Ternary diffusion concentration profiles

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary case. 

Figure 6.2 Concentration profiles for Ni and Co in ternary diffusion couple fabricated...

The results of ternary diffusion experiments are often presented in the form of diffusion paths, which are plots of the concentrations measured across the diffusion zone. The diffusion path corresponding to the measurements in Fig. 6.2 is shown in Fig. 6.3 note the characteristic S-shape, due to the inflections in the CNi profile. 

Because the diffusion profiles C x) and C2(x) are known, the fluxes J and J2 can be determined at any x by inverting C x) and C2(x) and evaluating the integrals in Eqs. 6.53 for i = 1,2. Because dc /dx and dc2/dx are known at the selected x, two equations relating the four diffusivities are obtained. Therefore, if two ternary diffusion experiments are analyzed at a point of common concentration, four equations relating the four diffusivities at those common concentrations will be obtained and all four diffusivities can be determined. However, use of this method to derive diffusivities from experimental results is highly labor intensive and subject to significant error [3]. 

For mass transfer in a simple ternary system without chemical reaction, the solute concentration profiles near the interface are as shown in Fig. 3. The concentration in the bulk of each phase is uniform because of convective mixing effects, but very near the interface the rate of mass transfer depends increasingly on molecular diffusion. 

Consider one-dimensional, steady-state, ternary diffusion along a capillary tube as shown in part (a) of the accompanying figure. Two immiscible liquids occupy the two halves of the tube with the position of the interface between them taken as the origin of the coordinate system (z = 0). Compositions A and D at the ends of the tube are known. Compositions B and C at the interface are not known initially. But if local equilibrium is assumed at the interface, B and C must be at the ends of a tie-line on the (known) ternary phase diagram as shown in part (b) of the figure. The question is which tie-line Once the tieline is determined, the concentration profiles are known and the question of whether spontaneous emulsification occurs can be settled. 

Example 7.3-1 Fluxes for ternary free diffusion Find the fluxes and the concentration profiles in a dilute ternary free-diffusion experiment. In such an experiment, one ternary solution is suddenly brought into contact with a different composition of the same ternary solution. Find the flux and the concentrations versus position and time at small times. 

Because air is really a misture, the exact solution involves ternary diffusion coefficients that can be calculated from Table 7.1-1. Calculate the ternary concentration profile and compare it with the binary one (S. Gehrke). 

Values of a diffusion coefficient matrix, in principle, can be determined from multicomponent diffusion experiments. For ternary systems, the diffusivity matrix is 2 by 2, and there are four values to be determined for a matrix at each composition. For quaternary systems, there are nine unknowns to be determined. For natural silicate melts with many components, there are many unknowns to be determined from experimental data by fitting experimental diffusion profiles. When there are so many unknowns, the fitting of experimental concentration... 

Figure 3-21 Calculated diffusion profiles for a diffusion couple in a ternary system. The diffusivity matrix is given in Equation 3-102a. The fraction of Si02 is calculated as 1 - MgO - AI2O3. Si02 shows clear uphill diffusion. A component with initially uniform concentration (such as Si02 in this example) almost always shows uphill diffusion in a multi-component system.

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