Wertheim’s perturbation theory

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 ... 

Calculating the inhomogeneous fluid structure of associating molecules based on Wertheim s perturbation theory was first proposed by Chapman [18]. At that time, accurate molecular DFTs for non-polar spherical molecules (e.g., Tarazona s... 

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

The other two approaches divide the excess functional into a hard-core and an attractive part with different approximations for the two. Rosinberg and coworkers [126-129] have derived a functional from Wertheim s first-order perturbation theory of polymerization [130] in the limit of complete association. Woodward, Yethiraj, and coworkers [39,131-137] have used the weighted density approximation for the hard-core contribution to the excess free energy functional, that is,... 

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. 

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. 

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. 

The association term is based on Wertheim s first order perturbation theory and follows a group-contribution approach ... 

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. 

As can be seen, under the single bonding condition when treated as a perturbation theory, both Andersen s and Wertheim s approaches give the same result for homogeneous fluids. Indeed, Eq. (13) can be rewritten in terms of monomer fractions... 

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... 

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