A Mathematical Approach to Ostwald Ripening Due to Diffusion and Deformation in Liquid Bridge

From many experiments with mixtures of small and large grains, it can be concluded that during liquid phase sintering, smaller grains partially dissolve and a solid phase precipitates on the larger grains and grain coarsening occurs. The growth rate can be controlled either by the solid-liquid phase boundary reaction or by diffusion through the liquid phase. The microstructure may change either by larger grains growing during the Ostwald ripening process or by shape accommodation. In this study, two-dimensional mathematical approach for simulation of grain coarsening by grain boundary migration based on a physical and corresponding numerical modeling of liquid phase sintering will be considered. A combined mathematical method of analyzing viscous deformation and solute diffusion in liquid bridge between two grains with different sizes will be proposed. The viscous FE method will be used for calculating meniscus of the liquid bridge, with the interfacial tensions taken into consideration. The FE method for diffusion will be also implemented by using the same mesh as the deformation analysis.


Introduction
Numerous researchers have studied liquid phase sintering (LPS) during the past few decades, beginning with Lenel [1].In particular, there are a large number of theoretical and experimental studies of grain growth.LPS is an important process for the production of many ceramic materials.One important aspect of its application is that it enhances densification and affects microstructural development.The main characteristic of this process is that the composition of the powder and the firing temperature must be chosen such that a small amount of liquid forms between the grains.Accordingly, the powder compact must satisfy three general requirements: there is a liquid phase at the sintering temperature, the solid phase is soluble in the liquid, and the liquid wets the solid.This process is especially important for systems that are difficult to densify by solid state sintering or when the use of solid state sintering requires high sintering temperatures.Unfortunately, the liquid phase used to promote sintering in most cases remains as a glassy grain boundary phase that may lead to a deterioration of materials' properties.
From many experiments with mixtures of small and large single particles annealed in the presence of liquid phase, it was concluded that shrinkage is directly linked to grain growth.During LPS, small particles partially dissolved and the solid phase precipitated on the large solid particles.Due to the reprecipitation process that large particles form polyhedral shapes.Simultaneously, the number of small particles decreases also due to coarsening.The dissolution of small particles leads to further densification by rearrangement of small and large particles.The initial system geometry may change either by large particles growing during the Ostwald ripening process or by shape accommodation.
A particularly interesting approach which leads better understanding of LPS phenomena is the application of numerical procedures, because they have great flexibility and can be used to obtain solutions for any model system configuration.In recent years, a range of computer simulation models have been developed with the aim of simulating the detailed evolution of microstructure during grain growth.
Voorhees et al. [2] employed a boundary integral technique to determine the morphological evolution of a small number of particles during Ostwald ripening in two dimensions.The approach allows the bodies to change shape consistent with interparticle diffusional interactions and the interfacial concentrations as given by the Gibbs-Thomson equation.Through particle arrangements similar to those found in solid-liquid systems during LPS, it is shown that the formation of regions of flat interface between particles is completely consistent with an Ostwald ripening mechanism.Recently the results of a computer simulation of boundary migration during LPS have been reported [3].The grain boundary migration means that solid atoms that are dissolved on one side of the boundary transport across the liquid layer and deposit on the other side of the boundary.
A particularly interesting approach for investigation of diffusion phenomena during LPS would be also the application of numerical procedures for the determination the equilibrium distributions of liquid in different packing arrangements of particles of the same or different radius [4].
Shinagawa et al. [5,6] proposed microscopic modeling for viscoplastic finite element (FE) analysis of sintering processes.Taking into account the surface tension acting on the pore surface, and assuming that the grains during sintering are viscoplastic and the flow stress is proportional to the viscosity, they calculated deformation behaviors of the grains for slightly compressible materials.This method can be also applied for definition twodimensional FE mesh within (pendular) liquid bridge as a function of grain size, the contact angle, the normalized liquid volume, and the separation distance and applied for computer simulation of Ostwald ripening characterized by small amount of liquid located within liquid bridges only.Recent advances in modeling grain coarsening using FE mesh approach was reported by Nikolic and Shinagawa [7] and Shinagawa and Nikolic [8].To the best of our knowledge, there was no computer simulation study of grain coarsening within liquid bridge only before.
This paper describes two-dimensional (2-D) mathematical approach that has been developed for simulation and determination of a qualitative and a quantitative effect of a moving grain boundary on the solid-liquid interface during LPS in which a combined method of analyzing viscous deformation and solute diffusion in liquid bridge between two grains with different sizes is proposed.The viscous FE method will be used for calculating meniscus of the liquid bridge, with the interfacial tensions taken into consideration.The FE method for diffusion will be also implemented by using the same mesh as the deformation analysis.Simulation of the grain coarsening in a liquid bridge of W-Ni alloy will be demonstrated as a first step.

Model topology
It is convenient to use multi-grain models of regular shape because they need to store only the position, orientation and size of each grain.Taking into account that so called primary rearrangement involves the individual grains and liquid, which forms at additive grain sites causes grain repacking, a microstructure (model system) consisting of N spherical grains connected with liquid bridges will be used.Solid grains can be represented by threedimensional (3-D) domains of regular shape, i.e. )] sin( 1 [ sin and 1 ϕ and 2 ϕ are the angles subtended by the contact at the center of the grains related by The angles 1 ϕ and 2 ϕ can be computed by iterative procedure [12].

Process modeling 3.1 Modeling solid-liquid interface concentration profile
The main characteristic of the solution-precipitation process is that the smaller solid grains dissolve at solid-liquid interface (thermodynamically unstable), diffuse through the liquid, and precipitate on the larger grains.
Let us assume a system consisting of a dispersion of spherical grains with different radii in a liquid in which the solid phase has some solubility.Thus, the concentration of the dissolved solid, c, around a grain of radius R is given by the Gibbs-Thomson equation where o c is the equilibrium concentration of liquid in contact with the flat solid, SL γ is the solid-liquid interfacial energy, Ω is the molecular volume of the solid, and kT has its usual This equation is not valid for a very small grain because c Δ becomes infinite as its radius goes to zero.However, the number of the small grains at any simulation time is sufficiently small so that equation ( 3) can be assumed to be valid for all grains.Notice that during sintering most of the grains are no longer circular because the diffusion field between grains within liquid bridge becomes highly asymmetric.Thus, the grain model system defined by ( 1) should be now represented in 2-D by contours (closed boundaries) of solid phase as } ,..., The most important numerical consideration in performing an accurate computation is the determination of the curvature of noncircular contour.In this sense, the better way would be to calculate the local curvature numerically, for example, using the interpolation functions at each boundary point separately.If one needs a smooth function for describing the contourboundaries, then a cubic polynomial could be the simplest function of this type, as recognized by Saetre and Ryum [13] and applied by Cocks and Gill [14].In our simulation method the curvature at each point on the contour l G will be computed by fitting a quadratic polynomial to the point and its two neighbors.Notice that a sharp curvature on contour-boundary requires a very fine mesh too.
It can be seen from ( 5) that the concentration at an interface SL I with high curvature will be above that at an interface with low curvature, thus a higher concentration around a smaller grain gives rise to a net flux of matter from the smaller to the larger one.If L D is the concentration independent diffusivity of the solid in the liquid, then the flux vector is Thus, the effects of the dissolution and precipitation processes can be computed by this equation and by applying equation ( 5) at boundary points of solid-liquid interfaces, SL I .
This process is accompanied by considerable coarsening and by changes in the shape of particles.The sizes of the grains and their locations change as simulation time increases: smaller grains dissolve and the dissolving material deposits on the large grains in such a way that grain shape accommodation occurs.

Modeling Ostwald ripening
If the solid grains ) ,..., 2 , 1 ( N G = l l are dispersed in the liquid phase, the grain coarsening is called Ostwald ripening process.If the solid phase forms a dense polyhedral grain structure, grain growth is due by grain boundary migration that is characterized by dissolution of smaller grains in the liquid, by its transportation through the liquid, and by precipitation on the larger ones. Grain coarsening, defined by the transformation (4), is a typical multibody free boundary problem in which the domains alter their morphologies in response to the diffusion field.After solution-precipitation, the grains grow in supersaturated liquid phase, and after the supersaturation becomes small, large grains start to grow at the expense of small grains.This tendency for grains to grow or to shrink depends on the size of grains relative to a critical grain size (zero-growth, i.e., the radius of critical grain for which 0 = dt dR ), c R [15].If the diffusion of atoms between the grains is the slowest and thus rate determining step then diffusion controlled growth occurs and , where 〉 〈R is the average grain size.When the deposition or dissolution of atoms at the liquid-solid interface is the slowest then reaction controlled growth occurs and In that sense, the theoretical basis for mathematical modeling of this process is the assumption that at any given moment the contours smaller than a critical contour size will dissolve, surrounding themselves with a zone of excess solute that will migrate to the contours larger than c R , and these therefore grow.An average concentration c c can be defined, which is in equilibrium with grain of size c R , that neither grow nor shrink.Notice that the critical grain size is also time-dependent, ) ( c t R .

Initial liquid bridge concentration
During solution-precipitation process smaller grains dissolve at solid-liquid interface, diffuse through the liquid, and precipitate on the larger grains.These processes are followed up by grain coarsening on both smaller and larger grains.
The initial liquid phase concentration within liquid bridge can be taken as the concentration of pure liquid with no dissolved solid.However, because the dissolution process starts very quickly after the additive melts, the same results can be obtained with equilibrium liquid concentration or even with minimal liquid/solid interface concentration as the initial concentration [16].
In general, three initial concentration profiles can be defined by three submodels shown in Fig. 2 [15]: model for pure surface control (M1), model for fast diffusion or large grain spacing (M2) and model for slow diffusion or small grain spacing (M3), where

Simulation method 4.1 FE mesh generation
The geometry of the FE mesh for the liquid bridge is built by deforming the initial mesh of a bridge model with flat liquid-vapor surface, ) 0 ( LV = t I , as shown in Fig. 3, in the viscous FE simulation itself, with the forces due to surface tension LV γ imposed at the nodes on the liquid surface.Especially, to fulfill a wetting angle θ in equilibrium state, the force in radial direction due to the difference in angles between the current and the ideal equilibrium states, , is exerted at the end-point node of solid-liquid interface, as shown in Fig. 4, where x is the distance between the node and the symmetry axis, β is the current angle of liquid surface, and α is the angle of solid surface.To keep circular solid-liquid interfaces during the deformation, boundary conditions of velocity are introduced for the nodes on the solid-liquid interfaces as follows: (i) the velocities in axial direction are equal, (ii) the movements are along the solid surface.Finally, to accommodate the bridge geometry, a tensile axial force is imposed to the central nodes on one interface, while the central node on the other interface is fixed in space.Under these conditions, the step-wise calculation in the viscous FE simulation is performed until the distance between grains become a prescribed value.Therefore, FE mesh calculation can be now described by the transformation Note that appropriate values should be found for the initial dimensions, o h and o R in Fig. 3.
In the present study,

FE method and flux computation
In FE analysis the 2-D concentration of liquid phase inside an element within liquid bridge can be interpolated from data i c at four nodal points that define this element, i.e.
is the shape function [17].Thus the flux (6) in 2-D inside the element is given by The concentrations around the large and small grains at interfaces 1 SL I and 2 SL I are imposed as boundary conditions of the liquid bridge and can be computed according to equation ( 5), but the time-dependent concentrations at nodes within liquid bridge can be updated using where Thus the time-dependent diffusion flux at interfaces 1 SL I and 2 SL I will be computed by equation ( 8) directly determining the grain coarsening.Fig. 5 shows the concentration distribution and the diffusion flux vectors for the steady state diffusion for W-Ni system with liquid nickel as a bridge for the conditions: the composition of the liquid in contact with these alloys:

Remeshing
The diffusion flux vectors are converted into the velocities of nodes on the solidliquid interfaces due to solution and precipitation.The velocities of the interfaces are imposed to the viscous FE analysis again, as the nodal velocities, and the mesh of the next step is calculated with keeping the meniscus of liquid bridge with o 10 = θ .

Results and discussions
We will assume that morphological development is governed by diffusion through the liquid (bridge) between grains.Assuming that solid-liquid system is held isothermally and under interfacial equilibrium condition, coarsening occurs by the exchange of solute between grains.When taking the above mentioned facts as the starting assumptions and using 2-D FE method of simulation, a general algorithm will be as follows: 1. Definition of initial FE mesh.7) and ( 9) in all mesh points within liquid phase, assuming that the concentration of liquid is time dependent, i.e.We will simulate grain coarsening during LPS of a W-Ni system.As an example, we will use two unequally sized grains of tungsten connected with liquid nickel as a bridge.The simulation parameters used in this model are: the composition of the precipitated alloys, 99.55 at.%W [19].The composition of liquid in contact with pure is W will be treated as time dependent simulation parameter.All calculations will be performed on 16 11×  As was already mentioned, the concentration on solid-liquid interfaces will be computed by equation (5).The initial concentration of liquid phase within liquid bridge (Fig. 6b) and the flux of materials will be computed on the FE mesh (Fig. 6a), which must be remeshed (explained in section 4.3) always when solid-liquid interface on larger grain reaches mesh line.
The time-dependent concentration of liquid phase in contact with the smaller grain is greater than in contact with the larger one.Therefore the smaller one dissolves in the liquid matrix and dissolved atoms flow from the matrix to the larger grain.Pure tungsten dissolved into the liquid, transports through the liquid and precipitates as W(Ni) solid solution on the larger grain (Fig. 7).It can be seen that due to high concentration gradient in this area, dissolution and reprecipitation processes are very fast.Although the larger grain continues to grow, the separation distance remains essentially constant, but the centers of mass of both grains move to the right (Fig. 7c).It can be also seen that the larger grain undergoing a larger shape distortion than the smaller one.The simulation results are very similar to the results of Voorhees et al. [2], but our approach treats more realistic LPS case with limited amount of liquid in which liquid phase is located within liquid bridges only.Even more, this methodology can be applied for simulation of LPS of multi-grain model, so that solution-reprecipitation process will be computed within each separated liquid bridge.

Conclusion
This paper outlines FE method for the simulation of coarsening of two-grain model during LPS, as a result of matter transport by diffusion from the dissolving smaller grain toward the growing larger one.The time dependent model system geometry is determined by initial topology, by size and shape of precipitates, by growth kinetics and by transport properties of a liquid phase.The physical and numerical models consist of a few basic equations that establish concentration profiles of solid-liquid system at sintering temperature.

Fig. 1 .and 2 ρ
Fig. 1. 2-D topology of the liquid bridge whose meniscus is approximated by a circle.
represents coordinates of s-th boundary point of l -th contour, l SV I and l SL I are the interfaces solid-vapor and solid-liquid, and l SV n and l SL n are the numbers of boundary points on l SV I and l SL I , respectively.Therefore, the grain coarsening in 2-D can be now described by the transformation ) Thomson boundary condition (3) are valid on solid-liquid interfaces only.If l s r is the radius of curvature at boundary point )

Fig. 4 .
Fig. 4. Movement of endpoint node due to interfacial tension in FE simulation.

3 .
Computation of the initial concentration of the liquid phase within the liquid bridge (the models M1, M2 or M3)

. 5 .∈
Computation of the flux by equation (8) and the mass flow dt dM including current results of both dissolution process