Square well chain

Yethiraj, A. and Hall, C. K., Generalized Hory equations of state for square-well chains. /. Chem. Phys. 95, 8494-8506 (1991). 

Ye Z, Cai J, Liu H, Hu Y Density and chain conformation profiles of square-well chains confined in a sht by density-functional theory,/ Chem Phys 123(19) 194902, 2005. 

Zhang SL, CaiJ,LiuHL, Hu Y Density functional theory of square-well chain mixtures near sohd surface. Mol Simul 30(2-3) 143-147, 2004. 

Figure 4 Coexistence data for square-well chain molecules of 4, 8, 16, and 100 segments employing expanded Gibbs ensemble simulations...

Equations of state for square-well chains and for linear alkanes have been obtained using perturbation theory. Figure 6 depicts the equation of state of square-well chains from Monte Carlo simulations and perturbation theory. (In the square-well chain model the site-site interaction potential is given by u r) = oo for r < d, u r) = —s for d < r < kd, and u(r) =0 otherwise.) Three reduced temperatures, T = kT/e), are depicted in Figure 6, all for k = 1.5. Theoretical predictions were obtained using the GFD equation for the hard chain pressure and PRISM with the PY closure for the... 

Figure 6 Comparison of perturbation theory (lines) to Monte Carlo simulations for the equation of state of square-well chains for three different temperatures...

In another study, McCabe and Jackson considered TPTl as applied to square well chains in order to model the n-alkane series [255]. These authors proposed a linear correlation for the square well strength, e and range parameters, X. In the limit of infinitely long chains, their correlation yields e°°/kB = 266K and X°° = 1.64. Fortunately, for the square well chain Eq. (115) provides analytical results for the asymptotic critical temperature [271]. For the above range parameter, one finds = 5.03, so that the estimated critical temperature for polyethylene is found to be about 1339K. 

We use the off-lattice MC model described in Sec. IIB 2 with a square-well attractive potential at the wall, Eq. (10), and try to clarify the dynamic properties of the chains in this regime as a function of chain length and the strength of wall-monomer interaction. 

Second virial coefficients represent the first approximation to the system equation of state. Yethiraj and Hall [148] obtained the compressibility factor, i.e., pV/kgTn, for small stars. They found no significant differences with respect to the linear chains in the pressure vs volume behavior. Escobedo and de Pablo [149] performed simulations in the NPT ensemble (constant pressure) with an extended continuum configurational bias algorithm to determine volumetric properties of small branched chains with a squared-well attractive potential... 

The chain ion-radical mechanism of ter Meer reaction has been supported by a thorough kinetic analysis. The reaction is well-described by a standard equation of chain-radical processes (with square-law chain termination) (Shugalei et al. 1981). This mechanism also explains the nature of side products—aldehydes (see steps 13 and 14) as well as vicinal dinitroethylenes. Scheme 4.37 explains formation of vic-dinitroethylenes. 

FEN Feng, W., Wen, H., Xu, Z., and Wang, W., Comparison of perturbed hard-sphere-chain theory with statistical associating fluid theory for square-well fluids, Ind. Eng. Chem. Res., 39, 2559, 2000. 

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 ... 

Although the above-mentioned perturbation theories account for the formation of chains in the repulsive contribution, the dispersion is still considered as resulting from the attraction of unbonded chain segments. This assumption is especially not justified in the case of polymer molecules where the segments do not interact independently but are influenced by the neighboring segments of the same molecule. Several attempts have been made to overcome this deficiency. Various models were suggested which use the square-well sphere (see, e.g.,... 

Within this framework, it is assumed that molecules are made of a chain of hard spheres having a temperature-independent segment diameter, a. The chains interact via a modified square-well potential u(r) ... 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

EDMD and thermodynamic perturbation theory. Donev et developed a novd stochastic event-driven molecular dynamics (SEDMD) algorithm for simulating polymer chains in a solvent. This hybrid algorithm combines EDMD with the direa simulation Monte Carlo (DSMC) method. The chain beads are hard spheres tethered by square-wells and interact with the surrounding solvent with hard-core potentials. EDMD is used for the simulation of the polymer and solvent, but the solvent-solvent interaction is determined stochastically using DSMC. 

To match the results of molecular-dynamics computer simulations for square-well molecules, Alder et alP used a double-power-series expansion in reduced inverse temperature and volume for the attractive part of an equation of state. Modifications of their attractive term have been used in a number of equations of state, such as the augmented and perturbed-hard-sphere equations, the perturbed-hard-chain equation, and the BACK equations of state. 

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