## Measurement of equilibrium vacancy concentrations

Vacancies as equilibrium defects

In a crystal, by definition, there is a regular arrangement of atoms; however, this regular arrangement is disrupted by defects of various kinds; point defects, more specifically, vacancies, are one of them; see this wiki page in crystallographic defects for some schematics.

Vacancies are equilibrium defects; by that what we mean is that the thermodynamic equilibrium demands the existence of these defects (at any temperature above absolute zero) in crystals.

The necessity for the creation of vacancies at any (absolute) temperature T can be understood by the following argument [1]: Let us consider a vacancy; because of its presence in the lattice, some bonds, which would otherwise have been satisfied, are now broken, and this costs the system some energy, resulting in an increase in the internal energy and hence the enthalpy of the system. This increase is directly proportional to the fraction of sites that are left vacant; that is, proportional to the vacancy concentration: $\Delta H \approx X_v \Delta H_v$, where, $X_v$ is the mole fractions of vacancies, $\Delta H_v$ is the increase in enthalpy per mole of vacancies created and $\Delta H$ is the total increase. On the other hand, the creation of vacancy leads to an increase in the entropy of the system; this increase comes in two ways: one is called the configurational entropy which is due to the fact that there are many different ways in which the vacancies can be distributed in the lattice; the other is the thermal entropy due to the changes in vibrational frequencies of atoms around the vacancies. Thus, the total change in entropy is $\Delta S = X_v \Delta S_v - R [ X_v \ln{X_v} + (1-X_v) \ln{(1-X_v)}]$. Hence, using these two expressions, the molar free energy $\Delta G = \Delta H - T \Delta S$ can be expressed in terms of $X_v$. In the specific case where $X_v \ll 1$, by differentiation, one can obtain the equilibrium concentration of vacancies $X_v^e$ for which the free energy is a minimum at any given temperature T. Schematically (following [1]), this optimization is depicted as below:

Graphical representation of minimization of free energy and equilibrium vacancy concentration

The work of Simmons and Balluffi

Simmons and Balluffi, in a series of papers published in Physical Review between 1960 and 1963, measured and reported the equilibrium concentration of vacancies in aluminium, silver, gold and copper [2-5]. In this post, I would like to talk about these papers — more specifically, about the first one, in which the equilibrium concentration of vacancies in aluminium was reported.

As Cahn notes in his wonderful (and must-read for every materials scientist) The coming of materials science, (a) even in 1920s it was known that vacancies are equilibrium defects and that they should exist in every crystal at any temperature above absolute zero, and, (b)  the experimental approach of Simmons and Balluffi (that came nearly four decades afterwards) to measuring these quantities for metals at various temperatures by comparing dilatometric measurements with precision measurements of lattice parameter is one of the very fruitful ones [6].

A side note: Cahn, while talking about point defects, notes why modern materials science is not physical metallurgy, and how

... the gradual clarification of the nature of point defects in crystals (...) came entirely from the concentrated study of ionic crystals, and the study of polymeric materials after the Second World War began to broaden from being an exclusively chemical pursuit becoming one of the most fascinating topics of physics research.

In any case, the relevant sections of Cahn's book (3.2.3.1 on point defects. 4.2.2 on diffusion and 5.1.3 on radiation damage) are very informative, readable, enjoyable and are a must-read; they also serve as nice historical introduction to the development of the theoretical concepts and experimental methodologies.

The paper of Simmons and Balluffi begins with a statement of the state of things as it existed at that time:

The experimental determination of the predominant atomic defects present in thermal equilibrium in metals has proven to be a difficult problem. Even though the general thermodynamic theory of point defects is well developed, experiment has not yet established the nature and concentrations of the defects in a completely satisfactory way.

Another digression: One of the references at the end of the second sentence in the quote above is to the work of Feder and Nowick [7]; Cahn calls the approach of Feder and Nowick, "ingenious", and the plot in Figure 2 of Feder and Nowik is the same as that in Simmons and Balluffi's Figure 3; in that sense, Simmons and Balluffi were following up on the work of Feder and Nowick. However, while Feder and Nowick wrote

It must therefore be concluded that, in the case of Pb and Al, an unexpected increase in a physical property at high temperatures must be explained in terms of the anharmonicity of the lattice vibrations, rather than in terms of large vacancy concentrations[.]

Simmons and Balluffi conclude

... it is concluded that they are predominantly lattice vacancies.

In fact the number that Simmons and Balluffi quote for the number of vacancies at melting point is of the same order (but greater by a factor of 3) to that quoted by Feder and Nowick.

Is it that Feder and Nowick tried the methodology and (a) got the right results but wrongly concluded that it is not of great use, or, (b) did not make accurate measurements and hence were lead to the wrong conclusion? To me, it looks like the answer is (b) -- one of the important conclusions of Simmons and Balluffi (in their own words) is

[In the experiments of Feder and Nowick] ... aluminium gave a rather questionable indication that vacancies are the predominant defect at elevated temperatures.

...

It appears that the method may be sufficiently precise to serve as a tool for investigating point defects in many substances.

However, this is a question on which I would love to hear from an historian of science. 

The experimental approach used by Simmons and Balluffi is very special among the various methodologies used to measure equilibrium vacancy concentration; while other methods (based on the measurements of internal friction, electrical resistivity, specific heat etc) give indirect information,

There is one type of experiment, however, that appears to be capable of giving direct information about the nature of predominant defects.

This method, as we noted above,

… consists of measuring the differences between the fractional lattice parameter change, $\Delta a / a$, as measured by x-ray diffraction, and the linear dilatation of the specimen, $\Delta L/L$, as defects are generated in a crystal containing constant number of atoms.

Or, in other words, when a material is heated its dimensions change which is directly related to the change in the lattice parameter of the material. In addition, if there are more vacancies (or, lattice points) that are created, there will be a discrepancy between the two (since the dilatation will also measure the change in length due to the creation of these extra lattice points) and this difference can then be used to calculate the vacancy concentration. The idea is as simple as that!

Of course, one of the things I like a lot about Simmons and Balluffi’s experiment is that the theoretical basis for their actual calculation was provided by Eshelby, in a theorem:

Eshelby has shown that a uniform random distribution of cubically symmetric point centers of dilatation (point defects) in an elastic material with cubic elastic constants will produce a uniform elastic strain of the crystal without change of shape. When uniform straining without change of shape occurs, the reciprocal lattice undergoes a uniform strain equal and opposite to the uniform strain of the crystal lattice; and the fractional change of lattice constant is equal to the fractional change in linear dimensions of the crystal.

From Simmons and Balluffi’s paper, I see that Eshelby’s theorem was also tested experimentally; those are probably some of the papers along with that of Eshelby which deserve a post of their own, and I might do that some time!

The key aspects of Simmons and Balluffi’s paper, as far as experimental technique is concerned, are

[a] that the lattice parameter and dilatational measurements were carried out at the same temperature, and as a function of temperature, and

[b] that the measurements were highly accurate — to within 1 in 100,000 for both the quantities.

And, to satisfy the above conditions, Simmons and Balluffi built a special apparatus such that

… both $\Delta L/L$ and $\Delta a/a$ could be measured simultaneously on the same specimen and where all temperatures were measured with the same thermocouple.

The details of the apparatus, speimen, furnace, and the precautions taken during the experiment and the measurement make fascinating reading — to give a few examples:

… Recrystallisation and grain growth occurred, and a final bamboo-type grain structure was produced where single large grains, several cm in size, occupied the full width of the bar.

… The paramount considerations in the desing of the present measurements of the change in length were that (1) the speciment be quite free of external constraint during expansion and contraction, (2) the influence of creep at the highest temperatures be minimized, and (3) the necessary sensitivity be obtained with very direct means, avoiding any apparatus which would be in a temperature gradient and therefore subject to possible unknown error.

… Characteristics required of the x-ray lattice expansion measurement included (1) high accuracy, (2) short measurement time, (3) a specimen having properties identical to those of the length change measurement speciment, and (4) careful desing so that the temperature distribution would be negligibly perturbed by the necessary access port for the x-rays.

The results of Simmons and Balluffi are, in some sense, not unexpected; they did find vacancies to be most dominant defects; as they note in their abstract, theirs

is the first direct measurement of formation entropy; the value is near that expected from theoretical considerations.

This success lead to their during similar experiments on copper, silver and gold, and as the review by Kraftmakher [8] point out, of the many important papers on point defects in metals, those of Simmons and Balluffi continue to be of relevance nearly four-and-a-half decades after their publication.

Current state

Simmons and Balluffi’s paper is a classic that shows how far you can go with careful experimentation (which are themselves based on strong theoretical foundations); the values reported by them were regarded the most reliable for nearly three decades after they published their paper [8]; even now, though there are some discrepancies between the more modern results and those reported by Simmons and Balluffi, the issue is far from settled (against Simmons and Balluffi, that is). Further, as Kraftmakher notes in his review [8], some of the original ideas and suggestions in Simmons and Balluffi (like the ones about quenched in vacancies and the lattice anharmonicity contributions at high temperatures) remain very much relevant and continue to be active areas of study.

As that great poem The dance of the solids by John Updike notes,

Textbooks and Heaven only are ideal;
Solidity is an imperfect state.

And, Simmons and Balluffi’s series of papers are nearly perfect on the imperfectness of the solid state of metals (noble and otherwise) and are worth reading and pondering on — Have fun!

References:

1. D A Porter and K E Easterling, Phase transformations in metals and alloys, Chapman & Hall, Second Edition, pp. 43-44, 1992.
2. R O Simmons and R W Balluffi, Measurements of equilibrium vacancy concentrations in aluminium, Physical Review, 117, 52-61, 1960.
3. R O Simmons and R W Balluffi, Measurement of the equilibrium concentration of lattice vacancies in silver near the melting point, Physical Review, 119, 600-605, 1960.
4. R O Simmons and R W Balluffi, Measurement of the equilibrium concentrations of lattice vacancies in gold, Physical Review, 125, 862-872, 1962.
5. R O Simmons and R W Balluffi, Measurement of the equilibrium concentrations of vacancies in copper, Physical Review, 129, 1533-1544, 1963.
6. R W Cahn, The coming of materials science, Pergamon Materials Series, Pergamon, First Edition, pp. 105-109, 2001.
7. R Feder and A S Nowick, Use of thermal expansion measurements to detect lattice vacancies near the melting point of pure lead and aluminium, Physical Review, 109, 1959-1963, 1958.
8. Y Kraftmakher, Equilibrium vacancies and thermophysical properties of metals, Physics Reports, 299, pp. 79-188, 1998.