Percus-Yevick model

Recent work has supported early observations (e.g. Aggarwal 1976 Hashimoto et al. 1983) of a liquid micellar phase between the BCC micelle phase and the disordered phase. A representative TEM image from a spherical micellar liquid phase is shown in Fig. 2.18. Kinning and Thomas (1984) analysed SANS data obtained by Berney et al. (1982) on PS-PB diblocks and PS/PS-PB blends where the minority (PB) component formed spherical micelles with only liquid-like ordering. The Percus-Yevick model for liquids of hard spheres was used to obtain the interparticle contribution to the scattered intensity (Kinning and Thomas 1984). The ordering of an asymmetric PS-PI diblock was observed by Harkless... 

Fig. 4.8 Micelle volume fraction () versus polymer concentration at different temperatures for solutions of PEO26PPO39PEO26 in D20 (Mortensen 1993a). 4> was obtained from fits of the hard sphere Percus-Yevick model to neutron scattering profiles (see Fig. 3.9). At high concentration the asymptote = for hard sphere crystallization is reached.

Roe and co-workers (Nojima et al. 1990 Rigby and Roe 1984,1986 Roe 1986) have used SAXS to characterize micelles formed by PS-PB diblocks at low concentrations in blends with low-molecular-weight PB. Micellar dimensions and association numbers were determined for symmetric and asymmetric diblocks (Rigby and Roe 1984,1986). The effective hard sphere radius, the core radius and volume fraction of hard spheres were determined using the Percus-Yevick model (Rigby and Roe 1986). These results were compared (Roe 1986) to the predictions of the theory of Leibler et al (1983). The theory qualitatively reproduced the observed trend for the cmc to increase with temperature for blends containing a particular diblock. The cmc was found to decrease at a fixed temperature... 

In order to complete the MSA estimate of Iny,- one must add the hard-sphere contribution, which accounts for the fact that work must be done to introduce the ions as hard spheres into the solution. It is obtained from the Percus-Yevick model for non-interacting hard spheres. For the case that all ions (spheres) have the same radius, the result is (see equation (3.9.22))... 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

Cummings P T and Stell G 1984 Statistical mechanical models of chemical reactions analytic solution of models of A + S AS in the Percus-Yevick approximation Mol. Phys. 51 253... 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

Bulk phase fluid structure was obtained by solution of the Percus-Yevick equation (W) which is highly accurate for the Lennard-Jones model and is not expected to introduce significant error. This allows the pressure tensors to return bulk phase pressures, computed from the virial route to the equation of state, at the center of a drop of sufficiently large size. Further numerical details are provided in reference 4. 

Fig. 4.10 Temperature dependence of the micellar volume fraction for a 28wt% solution of F88 (PEO,PP039PEO, in D2(), as obtained by Percus-Yevick fits to SANS data (Mortensen 1993f>).The insets show models for the solution structure in the three regimes unimers in solution at low temperature, micellar liquid above the cmt, and at high temperatures a cubic micellar crystal.

S(90) was calculated from the Percus-Yevick hard sphere model as given by Ashcroft and Lekner (23) enabling the droplet radius R to be estimated assuming that the thickness of the surfactant layer surrounding the water cone, t, to be 0.4 nm (the size of a buten-l-ol molecule). 

Kalyuzhnyi, Yu.V., and Cummings, P.T. Solution of the polymer Percus-Yevick approximation for the multicomponent totally flexible sticky 2-point model of polymerizing fluid. Journal of Chemical Physics, 1995, 103, No. 8, p. 3265-3267. 

Packing fractions are conveniently measured in relative separations = (- 0c)/ to the glass transition point, which for this model of hard spheres lies at 0c = 0.516 [38, 72]. Note that this result depends on the static structure factor 5 (g), which is taken from Percus-Yevick theory, and that the experimentally determined value 0expt. somewhat higher [13, 14]. The wavevector integrals were dis-... 

For separations outside the hard core, the direct correlation functions have to be approximated. Classic closure approximations recently applied to QA models axe the Percus-Yevick (PY) closure [301], the mean spherical approximation (MSA) [302], and the hypernetted chain (HNC) closure [30]. None of these relations, when formulated for the replicated system, contains any coupling between different species, and wc can directly proceed to the limit n — 0. The PY closure then implies... 

A realistic theory of nematics should, of course, incorporate the attractive potential between the molecules as well as their hard rod features. There have been several attempts to develop such hybrid models. Equations of state have been derived based on the Percus-Yevick and BBGKY approximations for spherical molecules subject to an attractive Maier-Saupe potential.However, a drawback with these models is that they lead to y = 1 (see (2.3.18)). 

A. Ben-Naim, Statistical mechanics of waterlike particles in two dimensions. I. Physical model and application of the Percus-Yevick equation. J. Chem. Phys., 54 (1971), 3682-95. 

In the calculation, a model of the averaged structure factor for a hard-sphere (HS) interaction potential, S(g) is used [47, 48], which considers the Gaussian distribution of the interaction radius cr for individual monodisperse systems for polydispersity m, and a Percus-Yevick (PY) closure relation to solve Omstein-Zernike (OZ) equation. The detailed theoretical description on the method has been reported elsewhere [49-51]. 

The application of the Percus-Yevick equation to the BN2D model... 

An approximate version of the Percus-Yevick ( Y) equation has been applied for the pair potential based on the Bjerrum model [Ben-Naim (1970)]. The pair correlation function is written in... 

Lamperski, S. and Outhwaite, C.W., A non primitive model for the electrode/electrolyte interface based on the Percus-Yevick theory, J. Electroanal. Chem., 460, 135-143, 1999. 

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