{"id":544,"date":"2018-12-04T11:27:15","date_gmt":"2018-12-04T11:27:15","guid":{"rendered":"http:\/\/msp06.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=544"},"modified":"2018-12-05T06:39:05","modified_gmt":"2018-12-05T06:39:05","slug":"zone-melting-refining","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/msp06\/chapter\/zone-melting-refining\/","title":{"rendered":"zone melting& refining"},"content":{"raw":"
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32.1.<\/strong>\u00a0Introduction<\/strong> \r\n

Some basic background of crystal growth by zone melting has already been described. However, some more details are required to be given here about zone melting with more emphasis on how crystals get refined by this technique .The technique finds wide application in the industry dealing with solid state devices. In the last 50 years it has made immense contribution to the development of different fields of materials technology in particular and solid state science in general. Through this technique, it has been possible to produce a wide variety of both organic as well as inorganic materials of high purity. The semiconducting materials like germanium and silicon were purified with the help of this technique. However, its use was extended to several kinds of solid state devices. The applications of zone refining are many folds not only in the growth of materia ls as single crystals but also in the purification of several materials, elements, organic and inorganic compounds.<\/p>\r\n \r\n

It was Pfann who in 1952 described zone melting and zone refining processes. In 1953 Pfann and Olsen experimentally showed that a high degree of purification of a rod of germanium could be achieved by using zone refining process. Pure germanium is required for solid state devices like transistors and Pfann successfully demonstrated the potential of the technique for removing impurities from the solid state material.<\/p>\r\n \r\n\r\n \r\n\r\n32.2<\/strong>\u00a0Theory of Zone Melting<\/strong>\r\n\r\n \r\n

In the understanding of basic theory behind zone melting processes, it is necessary to have some background of the following terms:<\/p>\r\n \r\n\r\ni)\u00a0\u00a0\u00a0 The equilibrium distribution coefficient.\r\n\r\n \r\n\r\nii)\u00a0\u00a0\u00a0 Interface distribution coefficient\r\n\r\n \r\n\r\niii)\u00a0\u00a0\u00a0 Effective distribution coefficient\r\n\r\n \r\n\r\nLet us describe them one by one:\r\n\r\n \r\n\r\n \r\n\r\n32.2.1<\/strong>\u00a0Equilibrium Distribution Coefficient<\/strong>\r\n\r\n \r\n

Let us consider phase diagram of a binary system consisting of a solute in the form of soluble impurity and a solvent in the form of a host material. The solvent (a host material)-solute (a soluble impurity) phase diagrams are shown schematically in figure 32.1 (a, b). The figures\u00a0schematically represent portions of such diagrams near the melting points of solvents. Here we may define an equilibrium distribution coefficient \u03c30 (symbolized by k0 by several authors) as the ratio of the concentration of the solute in the solid (Cs) phase to that in the liquid (CL) phase when the solid and liquid phases are in equilibrium. So,<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 32.1: Solvent-solute phase diagrams for (a) \u03c3<\/em><\/strong>0<\/em><\/strong> < 1, (b) \u03c3<\/em><\/strong>0<\/em><\/strong> > 1<\/em><\/strong><\/p>\r\n \r\n

Figure 32.1 represents solvent-solute phase diagrams. While figure 32.1(a) is the phase diagram in case of (a) \u03c30 = CS\/CL<1, figure32.1 (b) is for \u03c30 = CS\/CL> 1<\/p>\r\n \r\n

The distribution coefficient \u03c30 as defined by equation 32.1 also represents the ratio of the increment in thermodynamic potential as a result of interaction and entropy of mixing of the solute with the solvent in the solid and liquid phases.<\/p>\r\n \r\n\r\nThe chemical potentials of the solute in the solid and liquid phases are given by:\r\n\r\n \r\n\r\n\u03bc = \u03bc0 \u2500 RT ln a\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.2\r\n\r\n<\/div>\r\n \r\n\r\nwhere \u03bc0 is a constant,<\/span>\r\n\r\nR is the gas constant and\u00a0<\/span>a represents activity of the solute.<\/span>\r\n\r\n \r\n\r\nLet us now represent the chemical potential of the solute in the solid phase as:<\/span>\r\n\r\n \r\n\r\n\u03bcS =\u00a0 \u03bc0(s) - RT ln aS<\/span>\r\n

\r\n\r\n \r\n\r\nSimilarly, the chemical potential of the solute in the liquid phase may be represented as:\r\n\r\n \r\n\r\n\u03bcL\u00a0\u00a0 =\u00a0 \u03bc0(L) \u2500 RT ln aL\r\n\r\n \r\n\r\nOn establishment of equilibrium \u03bc(S) = \u03bc (L) ,\r\n\r\n \r\n\r\nSo,\r\n\r\n \r\n\r\n\u03bc0(s) \u2500 RT ln aS= \u03bc0(L) \u2500 RT ln aL ,\r\n\r\n \r\n\r\nOr, aS<\/sub>\/aL<\/sub> = a constant.\r\n\r\n \r\n\r\n \r\n

According to the definition of activity that as the concentration of the solute in the solution tends to zero the activity tends to be equal to concentration. Decrease in the concentration of the solute in the solution amounts to saying that the solution is getting diluted.<\/p>\r\n \r\n\r\nThus, for dilute solutions:\r\n\r\n \r\n\r\naS<\/sub>\/aL<\/sub> = CS<\/sub>\/CL<\/sub> \u2026\u2026\u2026\u2026\u2026\u2026......32.3\r\n\r\n \r\n

Equation 32.3 suggests that the equilibrium distribution coefficient \u03c30 is generally a function of the concentration of the solute except for very dilute solutions.<\/p>\r\n \r\n

Looking at figure 32.1, it suggests that if the addition of the solute lowers the melting point then \u03c30< 1 but if the addition of the solute increasesthe melting point then \u03c30> 1.<\/p>\r\n \r\n

\u03c30 is often calculated from the phase diagram data. In the event of non-availability of the desired phase diagram at the desired concentration, one may deduce \u03c30 for dilute solutions from a plot of ln \u03c30 against T\u25001, where T is the absolute temperature.<\/span><\/p>\r\n \r\n

Hayes and Chipman (1939) has suggested a thermodynamic method with the application of which one can estimate \u03c30 for very dilute solutions through an equation:<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nWhere N (L) represents the mole fraction of the solute in the liquid, H is the heat of fusion of the solvent,\r\n\r\n\u0394H is the heat of fusion of the solvent,\r\n\r\n\u0394T is the difference in freezing point between the pure solvent and the solution,\r\n\r\nR is the gas constant, and\r\n\r\nT is the absolute temperature.\r\n\r\n \r\n

The equilibrium distribution coefficient becomes applicable to equilibrium solidification processes where the temperature gradient across the solid-liquid system is negligibly low and the rate of solidification very small so that impurity gradients get wiped out by diffusion processes.<\/p>\r\n \r\n

If we consider zone-melting and any other practical solid-melt techniques, solidification does not occur in equilibrium conditions because there are large temperature and concentration gradients present, rates of solidification are high and the mixing of the solute in the liquid imperfect. It, therefore, becomes important to understand about two other distribution coefficient and those are interface distribution coefficient and effective distribution coefficient.<\/p>\r\n \r\n\r\n \r\n\r\n32.2.2.<\/strong>\u00a0Interface Distribution Coefficient (\u03c3*)<\/strong>\r\n\r\n \r\n\r\n \r\n\r\nIt is defined as the ratio of the solute concentration in the solid to that in the liquid, actually at the interface:\r\n\r\n \r\n\r\ni.e.,\u00a0 \u00a0\u03c3* = Concentration of solute in the solid\u00a0 \u00a0(at the interface) \/Concentration of solute in the liquid\r\n\r\n \r\n\r\n=C<\/span>S<\/sub>\/C<\/span>L(0)<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.5<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 32.2: Schematic solute concentration profile and the diffusion layer at a solid -liquid interface<\/em><\/strong><\/p>\r\n \r\n

It is found that the interface distribution coefficient \u03c3*is different from the equilibrium distribution coefficient \u03c30. Figure 32.2 is a schematic diagram showing concentration profile and the diffusion layer at a solid-liquid interface. Since the rates of the chemical processes and the rates of the exchange reactions between the solid and the liquid at the interface is dependent upon the velocity of growth, \u03c3* should be a function of the velocity of solidification or growth. Measurements of \u03c3* have shown that it does actually vary with the growth velocity. The following equation has been derived on a model in which the solute incorporation in the solid is assumed to be through exchange reactions across solid-liquid interface:<\/p>\r\n \r\n\r\n\u03c3* = \u03c30 + vg\u00a0 ( 1 \u2500 \u03c3* )\/\u03b2vm\u2026\u2026\u2026\u2026\u2026\u202632.6\r\n\r\n \r\n\r\nWhere \u03c30\u00a0 is the equilibrium distribution coefficient,<\/span>\r\n\r\nvg is the velocity of growth ,<\/span>\r\n

\u03b2 represents the dissociation coefficient of the observed solute molecule at the interface (describing <\/span><\/p>\r\n

molecules leaving the interface and incorporating into the liquid)<\/span><\/p>\r\n

and vm is the molecular velocity of diffusion of the solute.<\/span><\/p>\r\n

The equilibrium distribution coefficient in this model is given as:<\/span><\/p>\r\n

\u03c30 = \u03b1\/\u03b2 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.7,<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

where \u03b1 is known as sticking coefficient of the solute molecule and describes the entrance of the solute in the crystal.<\/p>\r\n \r\n\r\n \r\n\r\n32.2.3<\/strong>\u00a0Effective Distribution Coefficient (\u03c3eff)<\/strong>\r\n\r\n \r\n

It is another important term used in the understanding of Zone melting as a means of refining the crystals. The interface differential coefficient is not the same as equilibrium segregation coefficient. So, the overall incorporation of solute into the solid in actually experimental crystal growth methods cannot be described by the equilibrium segregation coefficients. It is here that the use of effective distribution coefficient \u03c3eff which is related to interface distribution coefficient \u03c3* through a functional relationship as given below:<\/p>\r\n \r\n\r\n\u03c3eff\u00a0 = \u03c3* F\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.8\r\n\r\n \r\n\r\n \r\n\r\nwhere F is a function of parameters such as growth rate, the degree of mixing \/stirring,diffusion of the solute\u00a0 in the liquid and the concentration of the solute. The effective distribution coefficient \u03c3e ff can be\u00a0 calculated from the\u00a0 equilibrium distribution coefficient\u00a0 \u03c30. For\u00a0 this\u00a0 purpose, the\u00a0 following one dimensional steady-state diffusion equation defining conservation of\u00a0 the solute molecular across the interface is used:\r\n\r\n \r\n\r\nD\u00a0 d2<\/sup>C\/dx2<\/sup> \u2500 vxdC\/dx = 0\u2026\u2026\u2026\u2026\u2026\u2026\u202632.9\r\n\r\n \r\n\r\nWhere D is the diffusion coefficient of the solute,<\/span>\r\n\r\nC is the concentration of the solute,<\/span>and vx is the fluid velocity of liquid in the x- direction.<\/span>\r\n\r\nThe fluid velocity of liquid\u00a0 vx = vg\u00a0 + \u03c9,<\/span>\r\n\r\nwhere vg is the solidification velocity and\u00a0<\/span>\u03c9 is the normal fluid velocity.<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

Let us refer to figure 32.2. It assigns the following boundary conditions to the equation of effective distribution coefficient \u03c3e ff defined by equation32.8. The interface is positioned at x=0 and the positive x direction is extended into the melt.<\/p>\r\n \r\n\r\nThe concentration of the solute C = CL(0) at x = 0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.10(a)\r\n\r\nThe concentration of the solute C = CL at x = \u03a9\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.10(b)\r\n\r\n \r\n\r\nThe boundary condition suggests that for distance x > \u03a9 the concentration in the melt on account of\u00a0<\/span>fluid flow is equal to CL. However, within the diffusion layer (i.e., the layer of thickness \u03a9 at the\u00a0<\/span>interface) the fluid velocity of liquid is equal to the solidification velocity.<\/span>\r\n\r\n \r\n

Equation 32.9 is very much similar to heat flow equation. With the application of boundary conditions as are applicable in the present case and specified in equations 32.10 (a & b) the solution yields for effective distribution coefficient as given below:<\/span><\/p>\r\n \r\n

vx = vg and so, \u03c9 = 0\u2026\u2026\u2026\u2026\u202632.10(c)<\/span><\/p>\r\n \r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\nThe\u00a0 above equation at\u00a0 32.11 gives\u00a0 us\u00a0 the\u00a0 value\u00a0 of\u00a0 \u03c3eff\u00a0 as\u00a0 determined by making boundary layer\u00a0analysis which can be used for obtaining information on the morphological stability of the solid-liquid interface when convection is taking place.<\/span>\r\n\r\n \r\n

\u03a9\u00a0 is a function of the diffusion coefficient kinematic viscosity of the liquid and conditions of the flow of fluid. The layer thickness \u03a9 at the interface is dependent on the growth velocity as is clear from equations 32.12(a,b) which is derived below.<\/span><\/p>\r\n \r\n

Writing equation 32. 11 in the form as given below:<\/span><\/p>\r\n\r\n<\/div>\r\n

<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n \r\n

If one is able to measure \u03c3eff in crystals which are grown at different velocities under identical stirring conditions, curves of the term ln(1\/\u03c3e ff \u2500 1) or ln(1\u2500 1\/\u03c3e ff) could be drawn against the growth velocity vg which should be a straight line with the slope ( \u2500 \u03a9\/D). It is important to note here that interface should be stable. However, if for one reason or the other, the interface is unstable then measurement of ( \u2500 \u03a9\/D) may yield erroneous results. Instability of the interface arises if it melts back. In that case the thickness of diffusion layer fluctuates. In general, the dependence of layer thickness \u03a9 at the interface on the velocity of solidification vg is weak but is strongly dependent on the conditions of stirring. As for example, \u03a9 is hardly of the order of 10\u2500 1cm under negligible stirring but is found to be of the order of 103 cm if stirring is done vigorously. Theoretical solutions of \u03a9 for conditions of zone melting are rather difficult and the solutions that have been worked out are not found to be satisfactory.<\/p>\r\n \r\n

The diffusion layer is also dependent on the actual impurity distribution profile within it. Based on an assumption that the interface is planar, growth velocity remaining constant and\u00a0no stirring, Tiller and his co-workers in 1953 gave the following equation that connects the concentration of the solute in the liquid at a distance x from the interface which may be represented as CL(x) with the mean initial concentration C0 and concentration at the interface as CL(0):<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nCL(x<\/sub>) = CL(0)<\/sub> exp (\u2500 vg\/D. x) + C0<\/sub>\u2026\u2026\u2026\u2026\u2026\u2026\u202632.13\r\n\r\n \r\n\r\n \r\n\r\nWhere vg is the growth velocity and\r\n\r\nD is the diffusion coefficient of the solute in the liquid.\r\n\r\n \r\n

In the crystal growth from melt, there could be temperature fluctuations which may be either due to external variations of the heat source or due to inherent hydrodynamic effects associated with the convective motion of the melt. The presence of perturbations of the thermal field makes average interface distribution coefficient symbolized as \u03c3* to be different from the interface distribution coefficient in the absence of these fluctuations. The average interface distribution coefficient \u03c3* is taken as an average over a cycle of the perturbation. It can be expressed as:<\/p>\r\n \r\n\r\n\u03c3* \u2260 \u03c3*\u2026\u2026\u2026\u2026\u2026\u2026\u2026....32.14\r\n\r\n \r\n\r\nThe change in the effective distribution coefficient \u03c3eff in the presence of perturbation is given by:\r\n\r\n\"\"\r\n\r\nThe expression for equation 32.15 has been obtained by Hurle and Jakeman in 1969 which may be given here as:\r\n\r\n\"\"\r\n\r\nwhere\u00a0\u00a0 Ima {} represents the imaginary part of the bracketed function ,\r\n\r\n \r\n\r\nI = \u221a(\u25001) ,\r\n\r\n \r\n\r\n\u03c6= \u03c9D\/vg2<\/sup>(0),<\/span>\r\n\r\n \r\n\r\n= vg(0)\u03a9\/D and<\/span>\r\n\r\n \r\n\r\n\u03a9 = (D\/\u03bd) 1\/3 \u03a9m ;<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nD is the diffusion coefficient of the solute in the liquid,\r\n\r\n \r\n\r\nvg(0<\/sub>) is the mean growth velocity,\r\n\r\n \r\n\r\n\u03a9 is the thickness of the diffusion layer,\r\n\r\n \r\n\r\n\u03bd is the kinematic viscosity of the melt and\r\n\r\n \r\n

\u03a9m is the momentum boundary layer (the definition of which and its relation with \u03a9 is described in detail by V.G. Levich (1962) in Physico Chemical Hydrodynamics, Prentice -Hall Inc.).<\/p>\r\n \r\n

The above said authors (Hurle and Jakeman) who obtained expression as at 32.16 applied the same to the experimental results of Barthel and Eichler (1967) on measurement of effective distribution coefficient of tungsten in electron beam zone melted molybdenum as a function of the length of the molten zone, and succeeded in showing the validity of the expression.<\/p>\r\n \r\n

It has already been said that the application of equations 32. 12(a,b) for the experimental determination of \u03a9\/D will lead to erroneous results in the event of temperature fluctuations and hence interface instabilities that may be occurring in the experimental set- up. However, if there are low frequency fluctuations one may put the equation as under:<\/p>\r\n \r\n\r\n\u03b4\u03c3eff<\/sub>\/\u03c3eff<\/sub> = \u03b7 (1\u2500 \u03c30)vg(0)\u03a9\/D\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.17\r\n\r\n \r\n

Here, \u03b7 is a parameter which indicates the error in \u03a9\/D. It is because the experimentally measured distribution coefficient is actually the average effective distribution coefficient<\/p>\r\n \r\n\r\n\u03c3eff<\/sub> = \u03c3eff<\/sub>(1+\u03b7) =\u00a0 \u03c30 {1+(1+\u03b7)(1\u2500 \u03c30)vg(0)\u03a9\/D}.\r\n\r\n \r\n\r\nIt, therefore, becomes clear that the slope of the plot obtained from equations 32.12(a ,b)should be (1+\u03b7)\u03a9\/D and not \u03a9\/D.\r\n\r\n \r\n\r\nWe have discussed some important expressions concerning distribution coefficient in a steady state which may be put here as follows:<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n\u03c30= CS<\/sub>\/CL<\/sub>,\r\n\r\nvg = 0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.18(a)\r\n\r\n\u03c3* = CS\/CL(0)<\/span>,<\/span>\r\n\r\nvg = vg\u2026\u2026\u2026\u2026\u2026\u2026.....\u202632.18(b)<\/span>\r\n\r\n\u03c3eff =\u00a0 CS\/CL<\/span>,<\/span>\r\n\r\nvg = vg\u2026...\u2026\u2026\u2026\u2026\u2026\u2026.32.18(c)<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\nAt equilibrium\r\n\r\n\u03c3eff = \u03c3* = \u03c30<\/span>\u2026\u2026\u2026\u2026\u2026\u2026\u2026...\u2026\u2026\u2026\u2026\u2026.32.18(d)<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\nIf growth rates are high (i.e., vg \u2192 \u221e)\r\n\r\n \r\n\r\n\u03c3* \u2192 1\r\n\r\nas vg\u2192 \u221e\u00a0<\/span>\u2026\u2026\u2026\u2026\u202632.18(e)<\/span>\r\n\r\n \r\n\r\nand\u00a0<\/span>\u03c3eff \u2192 1<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n

It has been experimentally observed that there are variations with orientation of growing crystals. This is called facet effect which is not explained by the above described equation given at no. 32.11 and<\/p>\r\n \r\n

32.18. However, one can explain it provided \u03c3* is taken as a parameter which does get influenced by the orientation of the growing face. In other words, the exchange rates of solute atoms between the solid and liquid are treated as varying with orientation of the growing facet.<\/p>\r\n \r\n

Segregation influences sheet growth of a monolayer and there is effect on the motion of growth steps if impurities get incorporated. The concentration of impurity increases with the velocity of growth steps and decreases with an increase in the rate at which the impurity diffuses out of the growth steps into the melt. The higher impurity in the faceted regions is because of the higher velocity of growth steps.<\/p>\r\n \r\n\r\n32. 3<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Zone refining<\/strong>\r\n\r\n \r\n\r\n32.3.1<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>Introduction.<\/strong>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

Zone melting is the general title of a number of techniques used for the purification of elements and compounds and also for the preparation of materials of the required composition .In these techniques a short liquid region, or zone , is made to travel slowly through a relatively long charge , or ingot of solid. When the zone travels, it redistributes impurities along the charge. The final distribution depends on the following:<\/p>\r\n \r\n\r\ni)\u00a0\u00a0\u00a0\u00a0\u00a0 Impurity distribution in the starting charge,\r\n\r\nii)\u00a0\u00a0\u00a0 the distribution coefficient \u03c3 of an impurity between the liquid and solid of the charge material,\r\n\r\niii)\u00a0 the size of the zone,\r\n\r\niv) the number of the zones, and\r\n\r\nv)\u00a0\u00a0\u00a0 travel direction of the zones.\r\n\r\n \r\n

Zone melting is a novel method of using the freezing process to manipulate impurities. It is\u00a0 a well known fact\u00a0 that\u00a0 the composition of\u00a0 a freezing crystal is\u00a0 different\u00a0 from that of\u00a0 its\u00a0 liquid. Obviously, one could think of a simple idea of passing a very thin liquid zone along a lengthy solid. Following this approach the process of crystallization as a means to purify a solid or to control its composition received a lot of attention and has been investigated and put to practice. Zone refining is the most valuable zone melting method. This process of refining involves passing of a number of molten zones along the charge in one direction. Each zone carries with it a fraction of the impurities to the end of the charge , thereby leaving behind the purified material. In early 1950\u2019s zone refining was introduced for purification of germanium for use in transistors. The purity of the material obtained by the process was extraordinary which in quantitative terms could be placed as one part of detectable impurity in 1010 parts of germanium. This was a remarkable achievement that boosted the electronic industry and applications of solid state devices. The method of zone refining has been applied to several substances. It received so much of popularity that by 1973 almost one -third of the elements of the periodic table had been purified to the highest degree by zone refining. Since zone refining provided the means of making ultra-pure materials available, it opened new areas of research and commerce.<\/p>\r\n \r\n\r\n32.3.2<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Principles of Zone Refining<\/strong>\r\n\r\n \r\n

It is important to first describe the distribution coefficient or segregation coefficient \u03c3 followed by a very old crystallization method known as normal freezing, one-pass zone melting and then multi-pass zone melting or zone refining.<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n32.3.2.1<\/strong>\u00a0\u00a0\u00a0 Normal Freezing.<\/strong>\r\n\r\n \r\n

The process of solidification in which whole of a charge is melted initially and then gradually solidified unidirectional is classified as normal freezing process. Two examples of normal freezing methods that are used for the growth of crystals are (i) the Bridgman process in which a crucible containing a molten charge is moved slowly with respect to a stationary temperature gradient for unidirectional solidification and (ii) the Stober process in which a temperature gradient is moved through a crucible containing a molten charge.<\/p>\r\n \r\n

Here, it is important to be conversant with expressions for impurity distribution in normal freezing process. Normal freezing process may be schematically shown in figure 32.3. When a cylinder of a substance say A having in it an impurity say B is melted, and is slowly frozen from one end to the other, the impurity gets usually concentrated in the \u201clast-to-freeze\u201d region of the cylinder. This is what is actually called normal freezing.<\/p>\r\n\"\"\r\n\r\n \r\n

Figure 32.3: Schematic representation of normal freezing process<\/em><\/strong><\/p>\r\n\r\n<\/div>\r\n \r\n\r\nThe following assumptions are made in the derivation of an expression:<\/span>\r\n

\r\n\r\n \r\n\r\n(i) Diffusion of the solute under consideration is negligible in the solid, i.e., x2 >> DSt,\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.32.19.\r\n\r\n \r\n\r\nwhere x is the length of the fraction solidified in time t and DS stands for diffusion coefficient of the solute in the solid.\r\n\r\n \r\n\r\n(ii) The effective distribution coefficient \u03c3eff is constant.\r\n\r\n \r\n\r\n(iii)\u00a0 The density change of the solution during freezing is zero.\r\n\r\n \r\n\r\n \r\n

Component B gets redistributed because the atoms (or molecules) of B at the liquid-solid interface prefer the liquid phase to the solid phase. The factor behind this preference is the distribution coefficient or segregation coefficient \u03c3 which is defined as the ratio of the concentration of B in just-forming solid A to that in liquid A. If rate of freezing is very slow, the equilibrium distribution coefficient \u03c30 becomes applicable. If , however, the rates of freezing are moderate say ~<\/strong> 1-30cm h\u25001, an effective distribution coefficient, \u03c3eff , which lies between \u03c30 and unity, is applicable. It is because for \u03c30< 1, the rejected impurity B accumulates in the liquid just ahead of the advancing solid, so that the just forming solid \u201cfinds\u201d a liquid more impure than the bulk liquid. On the other hand, if freezing is rapid, effective distribution coefficient \u03c3eff may actually approach unity; there would be no zone refining and as a result the interface may probably become dendritic in shape.<\/p>\r\n \r\n\r\nThe solute concentration (i.e., solute atoms \/unit volume) in the solid immediately behind the interface is given by:\r\n\r\n \r\n\r\nCS = \u2500 dS<\/sub>\/dg<\/sub>\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.20,\r\n\r\n \r\n\r\nWhere CS stands for the solute concentration in the solid,\r\n\r\n \r\n\r\nS is the amount of solute in the liquid, and\r\n\r\n \r\n\r\ng is the solidified fraction of the original volume of unity.\r\n\r\n \r\n\r\nAccording to the definition of the distribution coefficient (as given in equation 32.1):\r\n\r\n \r\n\r\n\u03c3 = CS<\/sub>\/CL<\/sub>\u00a0 , or CS<\/sub>= \u03c3 CL<\/sub>, where CL stands for concentration of the solute in the liquid.\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n\r\n \r\n

Here, integration is carried out over the total amount of the solute and hence its amount taken initially in the liquid when g = 0 i.e., S0 and final amount of solute when the entire length is solidified (i.e., S).<\/p>\r\n \r\n\r\nSo, S = S0( 1 \u2500 g )\u03c3<\/sup>\r\n\r\n \r\n\r\nEquation no. 32.20 may be written as:\r\n\r\n \r\n\r\nCS = \u2500 dS\/dg = \u03c3 S0 (1\u2500g) \u03c3 \u25001\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026\u2026\u2026.32.23\r\n\r\n \r\n\r\nbecause of our assumption that the original volume was unity.\r\n\r\n \r\n\r\nS0<\/sub> = C0<\/sub>, where C0\u00a0 represents original concentration.\r\n\r\n \r\n\r\nSubstituting it in equation no. 32.23, we have:\r\n\r\n \r\n\r\nCS\u00a0\u00a0 = \u03c3 C0 (1 \u2500 g) \u03c3 \u25001 \u2026\u2026\u2026\u2026\u2026.............................32.24<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nOr, ln (CS\/C0) = ln\u03c3 + (\u03c3 \u2500 1) ln(1\u2500g) .................\u2026.32.25\r\n\r\n \r\n\r\n \r\n

If logarithmic plot of CS\/C0 against (1\u2500g) is drawn it would be a straight line whose slope and intercept at g = 0 will yield the value of \u03c3.<\/p>\r\n \r\n

The equation at number 32.23 becomes inaccurate for the value of g > ~<\/strong> 0.9, because it implies an infinite impurity concentration at g = 1.0. In practice, either \u03c3 varies with concentration, or a second phase, rich in the impurity, nucleates, as it happens in a eutectic system.<\/p>\r\n \r\n

A normal freezing distribution for an impurity having an effective distribution coefficient of 0.5 (which is assumed to remain constant) is shown in figure 32.4.<\/p>\r\n\"\"\r\n

Figure 32.4: Relative impurity concentration after (a) normal freezing and (b) on zone melting pass for initial mean concentration C0 and for an effective distribution coefficient of 0.5<\/em><\/strong><\/p>\r\n \r\n\r\nThe equation for the curve shown is of the form C = \u03c3 C0( 1 \u2500 g )\u03c3 \u2500 1 [ see equation 32.23], where C0\u00a0<\/span>is the mean impurity concentration in the original liquid , and g is the fraction of the liquid that has frozen.<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n32.3.2.2\u00a0<\/strong>\u00a0Solute Distribution in Zone Melting<\/strong>\r\n\r\n \r\n\r\nIn order to be able to understand the solute distribution in zone melting, the following variables will have to be specified:\r\n\r\n \r\n\r\ni)\u00a0 The zone length\u2018l\u2019\r\n\r\n \r\n\r\nii)\u00a0 Length of the charge \u2018L\u2019\r\n\r\n \r\n\r\niii)\u00a0 The initial concentration of the solute is assumed to be uniform throughout the ingot. The same may be\r\n\r\nrepresented by a constant C0.\r\n\r\n \r\n\r\niv) The traverse velocity of the zone or the zone velocity vg.\r\n\r\n \r\n

We shall first consider single-pass distribution. Here we are concerned with evaluation of distribution of the solute in the ingot from one end to the other and make the following assumptions:<\/p>\r\n \r\n\r\ni)\u00a0\u00a0\u00a0\u00a0\u00a0 The distribution coefficient \u03c3 is constant.\r\n\r\n \r\n\r\nii)\u00a0\u00a0\u00a0 The zone length l is held constant.\r\n\r\n \r\n\r\niii)\u00a0 The initial concentration C0 of the solute is uniform throughout the ingot.\r\n\r\n \r\n\r\niv) The densities of the liquid and the solid under consideration are the same.\r\n\r\n \r\n\r\nv)\u00a0\u00a0\u00a0 Diffusion of the solute in the solid is negligible, i.e.,\r\n\r\n \r\n

x2 >> DS.t , where x is the length of the fraction solidified in time t and DS is the diffusion coefficient of the solute in the solid.<\/p>\r\n \r\n\r\nSchematic diagram of a non-continuous zone-melting process is shown in figure 32.5.\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 32.5: A schematic diagram of zone refining process<\/em><\/strong><\/p>\r\n \r\n

We consider that the direction in which the zone is travelling is in the direction x=0 to x= L. As the liquid zone moves along the ingot it leaves behind the resolidified ingot. Solidification of the charge of an incremental volume dx as the zone advances would involve solute transfer. We are interested in finding an equation for this solute transfer. Suppose CL represents concentration of the solute in the liquid, then the amount of solute leaving the zone because of solidification = \u03c3 CLdx. The solute that enters the zone due to melting of the charge of the volume dx is C0dx . The net change in the total volume of solute in the liquid is given by:<\/p>\r\n \r\n\r\ndS = ( C0<\/sub> \u2500 \u03c3CL<\/sub>) dx\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.26\r\n\r\n \r\n

Let us assume that the cross-sectional area of the ingot and zone is unity. In that case the concentration in the liquid zone is given by:<\/p>\r\n \r\n\r\nCL = S\/l , \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026........32.27\r\n\r\n \r\n\r\nwhere S is the amount of the solute in the zone at a distance x.\r\n\r\n \r\n\r\nIn the light of equation 32.26, it becomes:\r\n\r\n \r\n\r\ndS = ( C0 \u2500 \u03c3 S\/l ) dx\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.28\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\nor dS\/dx + ( \u03c3\/l ) S = C0........................................... 32.29\r\n\r\n\"\"\r\n\r\n \r\n\r\nS0 = C0l\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.32.31\r\n\r\n \r\n\r\nSubstituting for S0 from equation 32.31 in equation 32.30, we have:\r\n\r\n \r\n\r\nSe (\u03c3\/l)x \u2500 C0l = C0l\/\u03c3 { e(\u03c3\/l).x\u2500 1}\u2026\u2026\u2026\u202632.32\r\n\r\n \r\n\r\nSe(\u03c3\/l). x = C0l [ 1 + 1\/\u03c3{ e(\u03c3\/l).x\u2500 1}]]\r\n\r\n \r\n\r\nS = C0l [ 1 + 1\/\u03c3 { e(\u03c3\/l).x \u2500 1}] e\u2500(\u03c3\/l). x\r\n\r\n \r\n\r\nThe solute concentration CS in the solid at any value of x is given by:\r\n\r\n \r\n\r\nCS = \u03c3S\/l\r\n\r\n \r\n\r\nOr,\u00a0\u00a0 S = lCS\/\u03c3\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.33\r\n\r\n \r\n\r\nSubstituting for S from equation 32.33in the above expression we get:\r\n\r\n \r\n\r\nl CS\/\u03c3 = C0 l [1 + 1\/\u03c3{e(\u03c3\/l)x\u2500 1}] e\u2500(\u03c3\/l).x\r\n\r\n \r\n\r\nCS \/C0 = \u03c3 [1+ 1\/\u03c3 {e (\u03c3\/l).x\u2500 1} ] e\u2500 (\u03c3\/l). x<\/span>\r\n\r\n<\/div>\r\n
\r\n\r\n \r\n\r\n=\u00a0\u00a0 {\u03c3 + e (\u03c3\/l).x\u2500 1} e\u2500 (\u03c3\/l).x\r\n\r\n \r\n\r\n=\u03c3e \u2500 (\u03c3\/l).x+ 1 \u2500 e\u2500 (\u03c3\/l). x\r\n\r\n \r\n\r\n \r\n\r\n=\u00a0 {1\u2500 e\u2500 (\u03c3\/l).x (1\u2500 \u03c3)}\u2026\u2026\u2026..................32.34\r\n\r\n \r\n\r\n \r\n

Figure 32.6 illustrates schematically profile of the solute distribution (\u03c3 < 1) in case of single pass. In this profile equation 32.34 shows distribution in the region marked X. It covers the part of ingot from x=0 to x = L-l, where L is the total length of the ingot. The last liquid zone to freeze out is represented by region Y in the figure. Actually, it is normally frozen and as such the distribution in this region is given by the normal freeze (equation 32.23).<\/p>\r\n \r\n\r\n \r\n

The normal freezing operation as described in the previous section is the first step in the repeated fractional crystallizations which is not something new but, in fact, is centuries old technique. However, the technique of fractional crystallization could not receive much of response because it involved time-consuming and troublesome operations which included partial freezing, crystal separation from mother liquor, remelting, recombining with other fractions, and several others.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n\"\"\r\n

Figure 32.6: Solute distribution shown schematically in case of a single-pass zone melting<\/em><\/strong><\/p>\r\n \r\n\r\n \r\n

Zone refining has made the task much simpler and very effective in achieving purification of materials. A series of molten zones traverses the ingot in the same direction, usually with the arrangement of multi heaters, as shown in figure 32.7.<\/p>\r\n\"\"\r\n

Figure 32.7: Schematic representation of zone refining process using multi heaters<\/em><\/strong><\/p>\r\n \r\n\r\n \r\n

Here in this case, each zone draws impurity at its melting interface, and freezes out solid purer than\u00a0the liquid (for those with segregation coefficient \u03c3 < 1) at its freezing interface. So, it does not involve the complicated and cumbersome process of separation and recombination of fractions and also eliminates the need to touch or move the charge at all. The distribution of impurity say B after one zone-pass for an ingot ten zone-lengths and for \u03c3 equal to 0.5 is shown in figure 32.4. The equation for the one-pass curve, except in the last zone-length is given by equation no. 32.34.<\/span><\/p>\r\n \r\n

As the number of zone-passes is increased the impurity concentration at the beginning of the ingot decreases more and more till a limit called ultimate distribution is reached. The ultimate distribution is given (except for the last zone-length) by the simple exponential expression as given by Pfann (1952):<\/span><\/p>\r\n \r\n

C = A eBx ,<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n

where x represents distance along charge , and A and B are constants and may be calculated from the following expressions:<\/p>\r\n\"\"\r\n

where L denotes charge length (total length of ingot) and C0 mean solute concentration per unit volume. Equation 32.36 does not hold for the last zone length where normal freezing takes place. It also cannot hold for the enhanced solute pile up or the back reflection of the solute in front of the last zone length. It may be noted here that the lowest concentration of impurity B (in the system A\u2013B as discussed in the previous section) is very small, less than 0.0001.<\/p>\r\n \r\n\r\n32.3.2.3<\/strong>\u00a0\u00a0Zoning Equipment<\/strong>\r\n\r\n \r\n\r\nZoning apparatus includes the following:\r\n\r\n1. a means of producing liquid zones, i.e., heaters and or coolers.\r\n\r\n2.\u00a0 a traverse mechanism for the transport of molten zones<\/span>\r\n\r\n3.\u00a0 a means of mounting or holding a charge<\/span>\r\n\r\n4.\u00a0 means of stirring a liquid and maintaining constant or controlled ambient atmosphere and temperature <\/span>\r\n\r\nconditions, specifically at the solid-liquid interface.<\/span>\r\n\r\n<\/div>\r\n

\r\n\r\n \r\n\r\n1.\u00a0 Zoning in a container\r\n\r\n \r\n\r\n2.\u00a0 Zoning without a container.\r\n\r\n \r\n\r\n \r\n\r\n32.3.2.4\u00a0<\/strong>\u00a0Zoning in a container<\/strong>\r\n\r\n \r\n

The usual container is one which does not contaminate the material. Glass, Vycor, fused silica, molybdenum, tantalum and graphite are examples of materials for containers. If zone refining is required to be done vertically, it is helpful to use a transparent container. If the container is a horizontal, semicircular-cross section boat, it can be oblique because the liquid zone can be readily distinguished from the solid. It is, however, necessary to take due care to prevent cracking of the filled container (be it horizontal or vertical), which is likely to occur on account of change in volume during freezing (or melting) or by differential thermal contraction (if the charge sticks to the container wall).<\/p>\r\n \r\n

The design of the container itself is important as the same is one of the deciding factors in achieving a good control over zone length and zone spacing .It is advantageous to have the walls of the container as thin as possible. The cross-sectional shape of the container has to be such as to have the surface of the charge as in minimum contact with the container as possible.<\/p>\r\n \r\n

The cross-sectional design has also to be such that the container (particularly, if the charge expands on solidifying) does not strain due to expansion or mass transfer and thus tending to crack the container.<\/p>\r\n \r\n

High melting-point materials or very reactive materials may be zone melted in a water cooled boat or a \u201ccold hearth\u201d made from high thermal conductivity material such as copper.<\/p>\r\n \r\n\r\n32.3.2.5\u00a0<\/strong>\u00a0Zone Refining Without Containers<\/strong>\r\n\r\n \r\n

Contamination of the charge by the container is a problem in all purification processes. It\u00a0is particularly so for such materials which are very effective solvents or are very reactive at their melting points. Such materials obviously cannot be refined or retained at high purity at the slightest\u00a0<\/span>contact with other materials. The solution to this problem lies in finding ways of by-passing the use of containers for such materials. A unique solution was found for zone refining in devising a technique that is known as \u201cfloat-zone\u201d melting (FZM). The method was developed by Keck and Golay (1953), Emeis (1954), and Theuerer (1956, 1962) for the preparation of ultra-pure silicon. In float zoning a vertical silicon rod is held by end-clamps, and a short molten zone is produced by induction heating\u00a0<\/span>and moved along the rod. The liquid (molten zone) is held in place by surface tension. Theoretical analysis by Heywang and Ziegler (1954) and Heywang (1956) led them to an expression for\u00a0<\/span>maximum stable zone-height as:<\/span><\/p>\r\n \r\n

lmax \u2248 2.8 (\u03b3\/\u03c1g) 1\/2\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.32.37<\/span><\/p>\r\n \r\n

where \u03b3 is the surface tension of the liquid, \u03c1 is the density of the liquid, and g is the gravitational acceleration.<\/span><\/p>\r\n\r\n<\/div>\r\n \r\n

This expression was derived on the basis of condition of the stability that the inward pressure on the planar surface of the liquid due to surface tension should at all points be equal and opposite to the pressure exerted by the liquid head. It yielded the above expression for the maximum height of the liquid, lmax that can be supported in a floating zone technique. Various ingenious induction-heating processes have been developed for stabilizing zones of height greater than lmax. In 1973 highly perfect single crystals of ultra pure silicon were available commercially in diameters greater than 7cm.The credit for achievement of success in obtaining ultra pure form of materials like silicon goes to zone melting technique.<\/p>\r\n \r\n\r\n\r\n\r\n
you can view video on zone melting& refining<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nReferences.<\/strong>\r\n\r\n \r\n
    \r\n \t
  1. Pfann,W.G. : \u201c Zone Melting \u201c in \u201c Crystal Growth & Characterization \u201c ed. Ueda,R.& Mullin, J.B.,North- Holland, Amsterdam\/Am Elsevier,N.Y., 1975.<\/li>\r\n \t
  2. Shah,J.S. : \u201c Zone Refining & Its Applications\u201d in \u201c Crystal Growth\u201d ,II edition,ed.byPamplin,B.R. Pergamon Press, Oxford ,1980.<\/li>\r\n \t
  3. Kittel,C. : \u201c Introduction to Solid State Physics\u201d,Wiley,N.Y., 1971.<\/li>\r\n \t
  4. Gilman,J.J. : \u201c The Art & Science of Growing Crystals\u201d, Wiley,N.Y.,1963.<\/li>\r\n<\/ol>\r\n \r\n\r\nSuggested Reading for Detailed\/Additional Information<\/strong>.\r\n\r\n \r\n
      \r\n \t
    1. Brice, J.C. : \u201c The Growth of Crystals From Melt\u201d,North Holland.1965.<\/li>\r\n \t
    2. Herington,E.F.G.: \u201c Zone Melting of Organic Compounds\u201d,Blackwell,Oxford,1963.<\/li>\r\n \t
    3. kirgintsev,A.N.: \u201cMathematical Theory of Zone Melting Processes\u201d,IZd,Co A N SSSR, Novosibirsk (in Russian).<\/li>\r\n \t
    4. Paar,N.L.: \u201cZone Refining & Allied Techniques\u201d.,George Newnes,London,1960.<\/li>\r\n \t
    5. Pfann,W.G. : \u201c Zone Melting \u201c, II ed.,Wiley,N.Y.,1966.<\/li>\r\n \t
    6. Schildknecht,H. \u201c Zone Melting\u201d, Academic Press,1966.<\/li>\r\n \t
    7. Shah,J.S. : in \u201cCrystal Growth\u201c, vol.1,ed.I,edited by Pamplin,B.R.,Pergamon Press,Oxford,1974.<\/li>\r\n \t
    8. Shah,J.S. : Ph.dThesis,Bath Univ.,England.,1967.<\/li>\r\n \t
    9. M. Zief :\u201d Purification of Inorganic and Organic Materials\u201d, Marcel DekkerInc.,1969.<\/li>\r\n \t
    10. R.A.Laudise: \u201c Growth of Single Crystals\u201d,Prentice Hall,Inc,1970.<\/li>\r\n<\/ol>\r\n ","rendered":"
      \"epgp<\/a>
      \n<\/span><\/div>\n
      \u00a0<\/span><\/div>\n
      <\/div>\n
      <\/div>\n
      <\/div>\n
      <\/div>\n
      32.1.<\/strong>\u00a0Introduction<\/strong> <\/p>\n

      Some basic background of crystal growth by zone melting has already been described. However, some more details are required to be given here about zone melting with more emphasis on how crystals get refined by this technique .The technique finds wide application in the industry dealing with solid state devices. In the last 50 years it has made immense contribution to the development of different fields of materials technology in particular and solid state science in general. Through this technique, it has been possible to produce a wide variety of both organic as well as inorganic materials of high purity. The semiconducting materials like germanium and silicon were purified with the help of this technique. However, its use was extended to several kinds of solid state devices. The applications of zone refining are many folds not only in the growth of materia ls as single crystals but also in the purification of several materials, elements, organic and inorganic compounds.<\/p>\n

       <\/p>\n

      It was Pfann who in 1952 described zone melting and zone refining processes. In 1953 Pfann and Olsen experimentally showed that a high degree of purification of a rod of germanium could be achieved by using zone refining process. Pure germanium is required for solid state devices like transistors and Pfann successfully demonstrated the potential of the technique for removing impurities from the solid state material.<\/p>\n

       <\/p>\n

       <\/p>\n

      32.2<\/strong>\u00a0Theory of Zone Melting<\/strong><\/p>\n

       <\/p>\n

      In the understanding of basic theory behind zone melting processes, it is necessary to have some background of the following terms:<\/p>\n

       <\/p>\n

      i)\u00a0\u00a0\u00a0 The equilibrium distribution coefficient.<\/p>\n

       <\/p>\n

      ii)\u00a0\u00a0\u00a0 Interface distribution coefficient<\/p>\n

       <\/p>\n

      iii)\u00a0\u00a0\u00a0 Effective distribution coefficient<\/p>\n

       <\/p>\n

      Let us describe them one by one:<\/p>\n

       <\/p>\n

       <\/p>\n

      32.2.1<\/strong>\u00a0Equilibrium Distribution Coefficient<\/strong><\/p>\n

       <\/p>\n

      Let us consider phase diagram of a binary system consisting of a solute in the form of soluble impurity and a solvent in the form of a host material. The solvent (a host material)-solute (a soluble impurity) phase diagrams are shown schematically in figure 32.1 (a, b). The figures\u00a0schematically represent portions of such diagrams near the melting points of solvents. Here we may define an equilibrium distribution coefficient \u03c30 (symbolized by k0 by several authors) as the ratio of the concentration of the solute in the solid (Cs) phase to that in the liquid (CL) phase when the solid and liquid phases are in equilibrium. So,<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure 32.1: Solvent-solute phase diagrams for (a) \u03c3<\/em><\/strong>0<\/em><\/strong> < 1, (b) \u03c3<\/em><\/strong>0<\/em><\/strong> > 1<\/em><\/strong><\/p>\n

       <\/p>\n

      Figure 32.1 represents solvent-solute phase diagrams. While figure 32.1(a) is the phase diagram in case of (a) \u03c30 = CS\/CL<1, figure32.1 (b) is for \u03c30 = CS\/CL> 1<\/p>\n

       <\/p>\n

      The distribution coefficient \u03c30 as defined by equation 32.1 also represents the ratio of the increment in thermodynamic potential as a result of interaction and entropy of mixing of the solute with the solvent in the solid and liquid phases.<\/p>\n

       <\/p>\n

      The chemical potentials of the solute in the solid and liquid phases are given by:<\/p>\n

       <\/p>\n

      \u03bc = \u03bc0 \u2500 RT ln a\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.2<\/p>\n<\/div>\n

       <\/p>\n

      where \u03bc0 is a constant,<\/span><\/p>\n

      R is the gas constant and\u00a0<\/span>a represents activity of the solute.<\/span><\/p>\n

       <\/p>\n

      Let us now represent the chemical potential of the solute in the solid phase as:<\/span><\/p>\n

       <\/p>\n

      \u03bcS =\u00a0 \u03bc0(s) – RT ln aS<\/span><\/p>\n

      \n

       <\/p>\n

      Similarly, the chemical potential of the solute in the liquid phase may be represented as:<\/p>\n

       <\/p>\n

      \u03bcL\u00a0\u00a0 =\u00a0 \u03bc0(L) \u2500 RT ln aL<\/p>\n

       <\/p>\n

      On establishment of equilibrium \u03bc(S) = \u03bc (L) ,<\/p>\n

       <\/p>\n

      So,<\/p>\n

       <\/p>\n

      \u03bc0(s) \u2500 RT ln aS= \u03bc0(L) \u2500 RT ln aL ,<\/p>\n

       <\/p>\n

      Or, aS<\/sub>\/aL<\/sub> = a constant.<\/p>\n

       <\/p>\n

       <\/p>\n

      According to the definition of activity that as the concentration of the solute in the solution tends to zero the activity tends to be equal to concentration. Decrease in the concentration of the solute in the solution amounts to saying that the solution is getting diluted.<\/p>\n

       <\/p>\n

      Thus, for dilute solutions:<\/p>\n

       <\/p>\n

      aS<\/sub>\/aL<\/sub> = CS<\/sub>\/CL<\/sub> \u2026\u2026\u2026\u2026\u2026\u2026……32.3<\/p>\n

       <\/p>\n

      Equation 32.3 suggests that the equilibrium distribution coefficient \u03c30 is generally a function of the concentration of the solute except for very dilute solutions.<\/p>\n

       <\/p>\n

      Looking at figure 32.1, it suggests that if the addition of the solute lowers the melting point then \u03c30< 1 but if the addition of the solute increasesthe melting point then \u03c30> 1.<\/p>\n

       <\/p>\n

      \u03c30 is often calculated from the phase diagram data. In the event of non-availability of the desired phase diagram at the desired concentration, one may deduce \u03c30 for dilute solutions from a plot of ln \u03c30 against T\u25001, where T is the absolute temperature.<\/span><\/p>\n

       <\/p>\n

      Hayes and Chipman (1939) has suggested a thermodynamic method with the application of which one can estimate \u03c30 for very dilute solutions through an equation:<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Where N (L) represents the mole fraction of the solute in the liquid, H is the heat of fusion of the solvent,<\/p>\n

      \u0394H is the heat of fusion of the solvent,<\/p>\n

      \u0394T is the difference in freezing point between the pure solvent and the solution,<\/p>\n

      R is the gas constant, and<\/p>\n

      T is the absolute temperature.<\/p>\n

       <\/p>\n

      The equilibrium distribution coefficient becomes applicable to equilibrium solidification processes where the temperature gradient across the solid-liquid system is negligibly low and the rate of solidification very small so that impurity gradients get wiped out by diffusion processes.<\/p>\n

       <\/p>\n

      If we consider zone-melting and any other practical solid-melt techniques, solidification does not occur in equilibrium conditions because there are large temperature and concentration gradients present, rates of solidification are high and the mixing of the solute in the liquid imperfect. It, therefore, becomes important to understand about two other distribution coefficient and those are interface distribution coefficient and effective distribution coefficient.<\/p>\n

       <\/p>\n

       <\/p>\n

      32.2.2.<\/strong>\u00a0Interface Distribution Coefficient (\u03c3*)<\/strong><\/p>\n

       <\/p>\n

       <\/p>\n

      It is defined as the ratio of the solute concentration in the solid to that in the liquid, actually at the interface:<\/p>\n

       <\/p>\n

      i.e.,\u00a0 \u00a0\u03c3* = Concentration of solute in the solid\u00a0 \u00a0(at the interface) \/Concentration of solute in the liquid<\/p>\n

       <\/p>\n

      =C<\/span>S<\/sub>\/C<\/span>L(0)<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.5<\/span><\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure 32.2: Schematic solute concentration profile and the diffusion layer at a solid -liquid interface<\/em><\/strong><\/p>\n

       <\/p>\n

      It is found that the interface distribution coefficient \u03c3*is different from the equilibrium distribution coefficient \u03c30. Figure 32.2 is a schematic diagram showing concentration profile and the diffusion layer at a solid-liquid interface. Since the rates of the chemical processes and the rates of the exchange reactions between the solid and the liquid at the interface is dependent upon the velocity of growth, \u03c3* should be a function of the velocity of solidification or growth. Measurements of \u03c3* have shown that it does actually vary with the growth velocity. The following equation has been derived on a model in which the solute incorporation in the solid is assumed to be through exchange reactions across solid-liquid interface:<\/p>\n

       <\/p>\n

      \u03c3* = \u03c30 + vg\u00a0 ( 1 \u2500 \u03c3* )\/\u03b2vm\u2026\u2026\u2026\u2026\u2026\u202632.6<\/p>\n

       <\/p>\n

      Where \u03c30\u00a0 is the equilibrium distribution coefficient,<\/span><\/p>\n

      vg is the velocity of growth ,<\/span><\/p>\n

      \u03b2 represents the dissociation coefficient of the observed solute molecule at the interface (describing <\/span><\/p>\n

      molecules leaving the interface and incorporating into the liquid)<\/span><\/p>\n

      and vm is the molecular velocity of diffusion of the solute.<\/span><\/p>\n

      The equilibrium distribution coefficient in this model is given as:<\/span><\/p>\n

      \u03c30 = \u03b1\/\u03b2 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.7,<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      where \u03b1 is known as sticking coefficient of the solute molecule and describes the entrance of the solute in the crystal.<\/p>\n

       <\/p>\n

       <\/p>\n

      32.2.3<\/strong>\u00a0Effective Distribution Coefficient (\u03c3eff)<\/strong><\/p>\n

       <\/p>\n

      It is another important term used in the understanding of Zone melting as a means of refining the crystals. The interface differential coefficient is not the same as equilibrium segregation coefficient. So, the overall incorporation of solute into the solid in actually experimental crystal growth methods cannot be described by the equilibrium segregation coefficients. It is here that the use of effective distribution coefficient \u03c3eff which is related to interface distribution coefficient \u03c3* through a functional relationship as given below:<\/p>\n

       <\/p>\n

      \u03c3eff\u00a0 = \u03c3* F\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.8<\/p>\n

       <\/p>\n

       <\/p>\n

      where F is a function of parameters such as growth rate, the degree of mixing \/stirring,diffusion of the solute\u00a0 in the liquid and the concentration of the solute. The effective distribution coefficient \u03c3e ff can be\u00a0 calculated from the\u00a0 equilibrium distribution coefficient\u00a0 \u03c30. For\u00a0 this\u00a0 purpose, the\u00a0 following one dimensional steady-state diffusion equation defining conservation of\u00a0 the solute molecular across the interface is used:<\/p>\n

       <\/p>\n

      D\u00a0 d2<\/sup>C\/dx2<\/sup> \u2500 vxdC\/dx = 0\u2026\u2026\u2026\u2026\u2026\u2026\u202632.9<\/p>\n

       <\/p>\n

      Where D is the diffusion coefficient of the solute,<\/span><\/p>\n

      C is the concentration of the solute,<\/span>and vx is the fluid velocity of liquid in the x- direction.<\/span><\/p>\n

      The fluid velocity of liquid\u00a0 vx = vg\u00a0 + \u03c9,<\/span><\/p>\n

      where vg is the solidification velocity and\u00a0<\/span>\u03c9 is the normal fluid velocity.<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      Let us refer to figure 32.2. It assigns the following boundary conditions to the equation of effective distribution coefficient \u03c3e ff defined by equation32.8. The interface is positioned at x=0 and the positive x direction is extended into the melt.<\/p>\n

       <\/p>\n

      The concentration of the solute C = CL(0) at x = 0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.10(a)<\/p>\n

      The concentration of the solute C = CL at x = \u03a9\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.10(b)<\/p>\n

       <\/p>\n

      The boundary condition suggests that for distance x > \u03a9 the concentration in the melt on account of\u00a0<\/span>fluid flow is equal to CL. However, within the diffusion layer (i.e., the layer of thickness \u03a9 at the\u00a0<\/span>interface) the fluid velocity of liquid is equal to the solidification velocity.<\/span><\/p>\n

       <\/p>\n

      Equation 32.9 is very much similar to heat flow equation. With the application of boundary conditions as are applicable in the present case and specified in equations 32.10 (a & b) the solution yields for effective distribution coefficient as given below:<\/span><\/p>\n

       <\/p>\n

      vx = vg and so, \u03c9 = 0\u2026\u2026\u2026\u2026\u202632.10(c)<\/span><\/p>\n

       <\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      The\u00a0 above equation at\u00a0 32.11 gives\u00a0 us\u00a0 the\u00a0 value\u00a0 of\u00a0 \u03c3eff\u00a0 as\u00a0 determined by making boundary layer\u00a0analysis which can be used for obtaining information on the morphological stability of the solid-liquid interface when convection is taking place.<\/span><\/p>\n

       <\/p>\n

      \u03a9\u00a0 is a function of the diffusion coefficient kinematic viscosity of the liquid and conditions of the flow of fluid. The layer thickness \u03a9 at the interface is dependent on the growth velocity as is clear from equations 32.12(a,b) which is derived below.<\/span><\/p>\n

       <\/p>\n

      Writing equation 32. 11 in the form as given below:<\/span><\/p>\n<\/div>\n

      <\/div>\n
      \n

      \"\"<\/p>\n

       <\/p>\n

      If one is able to measure \u03c3eff in crystals which are grown at different velocities under identical stirring conditions, curves of the term ln(1\/\u03c3e ff \u2500 1) or ln(1\u2500 1\/\u03c3e ff) could be drawn against the growth velocity vg which should be a straight line with the slope ( \u2500 \u03a9\/D). It is important to note here that interface should be stable. However, if for one reason or the other, the interface is unstable then measurement of ( \u2500 \u03a9\/D) may yield erroneous results. Instability of the interface arises if it melts back. In that case the thickness of diffusion layer fluctuates. In general, the dependence of layer thickness \u03a9 at the interface on the velocity of solidification vg is weak but is strongly dependent on the conditions of stirring. As for example, \u03a9 is hardly of the order of 10\u2500 1cm under negligible stirring but is found to be of the order of 103 cm if stirring is done vigorously. Theoretical solutions of \u03a9 for conditions of zone melting are rather difficult and the solutions that have been worked out are not found to be satisfactory.<\/p>\n

       <\/p>\n

      The diffusion layer is also dependent on the actual impurity distribution profile within it. Based on an assumption that the interface is planar, growth velocity remaining constant and\u00a0no stirring, Tiller and his co-workers in 1953 gave the following equation that connects the concentration of the solute in the liquid at a distance x from the interface which may be represented as CL(x) with the mean initial concentration C0 and concentration at the interface as CL(0):<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      CL(x<\/sub>) = CL(0)<\/sub> exp (\u2500 vg\/D. x) + C0<\/sub>\u2026\u2026\u2026\u2026\u2026\u2026\u202632.13<\/p>\n

       <\/p>\n

       <\/p>\n

      Where vg is the growth velocity and<\/p>\n

      D is the diffusion coefficient of the solute in the liquid.<\/p>\n

       <\/p>\n

      In the crystal growth from melt, there could be temperature fluctuations which may be either due to external variations of the heat source or due to inherent hydrodynamic effects associated with the convective motion of the melt. The presence of perturbations of the thermal field makes average interface distribution coefficient symbolized as \u03c3* to be different from the interface distribution coefficient in the absence of these fluctuations. The average interface distribution coefficient \u03c3* is taken as an average over a cycle of the perturbation. It can be expressed as:<\/p>\n

       <\/p>\n

      \u03c3* \u2260 \u03c3*\u2026\u2026\u2026\u2026\u2026\u2026\u2026….32.14<\/p>\n

       <\/p>\n

      The change in the effective distribution coefficient \u03c3eff in the presence of perturbation is given by:<\/p>\n

      \"\"<\/p>\n

      The expression for equation 32.15 has been obtained by Hurle and Jakeman in 1969 which may be given here as:<\/p>\n

      \"\"<\/p>\n

      where\u00a0\u00a0 Ima {} represents the imaginary part of the bracketed function ,<\/p>\n

       <\/p>\n

      I = \u221a(\u25001) ,<\/p>\n

       <\/p>\n

      \u03c6= \u03c9D\/vg2<\/sup>(0),<\/span><\/p>\n

       <\/p>\n

      = vg(0)\u03a9\/D and<\/span><\/p>\n

       <\/p>\n

      \u03a9 = (D\/\u03bd) 1\/3 \u03a9m ;<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      D is the diffusion coefficient of the solute in the liquid,<\/p>\n

       <\/p>\n

      vg(0<\/sub>) is the mean growth velocity,<\/p>\n

       <\/p>\n

      \u03a9 is the thickness of the diffusion layer,<\/p>\n

       <\/p>\n

      \u03bd is the kinematic viscosity of the melt and<\/p>\n

       <\/p>\n

      \u03a9m is the momentum boundary layer (the definition of which and its relation with \u03a9 is described in detail by V.G. Levich (1962) in Physico Chemical Hydrodynamics, Prentice -Hall Inc.).<\/p>\n

       <\/p>\n

      The above said authors (Hurle and Jakeman) who obtained expression as at 32.16 applied the same to the experimental results of Barthel and Eichler (1967) on measurement of effective distribution coefficient of tungsten in electron beam zone melted molybdenum as a function of the length of the molten zone, and succeeded in showing the validity of the expression.<\/p>\n

       <\/p>\n

      It has already been said that the application of equations 32. 12(a,b) for the experimental determination of \u03a9\/D will lead to erroneous results in the event of temperature fluctuations and hence interface instabilities that may be occurring in the experimental set- up. However, if there are low frequency fluctuations one may put the equation as under:<\/p>\n

       <\/p>\n

      \u03b4\u03c3eff<\/sub>\/\u03c3eff<\/sub> = \u03b7 (1\u2500 \u03c30)vg(0)\u03a9\/D\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.17<\/p>\n

       <\/p>\n

      Here, \u03b7 is a parameter which indicates the error in \u03a9\/D. It is because the experimentally measured distribution coefficient is actually the average effective distribution coefficient<\/p>\n

       <\/p>\n

      \u03c3eff<\/sub> = \u03c3eff<\/sub>(1+\u03b7) =\u00a0 \u03c30 {1+(1+\u03b7)(1\u2500 \u03c30)vg(0)\u03a9\/D}.<\/p>\n

       <\/p>\n

      It, therefore, becomes clear that the slope of the plot obtained from equations 32.12(a ,b)should be (1+\u03b7)\u03a9\/D and not \u03a9\/D.<\/p>\n

       <\/p>\n

      We have discussed some important expressions concerning distribution coefficient in a steady state which may be put here as follows:<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      \u03c30= CS<\/sub>\/CL<\/sub>,<\/p>\n

      vg = 0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.18(a)<\/p>\n

      \u03c3* = CS\/CL(0)<\/span>,<\/span><\/p>\n

      vg = vg\u2026\u2026\u2026\u2026\u2026\u2026…..\u202632.18(b)<\/span><\/p>\n

      \u03c3eff =\u00a0 CS\/CL<\/span>,<\/span><\/p>\n

      vg = vg\u2026…\u2026\u2026\u2026\u2026\u2026\u2026.32.18(c)<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      At equilibrium<\/p>\n

      \u03c3eff = \u03c3* = \u03c30<\/span>\u2026\u2026\u2026\u2026\u2026\u2026\u2026…\u2026\u2026\u2026\u2026\u2026.32.18(d)<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      If growth rates are high (i.e., vg \u2192 \u221e)<\/p>\n

       <\/p>\n

      \u03c3* \u2192 1<\/p>\n

      as vg\u2192 \u221e\u00a0<\/span>\u2026\u2026\u2026\u2026\u202632.18(e)<\/span><\/p>\n

       <\/p>\n

      and\u00a0<\/span>\u03c3eff \u2192 1<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      It has been experimentally observed that there are variations with orientation of growing crystals. This is called facet effect which is not explained by the above described equation given at no. 32.11 and<\/p>\n

       <\/p>\n

      32.18. However, one can explain it provided \u03c3* is taken as a parameter which does get influenced by the orientation of the growing face. In other words, the exchange rates of solute atoms between the solid and liquid are treated as varying with orientation of the growing facet.<\/p>\n

       <\/p>\n

      Segregation influences sheet growth of a monolayer and there is effect on the motion of growth steps if impurities get incorporated. The concentration of impurity increases with the velocity of growth steps and decreases with an increase in the rate at which the impurity diffuses out of the growth steps into the melt. The higher impurity in the faceted regions is because of the higher velocity of growth steps.<\/p>\n

       <\/p>\n

      32. 3<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Zone refining<\/strong><\/p>\n

       <\/p>\n

      32.3.1<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span>Introduction.<\/strong><\/p>\n<\/div>\n

      \n

       <\/p>\n

      Zone melting is the general title of a number of techniques used for the purification of elements and compounds and also for the preparation of materials of the required composition .In these techniques a short liquid region, or zone , is made to travel slowly through a relatively long charge , or ingot of solid. When the zone travels, it redistributes impurities along the charge. The final distribution depends on the following:<\/p>\n

       <\/p>\n

      i)\u00a0\u00a0\u00a0\u00a0\u00a0 Impurity distribution in the starting charge,<\/p>\n

      ii)\u00a0\u00a0\u00a0 the distribution coefficient \u03c3 of an impurity between the liquid and solid of the charge material,<\/p>\n

      iii)\u00a0 the size of the zone,<\/p>\n

      iv) the number of the zones, and<\/p>\n

      v)\u00a0\u00a0\u00a0 travel direction of the zones.<\/p>\n

       <\/p>\n

      Zone melting is a novel method of using the freezing process to manipulate impurities. It is\u00a0 a well known fact\u00a0 that\u00a0 the composition of\u00a0 a freezing crystal is\u00a0 different\u00a0 from that of\u00a0 its\u00a0 liquid. Obviously, one could think of a simple idea of passing a very thin liquid zone along a lengthy solid. Following this approach the process of crystallization as a means to purify a solid or to control its composition received a lot of attention and has been investigated and put to practice. Zone refining is the most valuable zone melting method. This process of refining involves passing of a number of molten zones along the charge in one direction. Each zone carries with it a fraction of the impurities to the end of the charge , thereby leaving behind the purified material. In early 1950\u2019s zone refining was introduced for purification of germanium for use in transistors. The purity of the material obtained by the process was extraordinary which in quantitative terms could be placed as one part of detectable impurity in 1010 parts of germanium. This was a remarkable achievement that boosted the electronic industry and applications of solid state devices. The method of zone refining has been applied to several substances. It received so much of popularity that by 1973 almost one -third of the elements of the periodic table had been purified to the highest degree by zone refining. Since zone refining provided the means of making ultra-pure materials available, it opened new areas of research and commerce.<\/p>\n

       <\/p>\n

      32.3.2<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Principles of Zone Refining<\/strong><\/p>\n

       <\/p>\n

      It is important to first describe the distribution coefficient or segregation coefficient \u03c3 followed by a very old crystallization method known as normal freezing, one-pass zone melting and then multi-pass zone melting or zone refining.<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      32.3.2.1<\/strong>\u00a0\u00a0\u00a0 Normal Freezing.<\/strong><\/p>\n

       <\/p>\n

      The process of solidification in which whole of a charge is melted initially and then gradually solidified unidirectional is classified as normal freezing process. Two examples of normal freezing methods that are used for the growth of crystals are (i) the Bridgman process in which a crucible containing a molten charge is moved slowly with respect to a stationary temperature gradient for unidirectional solidification and (ii) the Stober process in which a temperature gradient is moved through a crucible containing a molten charge.<\/p>\n

       <\/p>\n

      Here, it is important to be conversant with expressions for impurity distribution in normal freezing process. Normal freezing process may be schematically shown in figure 32.3. When a cylinder of a substance say A having in it an impurity say B is melted, and is slowly frozen from one end to the other, the impurity gets usually concentrated in the \u201clast-to-freeze\u201d region of the cylinder. This is what is actually called normal freezing.<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      Figure 32.3: Schematic representation of normal freezing process<\/em><\/strong><\/p>\n<\/div>\n

       <\/p>\n

      The following assumptions are made in the derivation of an expression:<\/span><\/p>\n

      \n

       <\/p>\n

      (i) Diffusion of the solute under consideration is negligible in the solid, i.e., x2 >> DSt,\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.32.19.<\/p>\n

       <\/p>\n

      where x is the length of the fraction solidified in time t and DS stands for diffusion coefficient of the solute in the solid.<\/p>\n

       <\/p>\n

      (ii) The effective distribution coefficient \u03c3eff is constant.<\/p>\n

       <\/p>\n

      (iii)\u00a0 The density change of the solution during freezing is zero.<\/p>\n

       <\/p>\n

       <\/p>\n

      Component B gets redistributed because the atoms (or molecules) of B at the liquid-solid interface prefer the liquid phase to the solid phase. The factor behind this preference is the distribution coefficient or segregation coefficient \u03c3 which is defined as the ratio of the concentration of B in just-forming solid A to that in liquid A. If rate of freezing is very slow, the equilibrium distribution coefficient \u03c30 becomes applicable. If , however, the rates of freezing are moderate say ~<\/strong> 1-30cm h\u25001, an effective distribution coefficient, \u03c3eff , which lies between \u03c30 and unity, is applicable. It is because for \u03c30< 1, the rejected impurity B accumulates in the liquid just ahead of the advancing solid, so that the just forming solid \u201cfinds\u201d a liquid more impure than the bulk liquid. On the other hand, if freezing is rapid, effective distribution coefficient \u03c3eff may actually approach unity; there would be no zone refining and as a result the interface may probably become dendritic in shape.<\/p>\n

       <\/p>\n

      The solute concentration (i.e., solute atoms \/unit volume) in the solid immediately behind the interface is given by:<\/p>\n

       <\/p>\n

      CS = \u2500 dS<\/sub>\/dg<\/sub>\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.20,<\/p>\n

       <\/p>\n

      Where CS stands for the solute concentration in the solid,<\/p>\n

       <\/p>\n

      S is the amount of solute in the liquid, and<\/p>\n

       <\/p>\n

      g is the solidified fraction of the original volume of unity.<\/p>\n

       <\/p>\n

      According to the definition of the distribution coefficient (as given in equation 32.1):<\/p>\n

       <\/p>\n

      \u03c3 = CS<\/sub>\/CL<\/sub>\u00a0 , or CS<\/sub>= \u03c3 CL<\/sub>, where CL stands for concentration of the solute in the liquid.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

       <\/p>\n

      Here, integration is carried out over the total amount of the solute and hence its amount taken initially in the liquid when g = 0 i.e., S0 and final amount of solute when the entire length is solidified (i.e., S).<\/p>\n

       <\/p>\n

      So, S = S0( 1 \u2500 g )\u03c3<\/sup><\/p>\n

       <\/p>\n

      Equation no. 32.20 may be written as:<\/p>\n

       <\/p>\n

      CS = \u2500 dS\/dg = \u03c3 S0 (1\u2500g) \u03c3 \u25001\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026\u2026\u2026\u2026\u2026.32.23<\/p>\n

       <\/p>\n

      because of our assumption that the original volume was unity.<\/p>\n

       <\/p>\n

      S0<\/sub> = C0<\/sub>, where C0\u00a0 represents original concentration.<\/p>\n

       <\/p>\n

      Substituting it in equation no. 32.23, we have:<\/p>\n

       <\/p>\n

      CS\u00a0\u00a0 = \u03c3 C0 (1 \u2500 g) \u03c3 \u25001 \u2026\u2026\u2026\u2026\u2026………………………..32.24<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      Or, ln (CS\/C0) = ln\u03c3 + (\u03c3 \u2500 1) ln(1\u2500g) ……………..\u2026.32.25<\/p>\n

       <\/p>\n

       <\/p>\n

      If logarithmic plot of CS\/C0 against (1\u2500g) is drawn it would be a straight line whose slope and intercept at g = 0 will yield the value of \u03c3.<\/p>\n

       <\/p>\n

      The equation at number 32.23 becomes inaccurate for the value of g > ~<\/strong> 0.9, because it implies an infinite impurity concentration at g = 1.0. In practice, either \u03c3 varies with concentration, or a second phase, rich in the impurity, nucleates, as it happens in a eutectic system.<\/p>\n

       <\/p>\n

      A normal freezing distribution for an impurity having an effective distribution coefficient of 0.5 (which is assumed to remain constant) is shown in figure 32.4.<\/p>\n

      \"\"<\/p>\n

      Figure 32.4: Relative impurity concentration after (a) normal freezing and (b) on zone melting pass for initial mean concentration C0 and for an effective distribution coefficient of 0.5<\/em><\/strong><\/p>\n

       <\/p>\n

      The equation for the curve shown is of the form C = \u03c3 C0( 1 \u2500 g )\u03c3 \u2500 1 [ see equation 32.23], where C0\u00a0<\/span>is the mean impurity concentration in the original liquid , and g is the fraction of the liquid that has frozen.<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      32.3.2.2\u00a0<\/strong>\u00a0Solute Distribution in Zone Melting<\/strong><\/p>\n

       <\/p>\n

      In order to be able to understand the solute distribution in zone melting, the following variables will have to be specified:<\/p>\n

       <\/p>\n

      i)\u00a0 The zone length\u2018l\u2019<\/p>\n

       <\/p>\n

      ii)\u00a0 Length of the charge \u2018L\u2019<\/p>\n

       <\/p>\n

      iii)\u00a0 The initial concentration of the solute is assumed to be uniform throughout the ingot. The same may be<\/p>\n

      represented by a constant C0.<\/p>\n

       <\/p>\n

      iv) The traverse velocity of the zone or the zone velocity vg.<\/p>\n

       <\/p>\n

      We shall first consider single-pass distribution. Here we are concerned with evaluation of distribution of the solute in the ingot from one end to the other and make the following assumptions:<\/p>\n

       <\/p>\n

      i)\u00a0\u00a0\u00a0\u00a0\u00a0 The distribution coefficient \u03c3 is constant.<\/p>\n

       <\/p>\n

      ii)\u00a0\u00a0\u00a0 The zone length l is held constant.<\/p>\n

       <\/p>\n

      iii)\u00a0 The initial concentration C0 of the solute is uniform throughout the ingot.<\/p>\n

       <\/p>\n

      iv) The densities of the liquid and the solid under consideration are the same.<\/p>\n

       <\/p>\n

      v)\u00a0\u00a0\u00a0 Diffusion of the solute in the solid is negligible, i.e.,<\/p>\n

       <\/p>\n

      x2 >> DS.t , where x is the length of the fraction solidified in time t and DS is the diffusion coefficient of the solute in the solid.<\/p>\n

       <\/p>\n

      Schematic diagram of a non-continuous zone-melting process is shown in figure 32.5.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure 32.5: A schematic diagram of zone refining process<\/em><\/strong><\/p>\n

       <\/p>\n

      We consider that the direction in which the zone is travelling is in the direction x=0 to x= L. As the liquid zone moves along the ingot it leaves behind the resolidified ingot. Solidification of the charge of an incremental volume dx as the zone advances would involve solute transfer. We are interested in finding an equation for this solute transfer. Suppose CL represents concentration of the solute in the liquid, then the amount of solute leaving the zone because of solidification = \u03c3 CLdx. The solute that enters the zone due to melting of the charge of the volume dx is C0dx . The net change in the total volume of solute in the liquid is given by:<\/p>\n

       <\/p>\n

      dS = ( C0<\/sub> \u2500 \u03c3CL<\/sub>) dx\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.26<\/p>\n

       <\/p>\n

      Let us assume that the cross-sectional area of the ingot and zone is unity. In that case the concentration in the liquid zone is given by:<\/p>\n

       <\/p>\n

      CL = S\/l , \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026……..32.27<\/p>\n

       <\/p>\n

      where S is the amount of the solute in the zone at a distance x.<\/p>\n

       <\/p>\n

      In the light of equation 32.26, it becomes:<\/p>\n

       <\/p>\n

      dS = ( C0 \u2500 \u03c3 S\/l ) dx\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u202632.28<\/p>\n<\/div>\n

      \n

       <\/p>\n

      or dS\/dx + ( \u03c3\/l ) S = C0……………………………………. 32.29<\/p>\n

      \"\"<\/p>\n

       <\/p>\n

      S0 = C0l\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.32.31<\/p>\n

       <\/p>\n

      Substituting for S0 from equation 32.31 in equation 32.30, we have:<\/p>\n

       <\/p>\n

      Se (\u03c3\/l)x \u2500 C0l = C0l\/\u03c3 { e(\u03c3\/l).x\u2500 1}\u2026\u2026\u2026\u202632.32<\/p>\n

       <\/p>\n

      Se(\u03c3\/l). x = C0l [ 1 + 1\/\u03c3{ e(\u03c3\/l).x\u2500 1}]]<\/p>\n

       <\/p>\n

      S = C0l [ 1 + 1\/\u03c3 { e(\u03c3\/l).x \u2500 1}] e\u2500(\u03c3\/l). x<\/p>\n

       <\/p>\n

      The solute concentration CS in the solid at any value of x is given by:<\/p>\n

       <\/p>\n

      CS = \u03c3S\/l<\/p>\n

       <\/p>\n

      Or,\u00a0\u00a0 S = lCS\/\u03c3\u2026\u2026\u2026..\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..32.33<\/p>\n

       <\/p>\n

      Substituting for S from equation 32.33in the above expression we get:<\/p>\n

       <\/p>\n

      l CS\/\u03c3 = C0 l [1 + 1\/\u03c3{e(\u03c3\/l)x\u2500 1}] e\u2500(\u03c3\/l).x<\/p>\n

       <\/p>\n

      CS \/C0 = \u03c3 [1+ 1\/\u03c3 {e (\u03c3\/l).x\u2500 1} ] e\u2500 (\u03c3\/l). x<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      =\u00a0\u00a0 {\u03c3 + e (\u03c3\/l).x\u2500 1} e\u2500 (\u03c3\/l).x<\/p>\n

       <\/p>\n

      =\u03c3e \u2500 (\u03c3\/l).x+ 1 \u2500 e\u2500 (\u03c3\/l). x<\/p>\n

       <\/p>\n

       <\/p>\n

      =\u00a0 {1\u2500 e\u2500 (\u03c3\/l).x (1\u2500 \u03c3)}\u2026\u2026\u2026………………32.34<\/p>\n

       <\/p>\n

       <\/p>\n

      Figure 32.6 illustrates schematically profile of the solute distribution (\u03c3 < 1) in case of single pass. In this profile equation 32.34 shows distribution in the region marked X. It covers the part of ingot from x=0 to x = L-l, where L is the total length of the ingot. The last liquid zone to freeze out is represented by region Y in the figure. Actually, it is normally frozen and as such the distribution in this region is given by the normal freeze (equation 32.23).<\/p>\n

       <\/p>\n

       <\/p>\n

      The normal freezing operation as described in the previous section is the first step in the repeated fractional crystallizations which is not something new but, in fact, is centuries old technique. However, the technique of fractional crystallization could not receive much of response because it involved time-consuming and troublesome operations which included partial freezing, crystal separation from mother liquor, remelting, recombining with other fractions, and several others.<\/p>\n<\/div>\n

      \n

      \"\"<\/p>\n

      Figure 32.6: Solute distribution shown schematically in case of a single-pass zone melting<\/em><\/strong><\/p>\n

       <\/p>\n

       <\/p>\n

      Zone refining has made the task much simpler and very effective in achieving purification of materials. A series of molten zones traverses the ingot in the same direction, usually with the arrangement of multi heaters, as shown in figure 32.7.<\/p>\n

      \"\"<\/p>\n

      Figure 32.7: Schematic representation of zone refining process using multi heaters<\/em><\/strong><\/p>\n

       <\/p>\n

       <\/p>\n

      Here in this case, each zone draws impurity at its melting interface, and freezes out solid purer than\u00a0the liquid (for those with segregation coefficient \u03c3 < 1) at its freezing interface. So, it does not involve the complicated and cumbersome process of separation and recombination of fractions and also eliminates the need to touch or move the charge at all. The distribution of impurity say B after one zone-pass for an ingot ten zone-lengths and for \u03c3 equal to 0.5 is shown in figure 32.4. The equation for the one-pass curve, except in the last zone-length is given by equation no. 32.34.<\/span><\/p>\n

       <\/p>\n

      As the number of zone-passes is increased the impurity concentration at the beginning of the ingot decreases more and more till a limit called ultimate distribution is reached. The ultimate distribution is given (except for the last zone-length) by the simple exponential expression as given by Pfann (1952):<\/span><\/p>\n

       <\/p>\n

      C = A eBx ,<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      where x represents distance along charge , and A and B are constants and may be calculated from the following expressions:<\/p>\n

      \"\"<\/p>\n

      where L denotes charge length (total length of ingot) and C0 mean solute concentration per unit volume. Equation 32.36 does not hold for the last zone length where normal freezing takes place. It also cannot hold for the enhanced solute pile up or the back reflection of the solute in front of the last zone length. It may be noted here that the lowest concentration of impurity B (in the system A\u2013B as discussed in the previous section) is very small, less than 0.0001.<\/p>\n

       <\/p>\n

      32.3.2.3<\/strong>\u00a0\u00a0Zoning Equipment<\/strong><\/p>\n

       <\/p>\n

      Zoning apparatus includes the following:<\/p>\n

      1. a means of producing liquid zones, i.e., heaters and or coolers.<\/p>\n

      2.\u00a0 a traverse mechanism for the transport of molten zones<\/span><\/p>\n

      3.\u00a0 a means of mounting or holding a charge<\/span><\/p>\n

      4.\u00a0 means of stirring a liquid and maintaining constant or controlled ambient atmosphere and temperature <\/span><\/p>\n

      conditions, specifically at the solid-liquid interface.<\/span><\/p>\n<\/div>\n

      \n

       <\/p>\n

      1.\u00a0 Zoning in a container<\/p>\n

       <\/p>\n

      2.\u00a0 Zoning without a container.<\/p>\n

       <\/p>\n

       <\/p>\n

      32.3.2.4\u00a0<\/strong>\u00a0Zoning in a container<\/strong><\/p>\n

       <\/p>\n

      The usual container is one which does not contaminate the material. Glass, Vycor, fused silica, molybdenum, tantalum and graphite are examples of materials for containers. If zone refining is required to be done vertically, it is helpful to use a transparent container. If the container is a horizontal, semicircular-cross section boat, it can be oblique because the liquid zone can be readily distinguished from the solid. It is, however, necessary to take due care to prevent cracking of the filled container (be it horizontal or vertical), which is likely to occur on account of change in volume during freezing (or melting) or by differential thermal contraction (if the charge sticks to the container wall).<\/p>\n

       <\/p>\n

      The design of the container itself is important as the same is one of the deciding factors in achieving a good control over zone length and zone spacing .It is advantageous to have the walls of the container as thin as possible. The cross-sectional shape of the container has to be such as to have the surface of the charge as in minimum contact with the container as possible.<\/p>\n

       <\/p>\n

      The cross-sectional design has also to be such that the container (particularly, if the charge expands on solidifying) does not strain due to expansion or mass transfer and thus tending to crack the container.<\/p>\n

       <\/p>\n

      High melting-point materials or very reactive materials may be zone melted in a water cooled boat or a \u201ccold hearth\u201d made from high thermal conductivity material such as copper.<\/p>\n

       <\/p>\n

      32.3.2.5\u00a0<\/strong>\u00a0Zone Refining Without Containers<\/strong><\/p>\n

       <\/p>\n

      Contamination of the charge by the container is a problem in all purification processes. It\u00a0is particularly so for such materials which are very effective solvents or are very reactive at their melting points. Such materials obviously cannot be refined or retained at high purity at the slightest\u00a0<\/span>contact with other materials. The solution to this problem lies in finding ways of by-passing the use of containers for such materials. A unique solution was found for zone refining in devising a technique that is known as \u201cfloat-zone\u201d melting (FZM). The method was developed by Keck and Golay (1953), Emeis (1954), and Theuerer (1956, 1962) for the preparation of ultra-pure silicon. In float zoning a vertical silicon rod is held by end-clamps, and a short molten zone is produced by induction heating\u00a0<\/span>and moved along the rod. The liquid (molten zone) is held in place by surface tension. Theoretical analysis by Heywang and Ziegler (1954) and Heywang (1956) led them to an expression for\u00a0<\/span>maximum stable zone-height as:<\/span><\/p>\n

       <\/p>\n

      lmax \u2248 2.8 (\u03b3\/\u03c1g) 1\/2\u00a0 \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.32.37<\/span><\/p>\n

       <\/p>\n

      where \u03b3 is the surface tension of the liquid, \u03c1 is the density of the liquid, and g is the gravitational acceleration.<\/span><\/p>\n<\/div>\n

       <\/p>\n

      This expression was derived on the basis of condition of the stability that the inward pressure on the planar surface of the liquid due to surface tension should at all points be equal and opposite to the pressure exerted by the liquid head. It yielded the above expression for the maximum height of the liquid, lmax that can be supported in a floating zone technique. Various ingenious induction-heating processes have been developed for stabilizing zones of height greater than lmax. In 1973 highly perfect single crystals of ultra pure silicon were available commercially in diameters greater than 7cm.The credit for achievement of success in obtaining ultra pure form of materials like silicon goes to zone melting technique.<\/p>\n

       <\/p>\n\n\n\n
      you can view video on zone melting& refining<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

       <\/p>\n

      References.<\/strong><\/p>\n

       <\/p>\n

        \n
      1. Pfann,W.G. : \u201c Zone Melting \u201c in \u201c Crystal Growth & Characterization \u201c ed. Ueda,R.& Mullin, J.B.,North- Holland, Amsterdam\/Am Elsevier,N.Y., 1975.<\/li>\n
      2. Shah,J.S. : \u201c Zone Refining & Its Applications\u201d in \u201c Crystal Growth\u201d ,II edition,ed.byPamplin,B.R. Pergamon Press, Oxford ,1980.<\/li>\n
      3. Kittel,C. : \u201c Introduction to Solid State Physics\u201d,Wiley,N.Y., 1971.<\/li>\n
      4. Gilman,J.J. : \u201c The Art & Science of Growing Crystals\u201d, Wiley,N.Y.,1963.<\/li>\n<\/ol>\n

         <\/p>\n

        Suggested Reading for Detailed\/Additional Information<\/strong>.<\/p>\n

         <\/p>\n

          \n
        1. Brice, J.C. : \u201c The Growth of Crystals From Melt\u201d,North Holland.1965.<\/li>\n
        2. Herington,E.F.G.: \u201c Zone Melting of Organic Compounds\u201d,Blackwell,Oxford,1963.<\/li>\n
        3. kirgintsev,A.N.: \u201cMathematical Theory of Zone Melting Processes\u201d,IZd,Co A N SSSR, Novosibirsk (in Russian).<\/li>\n
        4. Paar,N.L.: \u201cZone Refining & Allied Techniques\u201d.,George Newnes,London,1960.<\/li>\n
        5. Pfann,W.G. : \u201c Zone Melting \u201c, II ed.,Wiley,N.Y.,1966.<\/li>\n
        6. Schildknecht,H. \u201c Zone Melting\u201d, Academic Press,1966.<\/li>\n
        7. Shah,J.S. : in \u201cCrystal Growth\u201c, vol.1,ed.I,edited by Pamplin,B.R.,Pergamon Press,Oxford,1974.<\/li>\n
        8. Shah,J.S. : Ph.dThesis,Bath Univ.,England.,1967.<\/li>\n
        9. M. Zief :\u201d Purification of Inorganic and Organic Materials\u201d, Marcel DekkerInc.,1969.<\/li>\n
        10. R.A.Laudise: \u201c Growth of Single Crystals\u201d,Prentice Hall,Inc,1970.<\/li>\n<\/ol>\n

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