Square well

A monolayer can be regarded as a special case in which the potential is a square well however, the potential well may take other forms. Of particular interest now is the case of multilayer adsorption, and a reasonable assumption is that the principal interaction between the solid and the adsorbate is of the dispersion type, so that for a plane solid surface the potential should decrease with the inverse cube of the distance (see Section VI-3A). To avoid having an infinite potential at the surface, the potential function may be written... 

Figure Al.3.6. An isolated square well (top). A periodie array of square wells (bottom). This model is used iu the Krouig-Peimey deseriptiou of energy bauds iu solids.

Ic now serves to label the states iu the same sense n serves to label states for a square well. 

Figure A2.3.17 Theoretical (HNC) calculations of the osmotic coefficients for the square well model of an electrolyte compared with experimental data for aqueous solutions at 25°C. The parameters for this model are a = r (Pauling)+ r (Pauling), d = d = 0 and d as indicated in the figure.

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

Approximating the real potential by a square well and infinitely hard repulsive wall, as shown in figure A3.9.2 we obtain the hard cube model. For a well depth of W, conservation of energy and momentum lead [H, 12] to the very usefiil Baule fomuila for the translational energy loss, 5 , to the substrate... 

The uncertainty principle, according to which either the position of a confined microscopic particle or its momentum, but not both, can be precisely measured, requires an increase in the carrier energy. In quantum wells having abmpt barriers (square wells) the carrier energy increases in inverse proportion to its effective mass (the mass of a carrier in a semiconductor is not the same as that of the free carrier) and the square of the well width. The confined carriers are allowed only a few discrete energy levels (confined states), each described by a quantum number, as is illustrated in Eigure 5. Stimulated emission is allowed to occur only as transitions between the confined electron and hole states described by the same quantum number. 

While the smooth substrate considered in the preceding section is sufficiently reahstic for many applications, the crystallographic structure of the substrate needs to be taken into account for more realistic models. The essential complications due to lack of transverse symmetry can be dehneated by the following two-dimensional structured-wall model an ideal gas confined in a periodic square-well potential field (see Fig. 3). The two-dimensional lamella remains rectangular with variable dimensions Sy. and Sy and is therefore not subject to shear stresses. The boundaries of the lamella coinciding with the x and y axes are anchored. From Eqs. (2) and (10) one has... 

FIG. 3 Schematic of the two-dimensional square-well potential u x) of depth e, width d, and period I (from Ref. 48). 

P. A. Monson. The properties of inhomogeneous square-well mixtures in one dimension. Mol Phys 70 401—423, 1990. 

Since the results for thermodynamics from the Yukawa potential, with A = 1.8, are similar to the results of the LJ potential, it is quite possible that the DHH closure may be applicable to the Yukawa potential. With A = 1.5, the thermodynamics of the square well fluid are also similar. Here, too, the DHH closure may be useful. However, the DHH closure has not been applied to either of these potentials. 

Let us begin our discussion from the model of Cummings and Stell for heterogeneous dimerization a + P ap described in some detail above. In the case of singlet-level equations, HNCl or PYl, the direct correlation function of the bulk fluid c (r) represents the only input necessary to obtain the density profiles from the HNCl and PYl equations see Eqs. (6) and (7) in Sec. II A. It is worth noting that the transformation of a square-well, short-range attraction, see Eq. (36), into a 6-type associative interaction, see Eq. (39), is unnecessary unless one seeks an analytic solution. The 6-type term must be treated analytically while solving the HNCl... 

According to Wertheim, association of species is generated by a strong, square-well attraction between the sites either on the surface or embedded... 

If the associative site (a square-well, or defined by setting limits of appropriate angles) is embedded into a hard core of the molecule, there must be a hole in the infinite repulsive part of the nonassociative potential, to make the bond formation possible, i.e. 

In the above, ef is the energy of chemical binding of a fluid particle with the solid atom. In the calculations performed it was assumed that cr = a = I and the square mound satisfies the condition Qxp[—DJksT] 0. The bonding length of a fluid particle with the solid atom is and w is the width of the attractive square well (w was set to 0.1 in the calculations). 

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... 

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. 

Selecting the values of the parameters for the calculations we have in mind a 1 1 aqueous 1 m solution at a room temperature for which the Debye length is 0.3 nm. We assume that the non-local term has the same characteristic length, leading to b=. For the adsorption potential parameter h we select its value so that it has a similar value to the other contributions to the Hamiltonian. To illustrate, a wall potential with h = 1 corresponds to a square well 0.1 nm wide and 3.0 kT high or, conversely, a 3.0 nm wide square well of height 1.0 kT. 

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... 

In order to see whether the results are sensitive to the exact shape of the potential field, some calculations have been made in which the field w r) was replaced by a square well. The depth of the well was taken equal to the value (Eq. 31) of w(o) for an L-J-D- field, while the radius was taken equal to the value (at— a) valid for hard spheres. In this approximation the free volume is equal to m (a —or)3, and hence in formula 38... 

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... 

Einwohner T., Alder B. J. Molecular dynamics. VI. Free-path distributions and collision rates for hard-sphere and square-well molecules, J. Chem. Phys. 49, 1458-73 (1968). 

The sequence of levels shown in Figure 2 closely resembles the level diagram found by Mayer and Jensen by analysis of observed nuclear properties, with the help of the calculated level sequences for harmonic-oscillator and square-well potential func-... 

Barker, J. A. Henderson, D., Perturbation theory and equation of state for fluids the square-well potential, J. Chem. Phys. 1967, 47, 2856-2861... 

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]. 

Singh, J. K. Kofke, D. A. Errington, J. R., Surface tension and vapor-liquid phase coexistence of the square-well fluid, J. Chem. Phys. 2003,119, 3405-3412... 

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