Statistical associating fluid theory

Short description: Chemical theory

Statistical associating fluid theory (SAFT) [1][2] is a chemical theory, based on perturbation theory, that uses statistical thermodynamics to explain how complex fluids and fluid mixtures form associations through hydrogen bonds.[3] Widely used in industry and academia, it has become a standard approach for describing complex mixtures.[4][5][6][7] Since it was first proposed in 1990, SAFT has been used in a large number of molecular-based equation of state[8][9] models for describing the Helmholtz energy contribution due to association.

## Contents

* 1 Overview
* 2 Development
* 3 Variations
* 4 Association schemes
* 5 References

## Overview

SAFT is a Helmholtz energy term that can be used in equations of state that describe the thermodynamic and phase equilibrium properties of pure fluids and fluid mixtures. SAFT was developed using statistical mechanics. SAFT models the Helmholtz free energy contribution due to association, i.e. hydrogen bonding.[10] SAFT can be used in combination with other Helmholtz free energy terms. Other Helmholtz energy contributions consider for example Lennard-Jones interactions, covalent chain-forming bonds, and association (interactions between segments caused by, for example, hydrogen bonding).[2] SAFT has been applied to a wide range of fluids, including supercritical fluids, polymers, liquid crystals, electrolytes, surfactant solutions, and refrigerants.[5]

## Development

SAFT evolved from thermodynamic theories, including perturbation theories developed in the 1960s, 1970s, and 1980s by John Barker and Douglas Henderson, Keith Gubbins and Chris Gray, and, in particular, Michael Wertheim's first-order, thermodynamic perturbation theory (TPT1) outlined in a series of papers in the 1980s.[2][11]

The SAFT equation of state was developed using statistical mechanical methods (in particular the perturbation theory of Wertheim[12]) to describe the interactions between molecules in a system.[1][13][14] The idea of a SAFT equation of state was first proposed by Walter G. Chapman and by Chapman et al. in 1988 and 1989.[1][13][14] Many different versions of the SAFT models have been proposed, but all use the same chain and association terms derived by Chapman et al.[2][13][15] One of the first SAFT papers (1990) titled "New reference equation of state for associating liquids" by Walter G. Chapman, Keith Gubbins, George Jackson, and Maciej Radosz,[2] was recognized in 2007 by Industrial and Engineering Chemistry Research as one of the most highly cited papers of the previous three decades.[16] SAFT is one of the first theories to accurately describe (in comparison with molecular simulation) the effects on fluid properties of molecular size and shape in addition to association between molecules.[1][2][13][14]

## Variations

Many variations of SAFT have been developed since the 1990s, including HR-SAFT (Huang-Radosz SAFT),[6] PC-SAFT (perturbed chain SAFT),[17][18] Polar SAFT,[19] PCP-SAFT (PC-polar-SAFT),[20][21][22] soft-SAFT,[23] polar soft-SAFT,[24] SAFT-VR (variable range),[25] SAFT VR-Mie.[26] Also, the SAFT term was used in combination with cubic equations of state for describing the dispersive-repulsive interactions, for example in the Cubic-Plus-Association (CPA) equation of state model[27] and the SAFT + cubic model [28] and non-random-lattice (NLF) models based on lattice field theory.[3]

## Association schemes

The association term in SAFT (i.e. hydrogen bonding) is handled by assigning to each molecule a set of "association sites" that are permitted to form bonds with association sites on other molecules.[29] The strength of a bond between two sites is characterized by a bonding energy. In order to characterise different kinds of molecules and functional groups, a molecule or group is assigned one of many "association schemes",[29] which are summarised in the (non-exhaustive) table below. These schemes are used to differentiate between the number of association sites a molecule has, and which other sites each site is permitted to bond to. In this system, each association scheme consists of a number and a letter. The number indicates how many association sites the molecule (or group) has, while the letters are used to differentiate between different schemes based on the permitted bonds. In the "A" schemes, all association sites can form bonds with all other sites, while in the "B" and "C" schemes, only certain sites are permitted to bond to certain other sites.

While the appropriate association scheme for a given molecule or group can in principle be deduced from the electronic structure, it is common to treat the association scheme heuristically, and to determine the best scheme by fitting to experimental data.[30][31]

Scheme  | No. of sites  | Permitted bonds  | Example

|
|
|

1A  | 1  | A-A  | Acid, tertiary amine
2A  | 2  | A-A, A-B, B-B  |
2B  | 2  | A-B  | Secondary amine
2C  | 2  | A-A, A-B  |
3A  | 3  | A-A, A-B, A-C,

B-B, B-C, C-C

3B  | 3  | A-C, B-C  | Primary amine
4A  | 4  | A-A, A-B, A-C, A-D,

B-B, B-C, B-D, C-C, C-D, D-D

4B  | 4  | A-D, B-D, C-D  | Ammonia
4C  | 4  | A-C, A-D, B-C, B-D  | Water

## References

1. ↑ 1.0 1.1 1.2 1.3 Chapman, W.G.; Gubbins, K.E.; Jackson, G.; Radosz, M. (December 1989). "SAFT: Equation-of-state solution model for associating fluids" (in en). Fluid Phase Equilibria 52: 31–38. doi:10.1016/0378-3812(89)80308-5. Bibcode: 1989FlPEq..52...31C. https://linkinghub.elsevier.com/retrieve/pii/0378381289803085.
2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 Chapman, Walter G.; Gubbins, Keith E.; Jackson, George; Radosz, Maciej (August 1990). "New reference equation of state for associating liquids". Industrial & Engineering Chemistry Research 29 (8): 1709–1721. doi:10.1021/IE00104A021. ISSN 0888-5885.
3. ↑ 3.0 3.1 Kontogeorgis, Georgios M.; Folas, Georgios K. (2010). "The Statistical Associating Fluid Theory (SAFT)". Thermodynamic Models for Industrial Applications. John Wiley & Sons, Ltd. pp. 221–259. doi:10.1002/9780470747537.ch8. ISBN 9780470747537.
4. ↑ Müller, Erich; Gubbins, Keith (2001). "Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches". Industrial & Engineering Chemistry Research 40 (10): 2193–2211. doi:10.1021/ie000773w. https://pubs.acs.org/doi/10.1021/ie000773w.
5. ↑ 5.0 5.1 Economou, Ioannis G. (4 October 2001). "Statistical Associating Fluid Theory: A Successful Model for the Calculation of Thermodynamic and Phase Equilibrium Properties of Complex Fluid Mixtures". Industrial & Engineering Chemistry Research 41 (5): 953–962. doi:10.1021/ie0102201. ISSN 0888-5885.
6. ↑ 6.0 6.1 Huang, Stanley H.; Radosz, Maciej (November 1990). "Equation of state for small, large, polydisperse, and associating molecules". Industrial & Engineering Chemistry Research 29 (11): 2284–2294. doi:10.1021/ie00107a014. ISSN 0888-5885.
7. ↑ Tan, Sugata P.; Adidharma, Hertanto; Radosz, Maciej (2008-11-05). "Recent Advances and Applications of Statistical Associating Fluid Theory" (in en). Industrial & Engineering Chemistry Research 47 (21): 8063–8082. doi:10.1021/ie8008764. ISSN 0888-5885. https://pubs.acs.org/doi/10.1021/ie8008764.
8. ↑ Nezbeda, Ivo (2020-09-29). "On Molecular-Based Equations of State: Perturbation Theories, Simple Models, and SAFT Modeling". Frontiers in Physics 8: 287. doi:10.3389/fphy.2020.00287. ISSN 2296-424X. Bibcode: 2020FrP.....8..287N.
9. ↑ Gubbins, Keith E. (2016-05-25). "Perturbation theories of the thermodynamics of polar and associating liquids: A historical perspective" (in en). Fluid Phase Equilibria. Special Issue: SAFT 2015 416: 3–17. doi:10.1016/j.fluid.2015.12.043. ISSN 0378-3812. Bibcode: 2016FlPEq.416....3G.
10. ↑ Dufal, Simon; Lafitte, Thomas; Haslam, Andrew J.; Galindo, Amparo; Clark, Gary N.I.; Vega, Carlos; Jackson, George (19 May 2015). "The A in SAFT: developing the contribution of association to the Helmholtz free energy within a Wertheim TPT1 treatment of generic Mie fluids". Molecular Physics 113 (9–10): 948–984. doi:10.1080/00268976.2015.1029027. ISSN 0026-8976. Bibcode: 2015MolPh.113..948D.
11. ↑ Jiang, Shaoyi; Hall, Carol (24 October 2017). "Preface to the Tribute to Keith E. Gubbins, Pioneer in the Theory of Liquids Special Issue". Langmuir 33 (42): 11095–11101. doi:10.1021/acs.langmuir.7b03390. ISSN 0743-7463. PMID 29061054.
12. ↑ Wertheim, M. S. (April 1984). "Fluids with highly directional attractive forces. I. Statistical thermodynamics". Journal of Statistical Physics 35 (1–2): 19–34. doi:10.1007/bf01017362. ISSN 0022-4715. Bibcode: 1984JSP....35...19W. http://dx.doi.org/10.1007/bf01017362.
13. ↑ 13.0 13.1 13.2 13.3 Chapman, Walter G. (1988). "Theory and Simulation of Associating Liquid Mixtures" (in en). Doctoral Dissertation, Cornell University.
14. ↑ 14.0 14.1 14.2 Chapman, Walter G.; Jackson, G.; Gubbins, K.E. (11 July 1988). "Phase equilibria of associating fluids: Chain molecules with multiple bonding sites" (in en). Molecular Physics 65: 1057–1079. doi:10.1080/00268978800101601.
15. ↑ Gil-Villegas, Alejandro; Galindo, Amparo; Whitehead, Paul J.; Mills, Stuart J.; Jackson, George; Burgess, Andrew N. (1997). "Statistical associating fluid theory for chain molecules with attractive potentials of variable range". The Journal of Chemical Physics 106 (10): 4168–4186. doi:10.1063/1.473101. Bibcode: 1997JChPh.106.4168G.
16. ↑ "One of the most cited pieces of research gets its due". Imperial News. 12 October 2007. https://www.imperial.ac.uk/news/19654/one-most-cited-pieces-research-gets/.
17. ↑ Gross, J.; Sadowski, G. (2004). "Perturbed-Chain-SAFT". Supercritical Fluids as Solvents and Reaction Media. Elsevier. pp. 295–322. doi:10.1016/B978-044451574-2/50012-2. ISBN 9780444515742.
18. ↑ Gross, Joachim; Sadowski, Gabriele (2002-10-01). "Application of the Perturbed-Chain SAFT Equation of State to Associating Systems" (in en). Industrial & Engineering Chemistry Research 41 (22): 5510–5515. doi:10.1021/ie010954d. ISSN 0888-5885. https://pubs.acs.org/doi/10.1021/ie010954d.
19. ↑ Jog, Prasana; Sauer, Sharon G.; Ghosh, Auleen; Chapman, Walter G. (September 2001). "Application of Dipolar Chain Theory to the Phase Behavior of Polar Fluids and Mixtures" (in en). Industrial & Engineering Chemistry Research 40 (21): 4641–4648. doi:10.1021/ie010264+. https://doi.org/10.1021/ie010264+.
20. ↑ Gross, Joachim (September 2005). "An equation-of-state contribution for polar components: Quadrupolar molecules" (in en). AIChE Journal 51 (9): 2556–2568. doi:10.1002/aic.10502. ISSN 0001-1541. Bibcode: 2005AIChE..51.2556G. https://onlinelibrary.wiley.com/doi/10.1002/aic.10502.
21. ↑ Gross, Joachim; Vrabec, Jadran (March 2006). "An equation-of-state contribution for polar components: Dipolar molecules" (in en). AIChE Journal 52 (3): 1194–1204. doi:10.1002/aic.10683. ISSN 0001-1541. Bibcode: 2006AIChE..52.1194G. https://onlinelibrary.wiley.com/doi/10.1002/aic.10683.
22. ↑ Vrabec, Jadran; Gross, Joachim (2008-01-01). "Vapor−Liquid Equilibria Simulation and an Equation of State Contribution for Dipole−Quadrupole Interactions" (in en). The Journal of Physical Chemistry B 112 (1): 51–60. doi:10.1021/jp072619u. ISSN 1520-6106. PMID 18072758. https://pubs.acs.org/doi/10.1021/jp072619u.
23. ↑ FELIPE J. BLAS and LOURDES F. VEGA (September 1997). "Thermodynamic behaviour of homonuclear and heteronuclear Lennard-Jones chains with association sites from simulation and theory" (in en). Molecular Physics 92 (1): 135–150. doi:10.1080/002689797170707. ISSN 0026-8976. Bibcode: 1997MolPh..92..135F. https://www.tandfonline.com/doi/full/10.1080/002689797170707.
24. ↑ Alkhatib, Ismail I. I.; Pereira, Luís M. C.; Torne, Jordi; Vega, Lourdes F. (2020). "Polar soft-SAFT: theory and comparison with molecular simulations and experimental data of pure polar fluids". Physical Chemistry Chemical Physics 22 (23): 13171–13191. doi:10.1039/d0cp00846j. ISSN 1463-9076. PMID 32497165. Bibcode: 2020PCCP...2213171A.
25. ↑ McCabe, Clare; Jackson, George (1999). "SAFT-VR modelling of the phase equilibrium of long-chain n-alkanes". Physical Chemistry Chemical Physics 1 (9): 2057–2064. doi:10.1039/A808085B. ISSN 1463-9076. Bibcode: 1999PCCP....1.2057M.
26. ↑ Lafitte, Thomas; Apostolakou, Anastasia; Avendaño, Carlos; Galindo, Amparo; Adjiman, Claire S.; Müller, Erich A.; Jackson, George (2013-10-21). "Accurate statistical associating fluid theory for chain molecules formed from Mie segments" (in en). The Journal of Chemical Physics 139 (15): 154504. doi:10.1063/1.4819786. ISSN 0021-9606. PMID 24160524. Bibcode: 2013JChPh.139o4504L.
27. ↑ Kontogeorgis, Georgios M.; Michelsen, Michael L.; Folas, Georgios K.; Derawi, Samer; von Solms, Nicolas et al. (1 June 2006). "Ten Years with the CPA (Cubic-Plus-Association) Equation of State. Part 1. Pure Compounds and Self-Associating Systems". Industrial & Engineering Chemistry Research 45 (14): 4855–4868. doi:10.1021/ie051305v. ISSN 0888-5885.
28. ↑ Polishuk, Ilya (2011-12-21). "Implementation of SAFT + Cubic, PC-SAFT, and Soave–Benedict–Webb–Rubin Equations of State for Comprehensive Description of Thermodynamic Properties in Binary and Ternary Mixtures of CH 4, CO 2, and n -C 16 H 34" (in en). Industrial & Engineering Chemistry Research 50 (24): 14175–14185. doi:10.1021/ie201952n. ISSN 0888-5885. https://pubs.acs.org/doi/10.1021/ie201952n.
29. ↑ 29.0 29.1 Huang, Stanley H.; Radosz, Maciej (November 1990). "Equation of state for small, large, polydisperse, and associating molecules" (in en). Industrial & Engineering Chemistry Research 29 (11): 2284–2294. doi:10.1021/ie00107a014. ISSN 0888-5885. https://pubs.acs.org/doi/abs/10.1021/ie00107a014.
30. ↑ Aasen, Ailo; Jervell, Vegard G.; Hammer, Morten; Strøm, Bjørn A.; Skarsvåg, Hans L.; Wilhelmsen, Øivind (September 2024). "Bulk and interfacial thermodynamics of ammonia, water and their mixtures" (in en). Fluid Phase Equilibria 584. doi:10.1016/j.fluid.2024.114125. Bibcode: 2024FlPEq.58414125A. https://linkinghub.elsevier.com/retrieve/pii/S037838122400102X.
31. ↑ de Villiers, Adriaan J.; Schwarz, Cara E.; Burger, Andries J. (2011-07-20). "New Association Scheme for 1-Alcohols in Alcohol/Water Mixtures with sPC-SAFT: The 2C Association Scheme" (in en). Industrial & Engineering Chemistry Research 50 (14): 8711–8725. doi:10.1021/ie200521k. ISSN 0888-5885. https://pubs.acs.org/doi/10.1021/ie200521k.

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