非自发形核功与润湿角的数学模型

根据两个垂直相交平面基体上非自发形核功与润湿角数学模型,对临界形核半径和临界形核功进行了详细的推导,表明了临界形核功与润湿角的关系,完善了两个垂直相交平面基体上形核功与润湿角数学模型的研究.     

Marangoni效应对等曲率弯曲界面凝固过程的影响

用渐近方法和相场数值模拟分别研究了球状晶体和柱状晶体凝固过程中,固液界面表面张力随温度的变化对界面运动的影响．结果表明Marangoni效应将增大临界晶核半径,减缓界面运动速度．在静止熔体中,枝晶尖端生长速度随Marangoni数线性下降．渐近展开和相场模拟得到的结果定性是一致的．     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

Application of the string method to compute minimum energy paths for a chain of bi-stable elements using the finite element method and molecular dynamics

Activated processes are frequently found in solid state mechanics. The energy landscape of such processes show a non-convex behaviour, and therefore the computation of energy barriers between two stable minima is of importance. Such barriers are revealed by computing minimum energy paths. The string method is a simple and efficient algorithm to move curves over an energy landscape and to identify minimum energy paths. A hierarchical two-scale model recently introduced to the literature (molecular dynamics coupled with the finite element method) is used in this paper to investigate the string method in a model phase transition in a copper single crystal. To do so, bi-stable elements are constructed and the energetic behaviour of a two-elements chain is investigated. We identify successfully the minimum energy path between two local stable minima of the chain and demonstrate thereby the performance of the string method applied to a complex multiscale model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The morphological stability of dendritic growth from the binary alloy melt with an external flow

The morphological stability of dendritic growth from the binary alloy melt with an external flow is studied by means of the matched asymptotic expansion method and multiple variable expansion method. The uniformly valid asymptotic solution is obtained for the case of the large Schmidt number. The analytical result reveals that the stability of dendritic growth depends on a critical stability number above which dendritic growth is stable. The selection condition of dendritic growth determines the Peclet number, tip growth velocity, tip radius and oscillation frequency, which is significantly affected by the external flow. The stability mechanism of dendritic growth in the binary alloy melt with the external flow remains the same as that in pure melt. In the binary alloy melt with the external flow the solute concentration destabilizes the dendritic growth system. The numerical computation for various growth conditions demonstrates the variations of the critical stability number, tip growth velocity, tip radius, and oscillatory frequency with the undercooling, external flow and morphological number.     

A Practical Protocol for Advantage Distillation and Information Reconciliation

Information-theoretic secret key agreement generally consists of three phases, namely, advantage distillation information reconciliation and privacy amplification. Advantage distillation is needed in the case when two legitimate users, Alice and Bob, start in a situation which is inferior to that of the adversary Eve. The aim for them is to gain advantage over Eve in terms of mutual information between each other. Information reconciliation enables Alice and Bob to arrive at a common string by error correction techniques. Finally they distill a highly secret string from the common string in the privacy amplification phase. For the scenario where Alice and Bob as well as Eve have access to the output of a binary symmetric source by means of (three) binary symmetric channels, there are several advantage distillation and information reconciliation protocols proposed.In this paper, we present a general protocol to implement both advantage distillation and information reconciliation. Simulation results are compared with known protocols. A connection between our protocol and the known protocols is given.     

Mono- and multimolecular nucleation in polymer crystallization

General relations for the nucleation rates are obtained with allowance for the contribution of the change of free energy in nucleation by two mechanisms — monomolecular, with the formation of folded nuclei, and multimolecular, i.e., nucleation with straightened polymer chains. It is shown that for flexible coiled molecules the formation of folded nuclei is more probable. However, as the degree of coiling decreases, whatever the reasons for the uncoiling of the macromolecules, the relative probability of formation of multimolecular nuclei increases, and at a certain critical value of the chain-folding parameter =h/L the situation is reversed.Institute of High-Molecular-Weight Compounds, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 351–353, March–April, 1974.     

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

We consider a diffuse interface model describing flow and phase separation of a binary isothermal mixture of (partially) immiscible viscous incompressible Newtonian fluids having different densities. The model is the nonlocal version of the one derived by Abels, Garcke and Grün and consists in a inhomogeneous Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. This model was already analyzed in a paper by the same author, for the case of singular potential and non-degenerate mobility. Here, we address the physically more relevant situation of degenerate mobility and we prove existence of global weak solutions satisfying an energy inequality. The proof relies on a regularization technique based on a careful approximation of the singular potential. Existence and regularity of the pressure field is also discussed. Moreover, in two dimensions and for slightly more regular solutions, we establish the validity of the energy identity. We point out that in none of the existing contributions dealing with the original (local) Abels, Garcke Grün model, an energy identity in two dimensions is derived (only existence of weak solutions has been proven so far).     

Phase Field Approach to Multiphase Flow Modeling

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals?? theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick??s law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation.     

Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.

     

Differences in microstructure of Pd77.5Au6Si16.5 alloy solidified under microgravity and gravity conditions

The microstructure of Pd_77.5Au₆Si_16.5alloy solidified both on board a Chinese Retrievable Satellite and on the earth is studied. Postmortem analyses of microstructure presented that the same types of phases, primary phase (Pd₃Si) and eutectics (Pd₃Si + Pd solid solution) were formed in both cases. But the phase morphologies were quite different. It was dendritic for the primary phase and lamellar for the eutectics under normal gravity condition. However, under microgravity condition the primary phase was granular and the eutectic was peculiar network. Detailed analysis showed that the differences in morphologies of the microstructure were due to the existence of gravity-induced buoyancy convection on the earth which increased the mass transport abilities and decreased the thickness of the solute boundary in front of the solid-liquid interface during solidification under normal gravity condition.     

The Singular Limit Problem to the Extended Cahn–Hillard Equation

A mathematical model with a small parameter, which describes the hardening process of the binary tin–lead alloy, is investigated on the basis of nonlinear asymptotic analysis. A singular limit problem, namely an extended Stefan problem in the case of short relaxation time in the phase transformation zone, is derived. We prove the existence of an asymptotic solution with any accuracy on the time interval where the solution to the singular limit problem exists. The phase-separation interface is determined uniquely by three leading approximations. We also show that the stability of the separation interface depends on the so-called dissipation condition obtained for the solutions of the interface problem. Nonsymmetry of the surface tension tensor leads to a situation where the limit values of concentration distributions are in dependence on the geometry of the interface. This provokes the dispersion of the interface problem solutions on the part of the interface that not is tangent to the main crystallographic axis.     

The vibrating string forced by white noise

Summary The equation of the vibrating string forced by white noise is formally solved, using stochastic integrals with respect to a plane Brownian motion, and it is proved that a certain process associated to the energy is a martingale. Then Doob's martingale inequality is used to furnish some probability bounds for the energy.Such bounds provide a solution for the double barrier problem for the class of Gaussian stationary processes which can be represented as linear functionals of the positions and the velocities of the string.     

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

Summary. A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly. Received September 28, 1995 / Revised version received May 6, 1996     

Fast parameterized matching with q-grams

Two strings parameterize match if there is a bijection defined on the alphabet that transforms the first string character by character into the second string. The problem of finding all parameterized matches of a pattern in a text has been studied in both one and two dimensions but the research has been centered on developing algorithms with good worst-case performance. We present algorithms that solve this problem in sublinear time on average for moderately repetitive patterns.     

Thermohaline convection with cross-diffusion in an anisotropic porous medium

Using normal mode technique it has been shown that (i) values of the anisotropy parameter are important in deciding the mode of convection in a doubly diffusive fluid saturating a porous medium. (ii) Depending on the values of the Soret and Dufour parameters, an increase in anisotropy parameter either promotes or inhibits instability, (iii) cross-diffusion induces instability even in a potentially stable set-up and (iv) for certain values of the Dufour and Soret parameters there is a discontinuity in the critical thermal Rayleigh number, which disappears if the porous medium has horizontal isotropy.     

Short chaotic strings and their behaviour in the scaling region

Coupled map lattices are a paradigm of higher-dimensional dynamical systems exhibiting spatio-temporal chaos. A special case of non-hyperbolic maps are one-dimensional map lattices of coupled Chebyshev maps with periodic boundary conditions, called chaotic strings. In this short note we show that the fine structure of the self energy of this chaotic string in the scaling region (i.e. for very small coupling) is retained if we reduce the length of the string to three lattice points.     

Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one, show that the nucleation time divided by its average converges to an exponential random variable, express the proportionality constant for the average nucleation time in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion.     

Computing mountain passes and transition states

The mountain-pass theorem guarantees the existence of a critical point on a path that connects two points separated by a sufficiently high barrier. We propose the elastic string algorithm for computing mountain passes in finite-dimensional problems and analyze the convergence properties and numerical performance of this algorithm for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems. We show that any limit point of the elastic string algorithm is a path that crosses a critical point at which the Hessian matrix is not positive definite.This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract W-31-109-Eng-38.