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SOME CONVENTIONAL SCIENCE, WITH REGARD TO CRYSTALLIZATION FROM AQUEOUS SOL UTIONS
To illustrate what fields of science and what general knowledge must be brought to bear for a reliable analysis of a crystal growth problem, a brief overview will be given here to indicate what controls a crystal’s shape during unconstrained crystallization of water. For a more expanded view, the reader should explore references 11 to 13. For this “overview,” we will be satisfied with a phenomenological description of the most important simultaneous processes involved in terms of “lumped” material parameters. Table 4 indicates the different areas of study necessary to be considered with the minimum number of involved materials parameters, macroscopic variables, and system constraints.
AH is the latent heat of fusion, T0 is the melting temperature of solvent, kg is the solute distribution coefficient, mL is the liquidus slope,
jthe thermal conductivity, and a is the heat diffusivity.
The conventional macroscopic variables that one either sets or controls are (1) the water chemical composition, C(X), (2) the water cooling rate, t, and (3) the shape of the container holding the fluid.
Let us proceed with the process description by stages.
1. As the liquid is being cooled, we need to know the magnitude of the thermodynamic driving force for solid formation AG at any bath temperature T. This can be expressed as where f1 refers to the appropriate mathematical functional relationship between the latent heat of fusion, AH, and the liquidus temperature TL(C). Thus we see that phase equilibria data is one prerequisite. The material parameters needed for this area of study are listed and defined in Table 4.
2. As the bath undercooling, AT, increases with time, t, we need to know the undercooling at which particles of solid begin to form and also their density. Thus we must evaluate the nucleation frequency I, which can be most simply expressed as where f2 represents the appropriate functional relationship, N0 is the number of atoms in contact with the foreign substrates that catalyzes the nucleation event, and ATC is a parameter that defines the potency of the catalyst (the undercooling at which solid formation is initiated).
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3. When the crystal illustrated in Figure 6 begins to grow at some velocity V, solute partitioning will occur at the interface since the equilibrium concentration of solute in the solid CS is different from the concentration in the liquid at the interface.
Left: Illustration of a crystal growing from a super-cooled liquid. Right: The important temperatures in a growth process. The magnitudes of the temperature differences indicate the degree of solute diffusion, capillarity, kinetic, or heat transport control.
Thus the concentration of solute in the liquid at the interface Ci must be determined and can be represented by a functional relationship of the form where ki refers to an interface solute partition coefficient that is generally different from the phase diagram value, D is the solute diffusion coefficient, 8C refers to the solute boundary layer thickness at the crystal surface, and S refers to the shape of the crystal.
4. In order to evaluate 8C in Equation 4c, it is necessary to consider the hydrodynamics of the fluid. The fluid will generally exist in some state of motion, whether the driving force is applied by external means or arises naturally due to density variations in the fluid. We can consider the fluid far from the crystal-liquid interface to be moving with some relative stream velocity u due to the average fluid body forces. The fluid motion will aid in the matter transport of solute away from the crystal into the bulk liquid and cause a lowering of Ci . We find that 8C can be expressed as where v is the kinematic viscosity of the fluid.
Because the growing crystal is small in size, has curved surfaces, and often contains nonequilibrium defects, the solid contains a higher free energy than the solid considered in generating a phase diagram that we use as our standard state in the overall treatment. Thus the equilibrium melting temperature for such a solid is lowered by an amount ATE compared to that for the equilibrium solid. We find that the portion of the total undercooling consumed in the production of nonequilibrium solid ATE can be expressed as where TE(Ci) is the equilibrium interface temperature for interface liquid concentration Ci, y is the solid-liquid interfacial energy, AS is the entropy of fusion, yif is the fault energy for defects of type i, and Nif is the number of type i. (See Figure 7.)
Figure 7: Solute and temperature distributions plus key temperatures for (a) unconstrained crystallization and (b) constrained crystallization.
6. Next, because the crystal is growing, a departure from the equilibrium temperature ATK must exist at the interface in order to produce a net thermodynamic driving force for molecular attachment to the growing solid. At sufficiently large departures from equilibrium, the molecules can attach at any interface site and lower the free energy of the system However, at small departures from equilibrium, molecular attachment at random interface sites generally leads to an increase in the free energy of the system; and thus, such interface attachment will not occur as a spontaneous process. Rather, in such an instance molecules become a part of the solid only by attachment at layer edge sites on the interface, and one must consider the various mechanisms of layer generation on the crystal surface. The portion of the total undercooling consumed in driving this interface process ATK can be expressed as where Ti is the actual interface temperature and whereand |i2 are lumped parameters needed to specify the interface attachment kinetics for the various attachment mechanisms.
7. Finally, since the crystal is growing, it must be evolving latent heat and the interface temperature Ti must be sufficiently far above the bath temperature T to provide the potential for heat dissipation to the bath. That portion of the total undercooling consumed in driving the heat dissipation, ATH can be expressed as where K refers to the thermal conductivity and a refers to the thermal diffusivity. The foregoing has been a description of the subdivision of the total bath undercooling, AT, into its four component parts, i.e.,