited. Here we introduce the physical design principles that allow for defining high leading-order elastic colloidal multipoles that emerge w…
Exploring the role of hydrodynamic interactions in spherically-confined drying colloidal suspensions
Auburn University（奥本大学）；Leibniz-Institut für Polymerforschung Dresden e.V.（德累斯顿莱布尼茨聚合物研究所）；Technische Universität Dresden（德累斯顿工业大学）；Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学）
We examine the response of a dense colloidal suspension to a local force applied by a small magnetic bead. For small forces, we find a linear relationship between the force and the displacement, suggesting the medium is elastic, even though our colloidal samples macroscopically behave as fluids. We interpret this as a measure of the strength of colloidal caging, reflecting the proximity of the samples' volume fractions to the colloidal glass transition. The strain field of the colloidal particles surrounding the magnetic probe appears similar to that of an isotropic homogeneous elastic medium.…
The effects of confinement on colloidal self-assembly in the case of fixed number of confined particles are studied in the one dimensional lattice model solved exactly in the Grand Canonical Ensemble (GCE) in [J. Pękalski et al. J. Chem. Phys. 142, 014903 (2015)]. The model considers a pair interaction defined by a short-range attraction plus a longer-range repulsion. We consider thermodynamic states corresponding to self-assembly into clusters. Both, fixed and adaptive boundaries are studied. For fixed boundaries, there are particular states in which, for equal average densities, the number o…
We report Monte Carlo studies of a two-dimensional soft colloidal crystal confined in a strip geometry by parallel walls. The wall-particle interaction has corrugations along the length of the strip. Compressing the crystal by decreasing the distance between the walls induces a structural transition characterized by the sudden appearance of a one-dimensional array of extended defects each of which span several lattice parameters, a "soliton staircase". We obtain the effective interaction between these defects. A Lindemann criterion shows that the reduction of dimensionality causes a finite per…
We propose a microscopic picture for understanding the nonlinear rheology of supercooled liquids with soft-repulsive potentials. Based on Brownian dynamics simulations of supercooled charge-stabilized colloidal suspensions, our analysis shows that the shear thinning of viscosity (eta) at large enough shear rates (sr), expressed as eta~sr^(-lambda), originates from the evolution of the localized elastic region (LER). An LER is a transient region composed of the first several coordination shells of a reference particle. In response to the external shear, particles within LER undergo nearly affin…
Dilute oil dispersions of fractal carbon black particles with attractive Van der Waals interactions display continuous shear thickening followed by shear thinning at high shear rates. The shear thickening transition occurs at $\dotγ_{c}\approx 10^{2}-10^{3}s^{-1}$ and is driven by hydrodynamic breakup of clusters. Pre-shearing dispersions at shear rates $\dotγ>\dotγ_{c}$ produces enhanced-modulus gels where $G' \sim σ_{pre-shear}^{1.5-2}$ and is directly proportional to the residual stress in the gel measured at a fixed sample age. The observed data can be accounted for using a simple scaling …
The complex behavior of confined fluids arising due to a competition between layering and local packing can be disentangled by considering quasi-confined liquids, where periodic boundary conditions along the confining direction restore translational invariance. This system provides a means to investigate the interplay of the relevant length scales of the confinement and the local order. We provide a mode-coupling theory of the glass transition (MCT) for quasi-confined liquids and elaborate an efficient method for the numerical implementation. The nonergodicity parameters in MCT are compared to…
We present numerical results for the tagged-particle dynamics by solving the mode-coupling theory in confined geometry for colloidal liquids (cMCT). We show that neither the microscopic dynamics nor the type of intermediate scattering function qualitatively changes the asymptotic dynamics in vicinity of the glass transition. In particular, we find similar characteristics of confinement in the low-frequency susceptibility spectrum which we interpret as footprints of parallel relaxation. We derive predictions for the localization length and the scaling of the diffusion coefficient in the superco…
Recent progress in approaches to determine the elastic constants of solids starting from the microscopic particle interactions is reviewed. On the theoretical side, density functional theory approaches are discussed and compared to more classical ones using the actual pair potentials. On the experimental side, video microscopy has been introduced to measure the elastic constants in colloidal solids. For glasses and disordered systems, the theoretical basis is given for this novel technique, and some challenges and recent advances are reviewed.
We consider the theory of fluctuations of a colloidal solid observed in a confocal slice. For a cubic crystal we study the evolution of the projected elastic properties as a function of the anisotropy of the crystal using numerical methods based on the fast Fourier transform. In certain situations of high symmetry we find exact analytic results for the projected fluctuations.
Colloids self-assemble into various organized superstructures determined by particle interactions. There is a tremendous progress in both the scientific understanding and applications of self-assemblies of single-type identical particles. Forming superstructures in which the colloidal particles occupy predesigned sites and remain in these sites despite thermal fluctuations represents a major challenge of the current state-of-the art. Here we propose a versatile approach to direct placement of colloids using nematic liquid crystals with spatially varying molecular orientation pre-imposed by sub…
The Green function method developed in Ref.[S. B. Chernyshuk and B. I. Lev, Phys. Rev. E \textbf{81}, 041707 (2010)] is used to describe elastic interactions between axially symmetric colloidal particles in the nematic cell in the presence of the external electric or magnetic field. General formulas for dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interactions in the homeotropic and planar nematic cells with parallel and perpendicular field orientations are obtained. A set of new results has been predicted: 1) \textit{Deconfinement effect} for dipole particles in the homeotropic …
We study mutual alignment and interactions between colloidal particles of dissimilar shapes and dimensions when dispersed in a nematic host fluid. Convex pentagonal and concave starfruit-shaped nanoprisms and microspheres induce dipolar or quadrupolar director structures. The ensuing elastic pair interactions between microspheres and nanoprisms are highly dependent on the nanoparticle shape being omnidirectionally attractive for convex prisms but strongly anisotropic for concave prisms. Elastic deformations due to spherical particles cause well-defined alignment of complex-shaped nanoparticles…
