Nearest-neighbor interaction

In our paper we are concerned with a potential energy function which is somewhat more realistic than one implied by the excluded volume effects. This potential function allows indirectly bonded chain elements which occupy adjacent lattice sites to interact with a finite energy . We call these interactions nearest-neighbor interactions, which should not be confused with interactions between near-neighbor pairs along the chain. 

The quasi-lattice theory is based on a quasi-lattice, which consists of two interlocking sub-lattices of cations and anions. Only the nearest neighbor interactions are taken into account and the energy of any nearest neighbor pair is assumed to be independent of its environment (additivity of pair bond interactions). Nearest neighbor interactions are ignored, which means that all the binary systems are ideal. 

One may consider a molecule in the surface region as being in a state intermediate between that in the vapor phase and that in the liquid. Skapski [11] has made the following simplified analysis. Considering only nearest-neighbor interactions, if n, and denote the number of nearest neighbors in the interior of the liquid and the surface region, respectively, then, per molecule... 

The next point of interest has to do with the question of how deep the surface region or region of appreciably unbalanced forces is. This depends primarily on the range of intermolecular forces and, except where ions are involved, the principal force between molecules is of the so-called van der Waals type (see Section VI-1). This type of force decreases with about the seventh power of the intermolecular distance and, consequently, it is only the first shell or two of nearest neighbors whose interaction with a given molecule is of importance. In other words, a molecule experiences essentially symmetrical forces once it is a few molecular diameters away from the surface, and the thickness of the surface region is of this order of magnitude (see Ref. 23, for example). (Certain aspects of this conclusion need modification and are discussed in Sections X-6C and XVII-5.)... 

If the interaction between atoms that are not nearest neighbors is neglected, then the ratios B/A are each equal to the ratio of the number of nearest neighbors to a surface atom (across the dividing plane) to the number of nearest neighbors for an interior atom. The calculation then reduces to that given by Eq. Ill-15. 

Make the following approximate calculations for the surface energy per square centimeter of solid krypton (nearest-neighbor distance 3.97 A), and compare your results with those of Table VII-1. (a) Make the calculations for (100), (110), and (111) planes, considering only nearest-neighbor interactions, (b) Make the calculation for (100) planes, considering all interactions within a radius defined by the sum... 

Taking into account only nearest-neighbor interactions, calculate the value for the line or edge tension k for solid argon at 0 K. The units of k should be in ergs per centimeter. 

Some studies have been made of W/O emulsions the droplets are now aqueous and positively charged [40,41 ]. Albers and Overbeek [40] carried out calculations of the interaction potential not just between two particles or droplets but between one and all nearest neighbors, thus obtaining the variation with particle density or . In their third paper, these authors also estimated the magnitude of the van der Waals long-range attraction from the shear gradient sufficient to detach flocculated droplets (see also Ref. 42). 

One restriction imposed by Huckel theory that is rather easy to release is that of zero overlap for nearest-neighbor interactions. One can retain a — as the diagonal elements in the secular matrix and replace p by p — EjS as nearest-neighbor elements where S is the overlap integral. Now,... 

You can choose to calculate all nonbonded interactions or to truncate (cut off) the nonbonded interaction calculations using a switched or shifted function. Computing time for molecular mechanics calculations is largely a function of the number of nonbonded interactions, so truncating nonbonded interactions reduces computing time. You must also truncate nonbonded interactions for periodic boundary conditions to prevent interaction problems between nearest neighbor images. 

Each lattice site is defined to have z nearest neighbors, and 0i and 02 > respectively, can be used to describe the fraction of sites which are occupied by solvent molecules and polymer segments. The following inventory of interactions can now be made for the mixture ... 

Especially for large values of Aw, there could be an additional entropy effect beyond that calculated in the last section which arises from the interaction of nearest neighbors. That is, reaction (8.A) might be characterized by both a AHp jj. and a ASp jj.. In this case Aw might be viewed as the pairwise contribution to a free energy ACpaj with... 

Utilize periodic boundary conditions, which permit reduction of the number of nonbonded interactions at greater distances by involving only the "nearest neighbor" atoms from copies of the system which are in different but adjacent cells. 

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

In their model they retained only the first- and second-nearest neighbor interactions, so that the Hamiltonian assumed the following form... 

In the above, U] ] is the nearest neighbor interaction energy, V is the adsorption energy and Fb is the boundary field acting on the particles located at the patch boundary... 

Assuming that the interaction between the adsorbed particles is confined to the first nearest neighbors, the Hamiltonian of the model reads... 

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

For the extension to two dimensions we consider a square lattice with nearest-neighbor interactions on a strip with sites in one direction and M sites in the second so that, with cyclic boundary conditions in the second dimension as well, we get a toroidal lattice with of microstates. The occupation numbers at site i in the 1-D case now become a set = ( ,i, /25 5 /m) of occupation numbers of M sites along the second dimension, and the transfer matrix elements are generalized to... 

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