Some Predicted Effects of Temperature Gradients on Diffusion in Crystals

Abstract

S HOCKLEY has recently proposed 1 that a continuous flow of defects will occur in a crystal in which there exists a temperature gradient, on account of the gradient of defect concentration which exists if the density of defects at any point has the equilibrium value characteristic of the local temperature at that point. Such a flow of defects, he suggests, will produce a mass transport through the lattice towards regions of higher temperature for vacancies and towards regions of lower temperature for interstitials and will be continuous except for limitation imposed by stresses that may be set up by the mass flow. Observations on the direction of such a mass transport, he then proposes, may provide a means of distinguishing between vacancies and interstitials.It is the purpose of this note to point out that a flow of defects may occur as a direct result of the temperature gradient as well as indirectly through the gradient of local equilibrium concentrations. We shall consider first the more interesting case of vacancies. Figure . 1 represents two adjacent lattice planes normal to the direction of the temperature gradient, n and T are, respectively, the numbers of vacancies per unit area and the temperature at plane (1), and n-\-An and T-\-AT are the corresponding quantities for plane (2). In order that a vacancy on plane (1) may move to plane (2), an atom on plane (2) must acquire at temperature T-j-AT an activation energy Q. The rate of flow of vacancies from plane (1) to (2) is therefore:where v is the vibration frequency of the atoms. Similarly, the rate of flow of vacancies from plane (2) to (1) is:The net flow of vacancies from plane (1) to (2) is therefore, for small dT/T,The second term gives the diffusion flow arising from the concentration gradient and is of course proportional to the gradient and directed towards regions of lower concentrations. The first termgives the "thermal diffusion'' flow and is directed towards regions of higher temperature. Now, if we assume with Shockley that the concentration of vacancies is and remains, at each point in the crystal, equal to that which would be found in equilibrium in a crystal at a uniform temperature the same as at that point, then we shall have An/n = (U/RT 2 )dT, where U is the energy of formation of a vacancy. Thus n increases in the same direction as does the temperature, and the normal diffusion flow is in a direction opposite to that of the thermal diffusion flow. If Q= U, then there will be no flow of vacancies at all. If QT± U, the net flow of vacancies will be towards the region of higher temperature when Q> U and towards regions of lower temperature when Q<U.In the case of interstitial diffusion, we find by reasoning similar to the above that the net flow of interstitials from plane (1) to (2) is /= Q(AT) An FIG. 1where Q is the activation energy for the jump of an interstitial, which in this case is to be acquired at the temperature of the plane on which the interstitial is situated, n is the number of interstitials and v their vibrational frequ...

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