The important renewable method of growing single crystals of La₂Sr₂CuO₄ (LSCO) and YBa₂Cu₂O_7−δ (YBCO) is the high temperature solution using LSCO-CuO [1] - [3] and BaO-CuO [4] - [6] as flux respectively. A satisfactory flux should have high solubility for the solvent and an appreciable change of solubility with temperature. The widely used flux is the LSCO-CuO mixed in the ratio 0.23:0.77 for LSCO [7] and BaO-CuO mixed in the ratio 0.28:0.72 for YBCO [8] . The binary phase diagram of the solute-solvent system is very essential to grow LSCO and YBCO crystals from flux technique. The high temperature solution growth techniques require the knowledge of the essential nucleation thermodynamical parameters like interfacial energy, metastable zone- width, volume free energy, critical energy barrier for nucleation and radius of crystal nucleus for the growth of good quality bulk single crystals. Up to now, there have been no reports in the literature regarding these key parameters. Here an attempt has been made to estimate these parameters theoretically to understand the role of these nucleation parameters for the crystallization of LSCO from LSCO-CuO flux and YBCO from BaO-CuO flux. According to the assumption of regular solution model, the nucleation process is considered to be homogeneous in nature and in the nucleation process solid and liquid phases coexist together. In this paper the regular solution model has been adopted to calculate the nucleation parameters of LSCO and YBCO for the first time.

The interfacial energy, metastable zone-width (supercooling temperature), volume free-energy, critical energy barrier and radius of critical nucleus of LSCO and YBCO has been calculated using the different formalism as discussed above and presented in Table 1 and Table 2.

The variation of interfacial energy with temperature is shown in Figure 3 for LSCO. From Figure 3, it can be observed that the interfacial energy increases, with the decrease of temperature of the solution and the trend exhibited with the experimental observation of high temperature solution growth of KTP crystals crystallizing from K₆P₄O₁₃ flux [19] and in the NdBa₂Cu₃O_7−d (Nd123) system [20] , the strong anistropy in the interfacial energy may be due to the inherent property of LSCO in LSCO-CuO solute- solvent system. The variation of interfacial energy with temperature exhibits similar trend in case of YBCO in BaO-CuO solute-solvent system as shown in Table 2 which is also similar the experimental observation of high temperature solution growth [21] [22] .

The calculated value of metastable zone-width for LSCO at different temperatures is shown in Table 1 and for YBCO at different temperature is shown in Figure 4. From Table 1 for LSCO and Figure 4 for YBCO, it can be concluded that the metastable zone-width is narrow at higher temperature and wide at lower temperature of the solute in the solution. This is very much conformable with the expected prediction. The degree of probability of nucleation depends on the intermolecular distances of the solute particles in the solution and therefore on its concentration. In the literature there is no specific value of metastable zone-width and crystallization tem-

perature. Different laboratories reported different values of metastable zone-width and crystallizing temperature ranging from 1550 - 1350 K [21] [23] - [25] which is comparable to theoretically calculated values.

The calculated value of metastable zone-width for LSCO at different mole-fraction is shown in Figure 5. From Figure 5, it is concluded that metastable zone-width is wide at low concentrations i.e. at low temperature, and narrow at high concentration i.e. at higher temperature of the solute in the solution. The variation of interfacial energy with mole fraction is shown in Table 1 and Table 2 for LSCO and YBCO. It is observed that the interfacial energy increases with the decrease of mole fraction of the solution, which may be due to the inherent property of the flux and trend exhibited in the graph is similar to the experimental observation of high temperature solution growth [22] .

The change in free energy on the formation of a single YBCO molecule at equilibrium has been calculated using the available thermodynamical data. Using the regular solution model the thermodynamical potential of the components associated with the reaction has been calculated. The critical energy, has been evaluated by calculating the excess free energy. The critical energy of YBCO with temperature is shown in Figure 6 and for LSCO in Table 1. It is noticed that critical energy decrease with increasing temperature which associated with the interface in terms of the energy of interactions between the molecules.

For the better understanding of the growth kinetics of LSCO and YBCO crystals, the other nucleation thermodynamical parameters of LSCO and YBCO such as volume free-energy, critical energy, radius of critical energy, energy of formation in the nucleus are calculated for various values of interfacial tension is given in Table 1 and Table 2. The present work provides a comprehensive picture of LSCO crystal from crystallizing LSCO-CuO flux and YBCO crystal from crystallizing BaO-CuO flux. The uncertainty in determining the above nucleation parameters of LSCO and YBCO crystals from high temperature solution growth by experimental method creates interest in determining the above parameters.

The nucleation parameters of LSCO crystal growth from the LSCO-CuO flux and YBCO from BaO-CuO flux at different crystallization temperatures have been calculated. As there are no reports in the literature for the nucleation thermodynamical perameters of LSCO and YBCO crystals, the present theoretically estimated values will be useful for the growth of large size good quality single crystals of LSCO and YBCO.

Shahida Akhter,Deba Prasad Paul, (2016) Estimation of Nucleation Thermodynamical Parameters of La₂Sr₂xCuO₄ (LSCO) and YBa₂Cu₂O_7–δ (YBCO) Crystallizing from High Temperature Solution. Crystal Structure Theory and Applications,05,17-23. doi: 10.4236/csta.2016.52002