Grain growth in porous two-dimensional nanocrystalline materials

Abstract

Grain growth in two-dimensional polycrystals with mobile pores at the grain boundary triple junctions is considered. The kinetics of grain and pore growth are determined under the assumption that pore sintering and pore mobility are controlled by grain boundary and surface diffusion, respectively. It is shown that a polycrystal can achieve full density in the course of grain growth only when the initial pore size is below a certain critical value which depends on kinetic parameters, interfacial energies, and initial grain size. Larger pores grow without limits with the growing grains, and the corresponding grain growth exponent depends on kinetic parameters and lies between 2 and 4. It is shown that for a polycrystal with subcritical pores the average grain size increases linearly with time during the initial stages of growth, in agreement with recent experimental data on grain growth in thin Cu films and in bulk nanocrystalline Fe.

Appendix

Dissolution of a circular pore at a triple junction

Let us consider a cylindrical pore of radius r located at a GB triple junction in a two-dimensional polycrystal. We will assume that pore dissolution is controlled by GB diffusion alone. The diffusion flux, j, along the GB (per unit length of the pore) is

$$ j=-\frac{D_{GB}\delta}{kT}\frac{\partial\sigma_{nn}}{\partial x}, $$

(A1)

where σ_nn is the normal stress at the GB, and x is the coordinate along the GB. D _GB is the GB-self diffusion coefficient and we assume for simplicity that the diffusional width of the GB is equal to that of the surface. In the middle between two identical pores (x = 0) this flux should vanish because of symmetry reasons. In the steady state sintering regime the drift velocities of two grains normal to the GB should be constant, which means that the divergence of the flux j given by Eq. A1 is also constant. In this case

$$ \sigma_{nn}(x) = ax^2 + b, $$

(A2)

where a and b are constants. The distance between neighboring pores is 2l ≈ 2πL/n − 2r (see Fig. 1). The chemical potential of the atoms on the pore surface and at the triple junction between the pore surface and a GB are μ₀ − γ_sΩ/r and μ₀ + σ_nnΩ, respectively (here, γ_s is the surface energy and μ₀ is the chemical potential of atoms in the crystal bulk). From the balance of these chemical potentials we get

$$ \left.\sigma_{nn}\right|_{x=l} = - \gamma_s/r. $$

(A3)

To find the unknown constants a and b we use the boundary condition A3 and the total force balance at the GB:

$$ -\gamma_s + \int\limits_0^l \sigma_{nn} (x)dx=0, $$

(A4)

which yields $a=-\frac{3\gamma_s(l+r)}{2rl^3}.$ For the atomic flux to the pore we obtain from Eqs. A1 and A2

$$ j_{x=l}=\frac{3\gamma_sD_{GB}\delta}{2kT}\frac{l+r}{rl^2}. $$

(A5)

The rate of pore dissolution can be found from the condition of mass balance:

$$ 2\pi r\frac{dr}{dt} = \left.-3\Upomega j\right|_{x=l} $$

(A6)

$$ \frac{dr}{dt} = -\frac{9D_{GB}\delta\gamma_s\Upomega}{4\pi kT}\frac{l+r}{r^2l^2}. $$

(A7)

For l ≫ r

$$ \frac{d(r^3)}{dt}=-\frac{27D_{GB}\delta\gamma_s\Upomega}{4\pi kT}\frac{1}{l}\,\approx\,-\frac{27nD_{GB}\delta\gamma_s\Upomega}{4\pi^2 kT} \frac{1}{L}. $$