Model Of Interface Growth Research Articles

Microstructure evolution during solidification in a low superheat casting process of the Al-Mg2Si composites having excess Si: A phase field study

A two-dimensional phase field (PF) model of solidification during low superheat casting has been developed to grab insight into the microstructure formation mechanism of the Al-xMg2Si-ySi composite, having extra Si as the grain refining agent. The developed PF model employs a seed undercooling based nucleation model to simulate the formation of primary Mg2Si and α-Al grains, wherein the interfacial energy of the Al-melt interface is taken from literature, and a Molecular Dynamics (MD) model is employed to calculate the interfacial energy of the Mg2Si-melt interface. The simulation study predicts the microstructural parameters such as grain size, and sphericity of the primary Al and primary Mg2Si phases, as well identifies the appropriate numerical parameters such as mobility, anisotropy parameters to study the microstructure evolution in the Al-xMg2Si-ySi composites. The kinetics of grain growth of both α-Al and primary Mg2Si have been studied. Novelty of the study lies in developing a near-accurate PF model capable of optimising the weight fraction of Mg2Si and excess Si in the low superheat cast Al-xMg2Si-ySi composite system, in view of obtaining the lower grain size and higher sphericity of both primary Al and primary Mg2Si grains, which is in line with the experimental observations. The proposed PF model is capable of predicting the faceted growth of Mg2Si particles in the Al-xMg2Si-ySi composite, in presence of extra Si and during low superheat casting, and the dendritic growth of Mg2Si in the binary Al-Mg2Si composite having high Mg2Si weight fraction. Following the Jackson’s model of interface growth, it has been demonstrated that the modification of kinetic coefficient leads to a transformation of morphology of Mg2Si particles from large dendritic to faceted ones.

Existence of stationary ballistic deposition on the infinite lattice

AbstractBallistic deposition is one of the many models of interface growth that are believed to be in the KPZ universality class, but have so far proved to be largely intractable mathematically. In this model, blocks of size one fall independently as Poisson processes at each site on the ‐dimensional lattice, and either attach themselves to the column growing at that site, or to the side of an adjacent column, whichever comes first. It is not hard to see that if we subtract off the height of the column at the origin from the heights of the other columns, the resulting interface process is Markovian. The main result of this article is that this Markov process has at least one invariant probability measure. We conjecture that the invariant measure is not unique, and provide some partial evidence.

Modeling NCA/C6-Si battery ageing

The rapidly growing demand for energy storage has spurred the development of advanced Li-ion batteries. Physical-based modeling techniques have emerged as a powerful tool to assist the development of battery technologies because of the advantages of relating battery performances to internal physical and (electro)chemical processes. In the present paper, a pseudo-two-dimensional (P2D) based model has been developed for Li-ion batteries with NCA cathodes and C6/Si blended anodes. Coupled with a solid electrolyte interface (SEI) growth model and a loss of active materials (LAM) model, the P2D-based model has been applied to simulate the electrochemical behavior of pristine cells and the degradation of aged cells. For pristine cells, a competing reaction occurs between C6 and Si during (dis)charge, resulting in a sequential (de)lithiation and diverse stress behavior. For aged cells, the SEI growth mainly causes the battery capacity to fade under different cycling conditions. LAM also plays an important role in battery capacity loss. SEI growth is sensitive to the temperature, cycle time, cycle number, and cycle current, while LAM is significantly influenced by temperature and current. During storage, the main factor causing battery capacity losses is the SEI growth, which is affected by the storage time, storage state-of-charge, and temperature.

Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation

The one-dimensional Kardar-Parisi-Zhang (KPZ) equationis becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening (despite previous expectations for systems with local interactions and time-dependent noise) and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation.

A Comparison of Interface Growth Models Applied to Rayleigh–Taylor and Richtmyer–Meshkov Instabilities

Abstract Sophisticated numerical models that contain fluid interfaces rely upon interface evolution models to approximate the transition to turbulence near interfaces, in the presence of Rayleigh–Taylor (RTI) or Richtmyer–Meshkov (RMI) instability. Semi-analytical models have been developed in recent decades to predict the interface growth from an initial state into the nonlinear regime. Two of these models are considered in this study. They are the Goncharov and the z-models. Both of these interface models have strengths and weaknesses, which are examined here. Both of them have been implemented in the xRAGE compressible flow solver, which models fluid interfaces. The flow solver provides the fluids acceleration as a body force to the interface model. The purpose of such interface model is to evolve the early times of interface position as a subgrid model within a compressible flow simulation in order to then initialize a turbulence model. In this work, the interface models are assessed and compared for their evolution of RTI and RMI. The z-model performed better than the Goncharov model for all cases that were explored.

Large Scale Stochastic Dynamics

The goal of this workshop was to explore the recent advances in the mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. There were 55 participants, including postdocs and graduate students, working in diverse intertwining areas of probability and statistical mechanics. During the meeting, 29 talks of 45 minutes were scheduled and an evening session was organised with 10 more short talks of 10 minutes, mostly by younger participants. These talks addressed the following topics: randomness emerging from deterministic dynamics, hydrodynamic limits, interface growth models and slow convergence to equilibrium in kinetically constrained dynamics.

Bonding mechanism of Al/steel interface formed by laser-TIG welding assisted riveting technology

Laser-tungsten inert gas (TIG) welding assisted riveting technology was used to join Q460 high strength steel and AA6061-T651 aluminum alloy. The effect of laser power on the mechanical properties of Al/steel joints was studied. In this study, the welding-riveting hybrid structure was used to weaken the adverse effect of thermal-physical properties differences between aluminum alloy and steel, and the distribution of Fe-Al intermetallic compounds was optimized by adjusting laser power to obtain the joint with the best mechanical properties. Due to the different laser power, there were two fracture modes of Al/steel joints, namely, button-type fracture and pull-out fracture. The failure processes of these two fracture modes were discussed. Under the appropriate welding parameters, the maximum tensile shear load of the joint could reach to 6.3 kN, which was 92.2 % of the Q460 base metal. The Al/steel interface of the joint was mainly composed of Al, Fe4Al13, Fe2Al5, and α-Fe phases. When the favorable orientation relationships of the Fe4Al13/Al system, Fe4Al13/Fe2Al5 system, and Fe2Al5/α-Fe system in the Al/steel interface were selected, there was a semi-coherent-coherent-coherent matching relationship between these phases. The Al/steel interface growth model of the Al/steel joint was discussed in this study.

Effects of a kinetic barrier on limited-mobility interface growth models.

The role played by a kinetic barrier originated by out-of-plane step edge diffusion, introduced by Leal etal. [J. Phys.: Condens. Matter 23, 292201 (2011)JCOMEL0953-898410.1088/0953-8984/23/29/292201], is investigated in the Wolf-Villain and Das Sarma-Tamborenea models with short-range diffusion. Using large-scale simulations, we observe that this barrier is sufficient to produce growth instability, forming quasiregular mounds in one and two dimensions. The characteristic surface length saturates quickly indicating a uncorrelated growth of the three-dimensional structures, which is also confirmed by a growth exponent β=1/2. The out-of-plane particle current shows a large reduction of the downward flux in the presence of the kinetic barrier enhancing, consequently, the net upward diffusion and the formation of three-dimensional self-assembled structures.

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension.

We report local roughness exponents, α_{loc}, for three interface growth models in one dimension which are believed to belong to the nonlinear molecular-beam-epitaxy (nMBE) universality class represented by the Villain-Lais-Das Sarma (VLDS) stochastic equation. We applied an optimum detrended fluctuation analysis (ODFA) [Luis et al., Phys. Rev. E 95, 042801 (2017)2470-004510.1103/PhysRevE.95.042801] and compared the outcomes with standard detrending methods. We observe in all investigated models that ODFA outperforms the standard methods providing exponents in the narrow interval α_{loc}^{}∈[0.96,0.98] quantitatively consistent with two-loop renormalization group predictions for the VLDS equation. In particular, these exponent values are calculated for the Clarke-Vvdensky and Das Sarma-Tamborenea models characterized by very strong corrections to the scaling, for which large deviations of these values had been reported. Our results strongly support the absence of anomalous scaling in the nMBE universality class and the existence of corrections in the form α_{loc}^{}=1-ε of the one-loop renormalization group analysis of the VLDS equation.

Hydrodynamic Limit and Viscosity Solutions for a Two‐Dimensional Growth Process in the Anisotropic KPZ Class

AbstractWe study a two‐dimensional stochastic interface growth model that is believed to belong to the so‐called anisotropic KPZ (AKPZ) universality class [4,5]. It can be seen either as a two‐dimensional interacting particle process with drift that generalizes the one‐dimensional Hammersley process [1,24], or as an irreversible dynamics of lozenge tilings of the plane [5,29]. Our main result is a hydrodynamic limit: the interface height profile converges, after a hyperbolic scaling of space and time, to the solution of a nonlinear first‐order PDE of Hamilton‐Jacobi type with nonconvex Hamiltonian (nonconvexity of the Hamiltonian is a distinguishing feature of the AKPZ class). We prove the result in two situations: (1) for smooth initial profiles and times smaller than the time Tshock when singularities (shocks) appear or (2) for all times, including t > Tshock, if the initial profile is convex. In the latter case, the height profile converges to the viscosity solution of the PDE. As an important ingredient, we introduce a Harris‐type graphical construction for the process. © 2018 Wiley Periodicals, Inc.

Two-dimensional anisotropic KPZ growth and limit shapes

A series of recent works focused on two-dimensional (2D) interface growth models in the so-called anisotropic KPZ (AKPZ) universality class, that have a large-scale behavior similar to that of the Edwards–Wilkinson equation. In agreement with the scenario conjectured by Wolf (1991 Phys. Rev. Lett. 67 1783–6), in all known AKPZ examples the function giving the growth velocity as a function of the slope ρ has a Hessian with negative determinant (‘AKPZ signature’). While up to now negativity was verified model by model via explicit computations, in this work we show that it actually has a simple geometric origin in the fact that the hydrodynamic PDEs associated to these non-equilibrium growth models preserves the Euler–Lagrange equations determining the macroscopic shapes of certain equilibrium 2D interface models. In the case of growth processes defined via dynamics of dimer models on planar lattices, we further prove that the preservation of the Euler–Lagrange equations is equivalent to harmonicity of with respect to a natural complex structure.

A CLASS OF GROWTH MODELS RESCALING TO KPZ

We consider a large class of$1+1$-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf–Cole solutions to the KPZ equation.

From dynamical scaling to local scale-invariance: a tutorial

Dynamical scaling arises naturally in various many-body systems far from equilibrium. After a short historical overview, the elements of possible extensions of dynamical scaling to a local scale-invariance will be introduced. Schr\"odinger-invariance, the most simple example of local scale-invariance, will be introduced as a dynamical symmetry in the Edwards-Wilkinson universality class of interface growth. The Lie algebra construction, its representations and the Bargman superselection rules will be combined with non-equilibrium Janssen-de Dominicis field-theory to produce explicit predictions for responses and correlators, which can be compared to the results of explicit model studies. At the next level, the study of non-stationary states requires to go over, from Schr\"odinger-invariance, to ageing-invariance. The ageing algebra admits new representations, which acts as dynamical symmetries on more general equations, and imply that each non-equilibrium scaling operator is characterised by two distinct, independent scaling dimensions. Tests of ageing-invariance are described, in the Glauber-Ising and spherical models of a phase-ordering ferromagnet and the Arcetri model of interface growth.

Noise-driven interfaces and their macroscopic representation.

We study the macroscopic representation of noise-driven interfaces in stochastic interface growth models in (1+1) dimensions. The interface is characterized macroscopically by saturation, which represents the fluctuating sharp interface by a smoothly varying phase field with values between 0 and 1. We determine the one-point interface height statistics for the Edwards-Wilkinson (EW) and Kadar-Paris-Zhang (KPZ) models in order to determine explicit deterministic equations for the phase saturation for each of them. While we obtain exact results for the EW model, we develop a Gaussian closure approximation for the KPZ model. We identify an interface compression term, which is related to mass transfer perpendicular to the growth direction, and a diffusion term that tends to increase the interface width. The interface compression rate depends on the mesoscopic mass transfer process along the interface and in this sense provides a relation between meso- and macroscopic interface dynamics. These results shed light on the relation between mesoscale and macroscale interface models, and provide a systematic framework for the upscaling of stochastic interface dynamics.

Bulk characterization in a Monte Carlo particle-deposition model with a novel adherence-potential barrier

The aim of this work is to analyze in more depth a model of particle deposition by characterizing different parameters such as profile density, bonds and perimeter, and substrate coverage, all being involved in the description of deposits as bulk. Thus, this study is an extension of a previous work on non-equilibrium interface-growth systems where two different interface-growth models, called Standard Adherence Rule Model and Potential Adherence Rule Model, were characterized. In this work, bulk characterization is implemented for the complete range of Peclet numbers. The zones of density profile (Near-Wall, Plateau, and Active-Growth) are studied by proposing an adjustment for each of them and determining the full-setting density profile depending on the Peclet number. The density profiles are compared with other one- and two-stage models. Furthermore, an algorithm is proposed to calculate the number of bonds of the particles and the perimeter that a substrate forms over time. Finally, to analyze the coating, its temporal behavior is adjusted to an exponential function by comparing the results with those found for Random Sequential Adsorption models which describe systems like colloidal particles on solid substrates, adsorption of proteins at mineral surfaces, or oxidation of one-dimensional polymer chains.

Stochastic Heat Equation Limit of a (2 + 1)d Growth Model

We determine a $q\to 1$ limit of the two-dimensional $q$-Whittaker driven particle system on the torus studied previously in [Corwin-Toninelli, arXiv:1509.01605]. This has an interpretation as a $(2+1)$-dimensional stochastic interface growth model, that is believed to belong to the so-called anisotropic Kardar-Parisi-Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the $(2+1)$-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in $(2+1)$-dimension is irrelevant.

Critical dynamics of the jamming transition in one-dimensional nonequilibrium lattice-gas models.

We consider several one-dimensional driven lattice-gas models that show a phase transition in the stationary state between a high-density fluid phase in which the typical length of a hole cluster is of order unity and a low-density jammed phase where a hole cluster of macroscopic length forms in front of a particle. Using a hydrodynamic equation for an interface growth model obtained from the driven lattice-gas models of interest here, we find that in the fluid phase, the roughness exponent and the dynamic exponent that, respectively, characterize the scaling of the saturation width and the relaxation time of the interface with the system size are given by the Kardar-Parisi-Zhang exponents. However, at the critical point, we show analytically that when the equal-time density-density correlation function decays slower than inverse distance, the roughness exponent varies continuously with a parameter in the hop rates, but it is one-half otherwise. Using these results and numerical simulations for the density-density autocorrelation function, we further find that the dynamic exponent z=3/2 in all cases.

Morphology transition at depinning in a solvable model of interface growth in a random medium

We propose a simple, exactly solvable, model of interface growth in a random medium that is a variant of the zero-temperature random-field Ising model on the Cayley tree. This model is shown to have a phase diagram (critical depinning field vs. disorder strength) qualitatively similar to that obtained numerically on the cubic lattice. We then introduce a specifically tailored random graph that allows an exact asymptotic analysis of the height and width of the interface. We characterize the change of morphology of the interface as a function of the disorder strength, a change that is found to take place at a multicritical point along the depinning-transition line.

Non-local representations of the ageing algebra in higher dimensions

The ageing Lie algebra and especially its local representations for a dynamical exponent z = 2 have played an important role in the description of systems undergoing simple ageing, after a quench from a disordered state to the low-temperature phase. Here, the construction of representations of for generic values of z is described for any space dimension d > 1, generalizing upon earlier results for d = 1. The mechanism for the closure of the Lie algebra is explained. The Lie algebra generators contain higher order differential operators or the Riesz fractional derivative. Co-variant two-time response functions are derived. Some simple applications to the exactly solvable models of phase separation or interface growth with conserved dynamics are discussed.

Water Assisted Growth of C60 Rods and Tubes by Liquid–Liquid Interfacial Precipitation Method

C60 nanorods with hexagonal cross sections are grown using a static liquid–liquid interfacial precipitation method in a system of C60/m-dichlorobenzene solution and ethanol. Adding water to the ethanol phase leads instead to C60 tubes where both length and diameter of the C60 tubes can be controlled by the water content in the ethanol. Based on our observations we find that the diameter of the rods/tubes strongly depends on the nucleation step. We propose a liquid-liquid interface growth model of C60 rods and tubes based on the diffusion rate of the good C60 containing solvent into the poor solvent as well as on the size of the crystal seeds formed at the interface between the two solvents. The grown rods and tubes exhibit a hexagonal solvate crystal structure with m-dichlorobenzene solvent molecules incorporated into the crystal structure, independent of the water content. An annealing step at 200 °C at a pressure <1 kPa transforms the grown structures into a solvent-free face centered cubic structure. Both the hexagonal and the face centered cubic structures are very stable and neither morphology nor structure shows any signs of degradation after three months of storage.