Moving Interfaces

The phase-field model and generalizations are now widely used for simulations of dendritic growth and solidification [71-76] and even hydro-dynamic flow with moving interfaces [78,79]. One can even use the phase-field model to treat the growth of faceting crystals [77]. More details will be given later. 

Fig. 18. A sequence of 100 x 100 nm images of the Cu(lll) surface in 55.0 m/o AlCl3-EtMeImCl as the potential was stepped from (a) 0.204 V to (b)-(d) 0.354 V revealing the moving interface/step demarking the boundary between aluminum desorption and tetrachloroaluminate adsorption. Reproduced from Stafford et al. [104] by permission of The Electrochemical Society.

The photoablation behaviour of a number of polymers has been described with the aid of the moving interface model. The kinetics of ablation is characterized by the rate constant k and a laser beam attenuation by the desorbing products is quantified by the screening coefficient 6. The polymer structure strongly influences the ablation parameters and some general trends are inferred. The deposition rates and yields of the ablation products can also be precisely measured with the quartz crystal microbalance. The yields usually depend on fluence, wavelength, polymer structure and background pressure. 

Chadam, J. Ortoleva, P. (1984). Moving interfaces and their stability Applications to chemical waves and solidification. In Dynamics of Non-Linear Systems, ed. V. 

Coulson, J. M. and Skinner, S. J. Chem. Eng. Sci. 1 (1952) 197. The mechanism of liquid-liquid extraction across stationary and moving interfaces. Part 1. Mass transfer into single dispersed drops. 

U. Czubayko, D. A. Molodov, B.-C. Petersen, G. Gottstein and L. S. Shvindlerman, An X-Ray Device for Continuous Tracking of Moving Interfaces in Crystalline Solids, Meas. Set. Technol. 6 947 (1995). 

As illustrated by (3.2.11), for m > 2 the first derivative of concentration at the boundary of support is discontinuous that is, a weak shock is formed at the zero concentration front. This stands in accord with the classical Rankine-Hugoniot condition that prescribes for any moving interface Xi(t)... 

Let us begin the discussion of the last example of solid state kinetics in this introductory chapter with the assumption of local equilibrium at the A/AB and AB/B interfaces of the A/AB/B reaction couple (Fig. 1-5). Let us further assume that the reaction geometry is linear and the interfaces between the reactants and the product AB are planar. Later it will be shown that under these assumptions, the (moving) interfaces are morphologically stable during reaction. 

Experiments have shown that Aoxide spinel formation is on the order of 10 4cm at ca. 1000°C [C.A. Duckwitz, H. Schmalzried (1971)]. Using Eqns. (10.45) and (10.46) with the accepted cation diffusivities (on the order of 10 10 cm2/s), one can estimate from j% that each A particle crosses the boundary about ten times per second each way. In other words, quenching cannot preserve the atomistic structure of a moving interface which developed during the motion by kinetic processes. This also means that heat conduction is slower than a structural change on the atomic scale, unless one quenches extremely small systems. 

Figure 10-11. Distribution of point defects near a moving interface during transformation a- 0.

Figure 10-14. Relaxation processes at a moving interface, a) Scheme of boundary a/0 during transformation, b) coupling of transport and relaxation processes, c) detailed structure element steps in the relaxation zone.

To summarize the structure of a moving interface on the atomic scale depends on the atomic mechanism which operates in the structure transformation. The mode selection depends on the driving force and thus on the interface velocity. The interface mobility itself is determined by its structure and depends therefore on the driving force. This means that interface controlled reactions are normally nonlinear functions of the driving force. 

Normally, it is not possible to obtain analytical solutions for this transport problem and so we cannot a priori calculate the reaction path. Kirkaldy [J. S. Kirkaldy, D. J. Young (1985)] did pioneering work on metal systems, based on investigations by C. Wagner and the later work of Mullins and Sekerka. They used the diffusion path concept to formulate a number of stability rules. These rules can explain the facts and are predictive within certain limits if applied properly. One of Kirkaldy s results is this. The moving interface in a ternary system is morphologically stable if... 

Conceptually, we thus have the same situation as introduced in Figure 11-2. Therefore, we conclude again that (since ub is directed towards the cathode) the moving interface is morphologically unstable if LB (BX)La(AX), in accordance with Figures 11-5 a and c. These predictions have been confirmed experimentally [S. Schimschal (1993)]. 

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