Energies and Forces of Interaction

We are treating interactions in terms of energies, but they can also be treated as forces, because all contributions to intermolecular interactions studied in Section 4.3.3 cause forces to act on the molecules the relation between force and potential energy is well known  

FIG U RE 6.4 The Derjaguin approximation geometry. The force between the two spheres is calculated from the interaction between two circular regions (dark segments), assumed flat, using the expression for interaction between plane surfaces. 

If one sphere is much larger than the other one (f 2 R ), it is found again in the expression for a sphere near a plane surface. Equation 6.20 if both spheres have the same radius, half that value is obtained Fn(d) = nRJJi2 (d). For two spheres in contact (d = d, the collision diameter), the interaction energy may be equated to -2y, where y is the surface energy (Chapter 2), 

---

In the following tabulation we have listed the energy and force of interaction as functions of the relative gap x A... 

We should emphasize once more the difference between the interaction of individual molecules and the molecular interaction of condensed systems, in relation to the magnitude of the gap separating them. For individual molecules, the interaction is inversely proportional to and the force to for condensed systems, it is proportional to H and, respectively, for two spherical particles (or a partile and a plane) and to and for two planes. On the basis of Eqs. (II.21)-(II.26), we can calculate the energy and force of interaction between two bodies as a function of the gap separating them. 

It follows from the data presented that the energy and force of molecular interaction depend greatly on the gap width (x = H/dp) separating the contiguous bodies. For a sphere and a plane the energy and force of interaction are greater than for two spheres. 

Table 1.3 Interaction Energies and Forces of Attraction Between Two Bodies with Different Geometries... 

One consequence of the continuum approximation is the necessity to hypothesize two independent mechanisms for heat or momentum transfer one associated with the transport of heat or momentum by means of the continuum or macroscopic velocity field u, and the other described as a molecular mechanism for heat or momentum transfer that will appear as a surface contribution to the macroscopic momentum and energy conservation equations. This split into two independent transport mechanisms is a direct consequence of the coarse resolution that is inherent in the continuum description of the fluid system. If we revert to a microscopic or molecular point of view for a moment, it is clear that there is only a single class of mechanisms available for transport of any quantity, namely, those mechanisms associated with the motions and forces of interaction between the molecules (and particles in the case of suspensions). When we adopt the continuum or macroscopic point of view, however, we effectively spht the molecular motion of the material into two parts a molecular average velocity u = (w) and local fluctuations relative to this average. Because we define u as an instantaneous spatial average, it is evident that the local net volume flux of fluid across any surface in the fluid will be u n, where n is the unit normal to the surface. In particular, the local fluctuations in molecular velocity relative to the average value (w) yield no net flux of mass across any macroscopic surface in the fluid. However, these local random motions will generally lead to a net flux of heat or momentum across the same surface. 

The energy and force of the van der Waals interaction between two gas molecules may be put into the form... 

For generality Good and Girifalco set their relationship to 4> so that possessed a maximum value of unity and was defined in terms of molecular geometries and forces of interaction across adjacent phases [49]. Thus when expressed in terms of surface free energies for two phases a and b, the interfacial free energy between the phases, yab, is given by ... 

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

Calculation of the energies and forces due to the long-range Coulomb interactions between charged atoms is a major problem in simulations of biological molecules (see Chapter 5). In an isolated system the number of these interactions is proportional to N-, where N is the number of charged atoms, and the evaluation of the electrostatic interactions quickly becomes intractable as the system size is increased. Moreover, when periodic... 

In a solution of a solute in a solvent there can exist noncovalent intermolecular interactions of solvent-solvent, solvent-solute, and solute—solute pairs. The noncovalent attractive forces are of three types, namely, electrostatic, induction, and dispersion forces. We speak of forces, but physical theories make use of intermolecular energies. Let V(r) be the potential energy of interaction of two particles and F(r) be the force of interaction, where r is the interparticle distance of separation. Then these quantities are related by... 

An example of this third solution is presented in this paper which shows how it is possible to achieve supercomputer speeds from a low cost laboratory computer. By placing the task on a local lab computer, it is now also possible to develop reasonable interactive molecular modelling tools which utilize energies and forces in real time. 

Features. GEMM is written in a host-independent manner and it has been run with an Apollo, a VAX, and a MicroVAX II as a host. GEMM can currently perform the following operations perform molecular dynamics, perform energy minimizations, compute the energy and forces for a structure, and update the nonbond list (nonbond lists are usually automatic for the other operations). In addition, a wide variety of I/O sequences are possible, such as what is needed for interactive modelling work. 

Liu, W. B. Wood, R. H. Doren, D. J., Hydration free energy and potential of mean force for a model of the sodium chloride ion pair in supercritical water with ab initio solute-solvent interactions, 7. Chem. Phys. 2003,118, 2837-2844... 

Experimental studies of the thermodynamic, spectroscopic and transport properties of mineral/water interfaces have been extensive, albeit conflicting at times (4-10). Ambiguous terms such as "hydration forces", "hydrophobic interactions", and "structured water" have arisen to describe interfacial properties which have been difficult to quantify and explain. A detailed statistical-mechanical description of the forces, energies and properties of water at mineral surfaces is clearly desirable. 

---