Force between two particles

The pair correlation fiinction has a simple physical interpretation as the potential of mean force between two particles separated by a distance r... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

The calculation of the contact force between two particles is actually quite involved. A detailed model for accurately computing contact forces involves complicated contact mechanics (Johnson, 1985), the implementation of which is extremely cumbersome. Many simplified models have therefore been proposed, which use an approximate formulation of the interparticle contact force. The simplest one was originally proposed by Cundall and Strack (1979), where a linear-spring and dashpot model is employed to calculate the contact forces (see Fig. 11 and 12). In this model, the normal component of the contact force between two particles a and b can be calculated by... 

The contact force between two particles is now determined by only five parameters normal and tangential spring stiffness kn and kt, the coefficient of normal and tangential restitution e and et, and the friction coefficient /if. In principle, kn and k, are related to the Young modulus and Poisson ratio of the solid material however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become unpractically small. The values for kn and k, are thus mainly determined by computational efficiency and not by the material properties. More on this point is given in the Section III.B.7 on efficiency issues. So, finally we are left with three collision parameters e, et, and which are typical for the type of particle to be modeled. 

This gives the relationship between the attractive force between two particles and the maximum surface field intensity. There is a limit to the magnitude that some type of discharge or emission. From data such as those of Earhart (El) one would conclude that Sma% cannot exceed some 100 to 200 V/micron before the particle would discharge... 

The attraction force between two particles is proportional to the distance of separation and a Hamaker constant (specific to the system). The magnitude of H is of the order of 10 2 erg (Adamson and Gast, 1997 Birdi, 2002). 

Dispersion in aqueous electrolyte. According to Derjaguin s approximation (Eq. 6.36) the force between two particles is... 

FIGURE 3.1 The electromagnetic or gravitational force between two particles lies along a vector between the particles. 

For dispersed particles (or droplets, etc.) the dispersion forces can be approximated by adding up the attractions between all interparticle pairs of molecules. When added this way the dispersion force between two particles decays less rapidly as a function of separation distance than is the case for individual molecules. For two spheres of radius a in vacuum, separated by a small distance H, the attractive energy VA can be approximated by ... 

In the DLVO-theory, the force between two particles is seen as a summation of the attractive force imposed by the van der Waals force and the repulsive force imposed by the electrostatic interaction. 

In the pendular state, bonding is localized at the points of particle contact and calculated values of the cohesive force between two particles may be substituted directly for H into eqn. (4) to yield the tensile strength of the assembly. For two particles in contact, H is given by [8,10] ... 

Region III. The motion of the fictitious particle in this region is influenced by the attractive force between two particles. Since the effect of collisions with the medium on the motion of the fictitious particle is assumed to be negligible, the distribution function is governed by the collisionless Boltzmann equation... 

Colloidal dispersions, in general, are rendered stable either by electrostatic stabilization or by steric stabilization. In the former case, the repulsive electrical double layer forces between two particles counteract the attractive van der Waals forces and generate a potential barrier between the primary and secondary minima. If the potential barrier is sufficiently higher than the... 

If the radius of the rigid core of a particle is fixed, the stability ratio increases with the density of fixed charge. This is expected, since the higher the latter, the greater the electrical repulsion force between two particles, and, therefore, the more stable a colloidal suspension. 

Fr electrical force between two particles per unit area... 

The expression van der Waals attraction is widely used and is here defined as the sum of dispersion forces [9], Debye forces [17] and the Keesom forces [18]. Debye forces are Boltzmann-averaged dipole-induced dipole forces, while Keesom forces are Boltzmann-averaged dipole-dipole forces. The interaction for all three terms decays as 1 /r6, where r is the separation between the interacting particles, and they are combined into one term with the proportionality constant denoted the Hamaker constant. In order to determine the van der Waals force there are at least two approaches, either to calculate the force between two particles assuming that the interaction is additive, (this is usually called the Hamaker approach) or to use a variant of Lifshitz theory. 

Based on Equations 9.1 and 9.2, colloidal stability (maximum dispersion) depends on maximum R, which describes the maximum repulsive energy between two planar colloidal surfaces. It also appears from these equations that Rf is controlled by or CECV. However, in addition to the repulsive force, there is an attraction force (Af) between soil colloidal particles. The force of attraction (van der Waals force) between two particles separated by a distance of 2d (d = particle diameter) is described by... 

Aggregation of liposomes both in vitro and in vivo is one of their main stability problems. According to the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, or theory of colloidal stability, a colloidal system is stable if the electrostatic repulsion forces between two particles are larger than the attraction van der Waals forces. Therefore charged liposomal formulations are highly desirable. Manipulation of... 

FIGURE 8.3 Calculation of the interaction force between two particles by integrating the excess osmotic pressure AH and the Maxwell stress T over an arbitrary surface 2 enclosing... 

FIGURE 5.17 Flotation (a, c, e) and immersion (b, d, f) lateral capillary forces between two particles attached to fluid interface (a) and (b) are two similar particles (c) is a light and a heavy particle (d) is a hydrophibc and a hydrophobic particle (e) is small floating particles that do not deform the interface (f) is small particles captured in a thin liquid film deforming the interfaces due to the wetting effects. 

For the simplest case, the net attractive force between two particles at small separation distance after neglecting the retardation correction is given as... 

The potential of mean force between two particles is the effective potential that emerges when integrating out all other particles. Apart from the limit of zero density it differs from the plain intermolecular potential. In a sense, one enters the realm of rigorous statistical mechanics when this distinction is clearly appreciated [30],... 

Gravitational constant (G) - The universal constant in the equation for the gravitational force between two particles, F = Gm mJF, where r is the distance between the particles and Wj and are their masses. [1]... 

Fig. 7 7- The force between two particles acts along the line joining their centres of mass, in accordance with Newton s third law...

Statistical thermodynamics relates the force between two particles dispersed in a polymer solution to their separation by... 

During a simulation, one maintains both sets of positions, r and r, for the molecules. The potentials and forces between two particles are calculated using the nearest image vector, i, pointing from particle / to particle i. The vec-... 

The binding forces between two particles in these two situations is different and is a function of the actual humidity in both cases. 

The potential of mean force between two particles of a liquid mixture of ionic and molecular species breaks up into their direct interaction and the solvent contribution which can be presented as a difference in the excess free energy of their solvation in a given relative arrangement and at infinite separation,... 

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