The local rate of change of height of the surface at any point is given by the sum of the flux away from the surface into the solid and the divergence of the surface flux

ay/at = ---12(aJjax) —12J₁ (6-23)

This is a conservation equation like Fick's Second Law, but the ac­cumulation of the pure material appears as a shape change, not a change in concentration. Substituting Eqs. (6-21) and (6-22) into this equation and canceling sin cox gives

da/dt = — [Bco⁴ + Cw³]a (6-24)

with"

B = D„.412/RT and C = Dryfi/RT

Since a pure solid is being considered, c_oil = 1 has been used in the equation for C. Eq. (6-24) can be integrated to give a(t). This can be used with Eq. (6-00). Since a pure solid is being considered, c0¹² = 1 has been used in the equation for C. Eq. (6-24) can be integrated to give a(t). This can be used with Eq. (6-17) to give the complete pro­file, y(x,t), but experimental observations are made on the decay of the amplitude with time. This equation is

din a/dt —(Bto⁴ + Cco³) = —Bw⁴(1 + D_IA/ 27rD_s8) (6-25)

where the last equality comes from replacing to with the wave length A using the equation 27r/A = co. Thus at small values of A (large co) the rate of decay in amplitude is controlled by surface diffusion and it varies as DNA'', while at larger wavelengths volume diffusion is

"It is easier for some to visualize this flow of atoms through the solid by consider the equal and opposite flow of vacancies.

'Mullins' expression is B = DJ/WORT. The difference is in his use of the term v = 81f2 where he calls v "the number of atoms per unit area" on the surface.

rate controlling and the rate varies as D₁ /A³. Surface and volume dif­fusion paths operate in parallel here. The thickness of the effective diffusion cross section in the lattice increases with A while the thick­ness of the surface layer transporting material, 8, is a constant inde­pendent of wavelength.

Other Shape Change Techniques. The surface diffusion coeffi­cient can be determined by following the rat of smoothing of a set of parallel grooves of spacing p that have been etched into the surface. D_s has also been measured by following the development of other shape changes.¹⁹ A more general equation for analyzing these changes is ob­tained by combining the flux equation (6-21) and the conservation equation to give the differential equation for pure surface diffusion

ay/at = Ba⁴y/ax⁴ (6-26)

As an example this equation can be used to describe the development of a groove where a grain boundary meets an initially flat surface. Local reduction of the surface free energy gives an equilibrium groove angle which remains constant as the groove develops. Atoms move away from the sharply curved region near the groove root to form ridges on the relatively flat surface on either side. The largest and most easily measured dimension of a groove is the width between the tops of the ridges. The increase in this width w_s with time is given by

w_s = 4.6 (B0¹¹⁴ (6-27)

Another example of a surface diffusion controlled process is the creep of a pure polycrystalline solid with an applied stress which is too low to move dislocations. The creep then occurs by the diffusion of atoms from grain boundaries with no normal stress to boundaries with a nor­mal stress acting on them. The diffusion can occur by lattice diffusion, in which case it is called Nabarro-Herring creep, or by grain boundary diffusion, in which case it is called Coble creep. To make the calcu­lation easier, consider the solid to be made of cubic grains of length L on a side as shown in Fig. 6-16. On boundaries normal to the applied load the chemical potential of atoms is reduced by 8_au = —sfi due to the work (stress times displacement) that the applied stress, s, does if an atom is added to these boundaries, i.e. the sample is extended. There is no strain in the direction of the applied stress if an atom is added or removed from the vertical boundaries, i.e. the sample- is not extended. Thus there will be a chemical potential difference and a dif­fusive flow through the lattice from one set of boundaries to the other.

T t^st t t

_s

Fig. 6-16—Diagram showing flux of atoms inside a square grain under the applied stress s. The flow of vacancies will be opposite that shown by the curved arrows.

There is an equal and opposite flow of vacancies. At steady-state, local stresses will develop near the corners to make the flux into or out of the boundary equal at each point along the boundary. The average flux caused by the applied stress s is then

Di 5/.1, Di sn J1= __________________ RTO 3y RTO L/3

(6-28)

where L/3 is taken to be the average diffusion distance between the boundaries under tension and those with zero normal stress. Now the rate at which a grain lengthens is dL/dt = 2J₁Q, so the overall strain rate caused by this shape change of the grains is

d€ dL 2./1n nD1sQ	(6-29)
- = =____
dt Ldt L RTL^2

the grain shape, and stress state.^2° Note that the strain rate varies as the first power of the stress, and the reciprocal of the grain size squared. Both of these distinguish this type of creep from creep involving dis­location glide, or grain boundary diffusion.

If the transport occurs not by lattice diffusion but along the grain boundaries, the expression for the strain rate is n(DO/LRT)(s(1/L²). Adding this to Eq. (6-29), gives an equation for the strain rate when transport occurs by both paths

de nD_b8sii [ D_LL]

1 + (6-30)

dt RTL³ 3D _b8

Note that the grain boundary contribution to the strain rate varies as 1/L³ so it will be more important at smaller grain sizes. Comparing Eqs. (6-30) and (6-26) one sees again the dimensionless variable (D_b5/ D₁L), which determines whether surface or lattice diffusion is con­trolling. It always appears in equations where the two paths work in parallel and indicates which process is dominant. It is an example of a general scaling law developed by Herring.²¹

Field Emission Spectroscopy. A quite different technique for studying surfaces involves field emission microscopy. In these instru­ments a very high electric field is applied to a sharply pointed metal wire (radius of curvature roughly 10-60 nm). In a high vacuum, the tip emits electrons in a pattern that reflects the atomic structure and composition of the surface (field electron microscopy). If a slight pres­sure of helium is admitted around the point, helium atoms are adsorbed on the surface of the point, ionized, and emitted from the surface. These emitted ions image the atomic details of the surface (field ion microscopy). The metal most commonly worked with is tungsten, pri­marily because it is strong enough to resists the high stresses developed in the tip by the electric field, but also because it can be easily cleaned by heating in a high vacuum. Two types of experiments have been performed using these techniques, one involving shape change and the other the diffusion of adsorbed atoms.

If the clean tip is heated slightly, the atoms rearrange themselves so the surface is bounded by flat, low index planes of the crystal. Using the atomic resolution of field ion microscopy the motion of individual atoms can be observed, and the diffusion coefficient of atoms across different faces and along different directions can be measured. This will be discussed in the next section.

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