Materials Science and Engineering A 412 (2005) 271–278 Anisotropic grain growth with pore drag under applied loads X.N. Jing a,b , J.H. Zhao a , G. Subhash b,∗ , X.-L. Gao c a CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei, Anhui 230027, PR China b Department of Mechanical Engineering and Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931-1295, USA c Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA Accepted 31 August 2005 Abstract In the final stage of sintering of ceramics, residual pores co-evolve with grain boundaries because of incomplete densification. Their interactions coupled with external loads are critical to the microstructural evolution of structural ceramics. A modified two-dimensional (2D) diffuse-interface phase field model, which differs from the boundary-tracking methods, is utilized to investigate the effects of stochastically distributed pore drag on grain growth kinetics and morphological evolution process of ceramics under applied loads. Contributions from both the boundary energy and elastic strain energy caused by pore drag forces and applied loading are incorporated in the modified phase field model to describe the isotropic or cubically anisotropic behaviors of polycrystalline materials. The temporal evolution of the spatially dependent grain orientation variables is determined by numerically solving non-linear Ginzburg–Landau equations using a semi-implicit Fourier-spectral method. Numerical results show that the anisotropic strain energy dominates the non-self-similar growth manner, leads to ordered grain morphologies and changes the growth rate. © 2005 Elsevier B.V. All rights reserved. Keywords: Phase field model; Grain growth; Pore drag; Morphological evolution 1. Introduction Mechanical properties of polycrystalline materials (e.g., ceramics) depend strongly on the size, distribution and mor- phology of grains. Thus, how to control grain growth during processing is a critical issue for the development of polycrys- talline materials [1–3]. Grain growth can be viewed as a process of grain boundary migration to decrease the total grain boundary areas and the total free energy of the material system, both of which are driven by mean curvatures of grain boundaries [1,2]. In a single-phase material, the only process which occurs dur- ing grain growth is local atomic re-arrangement. However, in a multi-phase material, the long-range diffusion dominates the migration of grain boundary. Hence, the kinetics of grain growth is strongly affected by the presence or absence of solute or impu- rity migration and segregation at grain boundaries. The pinning effect of pores is of great practical importance in the sintering of high-quality ceramics for structural applications, where den- ∗ Corresponding author. Tel.: +1 906 487 3161; fax: +1 906 487 2822. E-mail address: subhash@mtu.edu (G. Subhash). sification and small grain sizes are often required to obtain good strength and toughness. In the last two decades, many models have been proposed to predict the time dependence of average grain size and size distribution in polycrystalline materials [3–7]. Due to the complexity of topologies and coupled multiple interactions, the analytical modeling of microstructure evolution is often very difficult. As a result, computer simulation plays a key role in exploring the details of grain growth and validating the analytical models. The sharp-interface approaches, such as Vortex model [8], Potts models [9–11] and Cellular Automata method [12], have been used in simulations of grain growth. Recently, simulating grain growth using a continuum-based diffuse-interface phase field model has been developed by several researchers [13–19]. Chen and co-workers simulated grain growth in single-phase materials [13,14] and studied the microstructure evolution in volume-conserved two-phase mate- rial systems with finite interface thickness [15] by considering grains of different crystallographic orientations represented by a set of non-conserved order parameter fields. In their model, the effects of solute or precipitate drag on grain growth rate and size distribution are taken into account without ad hoc assumptions 0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.220