Square well system

In Fig. 11 we have plotted the fluidity of the hard-sphere fluid (from Alder, et al. ), together with some very recent data of Michels on the fluidity of the square-well liquid. The square-well model has a uniform attractive potential between a and 1.5 a of depth e. When we extrapolate linearly the fluidity of the square-well system, an interesting result is obtained that vindicates the inferences of the preceding section. 

As an adsorption geometry one considers a semi-infinite system with an impenetrable wall at z = 0, such that monomer positions are restricted to the positive half-space z > 0. At the wall acts a short-range attractive potential, either as a square well... 

Alder and Wainwright gave MD treatments of particles whose pair potential was very simple, typically the square well potential and the hard sphere potential. Rahman (1964) simulated liquid argon in 1964, and the subject has shown exponential growth since then. The 1970s saw a transition from atomic systems... 

Spin orbitals, 258, 277, 279 Square well potential, in calculation of thermodynamic quantities of clathrates, 33 Stability of clathrates, 18 Stark effect, 378 Stark patterns, 377 Statistical mechanics base, clathrates, 5 Statistical model of solutions, 134 Statistical theory for clathrates, 10 Steam + quartz system, 99 Stereoregular polymers, 165 Stereospecificity, 166, 169 Steric hindrance, 376, 391 Steric repulsion, 75, 389, 390 Styrene methyl methacrylate polymer, 150... 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

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... 

We have thus far described the properties of the exact eigenfunctions of the Hamiltonian for the system consisting of a particle moving under the influence of a finite square well. These eigenfunctions form a complete... 

Hydrogenic atoms (one electron bound by a nuclear charge Z) have 7 proportional to the seventh power of the orbital radius [29]. Square well 1-D potentials with infinitely high walls and an appropriate number of filled states give 7 proportional to the 5th power of the well width [29]. There is clearly a rapid increase expected in the second hyperpolarizability with system size for delocalized systems. 

The one-dimensional potential depicted in Fig. 7(a) provides an illustration of this effect. The Schrodinger equation can be solved with the method used for the square-well case above. Each well gives rise to a nearly independent progression of states. For fi = 2p = 1 and other potential parameters indicated in Fig. 7 one finds that the system has two bound states and a resonance state at 9.46 — i 0.11 localized above the deep outer well. There is also another resonance in the system, E = 9.8 - i 0.002. Its width is very small because this state belongs to the shallow inner well, which is separated from the continuum by a potential barrier. Suppose that we force — by varying a parameter in the Hamiltonian — the narrow state (denoted n) in the shallow well to move across the broader resonance (b) belonging to the deep minimum. The relative positions of the two states can be, for example, controlled by shifting the infinite wall at the... 

The properties of the above system at modest particle concentrations are relatively simple to model, because the grafted octadecyl layer is thin compared to the particle radius and because the particle-particle interactions are weak enough that the properties of the dispersion are not sensitive to the detailed shape of the particle-particle interaction potential. These considerations have motivated the use of a simple square-well potential as a model of the particle-particle interactions (Woutersen and de Kruif 1991) (see Fig. 7-3). This potential consists of an infinite repulsion at particle-particle contact (where D — 0), bounded by an attractive well of width A and depth e. There are no interactions at particle-particle gaps greater than A. Near the theta point, the well depth s depends on temperature as follows (Hory and Krigbaum 1950) ... 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

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