Phase field simulations of grain growth during sintering of two unequal-sized particles

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Abstract

Sintering is fundamentally a process of bonding between solid particles which typically occurs under high temperatures relative to the material's melting point. The sintering of real powders requires consolidation of unequal-sized particles. The consolidation of unequal particles involves sintering and simultaneous grain growth. In this work, sintering and simultaneous concurrent grain growth of two unequal-sized particles are modeled using a phase field simulation method. The sintering process is revealed to consist of three sub-processes: (1) neck growth, (2) coarsening with concurrent slow grain boundary migration, and (3) rapid grain boundary motion. The observed grain boundary migration with coarsening was not predicted in prior analytical models of sintering. The simulation results are analyzed based on a thermodynamic analysis of the driving forces for different sub-processes. Analytical expressions for grain boundary velocity in the second and the third sub-processes are derived. The slow grain boundary migration is sensitive to the sintering geometry.

Research highlights

▶ Sintering of unequal particles occurs in three sub-processes. ▶ The sub-processes are: neck growth, coarsening, grain boundary migration (GBM). ▶ Coarsening is responsible for initial increase of grain size during sintering. ▶ The sub-processes overlap significantly. GB migration starts during coarsening. ▶ Rapid GBM signifies the dominance of grain growth by GBM & completion of coarsening.

Introduction

Sintering, a heat treatment processing step of converting a powder compact into dense solid [1] has been used in both ancient civilizations and modern technology industries [2]. The theories of sintering, of both scientific and technological importance, have therefore attracted considerable research interests in the past decades. To date, however, most analytical solutions of sintering are based on simple geometries such as two equal-size spheres and tetrakaidecahedron [3]. These models do not take into account inter-particle diffusions that leads to coarsening and grain boundary migration, which is usually present in real powders due to differences in particle size and shape. In recent years, the sintering of nanosized powders has emerged as an important technological challenge [4], [5]. The effects of particle size and its unavoidable wide distributions at the nanoscale as well as the expected strong contribution of surface diffusion to the process render the conventional theories of sintering limited in value. For example, the sintering of nanosized powders involves rapid coarsening of particles and grain growth by grain boundary migration. It is not clear if the conventional scaling law [6], which stipulates that the kinetics of sintering is proportional to particle size ratios and assumes the particle geometry does not change, is still valid for modeling such rapid grain growth and densification processes. Furthermore, the dependence of the diffusion processes on temperature and particle size also add complexities to the matter [7], [8]. In order to accurately model and control grain growth during sintering of nanosized powders with wide particle size distributions, the mechanisms of sintering of unequal-sized particles must be understood.

Lange et al. [9], [10] proposed a thermodynamic model for two unequal-size particles. Similar analysis and view points were also discussed by Shi [11]. The model describes the sintering process involving three sub-processes: (1) neck growth, (2) coarsening until a critical particle size ratio, and (3) rapid grain boundary migration. Similar to equal-size particle model, the separation of the sub-processes is based upon the relative mass transport driven by the chemical potential. The three sub-processes are distinctively separated based upon the equilibrium values of dihedral angles and critical radius ratios. The neck between the particles grows in the first sub-process until the dihedral angle acquires the equilibrium value. Once the neck is formed, the coarsening causes the particle size ratio (large/small) to increase until it attains a critical value which also depends on the equilibrium dihedral angle. Once the critical particle size ratio is reached, the grain boundary migrates rapidly, across the smaller particle. If, however, the particle size ratio is larger than the critical size ratio at the end of the first sub-process, the grain boundary migration would occur without the second sub-process: coarsening.

The existing models, as described above, explain well the sintering behavior of coarse powders with negligible overlap of sub-processes. However, when the overlap of sub-processes is expected during the sintering of nanosized particles due to high chemical potentials and nonlinear diffusion [7], the applicability of the model is limited.

In this paper, a numerical computational approach is used to study the sintering of two unequal-sized particles and the concurrent grain growth. A phase field simulation is performed to gain insight regarding the mechanisms of the increase in grain size. The numerical approach enables simultaneous considerations of mass transport for the three sub-processes, thus understanding the overlapping effect of neck growth, coarsening, and grain boundary migration on the evolving microstructure.

Section snippets

Simulation variables

In phase field simulation method, thermodynamic quantities such as concentration, crystal orientation and temperature are represented as phase fields or fields [12]. A thermodynamic state variable, free energy per unit volume, can be written as a function of thermodynamic quantities. The total free energy of the microstructure is a function of the phase field variables, which the system tries to minimize. The temporal evolution of the fields is calculated using kinetic equations, which are

Microstructure evolution

Fig. 2 shows the evolution of the microstructure of the two unequal particles geometry on a [0.95 1] scale. The total time for the particles to coalesce following the equations in Section 2 was 26,765 in arbitrary time units; which was rescaled to [0 1] for describing the results. The evolution is quantitatively described in Fig. 3; showing the relative change of particle size, neck size and grain boundary location. The rescaling is performed to show the change in a range of 0–1 with arrow

Summary

The sintering and grain growth of two unequal-size particles are studied using the phase field simulation method. The sintering and grain growth are found to occur in three sub-processes: (1) neck formation, (2) coarsening with slow grain boundary migration and (3) rapid grain boundary migration. The second stage is the slowest and thus the rate determining step for the overall sintering process. The slow grain boundary migration during coarsening had not been predicted by the existing models.

Acknowledgements

The authors of this paper are thankful to Prof. D. Shetty of the Department of Materials Science and Engineering and Prof. H.Y. Sohn of the Department of Metallurgical Engineering at the University of Utah for multiple illuminating discussions that one of the authors (VK) had with them during the course of this work. Authors would also like to acknowledge funding from US Department of Energy, the Office of Industrial Technology Program, for experimental work on sintering of nanosized particles

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