Boundary motion

D. Fan, L.-Q. Chen. Diffuse-interface description of grain boundary motion. Phil Mag Lett 75 187, 1997. 

During the experiment the boundary displacement is recorded. Its derivative with regard to time is the velocity v of grain boundary motion, which is related to the driving force p by the boundary mobility m = v p. For convenience we use the reduced boundary mobility... 

K. Liicke, The orientation dependence of grain boundary motion and the formation of recrystallization textures, Can. Met. Quart. 13 261 (1974). 

K. Lucke and H. Stiiwe, The theory of grain boundary motion, in Recovery and Recrystallization of Metals, L. Hiuunel, ed.. Interscience, New York (1963). 

Denote boundary motion speed as u that may or may not depend on time. For crystal growth, the interface moves to the right with x = Xo>0. For crystal dissolution, the interface moves to the left with x = Xo<0. That is, u is positive during crystal growth and negative during crystal dissolution under our setup of the problem. The interface position can be found as... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

Growth or dissolution rate boundary motion velocity flow velocity... 

Figure 10-1. Field-driven cation flux and its effect on the boundary motion DA,DB>Dx,DY). a) Static interface, b) moving (without reaction), c) moving (with reaction, />., AY formation).

In a foregoing section, we mentioned that field forces (e,g., of the electric or elastic field) can cause an interface to move. If they are large enough so that inherent counterforces (such as interface tension or friction) do not bring the boundary to a stop, the interface motion would continue and eventually become uniform. In this section, however, we are primarily concerned with boundary motions caused by chemical potential changes. From irreversible thermodynamics, we know that the dissipated Gibbs energy of the discontinuous system is T-ab, where crb here is the entropy production (see Section 4.2). Since dG/dV = dG/dV = crb- T/ A < ), we have with Eqn. (4.8) at the boundary b... 

The main problem of the boundary motion, however, remains the description of relaxation processes that take place when supersaturated point defects are pumped into the boundary region A R. Outside the relaxation zone Asimple model of a relaxation box is shown in Figure 10-14c. The four exchange reactions 1) between the crystals a and /3, and 2) between their sublattices are... 

Thus, we conclude that the interface is morphologically unstable for negative vh if the flux of i indeed causes the boundary motion. (This flux, however, is not necessarily the rate determining one since all fluxes in the multicomponent system are coupled in one way or the other.)... 

Figure 11-5. Boundary velocity tib and the direction of fluxes (responsible for the boundary motion) in the contiguous phases a and 0. Figure 11-5 b corresponds to Figure ll-2a.

Let us finally comment on the morphological stability of the boundaries during metal oxidation (A + -02 = AO) or compound formation (A+B = AB) as discussed in the previous chapters. Here it is characteristic that the reaction product separates the reactants. 1 vo interfaces are formed and move. The reaction resistance increases with increasing product layer thickness (reaction rate 1/A J). The boundaries of these reaction products are inherently stable since the reactive flux and the boundary velocity point in the same direction. The flux which causes the boundary motion pushes the boundary (see case c) in Fig. 11-5). If instabilities are occasionally found, they are not primarily related to diffusional transport. The very fact that the rate of the diffusion controlled reaction is inversely proportional to the product layer thickness immediately stabilizes the moving planar interface in a one-... 

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