A discrete approach to grain growth based on pair interactions: Effect of local grain-boundary curvature
- PDF / 231,902 Bytes
- 9 Pages / 612 x 792 pts (letter) Page_size
- 89 Downloads / 221 Views
I. INTRODUCTION
IN
[1]
a previous article, it was shown that a discrete approach based on pair interactions can be formally used to describe grain growth in single-phase polycrystalline materials, with results similar to those from the mean-field model by Hillert[2] and with realistic predictions of the topological features of three-dimensional (3-D) polycrystals. Conversely, in similarity to the mean-field models, it still predicts a left-hand-skewed quasi-stationary grain size distribution (GSD), which is in contrast to the experimental evidence. All the analytical grain-growth models are based on the implicit assumption that the grain-boundary curvature appearing in the growth-rate equations is well approximated by the average curvature proportional to the reciprocal of the grain radius. In the present work, using the theoretical framework of the discrete model of grain growth based on pair interactions,[1] a generalized formulation of the local grain-boundary curvature is proposed, which enables the model to reproduce a right-hand-tailed quasi-stationary GSD, with a shape closely approaching the Rayleigh distribution proposed by Louat[3] and Pande[4] in the stochastic models for grain growth. An alternative calculation of the pinning force due to second-phase particles, extending the result by Gladman,[5] is also presented and discussed.
dvi 앚j ⫽ 0 for 앚Z앚 ⬎ ji ⫺ ij dt
where M is the grain-boundary mobility, ␥ is the interfacial energy. Aij is the effective exchange area between grains i and j, and Z is the inhibition due to second-phase particles, which is always opposed to the boundary motion. The quantity ij is the effective local curvature of an ithclass grain in contact with a jth-class grain. The difference between the effective curvatures of the pair of facing grains is proportional to the driving force of the local graingrowth process. For a grain in the ith class with respect to the jth class, the surface curvature can be formally defined as[6]
ij ⫽
dSi 앚j dvi
[2]
where Si is the overall boundary surface of the grain and vi is its volume. The grain size is characterized by a linear dimension (Ri) generally associated with the radius of the equivalent sphere, so that the grain volume is vi ⫽
4 3 Ri 3
[3]
In the original formulation of the model,[1] the average overall surface was defined as Si ⫽ 具Ai典 mi
The Local Curvature and the Grain-Boundary Velocity
[1b]
[4]
According to the discrete grain-growth model based on pair interactions[1] and accounting for the grain-boundary curvature ( ), the equation for the relative rate of volume change of a grain belonging to the ith-size class sharing a common face with a grain in the jth class, with Ri ⬎ Rj , can be written in a generalized form as
where 具Ai典 is the average surface per face and mi is the average number of faces of each grain in the ith-size class. The average number of faces can be expressed as[1]
dvi 앚j ⫽ M ␥ Aij (ji ⫺ ij ⫺ Z ) for 앚Z앚 ⱕ ji ⫺ ij dt
Substituting Eq. [5] in [4], the grain surface was written in the spheri
Data Loading...
