Sintering kinetics at final stage sintering: model calculation and map construction

Abstract

Ever since the development of a late stage sintering model by Coble in 1961, densification at the final stage of solid state sintering has been explained basically by Coble's model for lattice diffusion. For grain boundary diffusion, equations based on Herring's scaling law were later developed and utilized. The present paper examines previous sintering models, in particular, Coble's sintering model, and presents a new diffusion model and kinetic equations. The new model accounts for the role of grain boundaries and the diffusion area in densification in contrast to Coble's model. Sintering kinetics have been calculated and the relative contributions of lattice and grain boundary diffusion evaluated. It has been shown that the densification mechanism can vary with pore size change during sintering, unlike the conclusions of previous investigations. The results have been highlighted as a sintering diagram.

Introduction

The densification in the final stage of solid state sintering is characterized by the shrinkage of isolated pores located at grain corners and junctions. As a geometrical model of final stage sintering, Coble's model of bcc-packed tetrakaidecahedral grains with spherical pores at 24 corners of a grain is widely accepted [1]. For this model, Coble suggested concentric spherical lattice diffusion of atoms from a distance of r₂ to the surface of the pore with a radius of r₁. Since concentric spherical diffusion was assumed, the material flux passing through any imaginary spherical surface between the pore surface and the surface of the material source at r₂ is constant. Therefore, the total material flux J_total passing through the imaginary spherical surface towards a pore is expressed as

$$\text{J}{}_{\mathrm{<mtext>total</mtext>}}\text{=}\text{const}\text{=−4}\text{π}\text{r}{}^2\text{D}{}_{\mathrm{<mtext>l</mtext>}}\text{RT}\text{d}\text{σ}\text{d}\text{r}\text{,}$$

where D_l is the lattice diffusion coefficient, σ the sintering stress, R the gas constant and T the absolute temperature. Upon integration and rearrangement,

$$\text{J}{}_{\mathrm{<mtext>total</mtext>}}\text{=4}\text{π}\text{D}{}_{\mathrm{<mtext>l</mtext>}}\text{RT}\text{Δ}\text{σ}\text{r}{}_1\text{r}{}_2\text{r}{}_2\text{−r}{}_1\text{.}$$

If r₁≪r₂, the densification rate dρ/dt is expressed as

$$\text{d}\text{ρ}\text{d}\text{t}\text{=}\text{−}\text{24}\text{4}\text{J}{}_{\mathrm{<mtext>total</mtext>}}\text{V}{}_{\mathrm{<mtext>m</mtext>}}\text{1}\text{6}\text{π}\text{G}{}^3\text{=}\text{288D}{}_{\mathrm{<mtext>l</mtext>}}\text{γ}{}_{\mathrm{<mtext>s</mtext>}}\text{V}{}_{\mathrm{<mtext>m</mtext>}}\text{RTG}{}^3\text{,}$$

where γ_s is the specific surface energy, V_m the molar volume and G the grain size. Eq. (3), the Coble equation, indicates that the densification rate is inversely proportional to the cube of the grain size. This result is, in fact, the same as that found for the dependence of neck growth and shrinkage on particle size in the initial stage model [2].

So far, Coble's model has been a standard for interpreting and predicting the densification at final stage sintering governed by lattice diffusion. In Coble's model, however, a fundamental aspect is not taken into account, namely, the grain boundary as an atom source for densification. In addition, it is hard to accept Coble's flux equation (Eq. (2) under r₁≪r₂) that predicts a constant material flux from grain boundaries to the surface of a pore irrespective of the pore size [3]. In contrast to Coble's model for lattice diffusion, densification by grain boundary diffusion has been explained by kinetic equations derived from Herring's flux equations [4], [5], [6], [7]. The equations derived show a grain size dependence of densification similar to that in the initial stage model [4], [5], [6], [7]; densification rate is inversely proportional to the fourth power of grain size.

In predicting densification kinetics and analyzing densification data, evaluation of the grain size dependence of densification has been a standard method. Analyzing their densification data of alumina as a function of grain size, Harmer and coworkers [4], [5] concluded that the densification was governed by grain boundary diffusion at final stage sintering. Johnson and his coworkers also studied the densification of powder compacts using kinetic equations based on Herring's flux equations [6], [7]. They developed the concept of a master sintering curve that characterizes the sintering behavior of powder compacts. This concept is useful to explain densification behavior but it does not differentiate between the two densification mechanisms. In these previous investigations [4], [5], [6], [7], it was not possible to estimate absolute densification kinetics by different densification mechanisms, lattice and grain boundary diffusion, and thus to compare relative contributions between the two mechanisms. Furthermore, the previous equations used for lattice and grain boundary diffusion have a different physical basis, as explained above.

This investigation suggests a new model of final stage sintering which accounts for the grain boundary as an atom source and vacancy sink. Based on the new model, it has been possible to predict the relative contributions of lattice and grain boundary diffusion to final densification. The relative contributions have been visualized as a sintering diagram.