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A study on the composition of a binary mixed crystal
Journal of Crystal Growth 79 (1986) 1001,-1004 North-Holland, Amsterdam
1001
A S T U D Y O N T H E C O M P O S I T I O N OF A BINARY M I X E D CRYSTAL V. N A T A R A J A N , C. S U B R A M A N I A N , D. J A Y A R A M A N and P. R A M A S A M Y Crystal Growth Centre, Anna University, Madras-600 025, India
The formation of centres of a new phase in a one-component system has already been investigated in detail, both theoretically and experimentally. In the present work, for a two-component melt system, the free energy required for the formation of a spherical nucleus having a moleculesof component A and b moleculesof component B has been studied. At the saddle point the values of a*, b* and AG* have been computed applying the binomial approximation. From the graphical analysis, the composition of the melt for which the composition of the nucleus becomes equal is determined. The LiBr-LiC1 binary system has been chosen for the above study.
1. Introduction Determination of the nucleus parameters in a single component system is a very complicated process incorporating all the factors which influence the formation of a new phase. It is still more complicated in the case of a two-component system. In view of the increasing demand for crystals of complex compositions, comprising stoichiometric, non-stoichiometric and doped materials as well as solid solutions, a crystal grower has to tackle several parameters that contribute to the growth of a good single crystal of the required composition and specification. Hence a nucleation study is of immense importance in evaluating, understanding and utilizing the phase relations between the crystal constituents and other substances taking part in the growth process. An extensive study of binary systems has been made by Sigsbee [1], Reiss [2] and Shugard et al. [3]. Using the Lundager Madsen [4] one-layer adsorption model, Subramanian et al. [5] have studied nucleation phenomena of the A D P - K D P binary system. In the present work, for a binary melt system, a new approach has been adopted to find the composition of the critical nucleus for various melt compositions. The equation for the free energy change contains four unknownes, namely a, b, AG and o. The interfacial energy (o) is the work involved in creating an interface and it is this quantity which is of significance in determin-
ing the equilibrium in a two-phase system. In the present study it is assumed to be constant and independent of composition. By maximizing the equation, AG can be eliminated. Still we are left with two similar equations, equivalent to only one equation with two unknowns a and b. The binomial approximation has been applied and the values of a and b at the saddle point have been computed.
2. Theory The formation of a solid particle (nucleus) in the liquid demands the expenditure of a certain quantity of energy. The nucleus which is initially unstable grows into a stable one as it passes over the free energy barrier. The top of such a barrier is the saddle point and it is here that an unstable nucleus becomes a critical nucleus. The total free energy change associated with the process of homogeneous nucleation is represented as AG = ~G s + ACv,
(1)
where AGs is the surface term, which is a positive quantity, and aGv is the volume term which is a negative quantity. The free energy change or work of formation of a nucleus in the one-component melt system is ,aG = zaG ° - Nza~,
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
V. Natarajan et al. / Composition of binary mixed crystal
1002
where the first term is the surface term and the latter is the volume term N is the number of molecules in the nucleus and A/~ is the difference in chemical potential between the mother phase (liquid) and crystallite (solid). Let us consider a binary system of component a and component B in the melt phase. When the system is supercooled, a nucleus is born. When the nucleus grows and attains a critical size of favourabe composition, it has lower chemical potentials for both the components than the parent phase and it can grown by receiving both kinds of atoms. The enthalpy of mixing for the liquid and the solid alloys are assumed to be zero, considering the system to be an ideal one. The entropies of mixing in the two phases are taken care of in formulating the expression for the free energy change for the formation of a nucleus. For a spherical nucleus in a binary system consisting of a number of A species and b number of B species under constant temperature and pressure, the work of formation can be written as
AG = AG O--
a
/]]1 A -
b AIXB -- a k T In XA
+(aVA+bVB)
( - ~ - ] r1, p , ~
= - z ~ l ~ B - k T In XB + k T In N B
(5)
On dividing eq. (4) by eq. (5) and re-arranging, we get
( XA/NA) v, : e x p { VAVB [( A H A T (XB/NB)V, _ kT [[~)B
(6)
Raising both sides of eq. (6) to the reciprocal power of VB and writing N B in terms of NA, eq. (6) is modified as
XA
(1 --NA) v^/v"
( XB ) v*/v"
NA
[ ray. [[ AII T
= / e x p [ - - ~ - - [[ ~ ) B X B
A H AT
# i ( i = A, B) = ( A H AT X V )
X,
"
1-6
Tm
is the difference in chemical potential between the two phases, where X, V, AH, AT and Tm are the mole fraction, molecular volume, enthalpy of fusion, supercooling and melting point respectively. The terms a k T In NA and bkT In N B are due to the entropy of mixing in the nucleus, and the terms a k T In XA and b k T In XB are due to mixing of the two liquids. Maximizing eq. (3) with respect to a and b, we get
'3~" 1'2 w~ ~o
Q,8
0"4
L
0
100
200 AT
-- k T In XA + k T In
JAXA]).
Tm
XB
Applying the binomial approximation and calculating N ~ / N f f , the ratio between a* and b* is determined. Then the a* and b* values are corn-
of the nucleus, and
~ ~ ~ A
]/3
(4)
xo=-0.
where AG O is the energy required for the formation of the surface of area
OAG ] = 0a ] r,P,b
~ VA(36~r)
+ ( a VA + b V B) - t/3~ VB (36~r)1/3
f
( a V A + bVB)2/a(367r) 1/3
]/3 2
×o=0,
- b k T ln X B + a k T ln N A + b k T l n NB,
(3)
-
U.
300
(K)
Fig. 1. Variation of 1lAG* with AT for the composition 60%LIC1-40% LiBr.
V. Natarajan et al. / Composition of binary mixed crystal
1003
3. Results and discussion
Z8
2.&
2.0 300 K
20O I00 ×
1.2
0'8
O,L
i
02
0'4
06
0.8
LiBr
Fig. 2. Variation of 1lAG* with crystalline and melt compositions for different AT. puted using either one of the maximized equations. Finally the AG* value is also computed. The variation of 1 l A G * with supercooling ( A T ) for a particular composition and the variation of 1 l A G * with composition for various degrees of supercooling are shown in figs. 1 and 2, respectively. F r o m the graphs, the composition of the nucleus and composition of the melt when they are nearly equal are determined. The above theory was applied to the LiBr-LiC1 system and the following data are used for the calculation: AHLicl = 6.5576 × 10 9 erg, AHLiBr = 4.8359 × 109erg, VLicJ = 3.4033 × 10 -23 cm 3, VLiBr = 4.1627 × 10-23cm 3 and o = 100 e r g / c m 2 The melting temperature for a particular composition is taken from the phase diagram [6]. The calculations were carried out for different supercoolings. It was found to crystallize for the entire range of composition confirming the phase diagram. The influence of the degree of supercooling on the composition of the critical nucleus has been investigated (fig. 2).
The LiBr-LiC1 system has a continuous solid solution for the entire composition. The values of a*, b* and AG* for different compositions and for different supercoolings have been computed. The variation of the composition of the nucleus with respect to the variation of the composition of the melt along with ziG* are given in table 1. The composition at which the melt composition and nucleus composition are equal is determined with the help of figs. 1 and 2. A pronounced change in the value of l / z i G * for the supercooling range 100-300 K has been observed in fig. 1. Fig. 2 is drawn for the above supercooling beteen 1 l A G * and the composition of the nucleus. Also, for various melt compositions, the values of 1 l A G * have been plotted on the same graph. For a fixed supercooling of the phase transformation, the composition of the melt and the composition of the nuclei will become almost equal only for a particular minimum value of 1lAG*. In the case of LiBr-LiC1, for 47.5% LiBr the two compositions are found to be nearly equal for the supercooling A T = 100 K. The procedure is extended to the A1203-SIO 2 system for which the experimental values of the composition of the mixed crystal are available. The existence of a solid solution between SiO 2 and A1203 was reported for melt compositions ranging from 71.Swt%A1203/28.2wt%SiO 2 to 77.3%/22.7%
Table 1 The values of a*, b* and AG* for a LiC1-LiBr system of different compositions for AT = 200 K Composition LiC1-LiBr
NA a)
NB b)
1lAG* × 1011 (erg-1)
90-10 80-20 70-30 60-40 50-50 40-60 30-70 20-80 10-90
0.8262 0.7662 0.6963 0.6189 0.5368 0.4538 0.3736 0.3027 0.2414
0.1738 0.2337 0.3036 0.3810 0.4631 0.5462 0.6264 0.6972 0.7586
2.4481 1.3946 1.0207 0.8489 0.8003 0.8664 1.1006 1.7223 3.1748
a) NA = a * / ( a * + b*).
b) NB = b*/(a* + b*).
1004
v. Natarajan et al. / Composition of binary mixed crystal
[7]. By the a b o v e m e t h o d , when A T = 250 K a n d o = 100 e r g / c m 2 for 73% A I 2 0 3, the c o m p o s i t i o n of the nucleus c a l c u l a t e d is f o u n d to be 72.89%, w h i c h is very n e a r l y equal to the e x p e r i m e n t a l value r e p o r t e d b y G u s e [8]. T h e e n t h a l p y of form a t i o n a n d m e l t i n g t e m p e r a t u r e of the system are t a k e n f r o m the l i t e r a t u r e [6,9].
References [1] R.A. Sigsbee, in: Nucleation, Ed. A.C. Zettlemoyer (Dekker, New York, 1969) p. 151.
[2] H.W. Reiss, J. Chem. Phys. 18 (1950) 840. [3] W.J. Shugard, R.H. Heist and H. Reiss, J. Chem. Phys. 16 (1974) 5298. [4] H.E. Lundager Madsen, J. Crystal Growth 46 (1979) 495. [5] C. Subramanian, R. Dhansekarar;, P. Ramasamy and D. Elwell, J. Crystal Growth 70 (1984) 41. [6] E.M. Levin, C.R. Robbins and H.E. McMurdie, Phase Diagrams for Ceramists (American Ceramic Society, 1964). [7] W. Guse and D. Mateika, J. Crystal Growth 22 (1974) 237. [8] W. Guse, J. Crystal Growth 26 (1974) 151. [9] CRC Handbook of Chemistry and Physics, 61st ed. (CRC Press, Cleveland, OH, 1980-81).
1001
A S T U D Y O N T H E C O M P O S I T I O N OF A BINARY M I X E D CRYSTAL V. N A T A R A J A N , C. S U B R A M A N I A N , D. J A Y A R A M A N and P. R A M A S A M Y Crystal Growth Centre, Anna University, Madras-600 025, India
The formation of centres of a new phase in a one-component system has already been investigated in detail, both theoretically and experimentally. In the present work, for a two-component melt system, the free energy required for the formation of a spherical nucleus having a moleculesof component A and b moleculesof component B has been studied. At the saddle point the values of a*, b* and AG* have been computed applying the binomial approximation. From the graphical analysis, the composition of the melt for which the composition of the nucleus becomes equal is determined. The LiBr-LiC1 binary system has been chosen for the above study.
1. Introduction Determination of the nucleus parameters in a single component system is a very complicated process incorporating all the factors which influence the formation of a new phase. It is still more complicated in the case of a two-component system. In view of the increasing demand for crystals of complex compositions, comprising stoichiometric, non-stoichiometric and doped materials as well as solid solutions, a crystal grower has to tackle several parameters that contribute to the growth of a good single crystal of the required composition and specification. Hence a nucleation study is of immense importance in evaluating, understanding and utilizing the phase relations between the crystal constituents and other substances taking part in the growth process. An extensive study of binary systems has been made by Sigsbee [1], Reiss [2] and Shugard et al. [3]. Using the Lundager Madsen [4] one-layer adsorption model, Subramanian et al. [5] have studied nucleation phenomena of the A D P - K D P binary system. In the present work, for a binary melt system, a new approach has been adopted to find the composition of the critical nucleus for various melt compositions. The equation for the free energy change contains four unknownes, namely a, b, AG and o. The interfacial energy (o) is the work involved in creating an interface and it is this quantity which is of significance in determin-
ing the equilibrium in a two-phase system. In the present study it is assumed to be constant and independent of composition. By maximizing the equation, AG can be eliminated. Still we are left with two similar equations, equivalent to only one equation with two unknowns a and b. The binomial approximation has been applied and the values of a and b at the saddle point have been computed.
2. Theory The formation of a solid particle (nucleus) in the liquid demands the expenditure of a certain quantity of energy. The nucleus which is initially unstable grows into a stable one as it passes over the free energy barrier. The top of such a barrier is the saddle point and it is here that an unstable nucleus becomes a critical nucleus. The total free energy change associated with the process of homogeneous nucleation is represented as AG = ~G s + ACv,
(1)
where AGs is the surface term, which is a positive quantity, and aGv is the volume term which is a negative quantity. The free energy change or work of formation of a nucleus in the one-component melt system is ,aG = zaG ° - Nza~,
0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
V. Natarajan et al. / Composition of binary mixed crystal
1002
where the first term is the surface term and the latter is the volume term N is the number of molecules in the nucleus and A/~ is the difference in chemical potential between the mother phase (liquid) and crystallite (solid). Let us consider a binary system of component a and component B in the melt phase. When the system is supercooled, a nucleus is born. When the nucleus grows and attains a critical size of favourabe composition, it has lower chemical potentials for both the components than the parent phase and it can grown by receiving both kinds of atoms. The enthalpy of mixing for the liquid and the solid alloys are assumed to be zero, considering the system to be an ideal one. The entropies of mixing in the two phases are taken care of in formulating the expression for the free energy change for the formation of a nucleus. For a spherical nucleus in a binary system consisting of a number of A species and b number of B species under constant temperature and pressure, the work of formation can be written as
AG = AG O--
a
/]]1 A -
b AIXB -- a k T In XA
+(aVA+bVB)
( - ~ - ] r1, p , ~
= - z ~ l ~ B - k T In XB + k T In N B
(5)
On dividing eq. (4) by eq. (5) and re-arranging, we get
( XA/NA) v, : e x p { VAVB [( A H A T (XB/NB)V, _ kT [[~)B
(6)
Raising both sides of eq. (6) to the reciprocal power of VB and writing N B in terms of NA, eq. (6) is modified as
XA
(1 --NA) v^/v"
( XB ) v*/v"
NA
[ ray. [[ AII T
= / e x p [ - - ~ - - [[ ~ ) B X B
A H AT
# i ( i = A, B) = ( A H AT X V )
X,
"
1-6
Tm
is the difference in chemical potential between the two phases, where X, V, AH, AT and Tm are the mole fraction, molecular volume, enthalpy of fusion, supercooling and melting point respectively. The terms a k T In NA and bkT In N B are due to the entropy of mixing in the nucleus, and the terms a k T In XA and b k T In XB are due to mixing of the two liquids. Maximizing eq. (3) with respect to a and b, we get
'3~" 1'2 w~ ~o
Q,8
0"4
L
0
100
200 AT
-- k T In XA + k T In
JAXA]).
Tm
XB
Applying the binomial approximation and calculating N ~ / N f f , the ratio between a* and b* is determined. Then the a* and b* values are corn-
of the nucleus, and
~ ~ ~ A
]/3
(4)
xo=-0.
where AG O is the energy required for the formation of the surface of area
OAG ] = 0a ] r,P,b
~ VA(36~r)
+ ( a VA + b V B) - t/3~ VB (36~r)1/3
f
( a V A + bVB)2/a(367r) 1/3
]/3 2
×o=0,
- b k T ln X B + a k T ln N A + b k T l n NB,
(3)
-
U.
300
(K)
Fig. 1. Variation of 1lAG* with AT for the composition 60%LIC1-40% LiBr.
V. Natarajan et al. / Composition of binary mixed crystal
1003
3. Results and discussion
Z8
2.&
2.0 300 K
20O I00 ×
1.2
0'8
O,L
i
02
0'4
06
0.8
LiBr
Fig. 2. Variation of 1lAG* with crystalline and melt compositions for different AT. puted using either one of the maximized equations. Finally the AG* value is also computed. The variation of 1 l A G * with supercooling ( A T ) for a particular composition and the variation of 1 l A G * with composition for various degrees of supercooling are shown in figs. 1 and 2, respectively. F r o m the graphs, the composition of the nucleus and composition of the melt when they are nearly equal are determined. The above theory was applied to the LiBr-LiC1 system and the following data are used for the calculation: AHLicl = 6.5576 × 10 9 erg, AHLiBr = 4.8359 × 109erg, VLicJ = 3.4033 × 10 -23 cm 3, VLiBr = 4.1627 × 10-23cm 3 and o = 100 e r g / c m 2 The melting temperature for a particular composition is taken from the phase diagram [6]. The calculations were carried out for different supercoolings. It was found to crystallize for the entire range of composition confirming the phase diagram. The influence of the degree of supercooling on the composition of the critical nucleus has been investigated (fig. 2).
The LiBr-LiC1 system has a continuous solid solution for the entire composition. The values of a*, b* and AG* for different compositions and for different supercoolings have been computed. The variation of the composition of the nucleus with respect to the variation of the composition of the melt along with ziG* are given in table 1. The composition at which the melt composition and nucleus composition are equal is determined with the help of figs. 1 and 2. A pronounced change in the value of l / z i G * for the supercooling range 100-300 K has been observed in fig. 1. Fig. 2 is drawn for the above supercooling beteen 1 l A G * and the composition of the nucleus. Also, for various melt compositions, the values of 1 l A G * have been plotted on the same graph. For a fixed supercooling of the phase transformation, the composition of the melt and the composition of the nuclei will become almost equal only for a particular minimum value of 1lAG*. In the case of LiBr-LiC1, for 47.5% LiBr the two compositions are found to be nearly equal for the supercooling A T = 100 K. The procedure is extended to the A1203-SIO 2 system for which the experimental values of the composition of the mixed crystal are available. The existence of a solid solution between SiO 2 and A1203 was reported for melt compositions ranging from 71.Swt%A1203/28.2wt%SiO 2 to 77.3%/22.7%
Table 1 The values of a*, b* and AG* for a LiC1-LiBr system of different compositions for AT = 200 K Composition LiC1-LiBr
NA a)
NB b)
1lAG* × 1011 (erg-1)
90-10 80-20 70-30 60-40 50-50 40-60 30-70 20-80 10-90
0.8262 0.7662 0.6963 0.6189 0.5368 0.4538 0.3736 0.3027 0.2414
0.1738 0.2337 0.3036 0.3810 0.4631 0.5462 0.6264 0.6972 0.7586
2.4481 1.3946 1.0207 0.8489 0.8003 0.8664 1.1006 1.7223 3.1748
a) NA = a * / ( a * + b*).
b) NB = b*/(a* + b*).
1004
v. Natarajan et al. / Composition of binary mixed crystal
[7]. By the a b o v e m e t h o d , when A T = 250 K a n d o = 100 e r g / c m 2 for 73% A I 2 0 3, the c o m p o s i t i o n of the nucleus c a l c u l a t e d is f o u n d to be 72.89%, w h i c h is very n e a r l y equal to the e x p e r i m e n t a l value r e p o r t e d b y G u s e [8]. T h e e n t h a l p y of form a t i o n a n d m e l t i n g t e m p e r a t u r e of the system are t a k e n f r o m the l i t e r a t u r e [6,9].
References [1] R.A. Sigsbee, in: Nucleation, Ed. A.C. Zettlemoyer (Dekker, New York, 1969) p. 151.
[2] H.W. Reiss, J. Chem. Phys. 18 (1950) 840. [3] W.J. Shugard, R.H. Heist and H. Reiss, J. Chem. Phys. 16 (1974) 5298. [4] H.E. Lundager Madsen, J. Crystal Growth 46 (1979) 495. [5] C. Subramanian, R. Dhansekarar;, P. Ramasamy and D. Elwell, J. Crystal Growth 70 (1984) 41. [6] E.M. Levin, C.R. Robbins and H.E. McMurdie, Phase Diagrams for Ceramists (American Ceramic Society, 1964). [7] W. Guse and D. Mateika, J. Crystal Growth 22 (1974) 237. [8] W. Guse, J. Crystal Growth 26 (1974) 151. [9] CRC Handbook of Chemistry and Physics, 61st ed. (CRC Press, Cleveland, OH, 1980-81).