Asad Ullah

Topological correlations of three-dimensional grains

Topological correlations of three-dimensional grains were investigated by Monte Carlo-Potts model simulation. The result shows that, unlike first nearest neighbors (the Aboav-Weaire law [D. Aboav, Metallography 3, 383 (1970) and D. Weaire, Metallography 7, 157 (1974)] holds), there appears to be very little correlation between grains and their second and third nearest neighbors (on average), i.e., the average number of faces of second nearest neighbors, m2, and third nearest neighbors, m3, are independent of faces f of the center grain (nearly m2 = 14.984 and m3 = 14.489). This result indicates that long range correlations beyond first nearest neighbors may have negligible effect on the growth of center grains and thus provides support to a topologically averaged growth law that only considered the non-random first nearest-neighbor interactions.

On the sampling of three-dimensional polycrystalline microstructures for distribution determination

How to sample three‐dimensional microstructure and effectively reduce experimental error is a challenging problem. Taking seven single‐phase polycrystalline structures generated by 400×400×400 Potts Monte Carlo simulation as the study object, effects of sampling strategy on the determination of various characteristic parameters of the grain size distribution and grain topology distribution are studied. The mean voxel value (or volume) of individual grains in the three‐dimensional simulated microstructure varies from 4.56×104 to 1.0×103, and the number of grains contained in the structure varies from 63 901 to 1403. The results show that, a minimum of 200 sampled grains can ensure the relative error to be less than 5% for determination of the mean grain volume, the mean grain face number and the coefficient of variance of the distribution of grain size and the grain face number. Whereas for the coefficient of the skewness and the kurtosis of grain size distribution or grain face number distribution, a minimum of 800 sampled grains are required for the same error level. However, if some exceptional big grains appear, e.g. a grain larger than with eight multiples of mean grain volume and/or three multiples of mean grain face number, abnormal values of the two parameters would be resulted, even the number of examined grains is over 1000.

A note on grain topology-size relationship of three-dimensional polycrystalline microstructures

A grain topology-size equation for three-dimensional grains is suggested by considering the interactions of the nearest-neighbor grains, which indicates that the average grain size of a given topological class is determined by the difference between the grain face number and the average face number of nearest neighbors. In addition, a practical grain topology-size equation is presented. The two equations show good agreement with various experimental results. They are also verified by data of large-scale Potts model Monte Carlo simulation and Surface Evolver simulation.

Topological correlations of grain faces in polycrystal with experimental verification

How grain faces are arranged in three-dimensional grain structures is one of the outstanding problems in physics, biology and materials science. Based on the topological analysis of 477 real pure iron grains and 6093 Monte Carlo simulated grains, two forms of topological correlations are studied. The first correlation is Riviers relation (Philos. Mag. B, 52 (1985) 795), in which the average number of edges of faces neighboring to e-edged faces on f-faceted grains is related to e and f; and the second correlation is the one derived in the current paper, in which a typical e-edged face in a polycrystal is a function of the average number of edges of its neighboring faces M(e). Both correlations are verified by experimental and simulation data.