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In this paper, a pre-compressed one-dimensional granular crystal model is studied. The bright analytic single and multiple solitary wave solutions in more general forms than those obtained the KdV system in the previous studies are derived by using the homogeneous balance principle and Hirota’s bilinear method. The difference between the present solutions and those from the KdV system are investigated both analytically and numerically. By analyzing the dispersion relation and the collision process of solitary waves, we find that there are two types of double-solitary waves in the pre-compressed granular crystal model. The geometric and numerical analysis of dynamic behaviors of the solutions is presented with emphasis on the relation between the double-solitary waves and elastic collision between single-solitary waves. We find that the opposite collision between single-solitary waves may be stable and thus generate a stable double-solitary wave. It is concluded that the collision is a special stable double-solitary wave solution. We further propose a possible way to determine the stability of multiple solitary waves qualitatively. The results of this paper provide a theoretical basis for finding stable multiple solitary wave solutions.

The phase-field crystal equation is a sixth-order nonlinear parabolic equation and can be applied to simulate various phenomena such as epitaxial growth, material hardness, and phase transition. We propose a series of efficient modified stabilized invariant energy quadratization approaches with unconditional energy stability for the phase-field crystal model. Firstly, we propose a more suitable positive preserving function strictly in square root and consider a modified invariant energy quadratization (MIEQ) approach. Secondly, a series of efficient and suitable functionals H(ϕ) in square root are considered and the modified stabilized invariant energy quadratization (MSIEQ) approaches are developed. We prove the unconditional energy stability and optimal error estimates for the semi-discrete schemes carefully and rigorously. A comparative study of classical IEQ, MIEQ, and MSIEQ approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.

Shape instability belongs to one of significant types of violation for disposable structural elements under high-stress levels. Due to lack of fundamental data on materials, it is quite problematic to consider the shape instability in the design of disposable structural elements. The crystal plastic finite element method is proposed to investigate the dispersion of shape instability life data. It allows these data to be obtained from traditional material parameters. The shape instability behavior is described with the constitutive crystal model of plastic damage accumulation. Then, to improve the accuracy of life prediction, the new method is developed to construct the simulation model of true microstructure. A modeling algorithm based on the image processing technology is provided to reduce the virtual stresses from the transient crystal plastic modeling method. Comparison of experimental and predicted results shows good agreement at high stresses close to the elastic limit of the material.

Molecular dynamics simulation is employed to study the tension and compression deformation behaviors of magnesium single crystals with different orientations. The angle between the loading axis and the basal direction ranges from 0° to 90°. The simulation results show that the initial defects usually nucleate at free surfaces, but the initial plastic deformation and the subsequent microstructural evolutions are various due to different loading directions. The tension simulations exhibit the deformation mechanisms of twinning, slip, crystallographic reorientation and basal/prismatic transformation. The twinning, crystallographic reorientation and basal/prismatic transformation can only appear in the crystal model loaded along or near the a-axis or c-axis. For the compression simulations, the basal, prismatic and pyramidal slips are responsible for the initial plasticity, and no twinning is observed. Moreover, the plastic deformation models affect the yield strengths for the samples with different orientations. The maximum yield stresses for the samples loaded along the c-axis or a-axis are much higher than those loaded in other directions.

Phase-field crystal model with one mode is introduced to describe morphological transition. The relationship between growth morphology and smooth density distribution was investigated. The results indicate that the pattern selection of dendrite growth is caused by the competition between interface energy anisotropy and interface kinetic anisotropy based on the 2D phase diagram. When the calculation time increases, the crystal grows to secondary dendrite at the dimensionless undercooling equal to − 0.4. Moreover, when noise is introduced in the growth progress, the symmetry is broken in the growth mode, and there becomes irregular fractal-like growth morphology. Furthermore, the single crystal shape develops into polycrystalline when the noise amplitude is large enough. When the dimensionless undercooling is less than − 0.3, the noise has a significant effect on the growth shape. In addition, the growth velocity of crystal near to liquid phase line is slow, while the shape far away from the liquid adapts to fast growth. Based on the simulation results, the method was proved to be effective, and it can easily obtain different crystal shapes by choosing the different points in 2D phase diagram.

Crystallization of supersaturated liquids usually starts by epitaxial growth or by heterogeneous nucleation on foreign surfaces. Herein, we review recent advances made in modeling heteroepitaxy and heterogeneous nucleation on flat/modulated surfaces and nanoparticles within the framework of a simple dynamical density functional theory, known as the phase-field crystal model. It will be shown that the contact angle and the nucleation barrier are nonmonotonous functions of the lattice mismatch between the substrate and the crystalline phase. In continuous cooling studies for substrates with lattice mismatch, we recover qualitatively the Matthews–Blakeslee mechanism of stress release via the misfit dislocations. The simulations performed for particle-induced freezing will be confronted with recent analytical results, exploring thus the validity range of the latter. It will be demonstrated that time-dependent studies are essential, as investigations based on equilibrium properties often cannot identify the preferred nucleation pathways. Modeling of these phenomena is essential for designing materials on the basis of controlled nucleation and/or nano-patterning.

A lot of the physical and the numerical modeling of Czochralski crystal growth is done on the generic Rayleigh-B´enard system. To better approximate the conditions in a Czochralski puller, the influences of a rounded crucible bottom, deviations of the thermal boundary conditions from the generic case, crucible and/or crystal rotation, and the influence of magnetic fields are often studied separately. The present contribution reviews some of these topics while concentrating on studies of the flow and related temperature fluctuations in systems where a rotating magnetic field (RMF) was applied. The three-dimensional convective patterns and the resulting temperature fluctuations will be discussed both for the mere buoyant case and for the application of an RMF. It is shown that a system between a Rayleigh-Benard and a more realistic configuration, which is still cylindrical but whose surface is partially covered by a crystal model, behaves much the same as a Rayleigh-Benard system. An RMF can be used to damp the temperature fluctuations. Secondly, a more Czochralski-like system is examined. It turns out that the RMF does not provide the desired damping of the temperature fluctutions in the parameter range considered.

This paper re-examines our previous molecular dynamics (MD) study on cellulose IIII crystal models with finite dimensions solvated in explicit water molecules. Eight crystal models, differing in a constituent lattice plane and dimensions, were studied. One calculation allowed for O–H and C–H bond stretching, and had a small time step of 0.5 fs. The other calculation adopted non-scaling factors of the 1–4 non-bonded interactions. As in our previous study, in the former MD calculations, six of the eight crystal models exhibited structure conversion with cooperative chain slippages generated by a progressive fiber bend. This converted the initial non-staggered chain packing of cellulose IIII into a near one-quarter staggering and gave the crystal model a triclinic-like configuration. In contrast, in the non-1–4 scaling MD calculations, all of the eight crystal models retained the initial cellulose IIII crystal structure. Another series of non-1–4 scaling MD calculations were performed for the four crystal models containing chains with a degree of polymerization (DP) of 40 at 370 K, which simulated hot water treatment to convert cellulose IIII to Iβ. Some of the hydroxymethyl groups irreversibly rotated from gt into tg conformation. This accompanied exchange of the intrasheet hydrogen bonding scheme along the (1 −1 0) lattice plane from O2–O6 to O3–O6. The original corrugated (1 −1 0) chain sheet was partly converted into a cellulose I-like flat chain sheet.

The elastic modulus of natural cellulose crystal was estimated by the molecular simulation technique. Values between 124 and 155 GPa were derived for the reasonable cellulose Iβ crystal model that were nearly equal to the observed value of 138 GPa. While the second-generation force fields were found to be superior to the first-generation ones for the optimization of cellulose structure, neither of these was good enough to achieve the structural optimization. They were, however, adequate for estimating the mechanical properties of cellulose, especially the second-generation force fields. The lateral (that is, intermolecular) interactions between cellulose chains were found to play an important role in the expression of the mechanical properties of cellulose crystal.

The exact analytical solution for the plane S-wave, propagating along the axis of spirality in the simple 1-D anisotropic “simplified twisted crystal” model, is compared with four different approximate ray-theory solutions. The four different ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. The comparison is carried out numerically, by evaluating both the exact analytical solution and the analytical solutions of the equations of the four ray methods. The comparison simultaneously demonstrates the limits of applicability of the isotropic and anisotropic ray theories, and the superior accuracy of the coupling ray theory over a broad frequency range. The comparison also shows the possible inaccuracy due to the quasi-isotropic perturbation of travel times in the equations of the coupling ray theory. The coupling ray theory thus should definitely be preferred to the isotropic and anisotropic ray theories, but the quasi-isotropic perturbation of travel times should be avoided. Although the simplified twisted crystal model is designed for testing purposes and has no direct relation to geological structures, the wave-propagation phenomena important in the comparison are similar to those in the models of the geological structures.

The coupling ray theory bridges the gap between the isotropic and anisotropic ray theories, and is considerably more accurate than the anisotropic ray theory. The coupling ray theory is often approximated by various quasi-isotropic approximations.

Analytical expressions for the exact 2 × 2 one-way propagator matrix of a plane S wave, propagating along the axis of spirality in the simple 1-D anisotropic “simplified twisted crystal” model, are presented. The analytical equations are useful in testing the applicability and accuracy of various approximate wavefield modelling methods, especially of the coupling ray theory and of its various quasi-isotropic approximations and various numerical implementations. In addition to the exact analytical solution of the elastodynamic equation in the “simplified twisted crystal” model, the analytical solutions of the equations of the four ray methods are given. The ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. These four approximate solutions of the elastodynamic equation are roughly compared with the exact solution. Both the exact analytical solution and the analytical ray-theory solutions in the “simplified twisted crystal” model are also helpful in debugging computer codes for various approximate wavefield modelling methods, especially for the coupling ray theory.

A quenched disordered phase of solid nitrogen has been studied in the temperature region where a dynamical crossover is observed between nearly free rotations and activated tumbling motion. The orientational dynamics of the molecules is fast enough to be simulated using the Monte Carlo Metropolis algorithm and Molecular Dynamics. However, these methods become non-ergodic at very low temperature where formation of a glassy state could be observed. The crystal structure below the crossover temperature remains hcp with the molecular orientations frustrated due to the incompatibility of the strong quadrupolar interactions with the hexagonal lattice. For this reason, quenched nitrogen at low temperature seems to be a good candidate for a glassy crystal model.

For a model given previously by the authors describing a structural phase transition we compute theq-mode critical fluctuations of momentum and displacement as a function of the critical temperatures, the wave vectorq, and a fading-out external field. An explicit dependence on the rates of fading out is obtained. In order to define the critical fluctuation operators we prove a reconstruction theorem, which is of model-independent value. Finally we study the critical spectrum and get rigorous results on the soft modes and the central peak.

Large adiabatic polarons in anisotropic crystals in the presence of constant magnetic field have been studied within the Holstein molecular crystal model in the continuum approximation. It was shown that magnetic field directed along the symmetry axis induces transverse confinement which may stabilize large polarons. They represent localized (soliton-like) nonlinear structure uniformly propagating along the symmetry axis and rotating around it in the same time. Such objects exist in 3D lattice provided that coupling constant and magnetic field do not exceed certain critical values. In contrast with pure 1D systems existence of large polarons is possible in a quite wider region of the values of coupling constant which may attain considerably higher values than in the pure 1D media. Furthermore, polaron effective mass, depending on the intensity of the applied magnetic field, may be considerably lighter than that of the the pure 1D polarons for the same values of coupling constant. This is the most significant difference with respect to pure 1D systems in the absence of magnetic field and may have substantial impact on polaron transport properties.

We consider wall-crossing phenomena associated to the counting of D2-branes attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both from M-theory and matrix model perspective. Firstly, from M-theory viewpoint, we review that open BPS generating functions in various chambers are given by a restriction of the modulus square of the open topological string partition functions. Secondly, we show that these BPS generating functions can be identified with integrands of matrix models, which naturally arise in the free fermion formulation of corresponding crystal models. A parameter specifying a choice of an open BPS chamber has a natural, geometric interpretation in the crystal model. These results extend previously known relations between open topological string amplitudes and matrix models to include chamber dependence., 25 pages, 8 figures, published version

The topological string partition function for the neighbourhood of three spheres meeting at one point in a Calabi-Yau threefold, the so-called 'closed topological vertex', is shown to be reproduced by a simple Calabi-Yau crystal model which counts plane partitions inside a cube of finite size. The model is derived from the topological vertex formalism. This derivation can be understood as 'moving off the strip' in the terminology of hep-th/0410174, and offers a possibility to simplify topological vertex techniques to a broader class of Calabi-Yau geometries. To support this claim a flop transition of the closed topological vertex is considered and the partition function of the resulting geometry is computed in agreement with general expectations., Comment: 26 pages, 7 figures; references added, classical part of flop analysis corrected and expanded

Using a crystal model with an adsorbed layer, the propagation of vibrational excitation induced by an adsorbed particle is considered. The relaxation times of excitation have been estimated. The role of excitation transfer processes in adsorption-desorption phenomena is discussed.

With the help of a computer, the transformation curves of the lattice force, i.e., the degree of stretching, have been calculated at different values of compressive forces for the hexagonal model of a polymer crystal having a vacancy at the center. The configurations of the system of particles of this model in the positions of stable and unstable equilibrium have been determined.

Optical and photoemission measurements on Ag-Au and Cu-Au alloys over the whole composition range are reported. It is found that for these alloys, which represent the situation in many binary alloy systems, neither the commonly used virtual crystal model (and thus not that of the rigid band), nor the independent band model describes the observed results. Calculations in the recently developed coherent potential approximation however reproduce the main features of the observed density of states for the alloys.

Within a model of central and noncentral forces the exact change of the vibrational entropy, ΔS, is calculated, both as function of the degree of order and as a function of the diverse force constants. A linear relationship between ΔS and the short-range order parameter is found. Results of ΔS as a function of the force constants are compared with ΔS as calculated by an empirical formula based on a modified Einstein crystal model.

An x-ray diffraction method is described for determining thicknesses of thin films grown on single crystal substrates. The equations, based on the kinematical theory of x-ray diffraction and the mosaic crystal model, were developed. The thickness of the thin film was computed from the absorption of the integrated diffracted x-ray intensity from the single crystal substrate by the film. Since the diffracted intensity from the film is not required, the film does not have to be single crystal in nature. Thus, thicknesses of less ordered, polycrystalline or even amorphous films can be measured with high precision by this technique.