0 Introduction

Grain size and uniformity are key factors that influence mechanical properties of polycrystalline materials^[1]. One effective way to improve material performance is to control grain growth with heat treatment technology. Although in most cases grain refinement is beneficial to mechanical properties of polycrystalline materials,it is not always the case,especially for high temperature materials that require coarse grains to enhance high temperature strength and creep resistance^{[2, 3]}. Controlling the evolution of grain structure is crucial to the optimization of materials properties through processing^[4]. Recently,directional annealing has been successfully applied in the experiment to produce columnar grain structures for several metallic materials^{[5, 6]}. Columnar grain structures can improve fatigue resistance,enhance creep property and impart crack-stopping behavior^{[7, 8]}. Therefore,it is significant to study the growth process of columnar grain under directional annealing.

In recent years,several modeling techniques have been successfully applied to study grain growth during directional annealing,including Monte Carlo Potts,front-tracking and phase field. Holm et al^[9] used Monte Carlo method (Potts model) to simulate non-uniform grain growth,and analyzed grain growth kinetics during directional annealing. Badmos et al^{[10, 11]} used the front-tracking method to simulate the growth process of columnar grain and discussed the effect of variable boundary energy and mobility on the evolution of columnar grain in single-phase polycrystalline materials during directional annealing. By setting the grain boundary mobility as a Gaussian function,Wei and Li^[12] investigated the grain growth behavior under the influence of temperature gradient using the phase field method. Ohno et al^[13] studied the discontinuous grain growth leading to a coarse columnar austenite grain structure in as-cast peritectic carbon steels in two- and three-dimensional (2D&3D) multi-phase field models. The experimental study in Ref. [14] revealed that SPPs (i.e.,oxide particles) have a significant influence on columnar grain growth,whereas the above simulated studies did not take into consideration. Therefore,a good understanding of the effect of SPPs on columnar grain growth is essential.

The phase field method^{[15, 16]} has emerged as a versatile tool for simulating microstructure evolution phenomena,such as solidification^[17],recrystallization^[18],grain growth^[19] and crack propagation^{[20, 21]}. In our previous studies^{[22, 23]},a phase field model with a moving hot zone under directional annealing has been developed,which is used to study the effect of hot zone width on the formation and continuous propagation of columnar grain structure without SPPs during directional annealing. The objective of this study is to study the effect of SPPs on columnar grain growth in polycrystalline materials during directional annealing.

1 Phase field model with a moving hot zone 1.1 Phase field model for grain growth with SPPs

In the phase field model^{[15, 16]} for grain growth of single-phase polycrystalline materials,the microstructure of polycrystalline materials is described by a set of continuous variables,η₁(r,t),η₂(r,t),…,η_P(r,t),where P is the total number of possible orientations in space and η_i(i=1,2,…,P) are called orientation field variables which distinguish different orientations of grains and are continuous in space,r is the spatial coordinate and t is the time. According to Moelans et al^[24],a spatially dependent function φ(r) that equals 1 inside SPPs and 0 in the matrix grains is used to describe the distribution of SPPs,which is considered constant in time.

Total free energy of the system containing SPPs is described by^[24]

$ F=\int{_{V}\left\{ \sum\limits_{i=1}^{P}{\left( -\frac{a}{2}\eta _{i}^{2}+\frac{\beta }{4}\eta _{i}^{4} \right)+\gamma \sum\limits_{i=1}^{P}{\sum\limits_{j\ne i}^{P}{\eta _{i}^{2}\eta _{j}^{2}+\phi }}}\sum\limits_{i=1}^{P}{\eta _{i}^{2}+}\sum\limits_{i=1}^{P}{\frac{{{k}_{i}}}{2}{{\left( \nabla {{\eta }_{i}} \right)}^{2}}} \right\}{{\text{d}}^{3}}r}, $

(1)

where α,β and γ are positive constants with γ>β/2,and k_iare the gradient energy coefficients. For isotropic grain boundary energy,k_i=k for all i.

The spatial and temporal evolution of orientation field variables are described by the time-dependent Ginzburg-Landau equations for non-conserved order parameters^[24]

$ \frac{\partial {{\eta }_{i}}\left( r,t \right)}{\partial t}=-{{M}_{i}}\frac{\delta F}{\delta {{\eta }_{i}}\left( r,t \right)}=-{{M}_{i}}\left[ \left( 2\phi -a \right){{\eta }_{i}}+\beta \eta _{i}^{3}+2\gamma {{\eta }_{i}}\sum\limits_{j\ne i}^{P}{\eta _{i}^{2}-{{k}_{i}}{{\nabla }^{2}}{{\eta }_{i}}} \right],i=1,2,\cdots P, $

(2)

where M_iis the grain boundary mobility. The Laplacian in Eq.(2) is discretized by^[15]

$ {{\nabla }^{2}}{{\eta }_{i}}=\frac{1}{{{\left( \vartriangle x \right)}^{2}}}\left[ \frac{1}{2}\sum\limits_{j}{\left( {{\eta }_{j}}-{{\eta }_{i}} \right)+\frac{1}{4}\sum\limits_{n}{\left( {{\eta }_{n}}-{{\eta }_{i}} \right)}} \right], $

(3)

where Δx is the discretizing grid size,j represents the first nearest neighbours of site i,and n represents the second nearest neighbours. For discretization with respect to time,we used the simple explicit Euler equation^[15]

$ {{\eta }_{i}}\left( r,t+\Delta t \right)={{\eta }_{i}}\left( r,t \right)+\frac{\text{d}{{\eta }_{i}}\left( r,t \right)}{\text{d}t}\Delta t,i=1,2,\cdots ,P, $

(4)

1.2 Setting of grain boundary mobility

The grain boundary mobility has a direct influence on grain growth during annealing,and the influence of a non-uniform temperature field on grain growth is due to the influence of the non-uniform grain boundary mobility on grain growth. For simplicity,we set a hot zone of uniform temperature and infinite temperature gradient,as shown in Fig. 1. The temperature profile ensures a uniform grain growth within the hot zone and zero growth outside the hot zone. Thus,grain boundary mobility M is given by^[23]

$ M=\left\{ \begin{matrix} {{M}_{\text{h}}},\ \ in\ the\ hot\ zone, \\ 0,\ \ in\ other\ zones, \\ \end{matrix} \right.\ $

(5)

where M_h is the grain boundary mobility within the hot zone.

1.3 Model parameters

A model system is used instead of a specific material,and the parameters used in this study are given in dimensionless units. All calculations are performed on 2D lattice of 512×512 grid points (gp),and periodic boundary conditions are applied. The coefficients in Eqs.(2)-(5) are chosen as follows^[15]: α=β=γ=1,k=2,P=36,M_h=1,Δx=2.0,Δt=0.25. In all cases discussed in this study,the hot zone width W is 30 gp,and the hot zone moves along the positive direction of x-axis at a velocity,v=0.002 5 gp·(ts)^-1,as shown in Fig. 1.

To visualize the grain microstructure,the following function is defined^[15]

$ \varphi \left( r,t \right)=\sum\limits_{i=1}^{P}{\eta _{i}^{2}\left( r,t \right),} $

(6)

which displays using gray-levels with white representing φ=1 and black φ=0. Thus,grains,grain boundaries and SPPs are white,gray and black,respectively.

2 Simulation results and discussion 2.1 Effect of SPPs on grain morphology

Grain growth process is influenced if there exist stable^{[14, 25]} and dispersed SPPs in metals and alloys such as oxide particles,carbide particles^[26] which are not easy to decompose or refractory at high temperatures. Since SPPs exert a pinning effect on grain boundaries and thus retard grain growth. In order to study the effect of SPPs on the columnar grain growth,dispersed SPPs were randomly distributed in the initial fine equiaxed grain (the sample) shown in Fig. 2. Directional annealing was performed on the sample. The final grain morphologies after directional annealing are shown in Fig. 3.

The SPPs radii (r_p) in Fig. 3(a)-(d) and Fig. 3(e)-(h) are 3 gp and 1 gp,respectively. It is clear from Fig. 3(a) that the grain structures are almost columnar when the volume fraction of SPPs f_v is very low (f_v=0.5%). As f_v increases,a growing number of SPPs are located at grain boundaries and a mixed structure of elongated and equiaxed grains forms (Fig. 3(b) and (c)). At f_v=8%,most grain structures are equiaxed,since most of the grain boundaries are pinned by SPPs. Comparing Fig. 3(c) and (g),it is indicated that smaller SPPs have a stronger pinning effect than larger ones at the same volume fraction and that both grain refinement and uniformity increase with increasing volume fraction of SPPs.

2.2 Effect of SPPs on grain size

Figure 4 shows the grain size distributions of the system containing different volume fractions of SPPs at r_p=1 gp. In the figure,R and R represent grain radius and average grain radius,respectively. The dotted line in Fig. 4 indicates that the grain size is the average grain size R. Note that the greater f_v is,the smaller R is. As shown in Fig. 4,when the system does not contain SPPs (f_v=0%),most grains (major peak) located at the regions of smaller grain. As f_v=1%,the major peak of the grain size distribution is at R/R≈0.5,with a minor peak at R/R> 2 (large abnormal grains). The minor peak is much smaller than that in the case of f_v=0%. As f_v increases,the peak of the grain size distribution gradually shifts towards 1.0. As f_v=4%,the grain size distribution tends to a Gaussian distribution with peak value 1.0,which is consistent with the grain size distribution of normal grain growth^[15].

At three radii r_p the final grain aspect ratio D_x/D_y as functions of f_v are plotted in Fig. 5. It is clear that SPPs have an important influence on the final grain aspect ratio. The grain structure in Fig. 3(a) has a high grain aspect ratio (D_x/D_y≈4.8) for f_v=0%. As f_v increases,the final grain aspect ratio D_x/D_y decreases and gradually converges to 1. The aspect ratio of 1 denotes equiaxed grain. The required volume fractions of SPPs are 8%,5%,and 3% to get a final grain aspect ratio of approximately 1 for r_p values of 3,2,and 1 gp,respectively. It is clear that the smaller the SPPs size is,the smaller the SPPs volume fraction for forming an equiaxed grain structure is required. The above results show that the volume fraction of SPPs in polycrystalline materials should be as low as possible in order to obtain better columnar grain structures during directional annealing.

The pinning effect of SPPs on grain boundaries influences directly the final grain radius. According to the Zener relation^[27],the ratios of final grain radius R_limto SPPs radius r_p as functions of volume fractions f_v of the SPPs are shown in Fig. 6. The relations are given by

$ {{R}_{\lim }}/{{r}_{\text{p}}}=af_{\text{v}}^{-b}=\left\{ \begin{matrix} 46f_{\text{v}}^{-0.45},\ \ \text{for}\ {{r}_{\text{p}}}=1\text{gp}, \\ 28f_{\text{v}}^{-0.47},\ \ \text{for}\ {{r}_{\text{p}}}=2\text{gp}, \\ 23f_{\text{v}}^{-0.48},\ \ \text{for}\ {{r}_{\text{p}}}=3\text{gp}, \\ \end{matrix} \right. $

(7)

where parameter b is slightly smaller than the theoretical value of 0.5^[27],whereas parameter α is significantly greater than that in Refs.[24, 27],which illustrates the grain size of directional annealing is much greater than that of uniform annealing in the case of the system containing SPPs. The reason is that grain growth during directional annealing is selective growth and/or competitive migration of grain boundaries,which causes abnormal grain growth.

3 Conclusions

SPPs inhibit the formation of columnar grain structure and the inhibitory effect increases with increasing volume fraction of SPPs and decreasing size of SPPs. The smaller the size of SPPs,the smaller the volume fraction of SPPs required for forming an equiaxed grain structure. Under condition of directional annealing,the effect of SPPs on the final grain radius follows the Zener relation in which the fit index is between 0.45 and 0.48. In order to obtain better columnar grain structures during directional annealing,the volume fraction of SPPs and their dispersive distribution at grain boundaries in materials should be reduced as far as possible.