Grand potential formalism of interfacial thermodynamics for critical nucleus

Atsushi Mori, Yoshihisa Suzuki

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Institute of Technology and Science, The University of Tokushima, Tokushima, Japan.

DOI: 10.4236/ns.2013.55078                

Abstract

In nucleation theories, the work of formation of a nucleus is often denoted by W = ΔG. This convention misleads that the nucleation should be considered in the isothermal-isobaric system. However, the pressure in the system with a nucleus is no longer uniform due to Laplace’s equation. Instead, the chemical potential is uniform throughout the system for the critical nucleus. Therefore, one can consider the nucleation in the grand ensemble properly. Accordingly, W is found to be the grand potential difference and the interfacial tension is also turned to be an interfacial excess grand potential. This treatment is not entirely new; however, to explicitly treat in the grand potential formalism is for the first time. We have successfully given an overwhelmingly clear description.

Keywords

Gibbs Interfacial Thermodynamics; Grad Potential; Interfacial Tension; Work of Nucleus Formation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Gibbs, J.W. (1993) The scientific papers of J. Willard Gibbs, thermodynamics. Ox Bow, Woodbridge.
[2]	Tolman, R.C. (1948) Consideration of the gibbs theory of surface tension. The Journal of Chemical Physics, 16, 758-774. doi:10.1063/1.1746994
[3]	Hill, T.L. (1951) On Gibbs Theory of Surface Tension, The Journal of Chemical Physics, 19, 1203-1203. doi:10.1063/1.1748502
[4]	Buff, F.P. (1951) The spherical interface. I. Thermodynamics. The Journal of Chemical Physics, 19, 1591-1594. doi:10.1063/1.1748127
[5]	Hill, T.L. (1952) Statistical thermodynamics of the transition region between two Phases. I. Thermodynamics and quasi-thermodynamics. Journal of Physical Chemistry, 56, 525-531. doi:10.1021/j150496a027
[6]	Kondo, S. (1955) A statistical-mechanical thory of surface tension of curved surface Layer I. Journal of the Physical Society of Japan, 10, 381-386. doi:10.1143/JPSJ.10.381
[7]	Kondo, S. (1956) Thermodynamical fundamental equation for spherical interface. The Journal of Chemical Physics, 25, 662-669. doi:10.1063/1.1743024
[8]	Plesner, I.W. (1964) Statistical thermodynamics of spherical droplets. The Journal of Chemical Physics, 40, 15101517. doi:10.1063/1.1725355
[9]	Nishioka, K. (1977) Thermodynamics of a liquid microcluster. Physical Review A, 16, 2143-2152. doi:10.1103/PhysRevA.16.2143
[10]	Wilemski, G. (1984) Composition of the critical nucleus in multicomponent vapor nucleation. Journal of Chemical Physics, 80, 1370-1372. doi:10.1063/1.446822
[11]	Nishioka, K. (1987) Thermodynamics formalism for a liquid microcluster in vapor. Physical Review A, 36, 48454851. doi:10.1103/PhysRevA.36.4845
[12]	Voorhees, P.W. and Johnson, W.C. (1989) The thermodynamics of a coherent interface. The Journal of Chemical Physics, 90, 2793-2801. doi:10.1063/1.455928
[13]	Nishioka, K. and Kusaka, I. (1992) Thermodynamics formulas of liquid phase nucleation form vapor in multicomponent systems. The Journal of Chemical Physics, 96, 5370-5376. doi:10.1063/1.462721
[14]	Oxtoby, D.W. and Kashchiev, D. (1994) A general relation between the nucleation work and the size of the nucleus in multicomponent nucleation. The Journal of Chemical Physics, 100, 7665-7671. doi:10.1063/1.466859
[15]	Debeneditti, P.G. and Reiss, H. (1998) Reversible work of formation of an embryo of a new phase within a uniform macroscopic mother phase. The Journal of Chemical Physics, 108, 5498-5505. doi:10.1063/1.475938
[16]	Laaksonen, A., McGraw, R. and Vehkamaki, H. (1999) Liquid-drop formalism and free-energy surface in binary homogeneous nucleation theory. The Journal of Chemical Physics, 111, 2019-2027. doi:10.1063/1.479470
[17]	Kashchiev, D. (2003) Thermodynamically consistent description of the work to form a nucleus of any size. The Journal of Chemical Physics, 118, 1837-1851. doi:10.1063/1.1531614
[18]	Schmelzer, J.W.P., Baidakov, V.G. and Boltachev, G.S. (2003) Kinetics of boiling in binary liquid-gas solutions: Comparison of different approaches. The Journal of Chemical Physics, 119, 6166-6183. doi:10.1063/1.1602066
[19]	Kashchiev, D. (2004) Multicomponent nucleation: Thermodynamically consistent description of the nucleation work. The Journal of Chemical Physics, 120, 3749-3758. doi:10.1063/1.1643711
[20]	Rusanov, A.I., Shchekin, A.K. and Tatyanenko, D.V. (2009) Grand potential in thermodynamics of solid bodies and surfaces. The Journal of Chemical Physics, 131, 161104. doi:10.1063/1.3254324
[21]	Corti, D.S., Kerr, K.J. and Torabi, K. (2011) On the interfacial thermodynamics of nanoscale droplets and bubbles. The Journal of Chemical Physics, 135, 024701. doi:10.1063/1.3609274
[22]	Tolman, R.C. (1949) The effect of droplet size on surface tension. The Journal of Chemical Physics, 17, 333-337. doi:10.1063/1.1747247
[23]	Koenig, F.O. (1950) On the thermodynamic relation between surface tension and curvature. The Journal of Chemical Physics, 18, 449-459. doi:10.1063/1.1747660
[24]	Buff, F.P. and Kirkood, J.G. (1950) Remarks on the surface tension of small droplets. The Journal of Chemical Physics, 18, 991-992. doi:10.1063/1.1747829
[25]	Buff, F.P. (1955) Spherical interface. II. Molecular theory. The Journal of Chemical Physics, 23, 419-427. doi:10.1063/1.1742005
[26]	Bogdan, A. (1997) Thermodynamics of the curvature effect on ice surface tension and nucleation theory. The Journal of Chemical Physics, 106, 1921-1929. doi:10.1063/1.473329
[27]	McGraw, R. and Laaksonen, A. (1997) Interfacial curvature free energy, the Kelvin relation, and vapor-liquid nucleation rate. The Journal of Chemical Physics, 106, 5284-5287. doi:10.1063/1.473527
[28]	Koga, K., Zeng, X.C. and Shchekin, A.K. (1998) Validity of Tolman’s equation: How large should a droplet be? The Journal of Chemical Physics, 109, 4063-4070. doi:10.1063/1.477006
[29]	Bartell, L.S. (2001) Tolman’s δ, surface curvature, compressibility effects, and free energy of drops. Journal of Physical Chemistry B, 105, 11615-11618. doi:10.1021/jp011028f
[30]	Blokhuis, E.M. and Kuipers, J. (2006) Thermodynamic expressions for the Tolman length. The Journal of Chemical Physics, 124, 074701. doi:10.1063/1.2167642
[31]	Tr?ster, A., Ottel, M., Block, B., Virnau, J.P. and Binder, K. (2012) Numerical approaches to determine the inter facial tension of curved interface from free energy calculations. The Journal of Chemical Physics, 136, 064709. doi:10.1063/1.3685221
[32]	Nishioka, K., Tomino, H., Kusaka, I. and Takai, T. (1989) Curvature dependence of the interfacial tension in binary nucleation. Physical Review A, 39, 772-782. doi:10.1103/PhysRevA.39.772
[33]	Yang, A.J.-M. (1983) Free energy for the heterogeneous systems with spherical interfaces. The Journal of Chemical Physics, 79, 6289-6293. doi:10.1063/1.445734
[34]	Mutaftschiev, B. (1993) Nucleation theory. In: Hurle, D.T.J., Ed., Handbook of Crystal Growth, Part 1a, Chapter 4, Elsevier, Amsterdam.
[35]	Saito, Y. (1996) Statistical physics of crystal growth. World Scientific, Singapore.
[36]	Markov, I.V. (2003) Crystal growth for beginners: Fundamentals of nucleation, crystal growth and epitaxy. 2nd Edition, World Scientific, Singapore. doi:10.1142/9789812796899
[37]	Doyle, G.J. (1961) Self-nucleation in the sulfuric acidwater system. The Journal of Chemical Physics, 35, 795799. doi:10.1063/1.1701218
[38]	Kashchiev, D. (1982) On the relation between nucleation work, nucleus size, and nucleation rate. The Journal of Chemical Physics, 76, 5098-5102. doi:10.1063/1.442808
[39]	Laaksonen, A., Kulmala, M. and Wanger, P.E. (1993) On the cluster compositions in the classical binary nucleation theory. The Journal of Chemical Physics, 99, 68326835. doi:10.1063/1.465827
[40]	Vilsanen, Y. and Strey, R. (1994) Homogeneous nucleation rates for n-butanol. The Journal of Chemical Physics, 101, 7835-7843. doi:10.1063/1.468208
[41]	Schmelzer, J.W.P., Schmelzer Jr., J. and Gutzow, I.S. (2000) Reconciling Gibbs and van der Waals: A new approach to nucleation theory. The Journal of Chemical Physics, 112, 3820-3831. doi:10.1063/1.481595
[42]	Bowels, R.K., Reguera, D., Djikaev, Y. and Reiss, H. (2001) A theorem for inhomogeneous systems: The generalization of the nucleation theorem. The Journal of Chemical Physics, 115, 1853-1866; errata (2002), 116, 2330-2330.
[43]	Auer, S. and Frenkel, D. (2001) Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature, 409, 1020-1023. doi:10.1038/35059035
[44]	Gasser, S., Weeks, E.R., Schofield, A., Pusey, P.N. and Weitz, D.A. (2001) Real-space imaging of nucleation and growth in colloidal crystallization. Science, 292, 258-262. doi:10.1126/science.1058457
[45]	Kawasaki, T. and Tanaka, H. (2010) Formation of a crystal nucleation from liquid. Proceedings of the National Academy of Sciences of the United States of America, 107, 14036-14041; correction (2011) 108, 6335-6336.
[46]	Nishioka, K., Mori, A., Takano, K.J., Kaishita, Y. and Narimatsu, S. (1999) Pressure-dependence of the interfacial tension of a critical nucleus in the binary-ideal solution. Journal of Crystal Growth, 200, 592-598. doi:10.1016/S0022-0248(98)01391-8
[47]	Callen, H.B. (1985) Thermodynamics and an introduction to thermostatistics. 2nd Edition, Wiley, New York.
[48]	Landau, L.D. and Lifshitz, E.M. (1989) Statistical physics. 3rd Edition, Pt. 1, Pergamon, Oxford.
[49]	Yang, A.J.-M. (1985) The thermodynamical stability of the heterogeneous system with a spherical interface. The Journal of Chemical Physics, 82, 2082-2085. doi:10.1063/1.448344
[50]	Barrett, J. (1999) First-order correction to classical nucleation theory: A density functional approach. The Journal of Chemical Physics, 111, 5938-5946. doi:10.1063/1.479889
[51]	Hurby, J., Labetski, D.G. and van Dongen, M.E.H. (2007) Gradient theory computation of the radius-dependent surface tension and nucleation rate for n-nonane clusters. The Journal of Chemical Physics, 127, 164720. doi:10.1063/1.2799515
[52]	Mori, A. and Suzuki, Y. (2013) Vanishing linear term in chemical potential difference in volume term of work of critical nucleus formation for phase transition without volume change. Journal of Crystal Growth, 375, 16-19.