Heat Conduction-Limited Growth

At this point, the interface position, x(f), is an unknown function of time. However, it can be determined by imposing the requirement that the net rate at which heat flows into the boundary must be equal to the rate at which heat is delivered to supply the latent heat needed for the melting. Any small difference between the densities p3 and pL may be neglected and the resulting uniform density is represented by p. Then, if the boundary advances a distance Sx, an amount of heat pHm Sx must be supplied per unit area. Also, if the time required for this advance is St, the heat that has entered the boundary from the liquid is JLSt (at x = x) and the heat that has left the interface through the solid is JsSt (at x = x)- Therefore, 

This type of relationship, accounting for the flux into and out of the interface, is generally known as a Stefan condition [1]. The Stefan condition introduces a new variable, the interface position, and one new equation. 

Using similar procedures, the solution of the conduction equation in the solid is 

Note that for these solutions no liquid exists at t = 0 and Ts = TSoc everywhere. Finally, an equation for determining the constant A can now be obtained by substituting Eqs. 20.7, 20.8, and 20.9 and K = cpK into the Stefan condition (Eq. 20.4), 

CHAPTER 20 GROWTH OF PHASES IN CONCENTRATION AND THERMAL FIELDS 

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FIGURE 6.24 Schematic illustration of heat-transport-limited growth. The rate of growth of a slice of transformed phase of thickness dx is determined by the rate at which the heat absorbed (or released) by this transformed volume can be conducted to (or away from) the interface. 

Growth Limited by Heat Conduction and Mass Diffusion Simultaneously... 

Here t Is the time and R the specific gas constant of the vapor. For r > X macroscopic thermodynamic processes of heat removal by conduction limit the growth rate... 

Besides being determined by processes at the interface of a crystal, the growth rate can also be determined by transport phenomena in the bulk of the mother phase. In the case of growth from solution, the discussion of transport phenomena can be limited to the transfer of mass. The heat liberated by the crystallization is small compared to the heat conductivity of the mother phase. Conversely, for growth from the vapor phase and from the melt, heat transfer can become limiting. 

Similarly, impervious yttria-stabilized zirconia membranes doped with titania have been prepared by the electrochemical vapor deposition method [Hazbun, 1988]. Zirconium, yttrium and titanium chlorides in vapor form react with oxygen on the heated surface of a porous support tube in a reaction chamber at 1,100 to 1,300 C under controlled conditions. Membranes with a thickness of 2 to 60 pm have been made this way. The dopant, titania, is added to increase electron How of the resultant membrane and can be tailored to achieve the desired balance between ionic and electronic conductivity. Brinkman and Burggraaf [1995] also used electrochemical vapor deposition to grow thin, dense layers of zirconia/yttria/terbia membranes on porous ceramic supports. Depending on the deposition temperature, the growth of the membrane layer is limited by the bulk electrochemical transport or pore diffusion. 

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