## Classics in Materials Science: Harper’s experiments with Snoek pendulum

### Introduction

Dislocations are one type of defect in a crystalline solid; they distort the crystalline lattice around them; these distortions around a dislocation in a crystal could be dilatational (the distance between planes is more than what it should be) or compressive (the distance between planes is less than what it should be). Such distortions (strains) cost the system energy.

Consider a solid solution; let us say it is made up of two components; typically, there is a size difference between the different types of atoms. So, in a crystalline lattice of a solid solution, there are also strains associated with the different components.

The strain fields around the solutes/solvents and the dislocations in the crystal of a solid solution can interact — leading to an overall reduction in the energy. One such interaction was pointed out by Cottrell. Imagine a dislocation in a crystal of a solid solution; if the larger atoms of the solution sit in the dilatational regions around a dislocation and the smaller ones near the compressive regions, then, the elastic stresses can partly be relieved. This kind of migration of atoms to regions in a crystal where they can relieve the distortions in a lattice the most, leads to what is known as Cottrell atmosphere — a clustering of atoms around a dislocation.

Further, Cottrell, along with Bilby, in 1949, wrote a classic paper (which deserves a post of its own — and will be given its due in one of the posts in the future), describing the yield and strain ageing of iron by considering the ‘condensation’ of the atmosphere on dislocations; in addition, in their paper,  Cottrell and Bilby also showed that the kinetics of re-condensation of the atmosphere on a dislocation that is torn from the one it initially had, can be described by a $t^{2/3}$ law, where, $t$ is the time of strain-ageing after a steel specimen has been plastically deformed.

Around the middle of the last century, an experimental verification of the predictions of Cottrell and Bilby was one of the challenging problems that faced the materials scientists (more specifically, the physical metallurgists). What was needed was a measurement of the change with time of the free carbon dissolved in iron at ambient temperatures — at temperatures where only a minute fraction of 1% of carbon can be dissolved in iron. As Cahn notes in his classic [2],

Harper performed this apparently impossible task and obtained [the required data] by using a torsional pendulum, invented just as the War began by a Dutch physicist, Snoek, …

In this post, let us take a look at the classic paper of Harper [3], how he managed to achieve what he did achieve, and discuss the relevance of Harper’s experiments then, and their continuing relevance now.

### Internal friction and stress induced interstitial diffusion

Harper, begins his paper by describing some of the earlier studies on strain ageing and in the very second paragraph sets forth how his experiment is different from the earlier ones:

By measuring the internal friction arising from the stress-induced interstitial diffusion of the solute atoms it is possible to measure the amount of dissolved solute at any time and thus conveniently and quantitatively study the precipitation process.

The key words are: internal friction (a mechanism of damping of oscillations in solid materials by converting the vibrational energy into heat) and stress-induced interstitial diffusion (the movement of atoms  in interstitial positions — that is, positions in a crystalline lattice where in general there are no atoms) due to applied stress. How are these two connected? Cahn, in his commentary on the paper, explains in lucid manner, as is his wont:

Roughly speaking, the dissolved carbon atoms, being small, sit in interstitial lattice sites close to an edge of the cubic unit cell of iron, and when the edge is elastically compressed and one perpendicular to it is stretched by an applied stress, then the equilibrium concentrations of carbon in sites along the two cube edges become slightly different: the carbon atoms “prefer” to sit in sites where the space available is slightly enhanced. After half a cycle of oscillation, the compressed edge becomes stretched and vice versa. When the frequency of oscillation matches the most probable jump frequency of carbon atoms between adjacent sites, then the damping is maximum.

A carbon atoms situated in an ‘atmosphere’ around a dislocation is locked to the stress-field of the dislocation and thus cannot oscillate between sites; it therefore does not contribute to peak damping.

Of course, the next question is as to how to probe this peak; for that, Harper used the Snoek’s apparatus and the theories put forth by Zener in his classic text [4]. Let us digress a bit to look at Snoek’s apparatus before we return to the results of Harper.

### Snoek’s apparatus

Snoek is a Dutch physicist with varying contributions; according to this biographical sketch [5] of Snoek,

From a purely physical point of view, his finest work might be the discovery and explanation of the so-called diffusion damping in solids: the Snoek effect, …

Snoek’s paper describing the damping effect and its measurement using the apparatus (which is nothing but a torsional pendulum), not surprisingly, deals with carbon and nitrogen in alpha-iron [6], though, according to [5]

Nowadays, the term “the Snoek effect” is used in a wider sense, implying the relaxation phenomenonassociated with reorientation of siolated solute atoms in any crystal lattice or even in amorphous structure.

For studying the damping effect, Snoek used a torsional pendulum and its oscillations:

The simplest way of determining the elastic after effect of a given material seems to make it part of a freely oscillating system and measuring the logarithmic decrement $\Delta$ of the system.

A detailed description of the pendulum arrangement is described both in Snoek’s paper and in Cahn’s book (with a schematic). The overall idea is that a wire, held slightly in tension, when set into torsional vibrations, the amplitude of its vibration will slowly decrease, and this decrease can then be related to the internal friction associated with the movement of the dissolved carbon. In effect, the variation of the temperature of peak damping with the pendulum frequency, one can determing the jump frequency, and hence the diffusion coefficient — even at very low temperatures. Further, the magnitude of the peak damping is proportional to the amount of carbon in solution.

### Harper’s experiment

Harper used the same torsional arrangement except that his specimen of steel wire was stretched beyond its yield point — that is, to such stress levels as would make the material undergo permanent (plastic) deformation by introducing and moving dislocations inside the material. Now, if such a wire is clamped to a torsional pendulum of the type used by Snoek and set into oscillations, the decay of the damping coefficient with the passage of time can be followed, from which the amount of dissolved carbon as a function of time can be calculated. If ‘f’ be the fraction of carbon atoms that form the ‘atmosphere’ around dislocations, the logarithm of (1-f) as a function of time raised to the exponent (2/3) was shown by Harper to follow the exact linear relationship that Cottrell and Bilby predicted! See the figure below:

Nearly fifty years after the experimental proof offered by Harper for strain ageing, Cottrell atmosphere can now be directly imaged, as Cahn points out, using atom-probe [7].

### The relevance: then and now

Harper’s experiment, according to Cahn,

… encapsulates very clearly what was new in physical metallurgy in the middle of the [last] century. The elements are: an accurate theory of the effects in question, preferably without disposable parameters; and, to check the theory, the use of a technique of measurement (the Snoek pendulum) which is simple in the extreme in construction and use but subtle in its quantitative inprepretation, so that theory ineluctably comes into measurement itself.

On the other hand, the idea of relaxations and means of measuring them continue to be of interest and relevance. As Koiwa notes [5] in his sketch, the Snoek effect

… is probably the most fully investigated and well understood among a variety of relaxation phenomena.

And, finally, these ideas are of use in studying a material, which, to quote Kipling

Gold is for the mistress — silver for the maid —
Copper for the craftsman cunning at his trade.
“Good!” said the Baron, sitting in his hall,
“But Iron — Cold Iron — is master of them all.”

What more can one ask for!

### References

[1] A H Cottrell and B A Bilby, Dislocation theory of yielding and strain ageing of iron, Proceedings of Physical Society A, 62, pp. 49-62, 1949.

[2] R W Cahn, The coming of materials science, Pergamon Materials Series, Pergamon Press, pp. 191-195, 2001.

[3] S Harper, Precipitation of carbon and nitrogen in cold-worked alpha-iron, Physical Review, 83, pp. 709-712, 1951.

[4] C Zener, Elasticity and anelasticity of metals, The University of Chicago Press, Chicago, 1948.

[5] M Koiwa, A note on Dr. J. L. Snoek, Materials Science and Engineering A, 370, pp. 9-11, 2004.

[6] J L Snoek, Effect of small quantities of carbon and nitrogen on the elastic and plastic properties of iron, Physica, 8, pp. 711-733, 1951.

[7] J Wilde, A Cerezo, and G D W Smith, Three-dimensional atomic scale mapping of a Cottrell atmosphere around a dislocation in iron, Scripta Materialia, 43, pp. 39-48, 2000.