Crystallization equilibrium

When the crystallized crystals always stay with the magma and reach chemical equilibrium, the compositions of the crystals are related to the composition of the melt by the bulk partition coefficient, 

The eqaadon for the mean enrichment of a trace element in the cumulate relative to the original liquid, i.e. the total residual solid is 

In reality, of course, crystallization processes probably operate between the two extremes of equilibrium crystallization and fractional crystallization, which should. be seen as limiting cases. 

Mq is the initial mass of the magma chamber and Mi is the mass of liquid so that the ratio Mi /Mq is equivalent to the term F — the fraction of melt remaining and used in equilibrium and fractional crystallization models /is the fraction of magma allocated to the solidification zone which is returned to the magma chamber. Equation [4.21] is similar in form to the Rayleigh fractionation equation but with a more complex exponent. 

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Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the... 

C. Relative Surface Tensions from Equilibrium Crystal Shapes... 

Referring to Problem 3, what should the ratio 710/711 be if the equilibrium crystal is to be a regular octahedron, that is, to have (10) and (11) edges of equal length ... 

An enlarged view of a crystal is shown in Fig. VII-11 assume for simplicity that the crystal is two-dimensional. Assuming equilibrium shape, calculate 711 if 710 is 275 dyn/cm. Crystal habit may be changed by selective adsorption. What percentage of reduction in the value of 710 must be effected (by, say, dye adsorption selective to the face) in order that the equilibrium crystal exhibit only (10) faces Show your calculation. 

At equilibrium, crystal growth and dissolving rates become equal, and the process of Ostwald ripening may now appear, in which the larger crystals grow at the expense of the smaller ones. The kinetics of the process has been studied (see Ref. 103). 

A somewhat subtle point of difficulty is the following. Adsorption isotherms are quite often entirely reversible in that adsorption and desorption curves are identical. On the other hand, the solid will not generally be an equilibrium crystal and, in fact, will often have quite a heterogeneous surface. The quantities ys and ysv are therefore not very well defined as separate quantities. It seems preferable to regard t, which is well defined in the case of reversible adsorption, as simply the change in interfacial free energy and to leave its further identification to treatments accepted as modelistic. 

In the next section we describe a very simple model, which we shall term the crystalline model , which is taken to represent the real, complicated crystal. Some additional, more physical, properties are included in the later calculations of the well-established theories (see Sect. 3.6 and 3.7.2), however, they are treated as perturbations about this basic model, and depend upon its being a good first approximation. Then, Sect. 2.1 deals with the information which one would hope to obtain from equilibrium crystals — this includes bulk and surface properties and their relationship to a crystal s melting temperature. Even here, using only thermodynamic arguments, there is no common line of approach to the interpretation of the data, yet this fundamental problem does not appear to have received the attention it warrants. The concluding section of this chapter summarizes and contrasts some further assumptions made about the model, which then lead to the various growth theories. The details of the way in which these assumptions are applied will be dealt with in Sects. 3 and 4. 

Furthermore, even if we accept the use of A s and B s as rate constants with a ratio determined by detailed balance, there are many problems associated with evaluating the free energy changes (see e.g. Ref. [26]) and it is unlikely that they are simply related to equilibrium crystals. 

During equilibrium crystal growth from a melt a U-series parent and daughter will be incorporated according to their equilibrium partition coefficients, Dp and D respectively ... 

Batch partial melting will hereafter be understood as equilibrium melting, which is in contrast to fractional melting discussed in Section 9.3.3. The foundation of this model is remarkably simple and was first laid down by Schilling and Winchester (1967). A number of more or less complex modifications enabling useful information to be extracted from the data were later introduced by Gast (1968), Shaw (1970) and Albarede (1983). Bulk equilibrium crystallization of a liquid batch can be handled with equations identical to those for batch-melting. 

Ghiorso, M. S. (1985b). Chemical mass transfer in magmatic processes. II. Applications in equilibrium crystallization, fractionation and assimilation. Contrib. Mineral. Petrol., 90, 1021-41. 

In the liquid state sulfur and selenium are known to mix in all proportions. The provisional phase diagram shows an eutectic point at 40 mol-% of selenium (m.p. 105 °C). Mixtures with lower selenium content should show freezing points between 105 and 118 °C while those with higher selenium content are expected to have their freezing points at considerably higher temperatures. In practice equilibrium crystallization of the melt is hindered by supercooling and therefore only the melting points can be studied. 

By repeating the above procedure, but now eliminating La instead of L3, we obtain fj- = Therefore, based on only thermodynamic arguments, the dimensions of an equilibrium crystal in different orthogonal directions are proportional to the surface free energies of the perpendicular surfaces ... 

Networks of steps, seen in STM observations of vicinal surfaces on Au and Pt (110), are analyzed. A simple model is introduced for the calculation of the free energy of the networks as function of the slope parameters, valid at low step densities. It predicts that the networks are unstable, or at least metastable, against faceting and gives an equilibrium crystal shape with sharp edges either between the (110) facet and rounded regions or between two rounded regions. Experimental observations of the equilibrium shapes of Au or Pt crystals at sufficiently low temperatures, i.e. below the deconstruction temperature of the (110) facet, could check the validity of these predictions. 

As is well known (see, for instance, Van Beijeren and Nolden, 1987), the equilibrium crystal shape is the shape that minimizes the total surface free energy at a given fixed volume. From the minimization of the free energy calculated above we can construct the equilibrium shape of the crystal around the (110) facet. This shape depends crucially... 

For the description of die temperature and stress-related behavior of the crystal we used die method of consistent quasi-hannonic lattice dynamics (CLD), which permits die determination of the equilibrium crystal structure of minimum free energy. The techniques of lattice dynamics are well developed, and an explanation of CLD and its application to the calculation of the minimum free-energy crystal structure and properties of poly(ethylene) has already been presented. ... 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

With the advances of the non-equilibrium crystal growth techniques, other 1I1-V magnetic semiconductors than (Ga,Mn)As and (In.Mn)As with different host semiconductors and different transition metals have appeared and the investigation of properties of these new materials are underway. 

Near the transition temperature, SMAs also exhibit the curious effect of pseudoelasticity, in which the metal recovers (apparently in the usual manner) from an isothermal bending deformation when the stress is removed. However, the elasticity is not due to the usual elastic modulus of a fixed crystalline form, but instead results from strain-induced solid-solid phase transition to a more deformable crystalline structure, which yields to the stress, then spontaneously returns to the original equilibrium crystal structure (restoring the original macroscopic shape) when the stress is removed. 

In this first chapter, we will outline the scope of this book on the kinetics of chemical processes in the solid state. They are often different from the kinetics of processes in fluids because of structural constraints. After a brief historical introduction, typical situations of non-equilibrium crystals will be described. These will illustrate some basic concepts and our approach to understanding solid state kinetics. 

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