General theory of multi-phase melt crystallization

Journal of Crystal Growth 234 (2002) 762–772

General theory of multi-phase melt crystallization N.A. Charykova, V.V. Sherstnevb, A. Krierb,* b

a St. Petersburg State Technological Institute (Technical University), St. Petersburg, 198013 Russia Physics Department, School of Physics and Chemistry, Adv. Materials and Photonics Group, Lancaster University, Lancaster LA1 4YB, UK

Received 1 December 2000; accepted 22 September 2001 Communicated by D.T.J. Hurle

Abstract A new approach is presented which has wide-ranging application for the modeling and understanding of both open multi-phase crystallization from the melt while the temperature is decreasing, and isothermal crystallization from solution while solvent is evaporating. The results are widely applicable. Examples are given which relate to laboratory situations such as the liquid-phase epitaxial growth of complex III–V alloys and also to crystallization processes in oceans or sea water. r 2002 Elsevier Science B.V. All rights reserved. Keywords: A2. Growth from melt; A2. Growth from solution

1. Open crystallization from multi-component melts Diagrams of open crystallization from multi-component melts while the temperature is increasing or decreasing are more interesting for technology and modeling of natural processes than phase equilibrium diagrams. Consequently, the thermodynamics of such phase processes (and processes of open evaporation) have been elaborated in detail for solid phases with invariable (components and compounds) and variable (solid solutions) compositions (see for example Refs. [1–5]). But in these works the phase equilibrium of only two phases (melt and solid) has been considered. At present, there is no comprehensive thermodynamic theory, which describes the crystallization processes of an arbitrary number of solid phases (kX1) with invariable and variable compositions within the conditions of multi-variant equilibrium (number of thermodynamic degrees of freedom f X1). This is because multi-variant crystallization takes place only in systems with a large number of components nX4; but till date these systems have been insufficiently studied (i.e. if n ¼ 3 crystallization of two solid phases only causes the movement of melt composition along the mono-variant crystallization curve corresponding to f ¼ 1; and such a consideration is trivial). There are two peculiarities of open crystallization: 1. The open-phase process demands that ‘‘new’’ phases, forming from the ‘‘old’’ phase, must be removed from the sphere of phase equilibrium [1], so only an infinitely small mass of ‘‘new’’ phases must be in the *Corresponding author. Tel.: +44-1524-593-651; fax: +44-1524-744-037. E-mail address: [email protected] (A. Krier). 0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 7 5 4 - 7

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equilibrium with the ‘‘old’’ phase. But this condition is not obligatory for the crystallizing phases with constant composition. In the case of crystallization of solid solutions (under the conditions of very slow diffusion in the solid phase) only the surface is in equilibrium with the melt, and this also does not limit our consideration. 2. During crystallization some solid phases may crystallize from the melt and some solid phases may dissolve in it. Later, we shall show that this is also not restrictive. 1.1. General differential equations Let us consider an n-component melt (nX4) at a constant pressure P; from which r solid phases (s1; s2y; srðrX2Þ) are crystallizing while the temperature is decreasing (under the conditions of (r þ 1)phase equilibrium). Let us ‘‘form’’ (‘‘form’’ is mathematical termFmeaning crystallization or dissolution) ~ðlÞ (which maintains 1 mol of components) r from the melt with composition in concentration spaceFX ðsiÞ ~ (they maintain dmðsiÞ moles of components), dmðsiÞ may solid equilibrium phases with composition X have arbitrary sign, but even for one phase dmðsiÞ must be >0. ~ðlÞ and the differential equation of mass In this process, the vector of melt composition shifts by vector dX balance is as follows: r r X X ~ðsiÞ dmðsiÞ ; ~ðlÞ þ dX ~ðlÞ Þð1  ~ðlÞ ¼ ðX X dmðsiÞ Þ þ ð1Þ X i¼1

i¼1

r ðlÞ P

~ and if we neglect the terms dX

dmðsiÞ :

i¼1 ðlÞ

~ ¼ dX

r X

ðsiÞ

~ ðX

ðlÞ

~ Þ dmðsiÞ : X

ð2Þ

i¼1

Vector Eq. (2) is equivalent to (n  1) independent scalar differential equations (for all elements of vector composition). One can see that the system of differential equations of open crystallization (Eq. (2)) has a structure analogous to the system of differential equations of open crystallization of a single solid phase [1] and turns into it at r ¼ 1: When r ¼ 1 it is easy to integrate Eq. (2) for the solid phase with invariable composition (in the whole region of two-phase equilibrium) and for a solid solution in the vicinity of specific points in the phase diagram [1–5]. In all other cases, one can easily numerically integrate system (2) for r ¼ 1: When rX2 we cannot solve system (2) even for the crystallization of solid phases with invariable composition. System (2) is of no use for us, because we do not know the ratios between dmðsiÞ ð1piprÞ: Points of all (r þ 1)-phases in concentration space may arbitrarily move along the multi-variant hyper-surface of (r þ 1)-phase equilibrium. But in all cases (at r ¼ 1 and at rX2) consideration of open crystallization results in a mono-variant curve, which must be determined in the hyper-surface of (r þ 1)-phase equilibrium. Therefore, to form the principal system of differential equations for solving open crystallization one must determine the derivatives ðdmðsiÞ =dmðsjÞ ÞP;cryst (1pi; jpr; iaj; ‘‘cryst’’Fopen crystallization) or exclude differentials dmðsiÞ ð1piprÞ from the system (2), for case r ¼ 1 [1]. Let us consider the system of common differential Van-der-Vaal’s equations for the equilibrium (l  si) at P¼ const [5] with the help of the variables of the liquid phase. Otherwise, for the case of solid phases with invariable composition these equations will be senseless, and we do not want to decrease the universality of our treatment. So ðsiÞ

~ ðX

~ðlÞ ÞG# ðlÞ dX ~ðlÞ ¼ S ðsi-lÞ dT; X

ð1piprÞ;

ð3Þ

where S ðsi-lÞ ð Qðsi-lÞ =TÞ is the differential molar entropy of the solid phase ðsiÞ in the melt [1, p. 155]; ~ðtÞ is a vector ~ðtÞ is a vector characterizing a figurative point of phase t in concentration space ðt ¼ l; siÞ; dX X ~ðtÞ while the (r þ1)-phase equilibrium shifts; G# ðlÞ is an operator, which is characterizing the movement of X

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determined by the matrix of second derivatives: GðlÞ  ðq2 GðlÞ =qXiðlÞ qXjðlÞ ÞT;P;Xkaj;n ; GðlÞ is the molar ij ðlÞ # Gibbs potential of phase l. The matrix of operator G and minors of its main diagonal are determined positively, according to the criteria of stability. Substitution of equations from system (2) into system (3) givestptxt 6pt r X ~ðsiÞ  X ~ðlÞ ÞG# ðlÞ ~ðsjÞ  X ~ðlÞ Þ dmðsjÞ ¼ S ðsi-lÞ dT; ð1piprÞ: ðX ðX ð4Þ j¼1

Let us assume that the composition of equilibrium phases l; si (1pipr) are linearly independent and that ~ðlÞ Þ are linearly independent. The other case is trivialFthe temperature of phase ~ðsiÞ  X vectors ðX equilibrium is constant or crosses through a maximum or minimum (dT ¼ 0), which is formulated in the Gibbs rule [1] and in this case an open phase process with (r þ 1) equilibrium phases is impossible [1]. The main determinant of system (4) forms Gram’s determinant [6]:   ðs1Þ  ðX ~ðlÞ ÞG# ðlÞ ðX ~ðs1Þ  X ~ðlÞ ÞyðX ~ðsrÞ  X ~ðlÞ ÞG# ðlÞ ðX ~ðs1Þ  X ~ðlÞ Þ   ~ X   ðs1Þ  ~ ~ðlÞ ÞG# ðlÞ ðX ~ðs2Þ  X ~ðlÞ ÞyðX ~ðsrÞ  X ~ðlÞ ÞG# ðlÞ ðX ~ðs2Þ  X ~ðlÞ Þ   ðX  X ðrÞ ð5Þ Gr ¼  a0;   ? ? ? ?     ðs1Þ  ðX ~ X ~ðlÞ ÞG# ðlÞ ðX ~ðsrÞ  X ~ðlÞ ÞyðX ~ðsrÞ  X ~ðlÞ ÞG# ðlÞ ðX ~ðsrÞ  X ~ðlÞ Þ  where the index ‘‘r’’ shows the dimensions of determinants. According to the main properties of Gram’s determinants [6], determinant (5) is not zero because vectors which form scalar productions in its elements: n1 X n1 X ðlÞ ~ðsiÞ  X ~ðlÞ ÞG# ðlÞ ðX ~ðsjÞ  X ~ðlÞ Þ  Gri;j  Grj;i  ðX Gpq ðXpðsiÞ  XpðlÞ ÞðXqðsjÞ  XqðlÞ Þ; p¼1

are linearly independent   Gr1;1 y    Gr1;2 y  DðrÞ ¼ k  y y    Gr1;r y

q¼1

[6]. Let us also define the following determinants:  Grk1;1 S ðs1-lÞ Grkþ1;1 y Grr;1   Grk1;2 S ðs2-lÞ Grkþ1;2 y Grr;2  ; y y y y   Grk1;r S ðsr-lÞ Grkþ1;r y Grr;r 

ð6Þ

and solve system (4): dmðskÞ ¼ 

DðrÞ k dT GrðrÞ



dmðsiÞ dmðsjÞ

 ¼ P;cryst

DðrÞ i DðrÞ j

:

ð7Þ

So, the common vector equation of open phase processes will be the following: ðlÞ

~ GrðrÞ ¼ dX

r X

ðsiÞ

~ ðX

ðlÞ

~ ÞDðrÞ dT; X i

ð8Þ

i¼1

or in scalar form: dXjðlÞ dXkðlÞ

r P

! ¼ P;cryst

ðXjðsiÞ  XjðlÞ ÞDðrÞ i

i¼1 r P i¼1

:

ð9Þ

ðXkðsiÞ  XkðlÞ ÞDðrÞ i

One cannot a priori determine the sign of dmðsiÞ while temperature decreases (it depends on the type of crystallizationFeutectic, I-type peritectic, II-type peritectic and so on). Points, curves or surfaces of the

N.A. Charykov et al. / Journal of Crystal Growth 234 (2002) 762–772

765

change of phase process in relation to the solid phase (si), where mðsiÞ crosses through a maximum or minimum may be determined (DðrÞ i ¼ 0). One can determine the sign of the following sum: r X dmðsiÞ ; ð10Þ dmðshetÞ ¼ i¼1

~ðshetÞ : In where m is the mass of heterogeneous complex of solid phases (shet), which has composition X this case, the equation of open crystallization will have the usual form: ðshetÞ

~ðlÞ ¼ ðX ~ðshetÞ  X ~ðlÞ Þ dmðshetÞ ; dX or the scalar form: ! dXiðlÞ dXjðlÞ

P;cryst

¼

XiðlÞ  XiðshetÞ XjðlÞ  XjðshetÞ

;

ði; j ¼ 1ynÞ:

ð11Þ

ð12Þ

Putting Eq. (11) into common differential Van-der-Vaal’s equations for the equilibrium (l-shet) at P=constant [5] one can get   dT 1 ~ðshetÞ  X ~ðlÞ ÞG# ðlÞ ðX ~ðshetÞ  X ~ðlÞ Þgo0: ¼  ðshet-lÞ fðX ð13Þ ðshetÞ dm S P;cryst We shall not consider very rare processes of recurrent crystallization and practically without loss of universality one may confirm that the differential molar entropy of the solution of heterogeneous complex of solid phases (shet) in the melt is positive S ðshet-lÞ ð Qðshet-lÞ =TÞ > 0: The expression in brackets in Eq. (13) becomes positive according to Silvester’s theorem [6, p. 143], so one can say that while temperature decreases the total mass of solid phases increases; but this is not the case for the mass of each solid phase separately. System (12) cannot be integrated even for the phases with invariable composition ~ðshetÞ a~ because in the process of open crystallization dX 0; this system does not allow one to determine ~ðsiÞ ; dmðsiÞ ; important to the technologist or experimenter. ~ðsiÞ ; dX the functions X A real alternative to this method is the multiple determination of global conditional minimum of the Gibbs potential of a system of (r þ 1)-phases under the conditions of constancy of T; P; ni (molar numbers of components) [1] in all steps of the calculation of open phase process curve: GðhetÞ T;P;ni -min: However, this method is not absolutely correct. If the task is formalized correctly one must always prove that the calculated GðhetÞ T;P;ni minimum is global (in the condition of an indeterminate number of local minima, some of which may be not strict). In thermodynamics, one must prove that equilibrium is really stable, i.e. not metastable. It was shown earlier [7–9] that there is topological isomorphism between solubility and fusibility diagrams. So, an analogous system of differential equations may be obtained for the description of isotherm–isobaric open crystallization from the solutions while solvent is evaporating. Analogous curves of isothermal or isobaric open evaporation or condensation must have this topological isomorphism also. But the case of the formation of two or more gaseous phases simultaneously is very untypical (miscibility in vapor phases); although for open condensation from the vapor, miscibility in liquid phases is more typical, it does not accompany moving of liquid phases away from the sphere of phase equilibrium given finite diffusion velocities in the liquid.

2. Open isothermal crystallization from the solution while solvent is evaporating Diagrams of open isothermal–isobaric crystallization from multi-component solutions while solvent is evaporating are also very interesting in geochemistry, hydrochemistry, ecology, etc. Theory of the

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crystallization in the conditions of two-phase equilibrium (liquid–solid) was considered earlier [10]. In this work, we will consider multi-phase crystallization (rX2) in multi-component systems (nX5). If n ¼ 4 crystallization of two solid phases causes only the movement of the solution composition along monovariant crystallization curveFf ¼ 1; and such consideration is trivial. 2.1. General differential equations Let us consider an n-component solution (nX5) at P; T=constant, from which in the conditions of (r þ 1)-phase equilibrium r solid phases (s1; s2y; srðrX2Þ) are crystallizing while solvent is evaporating. We ~ to reduce the dimension of concentration space from the (n  1)-dimensional vector of molar fractions X ~ the (n  2)-dimensional vector of Yenecke indexes Y : Yi ¼ Xi =ð1  Xw Þ ¼ ni =

n X

nj ;

n1 X

jaw

Yi ¼ 1;

ð14Þ

i¼1

where nj is the number of moles of the j-th component, w the index of solvent. ~ðlÞ Let us ‘‘form’’ from the solution with composition in the reduced concentration spaceFY (which maintains 1 mol of components without solvent) r equilibrium solid phases with composition ~ðsiÞ (they maintain dm* ðsiÞ moles of components), dm* ðsiÞ may have arbitrary sign, but even for one phase Y ~ðlÞ and the dm* ðsiÞ must be >0. In this process, the vector of solution composition shifts by vector dY differential equation of mass balance will be as follows: ðlÞ

~ ¼ dY

r X

ðsiÞ

~ ðY

~ðlÞ Þ dm* ðsiÞ : Y

ð15Þ

i¼1

At r ¼ 1 it is also easy to integrate system (15) for the solid phase with invariable composition (in the whole region of two-phase equilibrium) and for the solid solution in the vicinity of the specific points in the phase diagrams (points of components, compounds, minimum or maximum of solvent chemical potentialFmw in the fields or volumes of crystallization of solid solutions) [10]. In all other cases, one can easily integrate system (15) numerically at r ¼ 1: At rX2 we cannot solve system (15) evidently even for the crystallization of solid phases with invariable composition, and this system is uncomfortable for us. Let us consider the system of common differential Van-der-Vaal’s equations for the equilibrium (l  si) at P; T=constant [11] with the help of the variables of the liquid phase in reduced concentration space in the ðlÞ matrics of the incomplete Gibbs potentialFG# w : Therefore, ðsiÞ

~ ðY

~ ÞG# dY ~ ¼ M ðsi-lÞ dmw ; Y ½w w ðlÞ

ðlÞ

ðlÞ

ð1piprÞ:

ð16Þ

where Mwðsi-lÞ is the number of moles of solvent change in isothermal–isobaric formation l-phase from infinitely ~ðtÞ Fvector, characterizing place large mass of phase (si), calculated on 1 mol of components without solvent;Y ~ðtÞ Fvector, characterizing of figurative point of phase t in reduced concentration space ðt ¼ l; siÞ;dY ðtÞ ðlÞ ~ while (r þ 1)-phase equilibrium shifts; G# Foperator, which is determined by the matrix movement of Y w ðlÞ ðlÞ ðlÞ of second derivatives: G½wij  ðq2 G½w =qYi ðlÞ qYjðlÞ ÞT;P;Ykaj;n1 ;mw ; GðlÞ ½w  G  mw nw Fthe molar incomplete ðlÞ Gibbs potential of phase l; matrix of operator G# w and minors of its main diagonal are also determined positively, according to the criteria of stability [11]. Let us assume that the composition of equilibrium ~ðlÞ ) ~ðsiÞ  Y phases l; sið1piprÞ are linearly independent in reduced concentration space and that vectors (Y are linearly independent too. The other case is trivialFchemical potential of solvent is constant or crosses through maximum or minimum (dmw ¼ 0) and in this case an open phase process with equilibrium (r þ 1) phases is impossible [12].

N.A. Charykov et al. / Journal of Crystal Growth 234 (2002) 762–772

Put equations of system (15) into the system (16):  ðsiÞ  ðrÞ * ðrÞ D D* i dm* dm* ðskÞ ¼  kðrÞ dmw ; ¼ ; * dm* ðsjÞ P;T;cryst D* ðrÞ Gr

767

ð17Þ

j

~ðlÞ Gr * ðrÞ ¼ dY

r X

ðsiÞ

~ ðY

~ðlÞ ÞD * ðrÞ dmw ; Y i

ð18Þ

i¼1

dYjðlÞ dYkðlÞ

r P

!

* ðrÞ ðYjðsiÞ  YjðlÞ ÞD i

i¼1

¼

r P

P;T;cryst

i¼1

ðYkðsiÞ  YkðlÞ ÞD* i

ðrÞ

ð19Þ

;

* ðrÞ is also not equal to zero [6]: where Gram’s determinantFGr   ðs1Þ  ~ ~ðlÞ ÞG# ðlÞ ðY ~ðs1Þ  Y ~ðlÞ Þ y ðY ~ðsrÞ  Y ~ðlÞ ÞG# ðlÞ ðY ~ðs1Þ  Y ~ðlÞ Þ   ðY  Y ½w ½w     ðs1Þ ðlÞ ðs2Þ ðlÞ ðsrÞ ðlÞ ðlÞ ~ Y ~ ÞG# ðY ~ Y ~ Þ y ðY ~ Y ~ ÞG# ðlÞ ðY ~ðs2Þ  Y ~ðlÞ Þ   ð Y ðrÞ ½w ½w a0;  * Gr ¼     y y y    ðs1Þ ðlÞ ðsrÞ ðlÞ ðsrÞ ðlÞ ðsrÞ ðlÞ  ðlÞ ðlÞ ~ Y ~ ÞG# ðY ~ Y ~ Þ y ðY ~ Y ~ ÞG# ðY ~ Y ~ Þ  ðY ½w ½w

* ðrÞ D k

  Gr *  1;1  *  Gr1;2 ¼   y   Gr * 1;r

* k1;1 y Gr * k1;2 y Gr

Mwðs1-lÞ

y

y

Mwðs2-lÞ y

y

* k1;r Gr

Mwðsr-lÞ

* kþ1;1 Gr * kþ1;2 Gr y * kþ1;r Gr

~ðsiÞ  Y ~ðlÞ ÞG# ðlÞ ðY ~ðsjÞ  Y ~ðlÞ Þ  * i;j  Gr * j;i  ðY Gr ½w

 * r;1  y Gr  * r;2  y Gr ; y y   * r;r  y Gr n2 X n2 X p¼1

ðsiÞ ðlÞ ðsjÞ ðlÞ GðlÞ ½wpq ðYp  Yp ÞðYq  Yq Þ:

ð20Þ

ð21Þ

ð22Þ

q¼1

The sign of dm* ðsiÞ also depends on the type of crystallizationFeutonic, I-type peritonic, II-type peritonic and so on. Points, curves or surfaces of change of phase process in relation to solid phase (si), * ðrÞ ¼ 0: Also, assume where m* ðsiÞ crosses through maximum or minimum may be determined: D i r P ðshetÞ ðsiÞ dm* ¼ dm* Fmass of heterogeneous complex of solid phases without solvent (shet), which has i¼1

ðshetÞ

~ the compositionFY ðlÞ

ðshetÞ

~ ¼ ðY ~ dY ! dYiðlÞ dYjðlÞ

P;T;cryst

¼

:

~ðlÞ Þ dm* ðshetÞ ; Y YiðlÞ  YiðshetÞ YjðlÞ  YjðshetÞ

ð23Þ

;

ði; j ¼ 1yn  1Þ:

ð24Þ

Putting Eq. (24) into the common differential Van-der-Vaals equation for two-phase equilibrium of the solution with heterogeneous complex of solid phases (l-shet), one can easily get:   dmw 1 ~ðshetÞ  Y ~ðlÞ ÞG# ðlÞ ðY ~ðshetÞ  Y ~ðlÞ Þgo0: ¼  ðshet-lÞ fðY ð25Þ ½w dm* ðshetÞ P;T;cryst Mw

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We shall not consider very rare systems, which have ‘‘forbidden type’’ diagrams of solubility [13]. In these systems, solid phases with greater concentration of solvent than the liquid are crystallizing. So, for a heterogeneous complex one may assume that Mwðshet-lÞ > 0: The expression in brackets in Eq. (25) is also positive, so one can say that while the chemical potential of the solvent decreases (solvent is evaporating) the sum mass of solid phases increases; but this is not true for the mass of each solid phase separately. In solubility and fusibility diagrams there exist crystallization processes, where in the beginning dm* ðsiÞ > 0 or dmðsiÞ > 0 ( the i-th solid phase crystallizes) and in the end dm* ðsiÞ o0 or dmðsiÞ o0 (the i-th solid phase dissolves in the solution or melt), and these result in the same crystallization curves [14]. Points, where dm* ðsiÞ ¼ 0 or dmðsiÞ ¼ 0 are formed by the points of contact of the surfaces of (r þ 1)-phase equilibrium and nodes (si  l) [14]. System (25) also cannot be integrated even for the phases with invariable composition ~ðshetÞ a~ 0: because dY Let us characterize the method of calculation as a whole: 1. One can use the equations, obtained earlier, if corresponding diagrams of phase equilibrium (solubility or fusibility) are known, in other words, if one has corresponding experimental data or an adequate thermodynamic model for equilibrium phases. In this case, one knows the dependence of chemical ~Þ and hence operators G# ðlÞ ; G# ðlÞ and functions potentials on the parameters of state: mi ¼ fi ðT; P; X ½w Sðsi-lÞ ; Mwðsi-lÞ : For example, [5]: ! qðmi  mn1 Þ ðlÞ ; ð26Þ G½wij  qYjðlÞ T;P;X ðlÞ ;m kaj;n1

ðsiÞ

~ S ðsi-lÞ ¼ S ðlÞ  S ðsiÞ þ ðX

ðlÞ

w

~ ÞrSðlÞ ; X

ð27Þ

where S ðtÞ is molar entropy of phase t; rS ðlÞ is the concentration gradient of the meltFvector with elements ðqS ðlÞ =qXjðlÞ ÞT;P;X ðlÞ and so on. This demand is absolutely common for all methods of kaj;n calculation of crystallization processes. 2. The differential equations are well behaved, their solution can be easily organized in a mathematical unconditional algorithm. Such a solution is simpler than the solution of conditional minimum: GðhetÞ T;P;ni -min: 3. The equations may be easily simplified in the case r ¼ 1 (in these cases only one scalar element remains in Gram’s operators). 4. The equations may also be easily used for calculation of the diagrams of phase equilibrium (fusibility and solubility diagrams). 2.2. Examples of application Examples of applications of the differential equations are given in Figs. 1–5. Fig. 1 shows simply calculations of open crystallization of the single solid solutions while temperature decreases in the A3B5 semiconductor quaternary system In–P–As–Sb. For the thermodynamic description of the solid phase we used the sub-regular solution model, for the thermodynamic description of the melt we used EFLCP-model (excess functions=linear combination of chemical potentials) [15]. Parameters of the respective models are given in Ref. [16]. In Figs. 2 and 3, the fusibility diagrams in the quaternary systems In–P–As–Sb and In–Ga–As–Sb are given. Our experimental data points for the systems In–P– As–Sb and In–Ga–As–Sb are also given in Figs. 1–3. One can see a good agreement between experimental and calculated data.

N.A. Charykov et al. / Journal of Crystal Growth 234 (2002) 762–772

769

22 20

TGr = 650-604 C

18 16

SbExp

14

%

12 10

SbCol

8 6

PCol

l

X As = 0.0318

2 0

PExp

l

X P = 0.00124

4

l

X Sb = 0.38 0

20

40

60

80

100

120

Distance, µm Fig. 1. Crystallization curves in the system In–P–As–Sb: curvesFcalculation; pointsFour experimental data.

0.4

0.18 T = 550 C

0.16 0.14

S

X Sb

l

X As:10

0.3

l X Ga 0.12

S

X Sb

l

X Sb 0.10

S X Ga 0.2

0.08

S

X Ga

0.06

0.1

0.04 0.02

l

X Ga

InGaAsSb/GaSb

0.3

l

0.4

0.0

X Sb Fig. 2. Fusibility diagram in the system In–Ga–As–Sb: curvesFcalculation; pointFour experimental data.

In Figs. 4 and 5, we present calculations of open multi-phase crystallization of the solid phases with invariable composition while temperature decreases (Fig. 4) or while water is evaporating at T=constant  2 (Fig. 5) in the multi-component water–salt system Na+, K+, Ca2+, Mg2þ 8Cl ; SO2 4 , HCO3 , CO3 –H2O, different compositions of which imitates waters of oceans and Black, Aral and Caspian seas. For the

770

N.A. Charykov et al. / Journal of Crystal Growth 234 (2002) 762–772 0.08

0.45 T = 590 C

0.40 l

X As:10

0.35

0.07

l

X P:100 0.06

0.30

S

X 0.25

0.05

P

l

0.04

0.20 0.15

X

0.03

S Sb

0.10

0.02

0.05 0.00

XP

InAsSbP/InAs 0.1

0.2

0.3

0.01

0.4

l

X Sb Fig. 3. Fusibility diagram in the system In–P–As–Sb: curvesFcalculation; pointsFour experimental data.

(1) (2) (3)

25

15

M

S l

(g)

20

10

5

0

-5

-10

-15

-20

-25

-30

Temperature (C) Fig. 4. Crystallization curves: dependence of mass of solid phases (Mis ) while temperature is decreasing in the process of crystallization of sea water (initial M1=1000 g, composition [20]): curvesFcalculation; pointsFexperimental data [20,21]. Solid phases: 1FNa2SO4 10H2O, 2FCaSO4 2H2O, 3FNaCl 2H2O.

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Fig. 5. Crystallization curves at 251C: dependence of mass of solid phases (Mis ) against logarithm mass of liquid phaseFlgMl (initial M1 ¼ 1000 g) while water is evaporating in the process of crystallization of water of Black sea (composition [20]): curvesFcalculation; pointsFour experimental data. Solid phases: 1FCaSO4 2H2O, 2FNaCl, 3FKCl MgSO4 3H2O.

thermodynamic description of the solution we used Pitzer’s model [17,18]. Parameters of models are given in Ref. [19]. Literature results [21,22] and our experimental data are also given in Figs. 4 and 5. Again one can see a good agreement between experimental and calculated data.

3. Conclusion A new approach has been used to develop a general theory of melt crystallization. The theory given here successfully deals with the two cases, (a) open crystallization from multi-component melts while the temperature is increasing or decreasing and (b) open isothermal–isobaric crystallization from multicomponent solutions while the solvent is evaporating. In this respect, the techniques developed here relate to more realistic practical situations than models previously given in the literature. The resulting formalism has been shown to have wide-ranging application and examples relevant to liquid-phase epitaxial growth of III–V compounds as well as multi-phase crystallization in sea water have been studied and shown to agree with experimental findings.

Acknowledgements The authors are thankful to the Russian fund for fundamental research (grant of president of Russia for young doctors of science N 00-15-99334).

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