Capillary Liquid Bridge and Grain Coarsening During Liquid Phase Sintering

The finite element method is employed to simulate the microstructural evolution through grain coarsening within capillary liquid bridge. Liquid and solid phase domains will be represented with curved interfaces defined by the discrete set of points. Numerical method for simulation of grain coarsening will be based on the interfacial concentration as given by the Gibbs-Thomson equation and on modeling of intergrain diffusional interactions. It will be shown that the strong intergrain diffusional interactions can induce large shape distortion of multi grain model. Simulation of the grain coarsening for W-Ni alloy will be demonstrated as a first step.


Introduction
The phenomenon of liquid phase sintering (LPS) has been studied extensively due to its wide applicability to engineering materials but also due to the fact that the presence of a liquid phase simultaneously increases both the density of the resulting compacts and the rate of particle coarsening.LPS 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.However, 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.
The main conditions for LPS are that the composition of the powder and the firing temperature must be chosen such that a small amount of liquid forms from between the grains and that 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.
Typical phenomenon of LPS is that the solid grains of different size dispersed in the liquid matrix show a tendency of grow of larger grains at the expense of smaller ones which dissolve and are their immediate neighbors.The explanations of this phenomenon are largely based on the empirically established laws [1][2][3][4][5].In that sense, the investigations of Yoon and Huppmann are particularly interesting [6,7].Studying the grain growth of single crystal W spheres of uniform sizes in the presence of liquid Ni, they concluded that the driving force of the process is not, as it usually assumed, equivalent to the difference in the grain size of the solid phase, but the difference of the chemical potential between solid grains which dissolve in the liquid phase and solid phase obtained by precipitation.
In recent years, many computer simulation models have been developed with the aim of simulating the detailed microstructural evolution during grain growth or coarsening.The grain coarsening process has been simulated by space-discretized models, including Monte Carlo Potts methods [8][9][10][11][12], cellular automata methods [13][14][15][16], vertex methods [17], and phase field methods [18].Voorhees and Glickman [19] developed a theory describing the simultaneous growth and shrinkage of a randomly dispersed phase in a matrix, with the second phase providing the only sources or sinks of solute.Voorhees et al. [20] employed a boundary integral technique to determine the morphological evolution of a small number of particles during Ostwald ripening in two dimensions (2D).
A particularly interesting approach for investigation of grain coarsening is the application of numerical procedures together with Monte Carlo Potts model because of its ability for describing temporal domain evolution.Tikare and Cawley [21,22] used a simulation technique based on 2D Monte Carlo Potts model for describing grain coarsening with a fully wetting condition.Similar methodology was applied for generating simplified three-dimensional, two phase microstructures from a physically based model of LPS [23].
Shinagawa et al. [24,25] proposed microscopic modeling for viscoplastic finite element (FE) analysis of sintering processes.Recent advances in modeling grain coarsening using FE mesh approach was reported in [26][27][28].To the best of our knowledge, it was the first computer study of grain coarsening from within liquid bridge only.
The objective of this paper is to perform FE method for simulation of grain coarsening inside the capillary liquid bridge during LPS.The method will be used for calculation the diffusion field and intergrain diffusional interactions.Multi grain model represented by interfaces will be defined by discrete sets of points.Taking into account that each point can evolve according to its local environment, shape distortion of 2D contours will be investigated.Simulation of the grain coarsening inside the capillary bridge for W-Ni alloy will be demonstrated as a first step.

Modeling capillary liquid bridge
The simulation procedure that will be used for generating 2D digital microstructure resulting from LPS has to begin with the construction of a discretized 2D simulation model based on the individual grains and liquid phase which forms at additive grain sites.In that sense, the geometry of any 2D two-phase system consisting of topologically distinct solid and liquid phase areas (domains) can be described by polygons representing the individual solid grains and the liquid phase [29,30].
An interface as a common boundary among two different phases of matter is ubiquitous in LPS.It is responsible for a number of phenomena encountered in LPS, e.g.dissolution and reprecipitation, coarsening, etc.
Generally speaking, an interface in 2D can be represented with curved closed or open contact (boundary) line.In that sense, the k-th interface will be defined by the set of points, i.e.
where the subscript represents solid-vapor, solid-liquid and liquid-vapor interfaces, respectively.Thus, solid grain (defined as an isolated grain) can be represented by 2D domain of regular or irregular shape, i.e.

LV or SL
where is the position of the center of the mass of k-th grain.

) , ( c c k k y x
When the liquid phase is dispersed from between solid grains, the (pendular or isolated) liquid bridges with a curved meniscus shape will form, where the interaction between the liquid and the grains is dependent on the amount of liquid present.Typical unit cell for LPS is usually represented by two grains of different size joined by a liquid bridge and defined by surface-to-surface (inter-grain) distance, D, contact angle measured between the solid and the liquid phases, and the liquid volume that depends on the scale of the system (usually defined as a fraction of the volume of the liquid, , compared to the volume of the solid grains, , ).These three variables mainly define the geometry of liquid meniscus (LV interface) [31].Taking into account mentioned liquid phase between two grains and the interface definition (1), the solid grain representation (2) has to be replaced by new one, i.e.

}, I
and the (pendular) liquid bridge itself can be represented by where and are liquid menisci between two grains i-th and j-th.Thus, according to definitions ( 3) and ( 4) the LPS unit cell can be modeled as

∪ ∪
Due to grain rearrangement, coalescence of liquid bridges and completely filled pores during LPS, liquid phase will be redistributed and two or more pendular bridges will form capillary liquid bridges.Note that three or more grains are necessary for the formation of the capillary liquid bridge.
Taking into account previously defined submodel (4), the capillary liquid bridge that connects N solid grains can be represented by closed boundary defined by recursive formula:

Modeling coarsening
The pioneering work on coarsening was published by Greenwood [32], Lifshitz and Slyozov [33] and Wagner [34].The process in which large grains grow and small ones disappear and which process is characterized by an overall reduction in the interfacial energy is known as grain coarsening or Ostwald ripening.For this process the steady-state diffusion equation with a Gibbs-Thomson boundary condition is generally used for theoretical and numerical investigations.As a matter of fact, Ostwald ripening is based on Ostwald's observations [35] on mercuric oxide precipitates in potassium bromide and sodium thiosulfate solutions.
Although pretty old and rather well understood qualitatively, grain coarsening is still far from having a close and satisfactory quantitative description.However, it should be taken into account as the mechanism which plays a key role in the later sintering stages.
Let the solid grains are dispersed in the liquid phase and let the solid phase forms a dense polyhedral grain structure.Thus grain growth and grain boundary migration occur due to dissolution of smaller grains at SL interfaces (thermodynamically unstable), transportation of dissolved atoms through the liquid, and their precipitation onto the larger grains.In general, grain coarsening occurs by mass transport through the liquid surrounding solid grains, with a corresponding driving force determined by the solubility of the grains.
Let be assumed 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 can be mathematically expressed by Thomson-Freundlich's equation [1], also known as Gibbs-Thomson's equation [36]: where is the equilibrium concentration of liquid in contact with the flat solid, o c SL γ is the solid-liquid interfacial energy, Ω is the molecular volume of the solid, k is the Boltzmann constant and T is the temperature of the system.Note that ( 5) is not valid for a very small grain because becomes infinite as its radius goes to zero.However, the number of the small grains at any simulation time is sufficiently small so that (5) can be assumed to be valid for all grains.

c Δ
The Gibbs-Thomson boundary condition ( 5) is valid on solid-liquid interfaces only.Thus, the concentration is where is the radius of curvature at boundary (SL interface) point and the is the curvature at .
According to (6) small precipitates are surrounded by relatively high solute concentration.Hence, the concentration at an interface 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 grain defined by Fick's first law [37] where is the concentration independent diffusivity of the solid in the liquid.This equation can be applied for computation of quantitative and qualitative effects of dissolution, diffusion and precipitation mechanisms that enable coarsening during LPS.All mentioned processes will be described by discrete (for small time interval

Numerical method
For computation of 2D microstructure initial model typically consists of circular grains connected with liquid bridges.However, during LPS most of the grains are no longer circular because the diffusion fields within joined liquid bridges become highly asymmetric.FE method will be applied as a model for computation of time-dependent interfacial areas changed due to a diffusional mass transfer process from the interfaces with high interfacial curvature to those ones with low interfacial curvature that will be increased.
The concentration within the capillary liquid bridge around the large and small grains, i.e. at their SL interfaces will be imposed as boundary conditions (6).Thus, the timedependent concentration at nodes within liquid bridge can be updated using: where the liquid concentration will be interpolated from data at three nodal points that i c define this element, i.e.
is the shape function [38], and The initial boundary condition for the concentration at LV interfaces can be represented by three initial concentration profiles [1]: Model A: the model for pure surface control, Model B: the model for fast diffusion or large grain spacing, Model C: the model for slow diffusion or small grain spacing.During coarsening the solid phase domains will change their SL interfaces through Gibbs-Thomson equation ( 5), so that the interface concentration will vary pointwise along the interfaces.Thus, the dissolution and interface precipitation reactions will be considered as limiting process in which the flux (7)

Simulation results
In this paper simulation of the grain coarsening inside the capillary liquid bridge during sintering of W-Ni alloy (which usually serves as a model system) will be demonstrated.After heating to a temperature of melt formation, the liquid wets the tungsten and provides a soluble diffusion network for rapid sintering.Starting with such initial microstructure, it will be assumed that morphological development is governed by diffusion through the liquid between grains and by the reduction of the total interfacial energy.It will be also assumed that solid-liquid system is held isothermally.Under interfacial equilibrium condition, grain coarsening will occur by the exchange of solute between grains.The sintered performance of W-Ni system depends on several factors, including the tungsten (solid) content and matrix composition (initially liquid nickel as a bridge).In our simulation next data will be used: the composition of the liquid in contact with these alloys: the equilibrium composition of the precipitated alloy, 99.55 at.%W [39];   Taking into account initial concentration on LV interfaces (obtained according to Model C), as well as initial concentration on SL interfaces determined by the boundary condition ( 6), 2D initial concentration profiles (Fig. 3) can be determined using model (8).Due to higher concentration around smaller grains (numbered 1, 2 and 3) a net flux of matter from smaller grains to larger one will be distributed within the capillary liquid bridge, as it shown in Fig. 4. The numerical coarsening model defined in previous sections incorporates much of the essential physics of the coarsening process in this context, including two phases (solid and liquid), important interfaces and a mechanism for mass transport.Although the interfacial energy is isotropic, it is expected that the shape of the grains becomes increasingly non-spherical as the amount of precipitates increases.
The defined simulation model includes diffusion in the liquid bridge (here capillary liquid bridge) only, and not other diffusional mechanisms.Mass transport induced by concentration gradient is incorporated directly into the SL interface dynamics.It was stated that the interfacial concentration depends on position along SL interface in a manner which is related to the curvature.The liquid concentration and the flux of matter will be computed on FE mesh points.It should be noted that the mesh must be remeshed always when at least one interface point at SL interface reaches mesh line.
During solution-precipitation process smaller grains dissolve at SL interfaces, dissolved atoms diffuse through the liquid, and precipitate onto the larger grains.These processes are followed up by grain coarsening on both smaller and larger grains.For applied W-Ni system, pure tungsten will dissolve into the liquid, transport through the liquid matrix and precipitate as W(Ni) solid solution onto the larger grain.
Computed microstructure with grain coarsening of four-grain model forming the capillary liquid bridge between solid grains shown in Fig. 5.The morphological evolution occurred due to initial concentration on the SL and LV (Model C) interfaces and corresponding flux field, but also due to strong local diffusional interactions (the highest concentration gradient) between grains.As it can be seen, after 4 min.pure dissolution occurred onto the grain numbered 3 and pure precipitation onto the grain numbered 4 (largest shape distortion) forming solid solution W(Ni).Although dissolved matter is redistributed between grains, two different morphologies can be recognized due to the local diffusional interactions.It can be seen that some medium size solid grains are characterized by partial dissolution and precipitation at the same time, which processes depend of location of solid grains and intergrain distances between neighboring grains.Thus, both solid grains numbered 1 and 2 partly dissolved due to diffusional interaction with the grain numbered 4. At the same time, due to the local interaction between grains 1 and 2, solid solution W(Ni) was formed on grain 1.Similar situation occurred between grains 2 and 3, but the local diffusional interaction was weaker than previous one.It can be seen that the grains 1 and 2 developed two different morphology regions: one in direction close to the growing grain 1 and another one in direction close to the dissolving grains 2 and 3, respectively.As much as we do know, there is no simulation method for investigation of grain coarsening inside the capillary liquid bridge.Presented simulation results for the W-Ni system are in very good agreement with the microstructural evolution published elsewhere [6,7,39,41].Even more, there is a fairly good agreement with the boundary integrals approach [20], but our approach treats LPS case with amount of liquid located inside the capillary liquid bridge only.It should be noted that the pendular liquid bridge can be also treated in our approach as special case of the capillary liquid bridge.

Conclusion
This paper outlines FE numerical method for computer simulation of grain coarsening during LPS.It can be seen that the morphological evolution very much depends on locations of solid grains, but also on intergrain distances between smaller and larger grains.Due to high concentration gradient within the capillary liquid bridge, dissolution and reprecipitation processes are usually very fast.Smaller grains dissolve and larger ones grow at the expense of dissolved matter.Dissolved pure tungsten transports through the liquid matrix and precipitates as W(Ni) solid solution onto the larger grains nearby the intergrain region.
Although the applied model is simple with small number of grains and their regular geometry, we found good agreement with similar theoretical models and computer simulations.It seems that the processes such as dissolution, diffusion, reprecipitation and coarsening are included in FE method in an appropriate manner.The accuracy and efficiency of the proposed simulation method depend on how well the compositions of the solid phase and the diffusion coefficient in the liquid are known.The qualitative and quantitative characteristics of model system should be sound and appear to agree with the expected physical behavior of real system.
Our approach can be used as a tool to investigate the effects that changes to individual physical and technological parameters have on the precipitate evolution predicted by coarsening process within the capillary liquid bridge, something that is often not possible experimentally.
Obtained computational results can be compared with available experimental results or with other similar analytical approaches.Most of them lack an analytical expression for the evolution of the grain size.However, as much as we do know there is no simulation method for investigation of grain coarsening inside the capillary liquid bridge.Hence, that makes the comparison with our results difficult.
of closed curved boundary filled by liquid (the capillary liquid bridge), defined as the union of SL and LV interfaces, i.e.)}.

Fig. 1 .
Fig. 1.Initial four grain model.Light gray colored area is the capillary liquid bridge.
For computer simulation of grain coarsening four grain model shown in Fig.1grain radius.Diffusional field within corresponding capillary liquid bridge will be simulated on generated FE mesh shown in Fig.2.

Fig. 5 .
Fig. 5.The morphology of SL interfaces after 4 min.Light gray colored area is liquid phase, dark gray colored areas are W(Ni) solid solution.
depends on the local curvature at each interface point that locally obeys (6): computation of the surface integrated mass flow dt dM