Wertheim thermodynamic perturbation theory

In the subsequent papers in the series, Wertheim extended his analysis to multiple association sites and to systems undergoing polymerization." His key contribution was to show that it is possible to obtain the properties of an associating or chain fluid based on knowledge of the thermodynamic properties (the Helmholtz energy and structure) of the monomer fluid. This is the basis of the now well-known Wertheim thermodynamic perturbation theory, and in turn, the basis of all SAFT equations of state. Interestingly, in this series of four papers, Wertheim did not present a single calculated result or any numerical tests of his proposed theories. 

Wertheim M S 1987 Thermodynamic perturbation theory of polymerization J Chem. Phys. 87 7323... 

L. G. MacDowell, M. Muller, C. Vega, and K. Binder (2000) Equation of state and critical behavior of polymer models A quantitative comparison between Wertheim s thermodynamic perturbation theory and computer simulations. J. Chem. Phys. 113, pp. 419-433... 

Wertheim, M.S. Thermodynamic perturbation theory of polymerization. J. Chem. Phys. 1987, 87, 7323. 

Jog, P.K. and Chapman, W.G., Application of Wertheim s thermodynamic perturbation theory to dipolar hard sphere chains, Mol. Phys., 97(3), 307-319, 1999. 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

NIE Nies, E., Li, T., Berghmans, H., Heenan, R.K., and King, S.M., Upper critical solution temperature phase behavior, composition fluctuations, and complex formation in poly(vinyl methyl ether)/D20 solutions Small-angle neutron-scattering experiments and Wertheim lattice thermodynamic perturbation theory predictions, J. 

Two UCSTs ate theoreticdly predicted for the low and high polymer concentrations using thermodynamic perturbation theory of Wertheim for saturation interactions (i.e. hydrogen bonds) [545-547], adapted to the lattice model [417]... 

Segura el al. combines Tarazona s WDA DFT for hard-spheres with Wertheim s thermodynamic perturbation theory and has been used in a number of studies of associating fluids in pores and with functionalized walls in the limit of complete association a DFT for polymeric fluids is obtained in this method. Based on these works, Chapman and co-workers have presented the interfacial-SAFT (iSAFT) equation, which is a DFT for polyatomic fluids formulated by considering the polyatomic system as a mixture of associating atomic fluids in the limit of complete association this approach allows the study of the microstructure of chain fluids. Interfacial phenomena in complex mixtures with structured phases, including lipids near surfaces, model lipid bilayers, copolymer thin films and di-block copolymers, have all been studied with the iSAFT approach. 

Following the original first-order thermodynamic perturbation theory of Wertheim heteronuclear models can be proposed where the segments in a given model molecule are arbitrarily different. The earliest works in this direction... 

Wertheim, M.S., 1984. Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations. J. Stat. Phys., 35 35. 

Vega, C., McBride, C., de Mibuel, E., J., B. F. Galindo, A. (2003). The phase diagram of the two center leimard-jones model as obtained from computer simulation and wertheim s thermodynamic perturbation theory, /. Chem. Phys. 118 10696. 

The statistical associating-fluid theory (SAET) developed by Chapman et al. [63, 64] is based on the thermodynamic perturbation theory of Wertheim [69]. Since it first appeared, many different versions of SAET have been published. The different SAFT versions and their application have been reviewed by Muller and Gubbins [70]. 

One of the main assumptions in the development of Wertheim s first-order thermodynamic perturbation theory (TPTl) is that association sites are singly bondable. Indeed, the entire multi-density formalism of Wertheim is constructed to enforce this condition. For the case of hydrogen bonding, the assumption of singly bondable sites is justified however for patchy colloids (see Section I for a background on patchy colloids), it has been shown experimentally [42, 43] that the number of bonds per patch (association site) is dependent on the patch size. It has been 30 years since Wertheim first published his two-density cluster... 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

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