Morphology of Silicon Nitride Grown from a Liquid Phase

Abstract

To explain the recent experimental observation of liquidgrown silicon nitride (Si 3 N 4 ) crystals with a concave depression in the center of the (0001) end face, we propose a new growth mechanism and develop an analytical solution for the steady state. The model allows for atoms that diffuse via the liquid to the side surface but demands that the majority of these atoms be transported to the end caps to feed axial growth. The analysis shows that, for a large radius crystal, the redistribution of atoms by surface diffusion on the end caps requires a long relaxation time; hence, a nonequilibrium shape results. For an isolated Si 3 N 4 crystal growing in a liquid environment, the shape of the end cap is largely determined by the ratio of the supersaturation to the equilibrium surface potential, which is inversely proportional to the crystal radius. A large shape distortion is predicted to occur during the growth stage for large-radius crystals and during the coarsening stage for a population of crystals with a large size distribution. This mechanism ceases to operate when the liquid flux to the side surface is blocked, as in silicon nitride ceramics, but is otherwise insensitive to factors such as radial growth kinetics and liquid diffusivity.

Formation of β‐Silicon Nitride Crystals from (Si,Al,Mg,Y)(O,N) Liquid: I, Phase, Composition, and Shape Evolutions

“…In the second article (Part II), we describe the crystal population dynamics to determine the growth velocity and to make a comparison with the theoretical predictions of crystal growth and coarsening based on a recent theory. 11 A further examination and modeling of the shape transition at the ends of a rod-shaped crystal 12 are reported elsewhere using the dimensional and velocity data obtained here.…”

Reply to “Comment on ‘Morphology of Silicon Nitride Grown from a Liquid Phase’”

“…(1) In their comment on our paper, 1 Kitayama et al 2 suggested that the weighted mean curvature method should have been used to obtain surface chemical potential. Their results are correct, but only for crystals that have a flat end.…”

“…They are certainly not useful for treating surface morphology that has undulations. 1 (2) To allow a more general discussion not restricted to grains with flat ends, we have derived the surface potential for a rod-shaped crystal having spherical end caps and a circular cross section. The derivation, which is consistent with the weight mean curvature concept, will be published elsewhere.…”

“…The derivation, which is consistent with the weight mean curvature concept, will be published elsewhere. It finds that the condition of local equilibrium at the end surface requires sin ϭ a/b ϭ ␥ Ќ /␥ (1) and the chemical potentials of the end surface and the side surface are, respectively,…”

Comment on “Morphology of Silicon Nitride Grown from a Liquid Phase”

“…I N THEIR recent paper, Wang et al 1 proposed a new growth mechanism, and developed an analytical solution for the steady state to explain their experimental observation of liquid-grown silicon nitride (Si 3 N 4 ) crystals with a concave depression in the center of the (001) end face. The key to their analysis is that the chemical potential at the center is lower than the edge as evidenced by the negative curvature at the center.…”