## Classics in Materials Science: The Bragg-Williams model of order-disorder transformations. II

Since Bragg and Williams (BW) wrote a series [1-3], I guess it is only fitting that my post on their classic should also be a series. So, here is the second part.

After the first post, Abi made an astute observation, namely, that while I mentioned the work of BW on the kinetics of the order-disorder transformation a couple of times, I never gave the details of how and what they actually did, but confined myself only to the thermodynamics aspects of their work, which, both by their own admission, was not their original contribution to the problem. He also felt that a comment on that aspect of the work probably should form a part of the discussions. I agreed with him; so, here is a short note on the kinetics aspects of the model.

Before I proceed further, a couple of comments:

1. The review paper of Nix and Shockley [4] is a nice place for a summary (with all the attendant mathematical details) on the kinetics of order-disorder transformations as described by BW [1], as well as for a summary of discussions on the various experiments on relaxation time. I have leaned heavily on the review in writing this post.
2. Though we did not explicitly identify the quantity in the previous paper, let us call the ordering energy $\Omega$ and it is given by $\Omega = 2 E_{AB} - E_AA - E_BB$.

### Kinetics of order-disorder transformations

BW begin their discussion on kinetics by comparing the experimental electrical resistivity measurements made in Copper-Gold alloys by Kurnakow and Ageev with that predicted by their own theory. These comparisons are nicely summarised in the figures 9 and 10 of their paper; I reproduce their tenth figure below for facilitating a detailed discussion:

The figure shows that the measured resistivities are not only in disagreement with the calculated values from the theory, but also that the rate of cooling of the alloys gives different values of resistivity. From this one may surmise that the degree of ordering has not achieved its equilibrium value; and, at every temperature, allowing the system to equilibrate might actually make the curves coincide with the ones based on the model. If that be the case, then, we are dealing with a thermally activated process; so, the relaxation time would be related to temperature exponentially; and, it would also involve an “activation energy” and a constant. So, the problem boils down to building a model using which we can estimate both the constant and the activation energy, so that, comparisons between experiments and theory can be made. Finally, based on the idea of relaxation time, it can  also be argued that the heating and cooling cycles will involve a hysteresis; the experimental data on hysteresis is discussed in greater detail in the review of Nix and Schockley.

Now, let us take a look at the question of determining the activation energy and the pre-exponential constant; since this is the problem of ordering that we are talking about, it is not difficult to see that the relevant energy is related to ordering energy. As far as the pre-exponential factor is concerned, it is of the same order as the frequency of the atomic vibrations of the atoms about their mean position. Of course, BW go on to give a more detailed model for the pre-exponential factor by assuming a couple different mechanisms of diffusion — ring mechanism involving three atoms and direct exchange.

I have to make two points before we complete this discussion; one is the fact that the ordering energy itself is a function of degree of order that exists in the system in BW model since their ordering energy was a coarse grained quantity (the way I have denoted the ordering energy has to do with the Bethe’s model that followed BW); hence, near the critical temperature, the slopes of the curves of the dependence of order on ordering energy and ordering energy on order plays a crucial role in their calculation. The second is the fact that the observation which BW make as a note added in proof on the discrepancy between the relaxation times calculated using their model and that calculated using the diffusion coefficient — and, this discrepancy was much greater than the experimental errors would account for; to quote BW:

The rate of inter-diffusion of two metals, in which the atoms of one diffuse through the single crystal blocks of the other, is intimately connected with the time of relaxation of an alloy of the two metals. Actually, the coefficient of diffusion, D, $\approx a^{2}/\tau$, where $a$ is the lattice spacing, and $\tau$ is the time of relaxation. Where both D and $\tau$ have been measured, we find that the coefficient of diffusion is of a different order of magnitude from $a^{2}/\tau$, being very much greater. We conclude either that the diffusion then takes place then not inside single crystals, but between single crystals in the metallic structure; or, as is less likely, that there occur atomic interchanges inside a crystal for which the potential barrier W is considerably less than that which applies to the atomic interchanges which alter the degree of order.

I believe this problem of diffusion mechanisms and diffusivities in intermetallic coumpounds continues to be an active area of study to this day!

### References

[1] W L Bragg and E J Williams, The effect of thermal agitation on atomic arrangement in alloys, Proc. Roy. Soc. London, 145A, 699-730 (1934).

[2] W L Bragg and E J Williams, The effect of thermal agitation on atomic arrangement in alloys. II, Proc. Roy. Soc. London, 151A, 540-566 (1935).

[3] E J Williams, The effect of thermal agitation on atomic arrangement in alloys. III, Proc. Roy. Soc. London, 152A, 231-252 (1935).

[4] F C Nix and W Shockley, Order-disorder transformations in alloys, Reviews of Modern Physics, 10, 1-71 (1938).