Temperature Dependence of Diffusion Coefficient

From diffusion experiments one learns that the temperature dependence of the diffusion coefficient can be written as = Z)f exp ( -QlRT), which is an Arrhenius-type relation. For example, in the case of carbon diffusion in iron or sodium diffusion in j5-alumina an Arr-henius-type relation is obeyed over many orders of magnitude of the diffusivities. According to eq. (5-19), Dj is the product of a point defect concentration and a diffusion coefficient of the point defects. Eq. (4-3) shows that the mole fraction of point defects is an exponential function of the reciprocal of the absolute temperature for low defect concentrations. Therefore, the individual jump frequency of a point defect, or of an atom or ion which is moved by the jump of a point defect, depends upon the temperature as exp - QjRT), 

There are two different ways to explain the experimentally observed temperature dependence. The first method employs the concept of the activated transition state in the absolute reaction rate theory [33]. Here it is assumed that the diffusion particle crosses an activation energy barrier between two equivalent lattice sites. One calculates the probability of the particle being on the saddle-point (transition state) and its velocity there. This implies that an equilibrium distribution of diffusing particles between normal lattice sites and the saddle-points exists. It is further assumed that the diffusing particles in the saddle-point configuration 

The diffusion coefficient /), was introduced in eqs. (5-4) and (5-13). From eq.(5-ll) it can be seen that is a measure of the mobility or of the jump frequency of the particles of type L Ever since sufficient quantities of stable or radioactive isotopes for most elements have become available, the so-called tracer method of measuring diffusion coefficients has been widely used. In this method, small quantities of isotope are permitted to diffuse into the system under investigation, and isotopic effects are neglected. In completely homogeneous material, the mean square displacement xf of the tracer atoms is experimentally determined. The following formula then applies [13]  

For purposes of illustration, a few situations will be briefly discussed. 

Correlation effects during vacancy diffusion in a metal A. As has already been shown, if the vacancy and the tracer atom exchange sites twice in succession, no net motion of the tracer atom takes place. We would like to be able to use the measured displacement of the tracer atom in order to calculate the jump frequency or the mobility of the vacancy. The jump frequency could then be used to calculate the diffusion coefficient in the metal A. In order to do this, then, we must calculate the probability of a repeated exchange of sites between a tracer and the same vacancy. The calculation is based upon the known geometry of the lattice and upon the assumption that the direction of a vacancy jump is independent of the direction of all previous jumps. This can be expressed quantitatively as follows Let xl (A ) be the mean square displacement of the tracer A and let xi (V) be the mean square displacement of the vacancy V after n jumps (lim n — oo). The correlation factor is then defined, by means of eqs. (5-11), (5-19), and (5-20), as  

---

This situation is termed pore-mouth poisoning. As poisoning proceeds the inactive shell thickens and, under extreme conditions, the rate of the catalytic reaction may become limited by the rate of diffusion past the poisoned pore mouths. The apparent activation energy of the reaction under these extreme conditions will be typical of the temperature dependence of diffusion coefficients. If the catalyst and reaction conditions in question are characterized by a low effectiveness factor, one may find that poisoning only a small fraction of the surface gives rise to a disproportionate drop in activity. In a sense one observes a form of selective poisoning. 

EXAMPLE 2.6 Temperature Dependence of Diffusion Coefficients. Suppose the diffusion coefficient of a material is measured in an experiment (subscript ex) at some temperature Tex at which the viscosity of the solvent is qgx. Show how to correct the value of D to some standard (subscript s) conditions at which the viscosity is j s. Take 20°C as the standard condition and evaluate D°20 for a solute that displays a D° value of 4.76-10 11 m2 s 1 in water at 40°C. The viscosity of water at 20 and 40°C is 1.0050-10 2 and 0.6560 -10 2 P, respectively. 

Most probable settling velocity from sedimentation data Particle-size determination from sedimentation equation Sedimentation in an ultracentrifuge Solvation and ellipticity from sedimentation data Diffusion and Gaussian distribution Temperature-dependence of diffusion coefficients... 

A full model of the charge transport in the electrolyte would require the detailed description of the ionic transport processes inside the electrolyte. However, for the orientating study pursued in this contribution, it seems more appropriate to choose a simpler model that is able to describe the temperature dependence of the electrolyte qualitatively. The temperature dependence of diffusion coefficients in molten electrolytes can be described by an Arrhenius function [1]. Therefore, the temperature dependence of the conductivity is assumed to be of an Arrhenius type, as suggested in [6]. 

A. Kishimoto and K. Matsumoto Concentration and temperature dependence of diffusion coefficients for systems polymethyl acrylate and n-alkyl acetates. Trans. Faraday Soc. 56, 424 (1960). 

Fig. 3.112. Temperature dependence of diffusion coefficient D of surface-adsorbed 5-N-(octadecanoyl)aminofluorescein in black foam films stabilised by DLPE ( ) DMPE (A) DPPE ( ) and DOPE (+) [492],...

Figure 14 Temperature dependence of diffusion coefficients for the Cr-N system obtained from layer growth and concentration profiles, / 5-CrNi, o/ S-CrzN, A a-Cr(N)...

If the mechanism of material transport in sintering is grain boundary diffusion, the viscosity can be expressed as a function of the temperature and the grain size. From a model for boundary diffusion controlled creep proposed by Coble [7] and the temperature dependence of diffusion coefficients, we obtain... 

Temperature dependence of diffusion coefficients in metals. Reprinted widi permission... 

Figure 7.2 Temperature dependence of diffusion coefficients for some common ceramic oxides. ...

Figure 1.4.28 shows plot of diffusion coefficient, D, (mutual diflusivity between PMAA and water) vs temperature. This figure shows the temperature dependence of diffusion coefficient obtained from the slope of the lines in Figs. 1.4.26 and 1.4.27 and using Eqs. (1.4.14-1.4.16). Since difffisivity vanishes at the spinodal temperature, by extrapolating the linear relationship, the spinodal temperature was estimated to be 63 C for a 7% PMAA/water system and about 62°C for the 10% PMAA/water system. 

The various results on diffusion coefficients are given in [108] with different models of diffusion processes in liquid crystals. The temperature dependences of diffusion coefficients and their anisotropy are estimated in [32]. 

Membrane Technology, Fig. 3 Temperature dependencies of diffusion coefficients surface exchange... 

It is also possible to estimate the spatial distribution of steady state concentration in chimney, giving temperature dependency of diffusion coefficient, D. The amounts of precipitation of minerals can be calculated from the saturation index that is obtained from the distribution of concentration of fluid in chimney. 

---