Chain fluids

To see that this is exact, consider an ideal chain fluid in a external field [Pg.122]

Budzien, J., McCoy, J.D., and Adolf, D.B. (2004) General relationships between the mobility of a chain fluid and various computed scalar metrics, J. Chem. Phys. 121, 10291... 

The remainder of this review is divided as follows. In Section II, methods for the calculation of free energies are discussed the focus of Section III is on methods for the simulation of phase equilibria of chain fluids Section IV briefly discusses algorithm for polarizable force fields and Section V offers some perspective. Simulation methods that are suitable for continuum-space simulations and standard molecular mechanics force fields are emphasized in this review. 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

Ha Lamellar phase with molten alkyl chains (fluid)... 

In his original works, Rosenfeld considered hard spheres, soft spheres, Lennard-Jones system, and one-component plasma [52,53]. Thereafter, the excess entropy scaling was applied to many different systems, including core-softened liquids [17,18,51,54,55], liquid metals [56,57], binary mixtures [58,59], ionic liquids [60,61], network-forming liquids [54,60], water [62], chain fluids [63], and bounded potentials [51,64,65]. 

Liu Y, Chen X, Liu H, Hu Y,JiangJ A density functional theory for Yukawa chain fluids in a nanoslit. Mol Simul 36(4) 291—301, 2010a. 

Spyriouni, T. Economou, I.G. Theodorou, D.N. Thermodynamics of chain fluids from atomistic simulation a test of the chain increment method for chemical potential. Macromolecules 1997, 30, 4744—4755. 

Yethiraj, A. Hall, C.K. Monte-Carlo simulation of the equilibrium partitioning of chain fluids between a bulk and a pore. Mol. Phys. 1991, 75 (3), 503-515. 

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. 

Johnson et aVsf proposed equation of state to treat LJ chains in which the free energy of the chain fluid was obtained using the free energy and radial distribution function of a monomer LJ fluid, the expressions for which were fitted to simulation data. This approach was extended to mixtures by Ghonasgi et al. and Bias and Vega, " who referred to the approach as the soft-SAFT equation of state. 

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. 

Diplock R, Sullivan DE, Jaffer KM, Opps SB. Nematic-isotropic phase transition in diblock fused-sphere chain fluids. Phys Rev E 2004 69 026701. 

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. 

Figure 2 Intermolecular pair correlation function, gir), for freely jointed tangent-hard-sphere chain fluids. Symbols are Monte Carlo Simulations and lines are predictions of the PRISM theory...

Accurate equations of state for hydrocarbons are of considerable interest to the petroleum and natural gas industry, and this has fueled active research in this area. Early equations of state have used lattice or cell model descriptions. Although some of these approaches are in good agreement with experimental data, they contain adjustable parameters that cannot be determined a priori and the physical insights are clouded by several qualitative concepts such as free volume. The development of molecularly based equations of state has focused on predicting the volumetric properties of hard chain fluids. The reason for this is that the repulsive part of the potential is expected to dominate liquid structure and the effect of attractions can normally be treated as a perturbation as has been done successfully for simple liquids. 

The GF theory contains two improvements. First, the probability of inserting a monomer into the chain fluid is assumed to be the same as that of inserting a monomer into a monomer fluid, and is obtained from an equation of state for hard spheres. Second, the probability of inserting a chain is obtained from that of inserting a monomer by noting that the volume required for chain insertion is different from that required for monomer insertion. The resulting insertion probability is... 

Chain contribution The free energy required to form the chain fluid from the reference fluid of unbonded monomers is given by ... 

Some molecular-based equations of state such as lattice models, chain fluid theories and SAFT-type approaches have also been used to model ionic liquids and their behaviour in mixtures. The advantage of building a model for the molecule describing the physics of the system is related to a major predictive ability, hence enhancing the possibility of extending the range of application of the equation. In the next sections of this chapter, some examples of successful applications and current limitations of the use of one of these tools, the soft-SAFT equation, will be presented and discussed. 

Ramies, J.C. (2003). Bulk and interfacial properties of chain fluids. Rh.D. Thesis, Universitat Rovira i Virgili, Tarragona, Spain. 

---