Grain size limit of nanocrystalline materials obtained by annealing glasses

A lower limit for the grain size of nanocrystalline solids obtained by crystallization of the glass and its dependency on the crystallization temperature are thermodynamically considered. It is found that the nanocrystalline materials have the smallest grain size when the crystallization temperature is roughly half of the melting temperature. At this temperature, the Gibbs free energy difference between the undercooled liquid and the crystal reaches the maximum. It is found that for polymorphous crystallization the lower bound of grain size is essentially dependent on melting entropy. The results are consistent with available experimental evidence. It is well known that nanocrystalline materials can be prepared by crystallization of glasses [1–3]. Experimentally it is found that the smallest grain size is obtained when the glasses are annealed at a crystallization temperature Txnear Tm/2 where Tm is the bulk melting temperature [1–3]. This has been interpreted in terms of transition kinetics, namely at Tm/2 the nucleation rate I is the largest while the growth rate u is relatively low [2]. Of course, besides the kinetic criteria, the crystallization process also needs to satisfy the energetic criterion that the total free energy diminishes upon crystallization. It is conceivable that such an energetic criterion can be of help in rationalizing experimentally observed grain size and its dependency on the materials parameters and on the temperature of crystallization. In this paper, we will consider a simple thermodynamic relation that suggests a general thermodynamic lower limit of the grain size of the metallic nanocrystalline materials at Tx. Obtaining nanocrystalline materials by crystallization generally requires that Tx is above or close to the galss transition temperature, Tg [2]. Except for a small difference due to the discontinuity in heat capacities at Tg, the free energy of the amorphous state is therefore that of the undercooled liquid. Thus, what we discuss below is the transition between the undercooled liquid and the nanocrystalline state. We shall first discuss the case of polymorphous crystallization, where the nanocrystalline product phase is uniform in composition. The consequence of decomposition into two phases during crystallization will be commented on at the end of the paper. 0953-8984/01/235503+04$30.00 © 2001 IOP Publishing Ltd Printed in the UK 5503