Measurement of solid state diffusion coefficients by a temperature-programmed method
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Measurement of solid state diffusion coefficients by a temperature-programmed method Rajat Kapoor and S. T. Oyamaa) Department of Chemical Engineering, Virginia Polytechnic, Blacksburg, Virginia 24061-0211 (Received 3 April 1996; accepted 25 October 1996)
This paper presents a method for determining diffusivities in solids where the diffusing species desorbs or reacts at the external surfaces, and where the diffusivity does not vary appreciably with concentration. The method involves measuring the flux of the diffusive species out of the solid under the influence of a temperature program. A general model is developed, based on nonisothermal Fickian diffusion, which is applicable to solid particles with slab or spherical geometry. The solution is presented both as an analytical expression and as correlation charts of experimentally observable quantities. These charts are contour diagrams of the temperatures of peak diffusion rate with ln(EyR) and ln(D0yh2 ) as the axes, where E and D0 are the activation energy and pre-exponential terms of the diffusivity expression D D0 exps2EyRT d, where R is the gas constant, and h the size of the particles. This paper deals exclusively with the case of oxygen diffusion in the vanadium oxide system. In this case, vanadium oxide was reduced in a reactive ammonia stream at conditions in which the surface reaction was fast compared to the diffusive transport process. Using this method the diffusion parameters were found to be D0 1.9 3 1025 cm2 s–1 and E 101 kJymol. The method was checked by varying the crystallite size of the vanadium oxide sample in the range 2h 0.14 –0.29 mm.
I. INTRODUCTION
Diffusion studies often involve measurement of diffusivity by “direct” or “indirect” methods. The direct methods require simultaneous measurement of diffusive flux and concentration gradients within the solid, such as depth profiling analysis,1 x-ray imaging,2 or NMR/ESR spin mapping,3 which are used directly in Fick’s equation for determination of diffusivity. The indirect methods rely on measurements at the external surface to give information on the internal gradients. For instance, the rate of loss of the diffusing species through a surface can be used to determine the flux, and applied to an appropriate solution of Fick’s equation to give the diffusivity and concentration gradients. Methods commonly applied here are either batch type (gravimetric, volumetric, frequency response, etc.)4 or flow type (chromatographic, zero length column, etc.).5,6 In this paper a temperature-programmed technique is used to determine oxygen diffusivity in vanadium pentoxide. The diffusivity is determined by matching an observable experimental quantity, namely the peak temperature of the species diffusing out of the solid, to a Fickian diffusion model for the process. The solution to the diffusion model is presented in a graphical form that correlates the pre-exponential diffusivity, D0 , to the activation energy, E, the temperature at maximum diffua)
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