Pairwise forces

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

L. L. Boyer and G. S. Pawley, J. Comput. Phys., 78, 405 (1988). Molecular Dynamics of Clusters of Particles Interacting with Pairwise Forces Using a Massively Parallel Computer. 

The coefBcient ( >2 is expressed as a function of the second and third virial coefficients in Eq. 2 from Ref 4. These coefficients depend on the magnitude of the pairwise and three-body interactions respectively. The value of becomes more negative as the pairwise attractive forces between molecules increase. These pairwise forces dominate the three-body interactions in the second term and the log of the compressibility factor, the third term of this equation. As the ln2, which is proportional to By, becomes more negative, becomes 1 and, being in the denominator ofEq. 1, causes the mole fi-action of the liquid in the SCF, 2. to increase. 

The use of accurate pairwise forces would greatly enhance our ability to use molecular dynamics simulations to describe melting and glass formation of the siliea polymorphs. Recent proposals for high pressure forms of silica which could be examined by accurate pairwise forces. 

The sueeessful application of pairwise forces to silica is counter intuitive. Si02 has traditionally been viewed as a covalent material. Tetrahedral units of Si04 dominate the structural properties for those structures containing four-fold coordinated silicon. Pairwise potentials would not be expected to... 

Since the interactions between particles in eFF are simply pairwise forces, the overall method is extremely fast and scales well computationally. We have simulated on a single processor tens of thousands of electrons, and on multiple processors millions of electrons [90]. 

Similar expansions using the same 16 terms can also be foimd for the forces. Obviously this has linear computation time with respect to the number of particles, as the only summations over all the particles occiu in the x -The infinite summation over (p, q) can be cut off at some value to achieve the desired accuracy. For a cutoff radius of R in the (uxpyUyq) space, an upper bound to the maximal pairwise force error is given by... 

By using Eq. (2.11) we find that the dynamical matrix D for central pairwise forces can be written as ... 

Practical Adaptation of the FM Method. The described FM method is implemented by fitting pairwise force field. The pair wise force field only depends upon the atom atom separations. The atom - atom forces are divided into a short range and a long range Coulomb part and can be written in the following form... 

We consider a d-dimensional system consisting of vortices and antivortices interacting via the pairwise force whose magnitude is proportional to R " with the intervortex distance R. We restrict ourselves to systems with weak long-range forces, i.e. the power exponent of the long-range force, n, satisfies the condition d — I [Pg.300]

Fig. 3.5 Implementation of surface tension effects by pairwise forces...

The attraction between solid primary particles is also parameterised using this approach. The solid-solid interaction strength is then set as a distinct fraction of the liquid-liquid one. The method of pairwise forces bears some peculiarities like a remaining net force on a flat surface. Nonetheless, as can be seen from Fig. 3.6, the method as has been used in our code relaxes to the correct static contact angle. Furthermore, it is easy to implement and proves to be robust.Please check if the sentence " It is sufficient that the force x2026 " is ok as edited.Just one shght modification added. 

Coulombic forces from net electrostatic charge and polarization-induced forces (e.g., van der Waals forces) play a prominent role even for particles larger than the micron scale, helow which Brownian motion dominates. A dimensionless form for all interparticle forces is not possible but, assuming pairwise forces, it is common to use a ratio of a characteristic value of the interparticle force near contact of a pair (Fq) with the representative shearing force at particle scale r ya to form the dimensionless interparticle force F = FQ/ r ya ). 

Let us go back to Equation 1.1, but now imagine that the Brownian particle diffuses in a colloidal dispersion formed by other N particles in the volume V with which it interacts by means of pairwise direct (i.e., conservative) interactions but in the absence of hydrodynamic interactions. The pairwise force that this tracer particle (T) exerts on particle / is given by Fj- = -V M( r - rj. ), so that Equation... 

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