Travelling-wave amplitudes as solutions of the phase-field crystal equation

The hodograph equation for slow and fast anisotropic interface propagation

Using the model of fast phase transitions and previously reported equation of the Gibbs-Thomson-type, we develop an equation for the anisotropic interface motion of the Herring-Gibbs-Thomson-type. The derived equation takes the form of a hodograph equation and in its particular case describes motion by mean interface curvature, the relationship 'velocity-Gibbs free energy', Klein-Gordon and Born-Infeld equations related to the anisotropic propagation of various interfaces. Comparison of the present model predictions with the molecular-dynamics simulation data on nickel crystal growth (obtained by Jeffrey J. Hoyt et al. and published in Acta Mater. 47 (1999) 3181) confirms the validity of the derived hodograph equation as applicable to the slow and fast modes of interface propagation. This article is part of the theme issue 'Transport phenomena in complex systems (part 1)'.

Traveling waves of the solidification and melting of cubic crystal lattices

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

Reversible signal transmission in an active mechanical metamaterial

Mechanical metamaterials are designed to enable unique functionalities, but are typically limited by an initial energy state and require an independent energy input to function repeatedly. Our study introduces a theoretical active mechanical metamaterial that incorporates a biological reaction mechanism to overcome this key limitation of passive metamaterials. Our material allows for reversible mechanical signal transmission, where energy is reintroduced by the biologically motivated reaction mechanism. By analysing a coarse-grained continuous analogue of the discrete model, we find that signals can be propagated through the material by a travelling wave. Analysis of the continuum model provides the region of the parameter space that allows signal transmission, and reveals similarities with the well-known FitzHugh-Nagumo system. We also find explicit formulae that approximate the effect of the time scale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.

Thermodynamics of rapid solidification and crystal growth kinetics in glass-forming alloys

Thermodynamic driving forces and growth rates in rapid solidification are analysed. Taking into account the relaxation time of the solute diffusion flux in the model equations, the present theory uses, in a first case, the deviation from local chemical equilibrium, and ergodicity breaking. The second case of ergodicity breaking may exist in crystal growth kinetics of rapidly solidifying glass-forming metals and alloys. In this case, a theoretical analysis of dendritic solidification is given for congruently melting alloys in which chemical segregation does not occur. Within this theory, a deviation from thermodynamic equilibrium is introduced for high undercoolings via gradient flow relaxation of the phase field. A comparison of the present derivations with previously verified theoretical predictions and experimental data is given. This article is part of the theme issue 'Heterogeneous materials: metastable and non- ergodic internal structures'.

From atomistic interfaces to dendritic patterns

Transport processes around phase interfaces, together with thermodynamic properties and kinetic phenomena, control the formation of dendritic patterns. Using the thermodynamic and kinetic data of phase interfaces obtained on the atomic scale, one can analyse the formation of a single dendrite and the growth of a dendritic ensemble. This is the result of recent progress in theoretical methods and computational algorithms calculated using powerful computer clusters. Great benefits can be attained from the development of micro-, meso- and macro-levels of analysis when investigating the dynamics of interfaces, interpreting experimental data and designing the macrostructure of samples. The review and research articles in this theme issue cover the spectrum of scales (from nano- to macro-length scales) in order to exhibit recently developing trends in the theoretical analysis and computational modelling of dendrite pattern formation. Atomistic modelling, the flow effect on interface dynamics, the transition from diffusion-limited to thermally controlled growth existing at a considerable driving force, two-phase (mushy) layer formation, the growth of eutectic dendrites, the formation of a secondary dendritic network due to coalescence, computational methods, including boundary integral and phase-field methods, and experimental tests for theoretical models-all these themes are highlighted in the present issue.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay-Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.