Lattice density functional theory

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.

## Description

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.

Lattice density functional theory has several advantages over more complicated free volume techniques such as Perturbation theory and the statistical associating fluid theory, including mathematical simplicity and ease of incorporating complex boundary conditions. Although this approach is known to give only qualitative information about the thermodynamic behavior of a system, it provides important insights about the mechanisms of various complex phenomena such as phase transition,[5][6][7] aggregation,[8] configurational distribution,[9] surface-adsorption,[10][11] self-assembly, crystallization, as well as steady state diffusion.

## References

1. ↑ Hill TL. Statistical Mechanics, Principles and Selected Applications. New York: Dover Publications; 1987.
2. ↑ Ising, Ernst (1925). "Beitrag zur Theorie des Ferromagnetismus" (in de). Zeitschrift für Physik (Springer Science and Business Media LLC) 31 (1): 253–258. doi:10.1007/bf02980577. ISSN 0044-3328. Bibcode: 1925ZPhy...31..253I.
3. ↑ Onsager, Lars (1944-02-01). "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition". Physical Review (American Physical Society (APS)) 65 (3–4): 117–149. doi:10.1103/physrev.65.117. ISSN 0031-899X. Bibcode: 1944PhRv...65..117O.
4. ↑ Hill TL. An introduction to statistical thermodynamics, New York, Dover Publications (1986).
5. ↑ Aranovich, G.L.; Donohue, M.D. (1997). "New approximate solutions to the Ising problem in three dimensions". Physica A: Statistical Mechanics and Its Applications (Elsevier BV) 242 (3–4): 409–422. doi:10.1016/s0378-4371(97)00258-6. ISSN 0378-4371. Bibcode: 1997PhyA..242..409A.
6. ↑ Aranovich, G. L.; Donohue, M. D. (1999-11-01). "Phase loops in density-functional-theory calculations of adsorption in nanoscale pores". Physical Review E (American Physical Society (APS)) 60 (5): 5552–5560. doi:10.1103/physreve.60.5552. ISSN 1063-651X. PMID 11970430. Bibcode: 1999PhRvE..60.5552A.
7. ↑ 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 (AIP Publishing) 124 (13): 134502. doi:10.1063/1.2185090. ISSN 0021-9606. PMID 16613456. Bibcode: 2006JChPh.124m4502C.
8. ↑ 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 (Royal Society of Chemistry (RSC)) 10 (38): 5840–7. doi:10.1039/b805241g. ISSN 1463-9076. PMID 18818836. Bibcode: 2008PCCP...10.5840C.
9. ↑ 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 (AIP Publishing) 127 (13): 134903. doi:10.1063/1.2780159. ISSN 0021-9606. PMID 17919050. Bibcode: 2007JChPh.127m4903C.
10. ↑ 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 (Elsevier BV) 211 (1): 61–80. doi:10.1006/jcis.1998.5971. ISSN 0021-9797. PMID 9929436. Bibcode: 1999JCIS..211...61H.
11. ↑ Wu, D.-W.; Aranovich, G.L.; Donohue, M.D. (1999). "Adsorption of Dimers at Surfaces". Journal of Colloid and Interface Science (Elsevier BV) 212 (2): 301–309. doi:10.1006/jcis.1998.6069. ISSN 0021-9797. PMID 10092359. Bibcode: 1999JCIS..212..301W.

* B. Bakhti, "Development of lattice density functionals and applications to structure formation in condensed matter systems". PhD thesis, Universität Osnabrück, Germany.

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