Transformation front

Ya.B. s studies of combustion and detonation are diverse and multidirectional. They include the chemical thermodynamics of combustion, propagation of exothermic chemical transformation fronts, deflagration and detonation theory, thermo-diffusion and chemo-kinetic processes in combustion and at high temperatures in general, and gasdynamics of flows in the propagation of non-uniform flame fronts and in detonation. 

Figure 7.62 Scanning transmission elec- trailing edge of the transformation front tron microscope (STEM) image of transfer- well-crystallised HAp is formed whereas at mation of ACP to crystalline phases on con- the leading edge nanocrystalline HAp pretact with revised simulated body fluid (r- vails (see text) (Heimann, 2007).

The morphology of the internal oxidation zone and stabihty of transformation front were studied in [29]. Solid-state reactions with formation of two-phase zones were analyzed in [3, 30, 31]. 

The obtained set of equations allows one calculating the diffusion path, to define the value and direction for the velocity of the transformation front depending on the initial compositions of diffusion couples and the values of the interdiffusion coefficient matrix. 

In the previous chapter, we considered the problem of selection of the path for the reactive diffusion when the choice is made among a finite number of phase formation modes. Let us now treat the problems allowing for an infinite set of solutions, all of which are compatible with the matter balance equations. These are mainly the problems of morphology choice when different modes are possible, and the task is to find the optimal one to be realized in practice. In the first two sections of the chapter, we consider the issue of low-temperature phase diffusion transformations, namely discontinuous precipitation and DIGM. At that, the evolution equations based on matter conservation laws allow an infinite number of solutions corresponding to different thicknesses of the phase formed and different velocities of phase transformation front movement. 

First, we treat a model of discontinuous precipitation of binary polycrystalline supercooled alloys at low temperatures as a result of DIGM. In the proposed approach, we independently determine the main parameters interlamellar distance, maximum velocity of the phase transformation front, and residual supersaturation at the front. This is achieved by using a set of equations for... 

In the model considered below, the role of both grain boundary and bulk diffusion in the transformation front and close to it, respectively, is analyzed within the problem of unambiguous determination of the discontinuous precipitation parameters in the binary Pb-Sn system at room temperature [9]. In order to complete this, we use the principle of maximum rate of free energy release and balance of entropy fluxes for the description of discontinuous precipitation kinetics for binary polycrystaUine alloys and independent determination of three basic parameters interlamellar distance, rate of phase transformation front, and concentration profile close to the transformation front. While solving the problem, we also find the optimal concentration distribution of components both along the precipitation lamella behind the transformation front and close to it, as well as the degree of the components separation. 

While analyzing the above-presented models, one realizes that the problem of mode choice cannot be unambiguously solved within the solution of mass transfer equation. This makes it necessary to consider thermodynamic or kinetic approaches to the analysis of transformation front stability and to choose a certain contact zone morphology. From the point of view of kinetics, the interphase boundary instability may be caused either by instabihty with respect to fluctuations of the boundary shape [15-17] or by the failure of balance equations for fluxes at the moving boundaries [16]. From a thermodynamic viewpoint, the problem of choice of one kinetically allowed mode can be solved using the variation principles of nonequilibrium thermodynamics [18-29],... 

Calculation of Energy Dissipation in the Transformation Front along the Precipitation Lamella... 

The main approximation employed here is the assumption of diffusion redistribution of components both inside the planar interphase boundary moving at constant velocity (Figure IZ.lbandc) and close to the transformation front because... 

Calculation of Energy Dissipation Close to the Transformation Front... 

To calculate the energy dissipation close to the phase transformation front due to bulk diffusion, we search for the concentration profile close to the front [9]. In order to do this, let us put down a steady-state equation of flux balance in the element of region R (Figure 12.6a) moving at constant velocity of the boundary v, in the reference frame of the transformation boundary ... 

To find the concentration profile close to the transformation front, we use both Equation 12.31 and the solution of Pick s equation for bulk diffusion close to the... 

Now, let us find the concentration distribution close to the transformation front at % > H from the solution of Equation 12.32 and taking into account the boundary... 

Assuming that just close the transformation front the concentration along the cell is c(z, x=h) = Cq, the free energy release rate Equation 12.37, using Equation 12.2, we may obtain... 

From the model calculations, it is clear that, by taking into account the energy dissipation induced by grain boundary diffusion in the transformation front and bulk diffusion close to iL the value of the tin fraction W is close to the experimental ones at average and large supersaturations. The obtained concentration profile along the a-phase ceU is presented in Figure 12.9b. Let us compare concentration dependencies for experimental and model values of interlamellar distance... 

Further, we can find Cay(Az, L) in Equation 12.48 after solving the optimi2ation problem (Equations 12.46 and 12.47). The triple product is used for the determination of the transformation front velocity also after the optimi2ation parameters 2 Sol jSol found. Therefore, the velocity of the transformation front is... 

First, we examine the behavior of the velocity of the front moving from Cq using the approximation (Equation 12.48) for shD. Lines 1 and 2 in Figure 12.10c show the dependence of the transformation front velocity taking into account the energy dissipation at the ao/a interface (only the term 4>pz(/ solj jjjjg 1, the sum of the terms dz(Az °, L ° ) and do(Az °, 1 ° ) - line 2). Here, the values of the velocity determined from the optimization procedure (i) are overestimated by an order for alloys with small supersaturation, (ii) coincide... 

Now, we consider the influence of concentration dependences (Equation 12.50) on the dependence of the transformation front velocity on the initial supersaturation. Figure 12.10d presents the mentioned concentration dependences of the transformation front velocity as compared to the experimental line 1. Line 2 describes the dependence of the velocity on Cq taking into account all the energy dissipation modes using the approximation Equation 12.48. In the case when the approximation Equation 12.50 is employed, we obtain lines 3 and 4, getting closer to the experimental values. Here, line 3 was obtained at curve 4 at... 

Az is the thickness of the formed solid solution AB the transformation front R moves at a constant velocity and coincides with the... 

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