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Studies on growth and nucleation kinetics of cadmium thiourea sulphate and magnesium cadmium thiourea sulphate
Materials Chemistry and Physics 114 (2009) 18–22
Contents lists available at ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Materials science communication
Studies on growth and nucleation kinetics of cadmium thiourea sulphate and magnesium cadmium thiourea sulphate Mekala Daniel a,∗ , M. Jeyarani Malliga b , R. Sankar c , D. Jayaraman d a
Quaid-e-Milleth College, Chennai 600002, India Bharathi Women College, Chennai 600108, India c Kings Engineering College, Irungatukottai, Sriperumbudhur, Chennai 602105, India d Presidency College, Chennai 600 004, India b
a r t i c l e
i n f o
Article history: Received 1 July 2007 Received in revised form 25 September 2008 Accepted 28 September 2008 Keywords: CTS and MCTS crystals Solution growth method Solubility Basic growth parameters Classical nucleation theory Capillarity approximation
a b s t r a c t Semiorganic materials, in general possess high non-linear coefficient and mechanical strength which will be more applicable for device fabrication. Cadmium thiourea sulphate (CTS) and magnesium cadmium thiourea sulphate (MCTS) are better semiorganic materials which find applications in the field of optoelectronics. Single crystals of CTS and MCTS have been successfully grown from aqueous solution by slow evaporation technique using predetermined solubility data. The basic growth parameters of the crystal nuclei of the grown crystals of CTS and MCTS were evaluated based on the classical theory of homogeneous nucleation. The classical nucleation theory makes use of capillarity approximation which has certain limitations. A correction has to be applied for it and the classical nucleation theory has been suitably modified in order to calculate the critical nucleus parameters. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Materials having good optical non-linearity with chemical flexibility and mechanical properties are in demand in recent times due to its wide applications. Semiorganic materials have attracted great interest as these materials have the potential to possess high nonlinearity, good mechanical hardness, low angular sensitivity and high resistance to laser induced damage [1,2]. Metal complexes of thiourea have now been extensively explored [3–7] as a result of its added advantages. The centrosymmetric thiourea molecule yields noncentrosymmetric materials when it is incorporated into inorganic salts, which is a general requirement of non-linear optics. In our investigations, growth of cadmium thiourea sulphate (CTS) and magnesium cadmium thiourea sulphate (MCTS) were carried out and basic growth parameters were evaluated. The various nucleation parameters essential for optimizing the growth conditions were calculated using classical nucleation theory which enable us to grow good quality crystals. The solubility data for the grown single crystals were determined experimentally at various temperatures. The classical nucleation theory has been
∗ Corresponding author at: Quaid-e-Milleth College, Kilpauk, Chennai 600002, India. Tel.: +91 4426414937. E-mail address: [email protected] (M. Daniel). 0254-0584/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2008.09.077
modified by applying correction to the crystal–solution interface in the microlevel in order to rectify the capillarity approximation. Based on the modified classical nucleation theory, the nucleation parameters such as radius of critical nucleus, Gibbs free energy change have been estimated and thereby the optimum nucleation rate fixed for the growth process. Cadmium thiourea sulphate possess orthorhombic structure with unit cell parameters a = 13.50 Å, b = 7.84 Å, c = 15.98 Å and V = 1692.58 Å3 . Magnesium cadmium thiourea sulphate has triclinic structure with lattice parameters a = 8.73 Å, b = 9.01 Å, c = 9.73 Å and V = 712.152 Å3 . 2. Solubility The solubility of CTS and MCTS have been determined at different temperatures 35 ◦ C, 40 ◦ C, 45 ◦ C, 50 ◦ C and 55 ◦ C. Recrystallized salt was dissolved in water and maintained at a constant temperature with continuous stirring to ensure homogeneous temperature and concentration throughout the entire region of the solution. On reaching saturation, the equilibrium concentration of the solute was analysed gravimetrically. The solubility curves for CTS and MCTS are shown in Fig. 1. 3. Experimental procedure CTS salt was synthesized from aqueous solution by incorporation of cadmium sulphate into thiourea in the molar ratio 1:1. The chemical reaction is given as
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
19
Fig. 1. Solubility of CTS and MCTS.
follows: Fig. 3. As grown MCTS crystals.
CdSO4 + CS(NH2 )2 → Cd[CS(NH2 )2 ]SO4 The purity of the salt was enhanced by successive recrystallization process. The solution was warmed in order to ensure homogenization throughout the solution without exceeding the temperature above 60 ◦ C to avoid decomposition. CTS single crystals grown by solvent evaporation were harvested after 4 weeks. MCTS salt was synthesized by mixing aqueous solution of cadmium sulphate, magnesium sulphate and thiourea in the molar ratio 0.5:0.5:1. Saturated solutions required for crystal growth was prepared in accordance with the solubility data using powdered synthesized salt and pure solvent. Evaporation of the solvent causes the driving force (supersaturation) to initiate nucleation resulting in the growth of the crystal. The nonhygroscopic and optically transparent grown single crystals of CTS and MCTS are shown in Figs. 2 and 3, respectively.
4. Nucleation kinetics Nucleation process is the initial and most important phenomenon in liquid–solid phase transition. Based on classical theory of homogeneous crystal nucleation, certain critical nucleation parameters like interfacial energy () between the solid and its mother solution, free energy of formation of the critical nucleus
and the radius of the nucleus in equilibrium with its solution can be calculated using solubility data which play a major role in fixing and controlling the initial growth conditions. Interfacial energy or the surface energy of the interface existing between a solid crystal and the surrounding supersaturated solution is an important parameter involved in the theories of nucleation and growth. It plays an important role in determining the rate of nucleation and the growth of the crystal. Bennema and Sohnel [8] have derived the expression for the relationship between the surface energy and solubility as =
kT a2
[0.173 − 0.248 ln(xm )]
(1)
where xm is the mole fraction of the solute, T is the temperature, a is the interionic distance and k is the Boltzmann constant. According to classical nucleation theory, the capillarity approximation is made use of in the evaluation of the nucleation parameters. In the capillarity approximation, the properties of the nucleus are considered to be the same as those in the bulk form for mathematical simplicity. When a crystal nucleus forms due to supersaturation of the solution, a certain quantity of energy is spent in the creation of a new phase. The overall excess free energy (G) between an embryo and the solute in the solution is equal to the sum of the surface excess free energy (GS ) and the volume excess free energy given by [9] G = GS + GV
(2)
where GS is the surface excess free energy and GV is the volume excess free energy. For a spherical nucleus, G = 4r 2 +
Fig. 2. As grown CTS crystal.
4 3
r 3 GV
(3)
The first term expresses the formation of a new surface and the second term represents the difference in chemical potential between the crystalline phase () and the surrounding mother liquid (o ). At the critical state, the free energy formation obeys the condition d(G/dr) = 0. By applying the above condition the radius of
20
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
critical nucleus is expressed as r∗ = −
2 GV
(4)
where GV = −/v, = −(kT) ln S, S = c/c*, c is the actual concentration and c* is the equilibrium concentration. Hence r∗ = −
2 v
where c is the constant of integration. The constant c is evaluated from the initial condition. When the size of the nucleus is equal to the size of the single monomer there is no interface at all. It means when r = ı, the size of the single monomer, = 0. Applying this initial condition an expression for as a function of the size of the cluster is obtained.
ı r
(5)
= 0
(6)
When r = ı, When r = ∞,
1−
(11)
The critical free energy barrier is given as G∗ =
16 3 v3 3()
2
The number of critical nuclei formed per unit time per unit volume is known as the rate of nucleation. The classical rate of nucleation is taken from [10] as
G∗
J = A exp −
(7)
kT
where A is a pre-exponential factor.
= 0 (the new phase just forms) = 0 (the nucleus attains macrolevel)
When the crystal nucleus just forms, there is no formation of interface between the crystal and the solution and hence the interfacial energy is 0. When the crystal nucleus attains considerably larger size, the interfacial energy becomes equal to the bulk value. This condition implies that interfacial energy depends upon the size of the nucleus. The critical radius is given as
0
ıGV 1− 0
1/2
5. Modified classical nucleation theory
r∗ =
The classical nucleation theory makes use of capillarity approximation. According to this approximation the physical properties of the nucleus at the microlevel are assumed to be the same as those in the macrolevel. A microcluster in a supersaturated system in the liquid phase possesses a structure which is different from that of a bulk condensed phase. Therefore the concept of constant interfacial tension in the microsystem and macrosystem is not valid. A correction has to be applied and hence the classical nucleation theory is suitably modified. The nucleation parameters are evaluated based on this concept. Let be the interfacial energy of the spherical shaped nucleus when the size is r. The interfacial energy at the microlevel will be less than that at the macrolevel. If the interfacial energy at the macrolevel is 0 , then the interfacial energy of a spherical shaped nucleus of size r can be written as
The free energy change associated with the critical nucleus is obtained as,
= 0 −
d dr
r
(8)
where d/dr is the surface energy gradient. dr d = 0 − r
(9)
Integrating, − ln(0 − ) = ln r + c
(10)
GV
1+
G∗ = 4r ∗ 0 (r ∗ − ı) −
1 3
(12)
r ∗2 GV
(13)
The rate of nucleation is
G∗
J = A exp −
(14)
kT
6. Results and discussions The solubility curves of CTS and MCTS are shown in Fig. 1. As CTS has a higher solubility than MCTS, the size of CTS crystal is larger than that of MCTS. The nucleation parameters of CTS and MCTS were calculated using classical nucleation theory and after rectifying the capillarity approximation. Tables 1 and 2 present the nucleation parameters at different supersaturations. It is observed that critical radius and critical free energy change are not affected very much due to correction in the classical theory. The rectification of capillarity approximation leads to a decrease in the values of nucleation parameters. Figs. 4–7 show the plots of nucleation rate against supersaturation. It is found that the nucleation rate is considerably enhanced with increase of supersaturation. After rectifying
Table 1 Nucleation parameters of CTS at different temperatures. Supersaturation ratio
1.032 1.034 1.036 1.038 1.040 1.042 1.044 1.046 1.048 1.050 1.052
323 K
318 K
´˚ r* (A)
G* (×10−12 ergs)
´˚ r* (A)
313 K G* (×10−12 ergs)
´˚ r* (A)
G* (×10−12 ergs)
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
62.44 58.82 55.61 52.73 50.15 47.80 45.68 43.73 41.95 40.31 38.80
61.94 58.32 55.11 52.23 49.64 47.30 45.17 43.23 41.44 39.80 38.29
3.372 2.993 2.675 2.405 2.175 1.976 1.804 1.654 1.522 1.405 1.302
3.210 2.841 2.531 2.269 2.045 1.853 1.686 1.541 1.414 1.301 1.202
64.71 60.97 57.63 54.65 51.97 49.54 47.34 45.32 43.48 41.78 40.21
64.21 60.46 57.13 54.15 51.47 49.04 46.83 44.82 42.97 41.27 39.70
3.696 3.280 2.931 2.636 2.384 2.166 1.978 1.813 1.668 1.540 1.427
3.525 3.119 2.779 2.492 2.247 2.036 1.853 1.694 1.554 1.430 1.321
67.23 63.34 59.88 56.78 53.99 51.47 49.18 47.09 45.17 43.40 41.77
66.73 62.83 59.37 56.28 53.49 50.97 48.67 46.58 44.66 42.90 41.27
4.079 3.620 3.062 2.909 2.631 2.391 2.183 2.001 1.841 1.700 1.575
3.891 3.449 3.074 2.756 2.485 2.252 2.050 1.874 1.719 1.583 1.462
CNT, classical nucleation theory; MCNT, modified classical nucleation theory.
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
21
Table 2 Nucleation parameters of MCTS at different temperatures. Supersaturation ratio
323 K
318 K
´˚ r* (A)
1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22
−12
G* (×10
ergs)
313 K
´˚ r* (A)
−12
G* (×10
ergs)
´˚ r* (A)
G* (×10−12 ergs)
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
46.20 43.31 40.79 38.56 36.58 34.80 33.20 31.71 30.44
45.70 42.81 40.28 38.05 36.07 34.29 32.70 31.25 29.93
3.395 2.984 2.646 2.365 2.128 1.926 1.753 1.604 1.474
3.176 2.779 2.453 2.182 1.954 1.761 1.596 1.454 1.330
48.27 45.25 42.61 40.28 38.21 36.36 34.69 33.18 31.81
47.77 44.75 42.11 39.78 37.71 35.85 34.18 32.67 31.30
3.812 3.350 2.971 2.655 2.389 2.163 1.969 1.801 1.655
3.576 3.130 2.763 2.458 2.203 1.986 1.800 1.639 1.500
50.80 47.62 44.85 42.39 40.21 38.26 36.51 34.92 33.47
50.29 47.12 44.34 41.89 39.71 37.76 36.00 34.41 32.96
4.373 3.843 3.408 3.046 2.741 2.481 2.259 2.066 1.899
4.116 3.603 3.181 2.831 2.531 2.288 2.074 1.890 1.730
CNT, classical nucleation theory; MCNT, modified classical nucleation theory.
Fig. 4. Nucleation rate as a function of supersaturation ratio for CTS based on classical nucleation theory.
Fig. 5. Nucleation rate as a function of supersaturation ratio for CTS based on modified classical nucleation theory.
Fig. 6. Nucleation rate as a function of supersaturation ratio for MCTS based on classical nucleation theory.
Fig. 7. Nucleation rate as a function of supersaturation ratio for MCTS based on modified classical nucleation theory.
capillarity approximation, even though there is only slight variation in the critical size and the free energy change of the crystal nucleus, the nucleation rate is enhanced appreciably. The supersaturation corresponding to J = 1 is called the critical supersaturation. The critical supersaturation is found to decrease with increase of temperature as shown in Figs. 8 and 9. Therefore, it is concluded that quality of the crystal will be improved if the supersaturation is fixed corresponding to unit nucleation rate. Figs. 2 and 3 show the as grown crystals of CTS and MCTS by slow evaporation technique. Single crystal X-ray diffraction analysis of CTS and MCTS crystals were carried out using Enraf CAD-4 Diffractometer with Ni ´˚ The crystallographic parameters of CTS filtered Cu K␣ ( = 1.5418 A). are a = 13.50 Å, b = 7.84 Å, c = 15.98 Å and V = 1692.58 Å3 and those of MCTS are a = 8.73 Å, b = 9.01 Å, c = 9.73 Å and V = 712.152 Å3 .
Fig. 8. Plot of supersaturation value at critical nucleation rate vs. temperature for CTS.
22
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
were estimated, so as to optimize them for the growth of good quality crystals. A new approach has been made to rectify the capillarity approximation in the classical nucleation theory. Single crystal Xray diffraction studies were utilized to determine its cell parameters and to confirm the crystalline nature of the grown materials. References
Fig. 9. Plot of supersaturation value at critical nucleation rate vs. temperature for MCTS.
7. Conclusion Growth of CTS and MCTS single crystals was grown by slow evaporation methods. Based on classical nucleation theory and modified classical nucleation theory the fundamental growth parameters
[1] P.R. Newman, L.F. Warren, P. Cunningham, T.Y. Chang, D.E. Cooper, G.L. Burdge, P. Polok Dingels, C.K. Lowe-Ma, in: C.Y. Chiang, P.M. Chalken, D.O. Cowan (Eds.), Proceedings of the Materials Research Society Symposium on Advanced Organic Solid State Materials-E, vol. 173, Materials Research Society, Pittsburgh, PA, 1990, p. 557. [2] H.O. Marcy, L.F. Warren, M.S. Webb, C.A. Ebbers, S.P. Velsko, G.C. Kennedy, G.C. Cutella, Appl. Opt. 31 (1992) 5051. [3] P.M. Ushasree, R. Muralidaran, R. Jayavel, P. Ramasamy, J. Cryst. Growth 218 (2000) 365. [4] V. Venkataraman, G. Dhanaraj, V.K. Wadhawan, J.N. Sherwood, H.L. Bhat, J. Cryst. Growth 154 (1995) 92. [5] S.G. Bhat, S.M. Dharmaprakash, Mater. Res. Bull. 33 (1998) 833. [6] S. Anie Roshans, Cyriac Joseph, M.A. Ittyachen, Mater. Lett. 49 (2001) 299. [7] V.K. Kannan, N.P. Rajesh, R. Bairava Ganesh, P. Ramasamy, J. Cryst. Growth 269 (2004) 565. [8] P. Bennema, O. Sohnel, J. Cryst. Growth 102 (1990) 547. [9] A.E. Nielsen, A. Sarig, J. Cryst. Growth 8 (1971) 1. [10] T. Kanagasekaran, M. Gunasekaran, P. Srinivasan, D. Jayaraman, R. Gopalakrishnan, P. Ramasamy, Cryst. Res. Technol. 40 (2005) 1128.
Contents lists available at ScienceDirect
Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Materials science communication
Studies on growth and nucleation kinetics of cadmium thiourea sulphate and magnesium cadmium thiourea sulphate Mekala Daniel a,∗ , M. Jeyarani Malliga b , R. Sankar c , D. Jayaraman d a
Quaid-e-Milleth College, Chennai 600002, India Bharathi Women College, Chennai 600108, India c Kings Engineering College, Irungatukottai, Sriperumbudhur, Chennai 602105, India d Presidency College, Chennai 600 004, India b
a r t i c l e
i n f o
Article history: Received 1 July 2007 Received in revised form 25 September 2008 Accepted 28 September 2008 Keywords: CTS and MCTS crystals Solution growth method Solubility Basic growth parameters Classical nucleation theory Capillarity approximation
a b s t r a c t Semiorganic materials, in general possess high non-linear coefficient and mechanical strength which will be more applicable for device fabrication. Cadmium thiourea sulphate (CTS) and magnesium cadmium thiourea sulphate (MCTS) are better semiorganic materials which find applications in the field of optoelectronics. Single crystals of CTS and MCTS have been successfully grown from aqueous solution by slow evaporation technique using predetermined solubility data. The basic growth parameters of the crystal nuclei of the grown crystals of CTS and MCTS were evaluated based on the classical theory of homogeneous nucleation. The classical nucleation theory makes use of capillarity approximation which has certain limitations. A correction has to be applied for it and the classical nucleation theory has been suitably modified in order to calculate the critical nucleus parameters. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Materials having good optical non-linearity with chemical flexibility and mechanical properties are in demand in recent times due to its wide applications. Semiorganic materials have attracted great interest as these materials have the potential to possess high nonlinearity, good mechanical hardness, low angular sensitivity and high resistance to laser induced damage [1,2]. Metal complexes of thiourea have now been extensively explored [3–7] as a result of its added advantages. The centrosymmetric thiourea molecule yields noncentrosymmetric materials when it is incorporated into inorganic salts, which is a general requirement of non-linear optics. In our investigations, growth of cadmium thiourea sulphate (CTS) and magnesium cadmium thiourea sulphate (MCTS) were carried out and basic growth parameters were evaluated. The various nucleation parameters essential for optimizing the growth conditions were calculated using classical nucleation theory which enable us to grow good quality crystals. The solubility data for the grown single crystals were determined experimentally at various temperatures. The classical nucleation theory has been
∗ Corresponding author at: Quaid-e-Milleth College, Kilpauk, Chennai 600002, India. Tel.: +91 4426414937. E-mail address: [email protected] (M. Daniel). 0254-0584/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2008.09.077
modified by applying correction to the crystal–solution interface in the microlevel in order to rectify the capillarity approximation. Based on the modified classical nucleation theory, the nucleation parameters such as radius of critical nucleus, Gibbs free energy change have been estimated and thereby the optimum nucleation rate fixed for the growth process. Cadmium thiourea sulphate possess orthorhombic structure with unit cell parameters a = 13.50 Å, b = 7.84 Å, c = 15.98 Å and V = 1692.58 Å3 . Magnesium cadmium thiourea sulphate has triclinic structure with lattice parameters a = 8.73 Å, b = 9.01 Å, c = 9.73 Å and V = 712.152 Å3 . 2. Solubility The solubility of CTS and MCTS have been determined at different temperatures 35 ◦ C, 40 ◦ C, 45 ◦ C, 50 ◦ C and 55 ◦ C. Recrystallized salt was dissolved in water and maintained at a constant temperature with continuous stirring to ensure homogeneous temperature and concentration throughout the entire region of the solution. On reaching saturation, the equilibrium concentration of the solute was analysed gravimetrically. The solubility curves for CTS and MCTS are shown in Fig. 1. 3. Experimental procedure CTS salt was synthesized from aqueous solution by incorporation of cadmium sulphate into thiourea in the molar ratio 1:1. The chemical reaction is given as
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
19
Fig. 1. Solubility of CTS and MCTS.
follows: Fig. 3. As grown MCTS crystals.
CdSO4 + CS(NH2 )2 → Cd[CS(NH2 )2 ]SO4 The purity of the salt was enhanced by successive recrystallization process. The solution was warmed in order to ensure homogenization throughout the solution without exceeding the temperature above 60 ◦ C to avoid decomposition. CTS single crystals grown by solvent evaporation were harvested after 4 weeks. MCTS salt was synthesized by mixing aqueous solution of cadmium sulphate, magnesium sulphate and thiourea in the molar ratio 0.5:0.5:1. Saturated solutions required for crystal growth was prepared in accordance with the solubility data using powdered synthesized salt and pure solvent. Evaporation of the solvent causes the driving force (supersaturation) to initiate nucleation resulting in the growth of the crystal. The nonhygroscopic and optically transparent grown single crystals of CTS and MCTS are shown in Figs. 2 and 3, respectively.
4. Nucleation kinetics Nucleation process is the initial and most important phenomenon in liquid–solid phase transition. Based on classical theory of homogeneous crystal nucleation, certain critical nucleation parameters like interfacial energy () between the solid and its mother solution, free energy of formation of the critical nucleus
and the radius of the nucleus in equilibrium with its solution can be calculated using solubility data which play a major role in fixing and controlling the initial growth conditions. Interfacial energy or the surface energy of the interface existing between a solid crystal and the surrounding supersaturated solution is an important parameter involved in the theories of nucleation and growth. It plays an important role in determining the rate of nucleation and the growth of the crystal. Bennema and Sohnel [8] have derived the expression for the relationship between the surface energy and solubility as =
kT a2
[0.173 − 0.248 ln(xm )]
(1)
where xm is the mole fraction of the solute, T is the temperature, a is the interionic distance and k is the Boltzmann constant. According to classical nucleation theory, the capillarity approximation is made use of in the evaluation of the nucleation parameters. In the capillarity approximation, the properties of the nucleus are considered to be the same as those in the bulk form for mathematical simplicity. When a crystal nucleus forms due to supersaturation of the solution, a certain quantity of energy is spent in the creation of a new phase. The overall excess free energy (G) between an embryo and the solute in the solution is equal to the sum of the surface excess free energy (GS ) and the volume excess free energy given by [9] G = GS + GV
(2)
where GS is the surface excess free energy and GV is the volume excess free energy. For a spherical nucleus, G = 4r 2 +
Fig. 2. As grown CTS crystal.
4 3
r 3 GV
(3)
The first term expresses the formation of a new surface and the second term represents the difference in chemical potential between the crystalline phase () and the surrounding mother liquid (o ). At the critical state, the free energy formation obeys the condition d(G/dr) = 0. By applying the above condition the radius of
20
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
critical nucleus is expressed as r∗ = −
2 GV
(4)
where GV = −/v, = −(kT) ln S, S = c/c*, c is the actual concentration and c* is the equilibrium concentration. Hence r∗ = −
2 v
where c is the constant of integration. The constant c is evaluated from the initial condition. When the size of the nucleus is equal to the size of the single monomer there is no interface at all. It means when r = ı, the size of the single monomer, = 0. Applying this initial condition an expression for as a function of the size of the cluster is obtained.
ı r
(5)
= 0
(6)
When r = ı, When r = ∞,
1−
(11)
The critical free energy barrier is given as G∗ =
16 3 v3 3()
2
The number of critical nuclei formed per unit time per unit volume is known as the rate of nucleation. The classical rate of nucleation is taken from [10] as
G∗
J = A exp −
(7)
kT
where A is a pre-exponential factor.
= 0 (the new phase just forms) = 0 (the nucleus attains macrolevel)
When the crystal nucleus just forms, there is no formation of interface between the crystal and the solution and hence the interfacial energy is 0. When the crystal nucleus attains considerably larger size, the interfacial energy becomes equal to the bulk value. This condition implies that interfacial energy depends upon the size of the nucleus. The critical radius is given as
0
ıGV 1− 0
1/2
5. Modified classical nucleation theory
r∗ =
The classical nucleation theory makes use of capillarity approximation. According to this approximation the physical properties of the nucleus at the microlevel are assumed to be the same as those in the macrolevel. A microcluster in a supersaturated system in the liquid phase possesses a structure which is different from that of a bulk condensed phase. Therefore the concept of constant interfacial tension in the microsystem and macrosystem is not valid. A correction has to be applied and hence the classical nucleation theory is suitably modified. The nucleation parameters are evaluated based on this concept. Let be the interfacial energy of the spherical shaped nucleus when the size is r. The interfacial energy at the microlevel will be less than that at the macrolevel. If the interfacial energy at the macrolevel is 0 , then the interfacial energy of a spherical shaped nucleus of size r can be written as
The free energy change associated with the critical nucleus is obtained as,
= 0 −
d dr
r
(8)
where d/dr is the surface energy gradient. dr d = 0 − r
(9)
Integrating, − ln(0 − ) = ln r + c
(10)
GV
1+
G∗ = 4r ∗ 0 (r ∗ − ı) −
1 3
(12)
r ∗2 GV
(13)
The rate of nucleation is
G∗
J = A exp −
(14)
kT
6. Results and discussions The solubility curves of CTS and MCTS are shown in Fig. 1. As CTS has a higher solubility than MCTS, the size of CTS crystal is larger than that of MCTS. The nucleation parameters of CTS and MCTS were calculated using classical nucleation theory and after rectifying the capillarity approximation. Tables 1 and 2 present the nucleation parameters at different supersaturations. It is observed that critical radius and critical free energy change are not affected very much due to correction in the classical theory. The rectification of capillarity approximation leads to a decrease in the values of nucleation parameters. Figs. 4–7 show the plots of nucleation rate against supersaturation. It is found that the nucleation rate is considerably enhanced with increase of supersaturation. After rectifying
Table 1 Nucleation parameters of CTS at different temperatures. Supersaturation ratio
1.032 1.034 1.036 1.038 1.040 1.042 1.044 1.046 1.048 1.050 1.052
323 K
318 K
´˚ r* (A)
G* (×10−12 ergs)
´˚ r* (A)
313 K G* (×10−12 ergs)
´˚ r* (A)
G* (×10−12 ergs)
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
62.44 58.82 55.61 52.73 50.15 47.80 45.68 43.73 41.95 40.31 38.80
61.94 58.32 55.11 52.23 49.64 47.30 45.17 43.23 41.44 39.80 38.29
3.372 2.993 2.675 2.405 2.175 1.976 1.804 1.654 1.522 1.405 1.302
3.210 2.841 2.531 2.269 2.045 1.853 1.686 1.541 1.414 1.301 1.202
64.71 60.97 57.63 54.65 51.97 49.54 47.34 45.32 43.48 41.78 40.21
64.21 60.46 57.13 54.15 51.47 49.04 46.83 44.82 42.97 41.27 39.70
3.696 3.280 2.931 2.636 2.384 2.166 1.978 1.813 1.668 1.540 1.427
3.525 3.119 2.779 2.492 2.247 2.036 1.853 1.694 1.554 1.430 1.321
67.23 63.34 59.88 56.78 53.99 51.47 49.18 47.09 45.17 43.40 41.77
66.73 62.83 59.37 56.28 53.49 50.97 48.67 46.58 44.66 42.90 41.27
4.079 3.620 3.062 2.909 2.631 2.391 2.183 2.001 1.841 1.700 1.575
3.891 3.449 3.074 2.756 2.485 2.252 2.050 1.874 1.719 1.583 1.462
CNT, classical nucleation theory; MCNT, modified classical nucleation theory.
M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
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Table 2 Nucleation parameters of MCTS at different temperatures. Supersaturation ratio
323 K
318 K
´˚ r* (A)
1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22
−12
G* (×10
ergs)
313 K
´˚ r* (A)
−12
G* (×10
ergs)
´˚ r* (A)
G* (×10−12 ergs)
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
CNT
MCNT
46.20 43.31 40.79 38.56 36.58 34.80 33.20 31.71 30.44
45.70 42.81 40.28 38.05 36.07 34.29 32.70 31.25 29.93
3.395 2.984 2.646 2.365 2.128 1.926 1.753 1.604 1.474
3.176 2.779 2.453 2.182 1.954 1.761 1.596 1.454 1.330
48.27 45.25 42.61 40.28 38.21 36.36 34.69 33.18 31.81
47.77 44.75 42.11 39.78 37.71 35.85 34.18 32.67 31.30
3.812 3.350 2.971 2.655 2.389 2.163 1.969 1.801 1.655
3.576 3.130 2.763 2.458 2.203 1.986 1.800 1.639 1.500
50.80 47.62 44.85 42.39 40.21 38.26 36.51 34.92 33.47
50.29 47.12 44.34 41.89 39.71 37.76 36.00 34.41 32.96
4.373 3.843 3.408 3.046 2.741 2.481 2.259 2.066 1.899
4.116 3.603 3.181 2.831 2.531 2.288 2.074 1.890 1.730
CNT, classical nucleation theory; MCNT, modified classical nucleation theory.
Fig. 4. Nucleation rate as a function of supersaturation ratio for CTS based on classical nucleation theory.
Fig. 5. Nucleation rate as a function of supersaturation ratio for CTS based on modified classical nucleation theory.
Fig. 6. Nucleation rate as a function of supersaturation ratio for MCTS based on classical nucleation theory.
Fig. 7. Nucleation rate as a function of supersaturation ratio for MCTS based on modified classical nucleation theory.
capillarity approximation, even though there is only slight variation in the critical size and the free energy change of the crystal nucleus, the nucleation rate is enhanced appreciably. The supersaturation corresponding to J = 1 is called the critical supersaturation. The critical supersaturation is found to decrease with increase of temperature as shown in Figs. 8 and 9. Therefore, it is concluded that quality of the crystal will be improved if the supersaturation is fixed corresponding to unit nucleation rate. Figs. 2 and 3 show the as grown crystals of CTS and MCTS by slow evaporation technique. Single crystal X-ray diffraction analysis of CTS and MCTS crystals were carried out using Enraf CAD-4 Diffractometer with Ni ´˚ The crystallographic parameters of CTS filtered Cu K␣ ( = 1.5418 A). are a = 13.50 Å, b = 7.84 Å, c = 15.98 Å and V = 1692.58 Å3 and those of MCTS are a = 8.73 Å, b = 9.01 Å, c = 9.73 Å and V = 712.152 Å3 .
Fig. 8. Plot of supersaturation value at critical nucleation rate vs. temperature for CTS.
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M. Daniel et al. / Materials Chemistry and Physics 114 (2009) 18–22
were estimated, so as to optimize them for the growth of good quality crystals. A new approach has been made to rectify the capillarity approximation in the classical nucleation theory. Single crystal Xray diffraction studies were utilized to determine its cell parameters and to confirm the crystalline nature of the grown materials. References
Fig. 9. Plot of supersaturation value at critical nucleation rate vs. temperature for MCTS.
7. Conclusion Growth of CTS and MCTS single crystals was grown by slow evaporation methods. Based on classical nucleation theory and modified classical nucleation theory the fundamental growth parameters
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