A crystal plasticity analysis of length-scale dependent internal stresses with image effects

Low angle grain boundaries can be modeled as arrays of line defects (dislocations) in crystalline materials. The classical continuum models for energetics and dynamics of curved grain boundaries are mainly based on those with equilibrium dislocation structures without the long-range elastic interaction, leading to a capillary force proportional to the local curvature of the grain boundary. The new continuum model recently derived by Zhu and Xiang (J. Mech. Phys. Solids, 69,175-194,2014) incorporates both the long-range dislocation interaction energy and the local dislocation line energy, and enables the study of low angle grain boundaries with non-equilibrium dislocation structures that involves the long-range elastic interaction. Using this new energy formulation, we show that the orthogonal network of two arrays of screw dislocations on a planar twist low angle grain boundary is always stable subject to both perturbations of the constituent dislocations within the grain boundary and the perturbations of the grain boundary itself.

In combination with the fundamental Lennard-Jones potential, basic equations for physisorption of rigid microparticles physisorbed on a solid bearing applied strains are derived. The results are then used to analyze long-range elastic interaction between two adsorbates mediated by the substrate. It is found that the interaction energy depends not only on the inter-adsorbate distance, but also on the adorbate size and the applied strains.

We investigate the effects of elastic coherency strains in graphite intercalation compounds. The long-range interaction energies of two-dimensional islands of intercalant are calculated. Using the domain model of Daumas and H\'erold, we show that these strains drive a mixed-stage or randomly staged crystal to pure-stage ordering.

The shapes of two-dimensional (2D) nanostructures on surfaces are determined by their boundary energies as well as by long-range elastic, electrostatic or magnetic interactions. Although it is well known that long-range interactions can give rise to shape bifurcation-an abrupt change in shape symmetry at a critical size-a general description of the evolution of shape with size, systematically incorporating both the azimuthal dependence of the boundary energy and long-range interactions, has been lacking. Here we show that unconstrained shape relaxation, including previously ignored boundary curvature, leads to a novel, continuous shape change from convex at small size to concave at large size. In addition to demonstrating a method to quantitatively determine the azimuthal dependence of the boundary energy, we show that the energy gain associated with boundary curvature relaxation is a key factor in stabilizing surface nanostructures. For 7 x 7 domains on Si(111), boundary curvature reduces the formation free-energy by up to 50%.

The investigation of the surface alignment of liquid crystal (LC) multilayers evaporated on photoaligned polyimide vertical alignment (PI-VA) film was carried out by means of the novel three-dimensional (3D) surface profiler. We report the first use of the surface profiler to visualize a microscopic image of the monolayer arrangement of LC molecules in contact with the surface of photo-treated PI-VA film. The photoinduced anisotropy of partially UV-exposed PI-VA film can be visualized as a topological image of LC multilayers. It seems that the topology of LC multilayers is indicating the orientational distribution of LC molecules on the treated film. It was found that the periodically photoaligned PI-VA film surface can align an adsorbed LC monolayer and the LC molecular alignment can be extended to the bulk via the epitaxylike LC–LC interaction, i.e. a short-range molecular interaction. With regard to the unexposed PI-VA film surface, noticeable anisotropy in the monolayer alignment was not observed, indicating that the long-range elastic interaction may be responsible for the bulk alignment. The appearance of small droplets in the masked region may be presumably related to the dewetting phenomena.

Self-organized epitaxial growth on spontaneously nano-patterned templates

A phase-field theory of dislocations, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for an arbitrary number and arrangement of dislocation lines over a slip plane; the long-range elastic interactions between dislocation lines; the core structure of the dislocations; the interaction between the dislocations and an applied resolved shear stress field; and the irreversible interactions with short-range obstacles, resulting in hardening, path dependency and hysteresis. We introduce a variational formulation for the statistical mechanics of dissipative systems. The influence of finite temperature as well as the mechanics in the phase-field theory are modeled with a Metropolis Monte Carlo algorithm and a mean field approximation. A chief advantage of the present theory is that at zero temperature it is analytically tractable, in the sense that the complexity of the calculations may be reduced, with the aid of closed form analytical solutions, to the determination of the value of the phase field at point-obstacle sites. The theory predicts a range of behaviors which are in qualitative agreement with observation, including hardening and dislocation multiplication in single slip under monotonic loading; the Bauschinger effect under reverse loading; the fading memory effect; the evolution of the dislocation density under cycling loading; temperature softening; strain rate dependence; and others. The model also reproduces the formation of dislocation networks observed in grain boundaries for different crystal structures and orientations. Simultaneously with the stable configurations the theory naturally predicts the equilibrium dislocation density independently of initial values or sources.

We propose a surface energy for conically degenerate anchorings of uniaxial liquid crystal mesogens by modifying tensorial Nobili-Durand surface energy that is usually employed for fixed anchoring orientations with preferred polar angles. By minimizing Landau-de Gennes free energy and the proposed surface energy, we obtain the equilibrium director configuration around a spherical colloid in the uniform nematic liquid crystal. Our calculations show that the proposed surface energy can cause boojum or/and Saturn-ring defect textures depending on the equilibrium conic angle. We also study the interactions between two spherical colloids with the equilibrium conic angle 45^{∘}, where the surface energy provides both boojum and Saturn-ring defects on the surface of particles. We compare the calculated anisotropic colloidal interactions with experimental observations [B. Senyuk et al., Nat. Commun. 7, 10659 (2016)2041-172310.1038/ncomms10659]. In agreement with experiment, our results show two stable angular assemblies in the close particle-particle separations. Also, the long-range elastic interactions are almost consistent with the hexadecapolar elastic distortion.

We study numerically the behavior of a two-dimensional elastic plate (a crustal plane) that terminates along one of its edges at a fault boundary. Slip-weakening friction at the boundary, inertial dynamics in the bulk, and uniform slow loading via elastic coupling to a substrate combine to produce a complex, deterministically chaotic sequence of slipping events. We observe a power-law distribution of small events and an excess of large events. For the small events, the moments scale with rupture length in a manner that is consistent with seismological observations. For the large events, rupture occurs in the form of narrow propagating pulses. [S0031-9007(96)00703-X] PACS numbers: 91.30.Bi, 03.40.Dz, 05.45. + b, 64.60.Ht The discovery of dynamic complexity in the uniform, one-dimensional Burridge-Knopoff (BK) model of an earthquake fault [1,2] has brought new urgency to some questions about models of seismic sources. Perhaps the most pressing of these questions concerns the role of elasticity in the crustal plane—an ingredient that is necessarily missing in any one-dimensional model but which must be important for an understanding of the dynamics of slipping events. Off-fault elasticity is relevant to many features of real earthquake faults such as stress concentrations at rupture fronts, long-range elastic interactions, and seismic radiation. Previous studies [3‐7] indicate the following: The completely uniform, one-dimensional BK model, with velocity-weakening stick-slip friction, is a deterministically chaotic dynamical system that exhibits a broad range of earthquake-like events. The frequency-magnitude distribution for these events includes a scaling region of small localized events that is qualitatively similar to a Gutenberg-Richter (GR) law [8], and a region of large delocalized events whose frequency exceeds that of the extrapolated GR law and which account for most of the moment release. The large events propagate along the fault at roughly the sound speed in the form of “Heaton pulses” [9]. In order to be well posed mathematically, the model requires a cutoff or an ad hoc mechanism for initiating rupture at very small length scales. The questions of whether such mechanisms imply inherent discreteness of these models and whether that discreteness, in turn, implies the need for small-scale heterogeneity in realistic fault models are beyond the scope of this investigation. The important point is that the large-scale properties—complexity of large events and existence of a GR regime—are independent of the discretization length or the heterogeneity scale, and therefore appear to be robust features of this class of models. Our purpose in the investigations reported here has been to test the above features of the one-dimensional BK models in a two-dimensional model that includes elasticity in the crustal plane. Like the one-dimensional models, our two-dimensional model is still a caricature of the physically realistic situation, but it brings us a large step closer to an understanding of the dynamic behavior of this class of systems. It remains unrealistic in at least two respects, both of which are dictated by computational feasibility. First, our crustal plane is an elastic plate that moves only normal to itself and is coupled elastically to a rigid substrate. In this way, we obtain an essential simplification of our equations of motion and retain a very important dynamic time scale associated with coupling

Based on the multiple scattering technique [K. F. Freed and M. Muthukumar, J. Chem. Phys. 69, 2657 (1978); 68, 2088 (1978); M. Muthukumar and K. H. Freed, J. Chem. Phys. 70, 5875 (1979)] previously applied to the study of suspensions of spheres and polymers, we propose an approach to the computation of the effective elastic properties of a composite material containing rigid, mono-sized, randomly dispersed, spherical particles. Our method incorporates the many-body, long-range elastic interactions among inclusions. The effective medium equations are constructed and numerically solved self-consistently. We have calculated the effective shear μ′ and Young E′ moduli, as well as the effective Poisson ratio σ′, as functions of the particle volume fraction Φ and of the Poisson ratio σ of the continuous phase. Comparisons with two sets of experimental data—glass beads in a polymer matrix and tungsten carbide particles in a cobalt matrix (Wc/Co)—and to a previous theoretical solution, are also presented. Our model can predict the effective Poisson ratio of the Wc/Co system for Φ⩽1 and for the glass/polymer system for Φ⩽0.5. In particular, the present work describes accurately composites with a high volume fraction of inclusions, where a percolation transition occurs. Very good agreement with the experimental data are obtained for E′ and μ′ when Φ⩽0.4, for both systems.

Some of the most exotic condensed matter phases, such as twist grain boundary and blue phases in liquid crystals and Abrikosov phases in superconductors, contain arrays of topological defects in their ground state. Comprised of a triangular lattice of double-twist tubes of magnetization, the so-called ‘A-phase’ in chiral magnets is an example of a thermodynamically stable phase with topologically nontrivial solitonic field configurations referred to as two-dimensional skyrmions, or baby-skyrmions. Here we report that three-dimensional skyrmions in the form of double-twist tori called ‘hopfions’, or ‘torons’ when accompanied by additional self-compensating defects, self-assemble into periodic arrays and linear chains that exhibit electrostriction. In confined chiral nematic liquid crystals, this self-assembly is similar to that of liquid crystal colloids and originates from long-range elastic interactions between particle-like skyrmionic torus knots of molecular alignment field, which can be tuned from isotropic repulsive to weakly or highly anisotropic attractive by low-voltage electric fields.

It is shown that the alloying of iron by Al (0 to 30 at%) shifts the carbon Snoek peak to a higher temperature and broadens it in the disordered state (quenched samples). Ageing of quenched samples with 19.6 at% Al shifts the peak to lower temperatures and makes it more narrow due to substitutional ordering of the DO 3 -type. The explanation of concentration dependence of internal friction and influence of ordering is done on the basis of computer simulation of energy distribution for carbon atoms in solid solution due to its interaction with Al atoms and changing of this distribution due to Al ordering. The model of the long-range strain-induced (elastic) C-Al interaction supplemented by the short-range chemical interaction is successfully used. The change of the calculated peak temperature and its width is in good agreement with experimental data. It is shown that the main factor which determines the effect of Al on the carbon Snoek peak in iron is the long-range elastic interaction.

A new package for accelerating large-scale phase-field simulations was developed by using GPU based on the semi-implicit Fourier method. The package can solve a variety of equilibrium equations with different inhomogeneity including long-range elastic, magnetostatic, and electrostatic interactions. Through using specific algorithm in Compute Unified Device Architecture (CUDA), Fourier spectral iterative perturbation method was integrated in GPU package. The Allen-Cahn equation, Cahn-Hilliard equation, and phase-field model with long-range interaction were solved based on the algorithm running on GPU respectively to test the performance of the package. From the comparison of the calculation results between the solver executed in single CPU and the one on GPU, it was found that the speed on GPU is enormously elevated to 50 times faster. The present study therefore contributes to the acceleration of large-scale phase-field simulations and provides guidance for experiments to design large-scale functional devices.

Chapter 1 - Introduction

Slip-discreteness-corrected strain gradient crystal plasticity (SDC-SGCP) theory