Surface diffusion and shape change

There are a variety of diffusion controlled phenomena familiar to a materials scientist which involve a change of shape and are driven by surface tension (surface free energy) or applied stresses too low to move dislocations. Examples are:

· creep at low stresses in polycrystalline materials can occur by dif­fusion through the lattice (Nabarro-Herring creep), or along grain boundaries (Coble creep).

· the sintering of powders occurs by the diffusion (lattice or boundary) from regions of sharp curvature to low curvature.

The principles which go into calculating the rate of each of these processes are understood. However, the actual calculation of the rate is often made difficult by the complicated geometry.

An example of surface tension driven diffusion with particularly simple geometry is the smoothing of a surface with a sinusoidal ripple in it. This problem is chosen because:

· A straight forward mathematical analysis exists.

· The analysis is borne out by experiment.

· The results provide a means for determining the surface diffusion coefficient.

· The analysis describes the relative contributions of competing trans­port processes.

· It provides a basis for treating other geometries by Fourier analysis.

Analysis of Surface Smoothing. `⁶ Consider the sinusoidally curved surface shown in Fig. 6-15 whose elevation y is given by the equation

ly

Surface

----- X= 2 r/w --I diffusion

-X

Volume diffusion in solid

Fig. 6-15—Transport processes causing the decay of a sine wave in a solid surface. (W. W. Mullins)

y(x,t) = a(t) sin(cox) (6-17)

The chemical potential of the atoms on the hills is higher (p, > 0) than that beneath a flat surface (ti, = 0), while in the valleys the chemical potential is lower (ti, < 0). The magnitude of this variation is related to the surface tension y and the curvature K by the equation

la = (6-18)

where 12 is the atomic volume. For a surface like this where the height y is a function only of x, K = y"/(1 + y¹²)¹/² where y' and y" are the first and second derivatives of y with respect to x. Thus

p(x,t) = --fiyy" = ilya(t)w² sin(wx) (6-19)

where Eq. (6-17) is used to obtain the final equality. As indicated in Fig. 6-15 transport from the hills to the valleys can occur by three paths, diffusion through the gas, the lattice, or along the surface. If we assume that surface diffusion occurs in a thin layer of thickness 8, the flux crossing unit length on the surface is

.1_s= (D_s8/RT)apilax (6-20)

where the derivative along the surface, a/as, has been set equal to a/ ax, an approximation good for small slopes. Combining Eqs. (6-19) and (6-20) gives

—(D_s8/RT)fly a(t)w³ cos(cox) (6-21)

The variation in chemical potential along the surface also gives rise to fluxes of atoms through the solid, and through the gas. The con­centration variation in each phase can be approximated by the steady-state solution to the diffusion equation, ❑²c = 0. There exists a so­lution c(x,y) which satisfies Eq. (6-19) as a boundary condition along the surface and decays to a constant value far from the interface. If this is differentiated to get the flux normal to the surface the equation is

.11 = —D1

a [Coyfito2 ay RT

exp(—wy)a sin(wx)

y=0

Or

J₁ = (D₁c₀/RT)yfico³ a sin(wx) (6-22)

This equation indicates that the net atom flux is out of the regions where sin(cox) is positive and into the regions where it is negative.'⁷ A similar equation holds for the gas phase, the only change being that D _lc_° is replaced by D and c for the gas phase. However, the vapor pressure is usually so low that transport through the gas phase is neg­ligible, and it can be neglected.

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