Square-well interaction

Fig. 11. The change in surface tension as a function of electrolyte concentration for (a) a square well interaction for anions, with d1 = 5 A and = 1 kT (b) a triangular well interaction for anions, with dl =...

Conical square well (CS W) association sites are commonly used as a primitive model for the association potential 0. First introduced by Bol [17] and later reintroduced by Chapman et al. [18, 19], CSWs consider association as a square well interaction which depends on the position and orientation of each molecule. Kem and Frenkel [20] later realized that this potential could describe the interaction between patchy colloids. For these CSWs the association potential is given by... 

Turning to molecular attractions, the molecular dynamics results of Alder et al (1972) indicate that, even the simple square-well interaction potential, leads to a much more complex expression for the attractive term, than those used in the van der Waals and Redlich-Kwong EoS. 

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... 

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... 

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. 

Although a square-well potential, with energy levels proportional to n2 is totally inappropriate to model the electronic levels of an atom, (E oc 1/n2), it provides a compelling qualitative rationalization of a whole class of chemical interactions. The simplest possible description of bond formation is in terms of two coalescing potential boxes [75]. 

Figure 2. (a) Interaction of small spherical molecules. The molecule at 2, on the inner edge of a square potential well of width x, is in contact with the central molecule 1. At 2 it is on the outer edge of the well. The radius 2r defines the exclusion volume around the central molecule the shell between radius 2r and (2r+x) defines its interaction volume. (6) Interaction of spherical shell-molecules. Atoms 1 and 2 on their respective shells are in contact on the inner edge of a square well of width x. Atoms 1 and 2 are beyond the outer edge of the well. If the shell-atoms are taken as uniformly smeared around the shells, the energy of interaction between the molecules should be approximately proportional to the overlap volume Va, the region in which the shells are closer than x. 

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. 

Fig. 9. Approximation of the interactions between ions and the interface (1) square well for anions (2) triangular well for anions (3) hard wall for cations.

Two cases have been investigated. First, the solution of the Poisson-Boltzmann equation was obtained for an interfacial square-well potential for the anions, with W1 = lkT and d1 = 5 A. Second, a triangular well was employed for the potential (the interaction potential varying linearly form AW, = — IMF at x=0 to AW =0 at... 

Plot the Leimard-Jones potentials for each of the gases studied. Obtain ft from Eqs. (16)-(18) by numerioal integration and compare the values from this two-parameter potential with those from the van der Waals and Beattie-Bridgeman equations of state. Optional A simple square-well potential model can also be used to eradely represent the interaction of two molecules. In place of Eq. (18), use the square-well potential and parameters of Ref. 6 to ealeulate /t. Contrast with the results from the Lennard-Jones potential and comment on the sensitivity of the calculations to the form of the potential.]... 

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) ... 

Sensitivity to the shape of W D) differentiates weakly from strongly interacting particles. For the former, the precise shape of the potential is not important we saw in Section 7.2.4 that even a simple square-well potential is an adequate approximation. But insensitivity to the shape of the potential can only be expected when the particles are only weakly bound by that potential, so that rapid, thermally driven changes in particle-particle separation average out the details of the shape of the potential. For strongly flocculated gels, the particle-particle separations remain trapped near the minimum in the potential well, and the shape of the well near this minimum matters much more. 

A square-well potential model for the interaction between protein molecules was used to derive a relation between the osmotic second virial coefficient B2 and the aqueous solubihty [33]. The following expression, which is valid at low solubilities, was obtained... 

Using the hard sphere adhesive state equation proposed by Baxter (16), it is possible to calculate the demixing line due to interactions. This state equation corresponds to the exact solution of the Percus-Yevick equation in the case of an hard sphere potential with an infinitively thin attractive square well. In our calculation we assum that the range of the potential is short in comparison to the size of the particles (in fact less than 10 %). 

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