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

The atomic arrangement in a crystal can be understood in terms of unit cells; for example, let us consider the metal copper; the arrangement of atoms in the copper crystal can be described using the face centred cubic (fcc) unit cells and their stacking in 3-dimensions. A schematic of the fcc unit cell is shown below — it consists of points on the eight corners and six face centres of a cube, and by placing a copper atom at each of these points, we obtain the fcc unit cell of  copper.

We can make bronze, which is an alloy of copper and tin, by removing some of the copper atoms and replacing them with tin atoms; such an alloy is called a random substitutional solid solution — indicating that the tin atoms substitute for copper atoms in the unit cells of the crystal in a random fashion, thus mixing the metals at the atomic scale to form a “solid solution” (which word, as we noted in one of my earlier posts, was probably used for the first time to describe bronze crystals).

On the other hand, if we mix gold atoms to copper to make a copper-gold alloy at the ratio of three copper atoms to one gold atom, then, the cube corners of the unit cell above are preferentially occupied by the gold atoms, while those at the face centres are occupied by copper. Such alloys are known as ordered alloys; and, it is also known that such alloys, on heating, lose this orderliness and become random, substitutional solid solutions at high temperatures. This transformation is known as the order-disorder transformation.

A model to describe these transformations was developed (as we shall see below, among others) by  W L Bragg and E J Williams in a series of papers [1-3]. In this post, I would like to talk about these papers, the model proposed therein and the order-disorder transformations themselves.

Apparently, the first indication that such ordered alloys exist came from some chemical experiments of G Tammann and X-ray studies of Johannson and Linde (although, unfortunately, I do not have access to either of these studies); see the review of Nix and Shockley [4] for a short and concise introduction to the historical aspects of the discovery of ordered structures and order-disorder transformation.

### Why ordering?

Ordering (and, its complementary arrangement of phase separation) can be understood using the following considerations [1]; let us consider an alloy that consists of two types of atoms, say A and B. In the alloy, as the schematic below indicates, there are three types of bonds that exist between the two types of atoms, namely, A-A, B-B, and A-B.

Let the interaction energies or the bond strengths for these three types of bonds be given, respectively, by $E_{AA}$, $E_{BB}$, and $E_{AB}$. Then, if $2E_{AB} - E_{AA}- E_{BB} < 0$, it is preferable for the A (B) atoms to be surrounded by B (A) atoms, which leads to ordering (on the other hand, if $2E_{AB} - E_{AA}- E_{BB} > 0$, then, it is preferable for the A (B) atoms to be surrounded by other A (B) atoms, which leads to phase separation).

However, this tendency to order (or phase separate) is spoiled by the configurational entropy — which prefers random arrangement of atoms at all lattice points with no specific preference at all; as temperatures increase, the contribution of the entropy to the free energy increases; and, hence, an alloy, which orders (or phase separates) at low temperatures will become a random, substitutional solid solution at high temperatures.

### Bragg-Williams model

The explanation of ordering in terms of the competition between the atomic interactions (bonding) and the configurational entropy was given by Bragg and Williams (though, probably, not for the first time — they themselves refer to earlier work by Dehlinger and others); to quote from their first paper in the series [1]:

The present paper is concerned with the effect of thermal conditions upon the manner of distribution of the atoms amongst the phase sites. If the metal atoms of either kind not only played essentially the same part in the structure, but were effectively identical, their distribution amongst the phase sites would be random. Actually the differences between them tend to cause atoms of the one kind to segregate into certain particular sites forming an orderly arrangement, because such an arrangement lower potential energy than one of disorder. Thermal agitation has the opposing effect of creating a random arrangement. The actual state of dynamical equilibrium of an alloy is one in which the two processes balance.

Bragg and Williams go on to discuss the structure of some of the ordered alloys known then using, primarily, the X-ray experiments. The first figure in the first paper (see below) shows a couple of ordered alloy crystal structures, the second of which, is the $Cu_3Au$ structure which I described above.

Bragg and Williams, are probably the first to consider the kinetics of order-disorder transformations, based on this idea of the competition between two opposing tendencies — the ordering one from the point of view of lowering of potential energy and the randomising one from the point of view of configurational entropy.

The crucial idea behind the model of Bragg and Williams is the introduction of an auxiliary parameter (known as the order parameter) to describe the degree of order in the alloy at various temperatures; the free energy of the system is then described in terms of this order parameter; finally, the equilibrium of the system is determined by minimising the free energy with respect to the order parameter. More specifically, the order parameter gives the probability of occupation of the given sites at the given temperature. In more complex ordered structures, more than one order parameter is needed to carry out the analysis.

The Bragg-Williams model is, in the modern statistical mechanical parlance, a mean field model; it is one of the possible mean field models; further, there is a close connection between this model for the treatment of the preference of atoms for particular sites to that of ferromagnetic order of spins; for example, see the lecture notes (Lectures 11-13) of this statistical mechanics course for more detailed and mathematical description of these connections and inter-relationship, or, the fourth chapter of Chaikin and Lubensky [6]. What is important to note here is the fact that none of this did escape Bragg and Williams:

The general conclusion that order sets in abruptly below a critical temperature $T_c$ has a close analogy in ferromagnetism, and there are, in fact, many points of similarity between the present treatment and the classical equations of Langevin and Weiss. We may compare the degree of order in the alloy with the intrinsic field of a ferromagnetic.

Bragg and Williams go on to use their model to study the effect of this transition on the properties of the alloy — resistivity and specific heat — and compare their results with the experimental ones. The final calculation that Bragg and Williams attempt in their paper is the calculation of the time of relaxation — how long does it take for the alloy to reach its equilibrium order at any given temperature.

I can do no better than to quote Bragg and Williams verbatim to summarise their second [2] and third [3] papers.

#### Summary of second paper:

The treatment given in a previous paper of the order-disorder transformation in alloys, and a recent more rigorous analysis along similar lines by Bethe, is compared with the formal treatments of Borelius, Gorky, and Dehlinger.

Measurements of specific heat by Sykes make it possible to estimate the changes in internal energy due to the transformation in the alloys CuZn and Cu$_{3}$Au. These measurements are compared with the results of theory.

Expressions are given for the dependence of critical temperature upon composition.

#### Summary of third paper:

The equilibrium equation for a general case, assuming the energy associated with the atomic arrangement to be uniquely determined by the superlattice order, is derived from the Boltzmann distribution formula and also from the principle of minimum free energy. The entropy and energy corresponding to the solution recently given by Bethe, assuming the energy of the atomic arrangement to be determined by the order of nearest neighbours, is also discussed, and a method of estimating the quantitative relation of Bethe’s type of treatment to that of Bragg and Williams is given. The paper concludes with a general discussion of existing theory and of the possible modifications for improving the relation to experiment.

### The classic

Order-disorder transformations have played an important role in materials science — specifically, in modern physical metallurgy; hence, it is not surprising that the Bragg-Williams model for its description is an important piece of work; and almost all materials scientists would be familiar with the model in some form or other. Thus, reading the original series is both less effortful and more edifying.

The crucial roles that the Bragg-Willaims model played in (a) describing a phase transformation of great practical importance, (b) early metallurgical education in the middle of the last century, and, (c) being a meeting point of solid state physics, statistical mechanics and physical metallurgy are described by Cahn, in his must-read, The coming of Materials Science, in a nice and concise fashion; I will end this post by quoting the same:

… in the 1930s it was recognised that the explanation was based on the Gibbsian competition between internal energy and entropy: at high temperature entropy wins and disorder prevails, while at low temperatures the stronger bonds between unlike atom pairs win. This picture was quantified by a simple application of statistical mechanics, perhaps the first application to a phase transformation, in a celebrated paper by Bragg and Williams (1934) … Both the experimental study of order-disorder transitions [...] and the theoretical convolutions jave attracted a great deal of attention, and ordered alloys, nowadays called intermetallics, have become important structural materials for use at high temperatures. The complicated way in which order-disorder transformations fit midway between physical metallurgy and solid-state physics has been surveyed by Cahn (1993,1998).

The Bragg-Williams calculation was introduced to metallurgical undergraduates (this was before materials science was taught as such) for the first time in a pioneering textbook by Cottrell (1948), … Bragg-Williams was combined with the Gibbsian thermodynamics underlying phase diagrams, … This book marked a watershed in the way physical metallurgy was taught to undergraduates, and had a long-lasting influence.

Have fun (till me meet again, next month, with another classic!).

### 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).

[5] D A Porter and K E Easterling, Phase transformations in metals and alloys, Chapman & Hall, 18-26 (1992).

[6] P M Chaikin and T C Lubensky, Principles of condensed matter physics, Cambridge University Press (2000).

[7] R W Cahn, The coming of materials science, Pergamon Press, 102-103 (2001).

I am an Assistant Professor in a Metallurgical Engineering and Materials Science Department; I also pursue research in the broad area of computational materials science.
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### 5 Responses to Classics in Materials Science: The Bragg-Williams model of order-disorder transformations

1. Abi says:

@Guru: Good stuff. A couple of questions:

Immediately after the figure for ordered structures (from the first BW paper), you say: “Bragg and Williams, are probably the first to consider the kinetics of order-disorder transformations

But the rest of the post doesn’t talk much about the kinetics; it’s all thermodynamics!

Towards the end, you return to kinetics again, and say, “The final calculation that Bragg and Williams attempt in their paper is the calculation of the time of relaxation — how long does it take for the alloy to reach its equilibrium order at any given temperature.” What was the nature of this calculation? How did they estimate the time?

It would be interesting if you could comment on this part of their work as well.

Thanks again, for another interesting post. Look forward to the next one — too bad we have to wait for a month!

2. Guru says:

Dear Abi,

Thanks.

Though, the way I signed off the post gives the impression that I will post only once in a month, that is not what I intended. I only meant that in this series, I will post only one per month (since it does take lots of time to identify a paper or a series, give a couple of readings and try to write something, even if cursory). Otherwise, I do hope to make regular posts.

Coming to the kinetics question, till you noted, I did not realise that I was emphasizing the contribution of Bragg and Williams to kinetics without talking about the details; it happened like this; after reading Cahn (and, having seen so may books and paper which call it Bragg-Williams), I went to their papers thinking that they are the first ones to think of this competition between the enthalpy and entropy; however, reading through the first two papers in the series dispelled that notion; they themselves acknowledge that several others has toyed with the idea before them; they only seems to have claimed that they will talk about kinetics, approach to equilibrium and how it is related to the quenching and annealing experiments, which were not dealt with in any good detail in the earlier papers. It was that impression which I was trying to convey when I wrote that sentence about their being probably the first to consider the kinetics question.

In any case, now that the question has come up, and, I do see that a discussion on those aspects should also be part of the post, I will have a second part for the post (with figures and all) sometime soon [since, as somebody else famously remarked in another instance, the comments section is too small to contain such a complete explanation :-) ]

3. Iti says:

hi Guru,
Thanks a lot. the article was very helpful.
Could you please write more stuff about the kinetics and the order parameters, and the quasichemical approach?