Lattice density functional theory (LDFT) is a statistical theory used in physics and thermodynamics to model a variety of physical phenomena with simple lattice equations.
Descriptionedit
Lattice models with nearest-neighbor interactions have been used extensively to model a wide variety of systems and phenomena, including the lattice gas, binary liquid solutions, order-disorder phase transitions, ferromagnetism, and antiferromagnetism.[1] Most calculations of correlation functions for nonrandom configurations are based on statistical mechanical techniques, which lead to equations that usually need to be solved numerically.
In 1925, Ising[2] gave an exact solution to the one-dimensional (1D) lattice problem. In 1944 Onsager[3] was able to get an exact solution to a two-dimensional (2D) lattice problem at the critical density. However, to date, no three-dimensional (3D) problem has had a solution that is both complete and exact.[4] Over the last ten years, Aranovich and Donohue have developed lattice density functional theory (LDFT) based on a generalization of the Ono-Kondo equations to three-dimensions, and used the theory to model a variety of physical phenomena.
The theory starts by constructing an expression for free energy, A=U-TS, where internal energy U and entropy S can be calculated using mean field approximation. The grand potential is then constructed as Ω=A-μΦ, where μ is a Lagrange multiplier which equals to the chemical potential, and Φ is a constraint given by the lattice.
It is then possible to minimize the grand potential with respect to the local density, which results in a mean-field expression for local chemical potential. The theory is completed by specifying the chemical potential for a second (possibly bulk) phase. In an equilibrium process, μI=μII.
^Hill TL. Statistical Mechanics, Principles and Selected Applications. New York: Dover Publications; 1987.
^Ising, Ernst (1925). "Beitrag zur Theorie des Ferromagnetismus" [Report on the theory of ferromagnetism]. Zeitschrift für Physik (in German). 31 (1). Springer Science and Business Media LLC: 253–258. Bibcode:1925ZPhy...31..253I. doi:10.1007/bf02980577. ISSN 0044-3328. S2CID 122157319.
^Onsager, Lars (1944-02-01). "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition". Physical Review. 65 (3–4). American Physical Society (APS): 117–149. Bibcode:1944PhRv...65..117O. doi:10.1103/physrev.65.117. ISSN 0031-899X.
^Hill TL. An introduction to statistical thermodynamics, New York, Dover Publications (1986).
^Aranovich, G.L.; Donohue, M.D. (1997). "New approximate solutions to the Ising problem in three dimensions". Physica A: Statistical Mechanics and Its Applications. 242 (3–4). Elsevier BV: 409–422. Bibcode:1997PhyA..242..409A. doi:10.1016/s0378-4371(97)00258-6. ISSN 0378-4371.
^Aranovich, G. L.; Donohue, M. D. (1999-11-01). "Phase loops in density-functional-theory calculations of adsorption in nanoscale pores". Physical Review E. 60 (5). American Physical Society (APS): 5552–5560. Bibcode:1999PhRvE..60.5552A. doi:10.1103/physreve.60.5552. ISSN 1063-651X. PMID 11970430.
^Chen, Y.; Aranovich, G. L.; Donohue, M. D. (2006-04-07). "Thermodynamics of symmetric dimers: Lattice density functional theory predictions and simulations". The Journal of Chemical Physics. 124 (13). AIP Publishing: 134502. Bibcode:2006JChPh.124m4502C. doi:10.1063/1.2185090. ISSN 0021-9606. PMID 16613456.
^Chen, Y.; Wetzel, T. E.; Aranovich, G. L.; Donohue, M. D. (2008). "Configurational probabilities for monomers, dimers and trimers in fluids". Physical Chemistry Chemical Physics. 10 (38). Royal Society of Chemistry (RSC): 5840–7. Bibcode:2008PCCP...10.5840C. doi:10.1039/b805241g. ISSN 1463-9076. PMID 18818836.
^Chen, Y.; Aranovich, G. L.; Donohue, M. D. (2007-10-07). "Configurational probabilities for symmetric dimers on a lattice: An analytical approximation with exact limits at low and high densities". The Journal of Chemical Physics. 127 (13). AIP Publishing: 134903. Bibcode:2007JChPh.127m4903C. doi:10.1063/1.2780159. ISSN 0021-9606. PMID 17919050.
^Hocker, Thomas; Aranovich, Grigoriy L.; Donohue, Marc D. (1999). "Monolayer Adsorption for the Subcritical Lattice Gas and Partially Miscible Binary Mixtures". Journal of Colloid and Interface Science. 211 (1). Elsevier BV: 61–80. Bibcode:1999JCIS..211...61H. doi:10.1006/jcis.1998.5971. ISSN 0021-9797. PMID 9929436.
^Wu, D.-W.; Aranovich, G.L.; Donohue, M.D. (1999). "Adsorption of Dimers at Surfaces". Journal of Colloid and Interface Science. 212 (2). Elsevier BV: 301–309. Bibcode:1999JCIS..212..301W. doi:10.1006/jcis.1998.6069. ISSN 0021-9797. PMID 10092359.
B. Bakhti, "Development of lattice density functionals and applications to structure formation in condensed matter systems". PhD thesis, Universität Osnabrück, Germany.