statistical associating fluid theory
{{Short description|Chemical theory}}
Statistical associating fluid theory (SAFT) {{Cite journal |last1=Chapman |first1=W.G. |last2=Gubbins |first2=K.E. |last3=Jackson |first3=G. |last4=Radosz |first4=M. |date=December 1989 |title=SAFT: Equation-of-state solution model for associating fluids |url=https://linkinghub.elsevier.com/retrieve/pii/0378381289803085 |journal=Fluid Phase Equilibria |language=en |volume=52 |pages=31–38 |doi=10.1016/0378-3812(89)80308-5|s2cid=53310898 |url-access=subscription }} 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.{{cite book | title = Thermodynamic Models for Industrial Applications | chapter = The Statistical Associating Fluid Theory (SAFT) | pages = 221–259 | publisher = John Wiley & Sons, Ltd |last1 = Kontogeorgis | first1 = Georgios M. | last2 = Folas | first2 = Georgios K.| year = 2010 | doi = 10.1002/9780470747537.ch8 | isbn = 9780470747537 | url = }} Widely used in industry and academia, it has become a standard approach for describing complex mixtures.{{Cite journal |last1=Müller |first1=Erich |last2=Gubbins |first2=Keith |title=Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches |url=https://pubs.acs.org/doi/10.1021/ie000773w |journal=Industrial & Engineering Chemistry Research |year=2001 |volume=40 |issue=10 |pages=2193–2211|doi=10.1021/ie000773w |url-access=subscription }}{{cite journal | last1 = Economou | first1 = Ioannis G. | title = Statistical Associating Fluid Theory: A Successful Model for the Calculation of Thermodynamic and Phase Equilibrium Properties of Complex Fluid Mixtures | journal = Industrial & Engineering Chemistry Research | date = 4 October 2001 | volume = 41 | issue = 5 | pages = 953–962 | issn = 0888-5885 | eissn = 1520-5045 | doi = 10.1021/ie0102201 | pmid = | url = }}{{cite journal | last1 = Huang | first1 = Stanley H. | last2 = Radosz | first2 = Maciej | title = Equation of state for small, large, polydisperse, and associating molecules | journal = Industrial & Engineering Chemistry Research | date = November 1990 | volume = 29 | issue = 11 | pages = 2284–2294 | issn = 0888-5885 | eissn = 1520-5045 | doi = 10.1021/ie00107a014 | pmid = | url = }}{{Cite journal |last1=Tan |first1=Sugata P. |last2=Adidharma |first2=Hertanto |last3=Radosz |first3=Maciej |date=2008-11-05 |title=Recent Advances and Applications of Statistical Associating Fluid Theory |url=https://pubs.acs.org/doi/10.1021/ie8008764 |journal=Industrial & Engineering Chemistry Research |language=en |volume=47 |issue=21 |pages=8063–8082 |doi=10.1021/ie8008764 |issn=0888-5885|url-access=subscription }} Since it was first proposed in 1990, SAFT has been used in a large number of molecular-based equation of state{{Cite journal |last=Nezbeda |first=Ivo |date=2020-09-29 |title=On Molecular-Based Equations of State: Perturbation Theories, Simple Models, and SAFT Modeling |journal=Frontiers in Physics |volume=8 |pages=287 |doi=10.3389/fphy.2020.00287 |bibcode=2020FrP.....8..287N |issn=2296-424X|doi-access=free }}{{Cite journal |last=Gubbins |first=Keith E. |date=2016-05-25 |title=Perturbation theories of the thermodynamics of polar and associating liquids: A historical perspective |journal=Fluid Phase Equilibria |series=Special Issue: SAFT 2015 |language=en |volume=416 |pages=3–17 |doi=10.1016/j.fluid.2015.12.043 |issn=0378-3812|doi-access=free }} models for describing the Helmholtz energy contribution due to association.
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.{{cite journal | last1 = Dufal | first1 = Simon | last2 = Lafitte | first2 = Thomas | last3 = Haslam | first3 = Andrew J. | last4 = Galindo | first4 = Amparo | last5 = Clark | first5 = Gary N.I. | last6 = Vega | first6 = Carlos | last7 = Jackson | first7 = George | title = The A in SAFT: developing the contribution of association to the Helmholtz free energy within a Wertheim TPT1 treatment of generic Mie fluids | journal = Molecular Physics | date = 19 May 2015 | volume = 113 | issue = 9–10 | pages = 948–984 | issn = 0026-8976 | eissn = 1362-3028 | doi = 10.1080/00268976.2015.1029027 | pmid = | bibcode = 2015MolPh.113..948D | s2cid = 93686586 | url = | doi-access = free }} 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).{{cite journal | last1 = Chapman | first1 = Walter G. | last2 = Gubbins | first2 = Keith E. | last3 = Jackson | first3 = George | last4 = Radosz | first4 = Maciej | title = New reference equation of state for associating liquids | journal = Industrial & Engineering Chemistry Research | date = August 1990 | volume = 29 | issue = 8 | pages = 1709–1721 | issn = 0888-5885 | eissn = 1520-5045 | doi = 10.1021/IE00104A021 | pmid = | url = }} SAFT has been applied to a wide range of fluids, including supercritical fluids, polymers, liquid crystals, electrolytes, surfactant solutions, and refrigerants.
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.{{cite journal | last1 = Jiang | first1 = Shaoyi | last2 = Hall | first2 = Carol | title = Preface to the Tribute to Keith E. Gubbins, Pioneer in the Theory of Liquids Special Issue | journal = Langmuir | date = 24 October 2017 | volume = 33 | issue = 42 | pages = 11095–11101 | issn = 0743-7463 | eissn = 1520-5827 | doi = 10.1021/acs.langmuir.7b03390 | pmid = 29061054 | url = | doi-access = free }}
The SAFT equation of state was developed using statistical mechanical methods (in particular the perturbation theory of Wertheim{{Cite journal|last=Wertheim|first=M. S.|date=April 1984|title=Fluids with highly directional attractive forces. I. Statistical thermodynamics|url=http://dx.doi.org/10.1007/bf01017362|journal=Journal of Statistical Physics|volume=35| issue=1–2|pages=19–34|doi=10.1007/bf01017362|bibcode=1984JSP....35...19W|s2cid=121383911|issn=0022-4715|url-access=subscription}}) to describe the interactions between molecules in a system.{{cite journal|last1=Chapman|first1=Walter G.|date=1988| title=Theory and Simulation of Associating Liquid Mixtures|journal=Doctoral Dissertation, Cornell University|language=en}}{{cite journal|last1=Chapman|first1=Walter G.|last2=Jackson|first2=G.| last3=Gubbins| first3=K.E.| date=11 July 1988|title=Phase equilibria of associating fluids: Chain molecules with multiple bonding sites| journal=Molecular Physics|language=en|volume=65|pages=1057–1079|doi=10.1080/00268978800101601}} The idea of a SAFT equation of state was first proposed by Walter G. Chapman and by Chapman et al. in 1988 and 1989. Many different versions of the SAFT models have been proposed, but all use the same chain and association terms derived by Chapman et al.{{Cite journal | doi=10.1063/1.473101|title = Statistical associating fluid theory for chain molecules with attractive potentials of variable range| journal=The Journal of Chemical Physics| volume=106| issue=10| pages=4168–4186|year = 1997|last1 = Gil-Villegas|first1 = Alejandro| last2=Galindo| first2=Amparo| last3=Whitehead| first3=Paul J.| last4=Mills| first4=Stuart J.| last5=Jackson| first5=George| last6=Burgess| first6=Andrew N.|bibcode = 1997JChPh.106.4168G}} 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, was recognized in 2007 by Industrial and Engineering Chemistry Research as one of the most highly cited papers of the previous three decades.{{cite news |title=One of the most cited pieces of research gets its due |url=https://www.imperial.ac.uk/news/19654/one-most-cited-pieces-research-gets/ |newspaper=Imperial News |access-date=3 October 2022 |date=12 October 2007}} 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.
Variations
Many variations of SAFT have been developed since the 1990s, including HR-SAFT (Huang-Radosz SAFT), PC-SAFT (perturbed chain SAFT),{{cite book | title = Supercritical Fluids as Solvents and Reaction Media | last1 = Gross | first1 = J. | last2 = Sadowski | first2 = G. | chapter = Perturbed-Chain-SAFT | date = 2004 | pages = 295–322 | publisher = Elsevier | doi = 10.1016/B978-044451574-2/50012-2 | isbn = 9780444515742 | url = }}{{Cite journal |last1=Gross |first1=Joachim |last2=Sadowski |first2=Gabriele |date=2002-10-01 |title=Application of the Perturbed-Chain SAFT Equation of State to Associating Systems |url=https://pubs.acs.org/doi/10.1021/ie010954d |journal=Industrial & Engineering Chemistry Research |language=en |volume=41 |issue=22 |pages=5510–5515 |doi=10.1021/ie010954d |issn=0888-5885|url-access=subscription }} Polar SAFT,{{Cite journal |last1=Jog |first1=Prasana |last2=Sauer |first2=Sharon G.|last3=Ghosh |first3=Auleen | last4=Chapman | first4=Walter G. |date=September 2001 |title=Application of Dipolar Chain Theory to the Phase Behavior of Polar Fluids and Mixtures | url=https://doi.org/10.1021/ie010264+ | journal=Industrial & Engineering Chemistry Research|language=en |volume=40 |issue=21 |pages=4641–4648 |doi=10.1021/ie010264+|url-access=subscription }} PCP-SAFT (PC-polar-SAFT),{{Cite journal |last=Gross |first=Joachim |date=September 2005|title=An equation-of-state contribution for polar components: Quadrupolar molecules |url=https://onlinelibrary.wiley.com/doi/10.1002/aic.10502 |journal=AIChE Journal |language=en |volume=51 |issue=9 |pages=2556–2568 |doi=10.1002/aic.10502 |issn=0001-1541|url-access=subscription }}{{Cite journal |last1=Gross |first1=Joachim |last2=Vrabec |first2=Jadran |date=March 2006 |title=An equation-of-state contribution for polar components: Dipolar molecules |url=https://onlinelibrary.wiley.com/doi/10.1002/aic.10683 |journal=AIChE Journal |language=en |volume=52 |issue=3 |pages=1194–1204 |doi=10.1002/aic.10683 |issn=0001-1541|url-access=subscription }}{{Cite journal |last1=Vrabec |first1=Jadran |last2=Gross |first2=Joachim |date=2008-01-01 |title=Vapor−Liquid Equilibria Simulation and an Equation of State Contribution for Dipole−Quadrupole Interactions |url=https://pubs.acs.org/doi/10.1021/jp072619u |journal=The Journal of Physical Chemistry B |language=en |volume=112 |issue=1 |pages=51–60 |doi=10.1021/jp072619u |pmid=18072758 |arxiv=0904.4637 |s2cid=17830580 |issn=1520-6106}} soft-SAFT,{{Cite journal |last=FELIPE J. BLAS and LOURDES F. VEGA |date=September 1997|title=Thermodynamic behaviour of homonuclear and heteronuclear Lennard-Jones chains with association sites from simulation and theory |url=https://www.tandfonline.com/doi/full/10.1080/002689797170707 |journal=Molecular Physics |language=en |volume=92 |issue=1 |pages=135–150 |doi=10.1080/002689797170707 |bibcode=1997MolPh..92..135F |issn=0026-8976|url-access=subscription }} polar soft-SAFT,{{cite journal | last1 = Alkhatib | first1 = Ismail I. I. | last2 = Pereira | first2 = Luís M. C. | last3 = Torne | first3 = Jordi | last4 = Vega | first4 = Lourdes F. | title = Polar soft-SAFT: theory and comparison with molecular simulations and experimental data of pure polar fluids | journal = Physical Chemistry Chemical Physics | date = 2020 | volume = 22 | issue = 23 | pages = 13171–13191 | issn = 1463-9076 | eissn = 1463-9084 | doi = 10.1039/d0cp00846j | pmid = 32497165 | bibcode = 2020PCCP...2213171A | s2cid = 219330886 | url = }} SAFT-VR (variable range),{{cite journal | last1 = McCabe | first1 = Clare | last2 = Jackson | first2 = George | title = SAFT-VR modelling of the phase equilibrium of long-chain n-alkanes | journal = Physical Chemistry Chemical Physics | date = 1999 | volume = 1 | issue = 9 | pages = 2057–2064 | issn = 1463-9076 | eissn = 1463-9084 | doi = 10.1039/A808085B | pmid = | bibcode = 1999PCCP....1.2057M | url = }} SAFT VR-Mie.{{Cite journal |last1=Lafitte |first1=Thomas |last2=Apostolakou |first2=Anastasia |last3=Avendaño |first3=Carlos |last4=Galindo |first4=Amparo |last5=Adjiman |first5=Claire S. |last6=Müller |first6=Erich A. |last7=Jackson |first7=George |date=2013-10-21 |title=Accurate statistical associating fluid theory for chain molecules formed from Mie segments |journal=The Journal of Chemical Physics |language=en |volume=139 |issue=15 |pages=154504 |doi=10.1063/1.4819786 |pmid=24160524 |bibcode=2013JChPh.139o4504L |hdl=10044/1/12859 |issn=0021-9606|doi-access=free |hdl-access=free }} 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{{cite journal |display-authors=5| last1 = Kontogeorgis | first1 = Georgios M. | last2 = Michelsen | first2 = Michael L. | last3 = Folas | first3 = Georgios K. | last4 = Derawi | first4 = Samer | last5 = von Solms | first5 = Nicolas | last6 = Stenby | first6 = Erling H. | title = Ten Years with the CPA (Cubic-Plus-Association) Equation of State. Part 1. Pure Compounds and Self-Associating Systems | journal = Industrial & Engineering Chemistry Research | date = 1 June 2006 | volume = 45 | issue = 14 | pages = 4855–4868 | issn = 0888-5885 | eissn = 1520-5045 | doi = 10.1021/ie051305v | pmid = | url = }} and the SAFT + cubic model {{Cite journal |last=Polishuk |first=Ilya |date=2011-12-21 |title=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 |url=https://pubs.acs.org/doi/10.1021/ie201952n |journal=Industrial & Engineering Chemistry Research |language=en |volume=50 |issue=24 |pages=14175–14185 |doi=10.1021/ie201952n |issn=0888-5885|url-access=subscription }} and non-random-lattice (NLF) models based on lattice field theory.