The Pre-exponential Factor,

There are currently two main theoretical approaches to determining the pre-exponential factor in the diffusion equation {equation (421)). The first is based upon elastic strain theory and will be discussed here; the second is based on lattice dynamics, the theory and calculations of which shall be the subject of the next chapter.

4.6.1 Zener*s Theory o f Vacancy Diffusion fo r obtaining D f.

The pre-exponential factor in the diffusion relations, Dq*, is a function of the entropy change in the system during a diffusion process. The entropy of a system may be obtained using statistical mechanics via:

(4.40)

However, Zener (1952) derived an approximation for AS for diffusion via elastic strain theory. For vacancy diffusion, an atom first has to reach the saddle point from the potential well in which it sits. Therefore isothermal work is done elastically to strain the atom around the saddle point moving it from position Xq to the saddle point

position X,. An elastic strain energy function is set up of the form:

AG^ = work done - j^ 'f d x - pe' (4.41)

where / is the restoring force and p is the appropriate elastic modulus for the material concerned and e is the strain.

Now: A5. d ^G .

or ;

but AGo=AHq and since e is independent of temperature, from equation (4.41) AG„ varies as p, therefore:

« -A H

(4.45) </r

Thus AS„ may be found graphically with a knowledge of the temperature variation of elastic constant data for the material under investigation. For all solids not undergoing

a phase change, it has been found experimentally that dp/dT is negative making AS,, always positive; therefore exp(ASg/R) > 0 and so Dq* in equation (422) must minimise at a ^ (typically » lO’^mV*).

However, if P is defined by:

P ---(4.46)

where T^en is the melting temperature, and if AHmo^AHg, (from experiment; Shewmon, 1963), then:

(4.47)

To test this, the following section shows how AS„ can be graphically obtained from elastic data for MgO.

4.6,2 Graphical Determination o f D f.

In order to obtain Dq* for MgO from equation (4.16), a least squares fit to experimental data of p as a function of temperature gave a value for po of 1.5725Mbar

(Table 4.4).

This value of po was used to obtain p/po as a function of T/T^^u (T„eir3150K) from which a value for p (equation (4.46)) was calculated to be 0.2621 (Table 45).

Table 4.4: p vs. T. T /K p* /Mbar 80 1.5673 90 1.5671 150 1.5643 160 1.5635 170 1.5627 180 1.5618 190 1.5609 200 1.5599

*Simmons and Wang, 1971.

Using Clyde’s equation, we have already calculated the migration enthalpies for diffusion in MgO to be 267kJmol * for magnesium ions and 286kJmol'* for oxygen ions {Table 43). Equation (4.47) therefore allows us to calculate an migration entropy for each species:

For magnesium: A5 = 22.22 JK'^mol'^ (4.48)

For oxygen: à S = 23.80 JK ^mol'^ (4.49)

We are now in a position to evaluate the pre-exponential factor, ftom equation (4.16). This equation was derived for simple cubic structures; for MgO the cell parameter is not equivalent to the jump distance and therefore a must be replaced by

(4.16) yields: For magnesium: Dq = 1^8x10"* m^s'^ (4.50) For oxygen: Dq = 1.55x10"* (4.51) Table 4.5: p/p# vs. T/T„. T /K p /M bar p/p« TAT„ 200 1.5599 0.9948 0.0635 220 1.5578 0.9935 0.0698 240 1.5554 0.9920 0.0762 260 1.5529 0.9904 0.0825 280 1.5503 0.9887 0.0889 300 1.5477 0.9871 0.0952 320 1.5451 0.9854 0.1016 340 1.5424 0.9837 0.1079 360 1.5396 0.9819 0.1143 380 1.5368 0.9801 0.1206 400 1.5339 0.9783 0.1270

This then gives us a phenomenologically derived extrinsic self-diffusion coefficient (in mV^) for each species (equation (4.17)):

For magnesium: = AT,,1.28xlO‘^ e x p |- ^ ^ ^ ^ j (4.52)

For oxygen: Dq^ = AT^1.55xlO"^exp|-?^^^j (4.53)

The comparison of these predicted diffusion coefGcients with experimental results is not straightforward because the values of Ny are not well constrained. The detailed discussion of the accuracy and utility of these phenomenological estimations of the diffusion coefficients will be presented in Chapter 5, where they will be compared with the lattice dynamical results.

4.7 Summary.

Investigations into diffusion processes yield several ways in which the defect parameters, AS and AH, may be obtained. In this chapter we have seen that phenomenologically, AS„ may be found via Zener’s theory for vacancy diffusion, as is a function of AH^,, which in turn may be obtained from Clyde’s relation given in

equation (428). Clyde’s theory appears to be limited in its applications, and may not be relevant to complex mineral phases such as perovskites since it was originally derived for cubic metals. However, when using a Debye temperature derived from Debye-Waller temperature factors, the activation energy for ionic migration in MgO calculated from Clyde’s relation is in good agreement with experiment. Such a phenomenological approach may therefore be applicable to the more simple Earth- forming structures.

Zener’s theory for vacancy diffusion allows values for the pre-exponential factor to be obtained for both ionic species in MgO. This may be related to the extrinsic diffusion coefficient when a vacancy concentration, Ny, is known. However, the theory

is not sufficiently sound for the calculation of the entropy of vacancy formation, ASf, and we therefore have to consider another method of obtaining this quantity in order to calculate the intrinsic diffusion coefficient for self-diffusion in MgO. In the next chapter we shall see that at the microscopic level ASf and AS„ may be found by studying the phonon frequency spectrum of the defected solid via the theory of lattice dynamics; AHf and AHg, may also be found using suitable interatomic potentials in computer simulations.

5. Computer Calculations for Absolute Ionic