Numerical simulation of a coupled nonlinear model for grain coarsening and coalescence

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Abstract

A kinetic nonlinear model of mass transfer, grain coarsening and coalescence with potential applications in sintering processes is studied. The model involves nonlinear ordinary differential equations that determine the transport of mass between grains. The rate of mass transfer is controlled by an Arrhenius factor leading to a nonlinear model of mass transfer and grain coarsening. The resulting dynamical system of coupled nonlinear differential equations with random initial conditions (i.e., initial grain mass configuration) is solved by means of the fourth order Runge–Kutta method. We conduct an analysis of the two-grain system and identify three dynamic regimes (diffusive, growth-decay and trapping). The same regimes are shown to persist in the multigrain system. We confirm the numerical performance of the Runge–Kutta method by means of a suitable convergence measure. The influence of the activation energy parameter on the dynamic regimes is investigated. It is shown that as the parameter grows the diffusive regime is progressively restricted to smaller values of the initial grain distribution. We introduce grain coalescence in the mass transfer equations, and we show that it accelerates the growth of the larger grains. Finally, we compare the dynamic evolution of the grain size distribution with the Ostwald ripening expression.

Introduction

Many technological materials, including ceramics, are produced by means of nonequilibrium physical processes that generate phase changes. A typical example is sintering, a process during which a powder is transformed into a monolithic material through atomic diffusion mechanisms. The modeling of sintering is a long-standing problem which remains poorly understood to date. Early studies of phase change kinetics have focused on interactions between grains that produce a crystal aggregate [1], [2]. Recent approaches use the Discrete Element Method (DEM) to relax assumptions regarding the particle kinematics [3], [4]. In DEM, grain coarsening is accounted by considering the overlapping volume of neighboring spherical particles, increasing the radius of the larger particle while decreasing the radius of the smaller by the same amount. By applying temperature and pressure the process of solid state sintering transforms a powder into a monolithic material [5]. Sintering involves diffusion and transport of atoms as well as plastic deformations. During the sintering process, the number of grains is reduced, whereas the average grain radius increases. Such process is known as Ostwald ripening [6], [7].

Modeling the sintering kinetics is a topic of continuing research. Recent computational techniques involve the Direct Multiscale Modeling [8], the DEM [3], [4] and generalized Monte Carlo simulations [9]. In DEM, grain coarsening is incorporated by transferring the overlapping volume of neighboring spherical particles from the smaller to the larger. Grain coarsening is also observed in diffusion processes such as the Cahn-Hilliard equation [10] that describes phase separation (reverse diffusion), and the phase-field models used in solidification [11]. Recently, a self-consistent, mean-field kinetic theory has been proposed to describe atomic diffusion in non uniform alloys [12].

The grain coarsening observed during sintering provides the motivation for this work. Our ultimate goal is to use the proposed model to predict the impact of key sintering parameters such as time and temperature as well as the powder grain size distribution and the activation energy on the final grain size distribution. Herein we investigate numerically a one-dimensional, nonlinear kinetic model of mass transfer, grain coarsening, and coalescence, which significantly extends the scope of the original model introduced in [13]. The model comprises a system of nonlinear ordinary differential equations (ODEs) and incorporates local fluctuations in mass transfer rates due to variations in the degree of grain amorphization. The initial conditions for the system are grain masses which are considered to be randomly distributed. The new contributions of this manuscript are briefly described below. We solve the initial value problem by means of the fourth-order Runge–Kutta method and compare the results with the explicit Euler scheme. We confirm the numerical performance of the two methods and the superiority of the Runge–Kutta method by studying suitable convergence measures. A detailed analysis of the two-grain system is presented, which leads to the identification of three dynamic regimes that include diffusion, growth-decay (reverse diffusion) and trapping. The first two regimes may be related, respectively, to the normal and abnormal growth regimes observed in [16]. Similar behavior is also observed in numerical solutions of the multigrain system. We also introduce a grain coalescence mechanism and study its impact on the evolution of the grain size distribution. Finally, we compare the dynamic evolution of the grain radii with the Ostwald ripening theoretical expression.

This manuscript is organized as follows. In Section 2 we present the nonlinear kinetic model of mass transfer between grains. The model comprises a system of nonlinear ordinary differential equations that allow the exchange of mass between grains. The grains are arranged along a line (open boundary conditions) or along a ring (periodic boundary condition). The initial values of the grain masses are random numbers drawn from the normal, lognormal or some empirical probability distribution. Section 3 is devoted to the analysis of the two-grain system, in particular a discussion of the different evolution regimes in the system. Numerical solution of the multigrain system using the Runge–Kutta method and comparison of the numerical results with the explicit Euler scheme is presented in Section 4. A computational algorithm involving grain coalescence technique is described in Section 5. Section 6 focuses on a numerical simulation of the multigrain system in which we permit coalescence of two neighboring grains based on their mass contrast. Numerical examples are illustrated. Here we also discuss about limitations and assumptions of the proposed techniques. Section 7 discusses the results obtained with the mass transfer model in relation to Ostwald ripening. Section 8 presents our conclusions and suggestions for further work based on the limitations of the current approach.

Section snippets

Nonlinear kinetic model

An ensemble of spherical grains is considered such that the surface of each grain (particle) is in contact with its nearest neighbors. We study a one-dimensional geometry in which N grains are located along a line, and each grain has two neighbors, one on each side, except for the two boundary grains (Fig. 1a). The latter also have two neighbors if periodic boundary conditions are assumed. The grains are indexed by the natural i(i=1,2,⋯,N).

We are interested in computations of grain masses at

Analysis of the two-grain system

In this section we analyze the case N=2, which will be helpful for investigating systems with N > 2. As we show, there are three different regimes (diffusive, growth-decay and trapping) that the system’s dynamical evolution can follow. Each regime corresponds to a distinct equilibrium state. We conduct numerical analysis of the two-grain system for different initial values of the grain masses. The latter are considered to take values at the nodes of a square lattice D=[0,4]×[0,4]. The two-grain

Numerical solution of the multigrain system

The kinetic model (1) with the initial conditions (2) is discretized using the fourth order Runge–Kutta method (RK4). An analogous scheme is obtained for the system of Eqs. (3) with the initial conditions (2). For the numerical implementation of the RK4 scheme we use the Matlab ordinary differential equation toolbox, and in particular the function (ode45). The statistical analysis of the resulting grain distributions is based on an ensemble of simulated states with initial configurations drawn

Grain coalescence model

In the preceding sections we investigated the problem of mass transfer between grains in a system that preserves the total number of grains. To make the model more realistic, we consider the possibility that grains coalesce if the mass of one grain significantly exceeds that of its neighbors. In this case the number of grains is progressively reduced, while the bigger grains tend to increase their mass at the expense of the smaller grains regardless of the value of u. We implement grain

Simulations of the multigrain system

In this section we use simulations to investigate the multigrain system (1), comprising N=10,000 grains, with initial conditions (2) for different values of u and δ. As a first step, we have verified that the numerical solution of the system conserves the total grain mass. We generate an initial configuration of grain masses from (i) the normal (Gaussian) distribution with mean μ=10 and standard deviation σ=2 and (ii) from the lognormal distribution with mean of the mass logarithm equal to 2.3

Comparison with Ostwald ripening

Ostwald ripening is a term used to refer to a thermodynamic process occurring in solutions that leads the growth of larger particles at the expense of smaller particles via the absorption of the smaller by the larger ones [6], [7]. The Ostwald ripening phenomenon is mathematically expressed as r(t)¯n−r(t0)¯n=kt,where r(t)¯ is given by (5), the exponent n≈2−3 and the rate constant k are experimentally determined parameters. The r(t)¯ denotes an average radius over the entire configuration at

Discussion and conclusions

We have investigated a system of coupled, nonlinear ordinary differential equations that describes mass transfer between grains of inhomogeneous sizes. This system of equations is motivated by potential application in sintering processes. A two-grain model was analyzed to determine the equilibrium points and to identify different dynamic regimes: a diffusive regime in which all the grains tend to the same mass at equilibrium, a growth-decay regime in which the bigger grains grow whereas the

Acknowledgments

This work has been funded by the project NAMCO: Development of High Performance Alumina Matrix Nanostructured Composites. NAMCO is implemented under the “THALIS” Action of the operational program “Education and Lifelong Learning” and is co-funded by the European Social Fund and National Resources. The authors would like also to thank Prof. Athena Tsetsekou (National Technical University of Athens) for helpful discussions.

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