Analytic Calculation of the Critical Temperature and Estimation of the Critical Region Size for a Fluid Model
Keywords:cell fluid model, Morse interaction potential, grand partition function, recurrence relations, critical temperature, critical region
An analytic procedure for calculating the critical temperature and estimating the size of the critical region for a cell fluid model is developed. Our numerical calculations are illustrated by the case of the Morse potential parameters characterizing the alkali metals (sodium and potassium). The critical temperatures found for liquid sodium and potassium as solutions of the resulting quadratic equation agree with experimental data. The expression for the relative temperature determining the critical region size is obtained proceeding from the condition for the critical regime existence. In the cases of sodium and potassium, the value of this temperature is of the order of a few hundredths.
J.M.H. Levelt Sengers, G. Morrison, R.F. Chang. Critical behavior in fluids and fluid mixtures. Fluid Phase Equilib. 14, 19 (1983).
S. Pittois, B. Van Roie, C. Glorieux, J. Thoen. Thermal conductivity, thermal effusivity, and specific heat capacity near the lower critical point of the binary liquid mixture n-butoxyethanol-water. J.Chem. Phys. 121, 1866 (2004).
Y.B. Melnichenko, G.D. Wignall, D.R. Cole, H. Frielinghaus, L.A. Bulavin. Liquid-gas critical phenomena under confinement: small-angle neutron scattering studies of CO2 in aerogel. J. Mol. Liq. 120, 7 (2005).
A.N. Vasil'ev, A.V. Chalyi. Critical parameters and pair correlations in confined multicomponent liquids. Condens. Matter Phys. 9, 65 (2006).
I.R. Yukhnovskii. Phase transitions in a vicinity of the vapor-liquid critical point. Ukr. J. Phys. Reviews 10, 33 (2015) [in Ukrainian].
I. Tsivintzelis, G.M. Kontogeorgis. Modelling phase equilibria for acid gas mixtures using the CPA equation of state. Part VI. Multicomponent mixtures with glycols relevant to oil and gas and to liquid or supercritical CO2 transport applications. J. Chem. Thermodyn. 93, 305 (2016).
P. de Castro, P. Sollich. Critical phase behavior in multicomponent fluid mixtures: Complete scaling analysis. J. Chem. Phys. 149, 204902 (2018).
T.J. Yoon, Y.-W. Lee. Current theoretical opinions and perspectives on the fundamental description of supercritical fluids. J. Supercrit. Fluids 134, 21 (2018).
L.F. Vega. Perspectives on molecular modeling of supercritical fluids: From equations of state to molecular simulations. Recent advances, remaining challenges and opportunities. J. Supercrit. Fluids 134, 41 (2018).
Y.X. Pang, M. Yew, Y. Yan et al. Application of supercritical fluid in the synthesis of graphene materials: a review. J. Nanopart. Res. 23, 204 (2021).
I.R. Graf, B.B. Machta. Thermodynamic stability and critical points in multicomponent mixtures with structured interactions. Phys. Rev. Res. 4, 033144 (2022).
I.V. Pylyuk, M.P. Kozlovskii, O.A. Dobush, M.V. Dufanets. Morse fluids in the immediate vicinity of the critical point: Calculation of thermodynamic coefficients. J. Mol. Liq. 385, 122322 (2023).
H. Okumura, F. Yonezawa. Liquid-vapor coexistence curves of several interatomic model potentials. J. Chem. Phys. 113, 9162 (2000).
J.K. Singh, J. Adhikari, S.K. Kwak. Vapor-liquid phase coexistence curves for Morse fluids. Fluid Phase Equilib. 248, 1 (2006).
E.M. Apfelbaum. The calculation of vapor-liquid coexistence curve of Morse fluid: Application to iron. J. Chem. Phys. 134, 194506 (2011).
A. Martinez-Valencia, M. Gonz'alez-Melchor, P. Orea, J. L'opez-Lemus. Liquid-vapour interface varying the softness and range of the interaction potential. Mol. Simul. 39, 64 (2013).
M.P. Kozlovskii, I.V. Pylyuk, O.A. Dobush. The equation of state of a cell fluid model in the supercritical region. Condens. Matter Phys. 21, 43502 (2018).
I.V. Pylyuk. Fluid critical behavior at liquid-gas phase transition: Analytic method for microscopic description. J. Mol. Liq. 310, 112933 (2020).
A.L. Rebenko. Cell gas model of classical statistical systems. Rev. Math. Phys. 25, 1330006 (2013).
V.A. Boluh, A.L. Rebenko. Cell gas free energy as an approximation of the continuous model. J. Mod. Phys. 6, 168 (2015).
I.V. Pylyuk, O.A. Dobush. Equation of state of a cell fluid model with allowance for Gaussian fluctuations of the order parameter. Ukr. J. Phys. 65, 1080 (2020).
I.V. Pylyuk, M.P. Kozlovskii. First-order phase transition in the framework of the cell fluid model: Regions of chemical potential variation and the corresponding densities. Ukr. J. Phys. 67, 54 (2022).
I.R. Yukhnovskii. Phase Transitions of the Second Order. Collective Variables Method (World Scientific, 1987) [ISBN-10: 9971500876, ISBN-13: 9789971500870].
I.R. Yukhnovskii, M.P. Kozlovskii, I.V. Pylyuk. Microscopic Theory of Phase Transitions in the Three-Dimensional Systems (Eurosvit, 2001) [in Ukrainian] [ISBN: 966-7343-26-X].
I.R. Yukhnovskii, M.P. Kozlovskii, I.V. Pylyuk. Thermodynamics of three-dimensional Ising-like systems in the higher non-Gaussian approximation: Calculational method and dependence on microscopic parameters. Phys. Rev. B 66, 134410 (2002).
M.P. Kozlovskii, I.V. Pylyuk, O.O Prytula. Microscopic description of the critical behavior of three-dimensional Ising-like systems in an external field. Phys. Rev. B 73, 174406 (2006).
M.P. Kozlovskii, I.V. Pylyuk, O.O Prytula. Free energy and equation of state of Ising-like magnet near the critical point. Nucl. Phys. B 753, 242 (2006).
F. Hensel. Critical behaviour of metallic liquids. J. Phys.: Condens. Matter 2, SA33 (1990).
L.D. Landau, E.M. Lifshitz. Statistical Physics, Part 1 (Nauka, 1976) (in Russian) [ISBN: 5922100548].
M.E. Lines, A.M. Glass. Principles and Application of Ferroelectrics and Related Materials (Clarendon Press, 1977) [ISBN-10: 0198512864, ISBN-13: 9780198512868].
K.G. Wilson, J. Kogut. The renormalization group and the ϵ expansion. Phys. Rep. 12, 75 (1974).
C.A. Eckert, B.L. Knutson, P.G. Debenedetti. Supercritical fluids as solvents for chemical and materials processing. Nature 383, 313 (1996).
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