ADVANCEMENTS IN NUCLEATION THEORY: EXTENSIONS TO THE XPFC MODEL AND APPLICATIONS IN PRECIPITATION

International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 200 ISSN 2394 - 7780 ADVANCEMENTS IN NUCLEATION THEORY: EXTENSIONS TO THE XPFC MODEL AND APPLICATIONS IN PRECIPITATION Sahil Sekhri Director Golden dots international Pvt Ltd International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 201 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 202 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 203 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 204 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 205 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 206 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 207 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 208 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 215 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 216 ISSN 2394 - 7780 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 217 ISSN 2394 - 7780 REFERENCES [1] E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev., vol. 106, pp. 620–630, May 1957. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.106.620 [2] E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev., vol. 40, pp. 749–759, Jun 1932. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.40.749 [3] J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev., vol. 44, pp. 31–37, Jul 1933. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.44.31 [4] G. E. Uhlenbeck and E. Beth, “The quantum theory of the non-ideal gas ii. behavior of the virial coefficients,” Physica, vol. 4, no. 8, pp. 729–745, 1937. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0031891438603462 [5] L. Landau and E. Lifshitz, “Chapter ii: the gibbs distribution,” in Statistical Physics (Third Edition, Revised and Enlarged), third edition, revised and enlarged ed., L. Landau and E. Lifshitz, Eds. Oxford: ButterworthHeinemann, 1980, pp. 79–110. [Online]. Available: http://www.sciencedirect.com/science/article/pii/B9780750626360500071 [6] W. Gibbs, Elementary principles in statistical mechanics, 1960. [7] R. Kubo, “Generalized cumulant expansion method,” Journal of the Physical Society of Japan, vol. 17, no. 7, pp. 1100–1120, 1962. [Online]. Available: http://dx.doi.org/10.1143/JPSJ.17.1100 [8] P. Espagnol and H. Löwen, “Derivation of dynamical density functional theory using the projection operator technique,” The Journal of Chemical Physics, vol. 131, no. 24, p. 244101, 2009. [Online]. Available: http://dx.doi.org/10.1063/1.3266943 [9] J.-P. Hansen and I. R. McDonald, “Appendix b: Two theorems in density functional theory,” in Theory of Simple Liquids (Fourth Edition), fourth edition ed., J.-P. Hansen and I. R. McDonald, Eds. Oxford: Academic Press, 2013, pp. 587–589. [Online]. Available: http://www.sciencedirect.com/science/article/pii/B9780123870322000272 [10] J. E. Mayer and E. Montroll, “Molecular distribution,” The Journal of Chemical Physics, vol. 9, no. 1, pp. 2–16, 1941. [Online]. Available: http://dx.doi.org/10.1063/1.1750822 [11] J.-P. Hansen and I. R. McDonald, “Chapter 6 - Inhomogeneous fluids,” in Theory of Simple Liquids (Fourth Edition), fourth edition ed., J.-P. Hansen and I. R. McDonald, Eds. Oxford: Academic Press, 2013, pp. 203–264. [Online]. Available: http://www.sciencedirect.com/science/article/pii/B9780123870322000064 [12] M. Oettel and S. A. Egorov, “Statistical mechanics of inhomogeneous fluids,” Annual Review of Chemical Physics, vol. 9, no. 7, pp. 514–526, 1941. [Online]. Available: http://dx.doi.org/10.1063/1.1750822 [13] M. Marconi and M. Yussouff, “First-principles order-parameter dynamics,” Phys. Rev. B, vol. 19, pp. 2775–2794, Mar 1979. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.19.2775 [14] T. Oxtoby, “The glass transition, the liquid-glass transition of the lennard-jones fluid,” Advances in Chemical Physics, vol. 40, pp. 1–39, 1969. [Online]. Available: http://dx.doi.org/10.1002/9780470143594.ch1 International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 218 ISSN 2394 - 7780 [15] J.-P. Hansen and L. Verlet, “Phase transitions of the lennard-jones system,” Phys. Rev., vol. 184, pp. 151–161, Aug 1969. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.184.151 [16] U. M. B. Marconi and P. Tarazona, “Dynamic density functional theory of fluids,” Journal of Physics: Condensed Matter, vol. 12, no. 8A, p. A413, 2000. [Online]. Available: http://stacks.iop.org/0953-8984/12/i= 8A/a=356 [17] A. J. Archer and M. Rauscher, “Dynamical density functional theory for interacting brownian particles: stochastic or deterministic?” Journal of Physics A: Mathematical and General, vol. 37, no. 40, p. 9325, 2004. [Online]. Available: http://stacks.iop.org/0305-4470/37/i=40/a=001 [18] K. R. Elder, N. Provatas, J. Berry, P. Stefanovic, and M. Grant, “Phase-field crystal modeling and classical density functional theory of freezing,” Phys. Rev. B, vol. 75, p. 064107, Feb 2007. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.75.064107 [19] M. Greenwood, N. Ofori-Opoku, J. Rottler, and N. Provatas, “Modeling structural transformations in binary alloys with phase field crystals,” Phys. Rev. B, vol. 84, p. 064104, Aug 2011. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevB.84.064104 [20] P. R. Subramanian and J. H. Perepezko, “The ag-cu (silver-copper) system,” Journal of Phase Equilibria, vol. 14, pp. 62 – 75, 1993. [21] M. Greenwood, N. Provatas, and J. Rottler, “Free energy functionals for efficient phase field crystal modeling of structural phase transformations,” Phys. Rev. Lett., vol. 105, p. 045702, Jul 2010. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.105. 045702 [22] H. A. Murdoch and C. A. Schuh, “Stability of binary nanocrystalline alloys against grain growth and phase separation,” Acta Materialia, vol. 61, no. 6, pp. 2121 – 2132, 2013. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1359645412008919 [23] N. D. Loh, S. Sen, M. Bosman, S. F. Tan, J. Zhong, C. A. Nijhuis, P. Kra´l, P. Matsudaira, and U. Mirsaidov, “Multistep nucleation of nanocrystals in aqueous solution,” Nat Chem, vol. 9, Jan 2017. [24] A. F. Wallace, L. O. Hedges, A. Fernandez-Martinez, P. Raiteri, J. D. Gale, G. A. Waychunas, S. Whitelam, J. F. Banfield, and J. J. De Yoreo, “Microscopic evidence for liquid-liquid separation in supersaturated caco3 solutions,” Science, vol. 341, no. 6148, pp. 885–889, 2013. [Online]. Available: http://science.sciencemag.org/content/341/6148/885 [25] J. F. Lutsko and M. A. Duran-Olivencia, “A two-parameter extension of classical nucleation theory,” Journal of Physics: Condensed Matter, vol. 27, no. 23, p. 235101, 2015. [Online]. Available: http://stacks.iop.org/0953-8984/27/i=23/a=235101 [26] D. Knezic, J. Zaccaro, and A. S. Myerson, “Nucleation induction time in levitated droplets,” The Journal of Physical Chemistry B, vol. 108, no. 30, pp. 10 672–10 677, 2004. [Online]. Available: http://dx.doi.org/10.1021/jp049586s [27] D. Erdemir, A. Y. Lee, and A. S. Myerson, “Nucleation of crystals from solution: Classical and twostep models,” Accounts of Chemical Research, vol. 42, no. 5, pp. 621–629, 2009, pMID: 19402623. [Online]. Available: http://dx.doi.org/10.1021/ar800217x [28] R. J. Davey, S. L. M. Schroeder, and J. H. ter Horst, “Nucleation of organic crystals—a molecular perspective,” Angewandte Chemie International Edition, vol. 52, no. 8, pp. 2166–2179, 2013. [Online]. Available: http://dx.doi.org/10.1002/anie.201204824 [29] L. Onsager and S. Machlup, “Fluctuations and irreversible processes,” Phys. Rev., vol. 91, pp. 1505– 1512, Sep 1953. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRev.91.1505 [30] M. Lax, “Fluctuations from the nonequilibrium steady state,” Rev. Mod. Phys., vol. 32, pp. 25–64, Jan 1960. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.32.25 [31] D. Ronis, I. Procaccia, and J. Machta, “Statistical mechanics of stationary states. vi. hydrodynamic fluctuation theory far from equilibrium,” Phys. Rev. A, vol. 22, pp. 714–724, Aug 1980. [Online]. International Journal of Advance and Innovative Research Volume 12, Issue 3: July - September 2025 219 ISSN 2394 - 7780 Available: http://link.aps.org/doi/10.1103/PhysRevA.22.714 [32] R. F. Fox and G. E. Uhlenbeck, “Contributions to non-equilibrium thermodynamics. i. theory of hydrodynamical fluctuations,” Physics of Fluids, vol. 13, no. 8, pp. 1893–1902, 1970. [Online]. Available: http://scitation.aip.org/content/aip/journal/pof1/13/8/10.1063/1.1693183 [33] M. Kardar, Statistical Physics of Fields, Jun. 2006.