Inter-particle interaction potential

As an example, it is useful to recall protein solutions. Protein molecules can be quite large for example, a lysozyme molecule is an ellipsoid with dimensions 3x4x5 nm, similar to nanoparticles (they are in fact nanoparticles). Protein aqueous solutions have seen significant smdy in the recent past because it is desirable to form protein crystals from solution for strucmral analysis. Such a system contains not only the protein molecule and the water solvent but usually dissolved ions which dissociate and a variable pH. Moreover, the protein molecule may have a variety of surface states that affect its interaction with other protein molecules as well as the water. The lesson here is that often the system is too complex and an effective inter-particle interaction potential must be prescribed. Such a procedure often works because the number concentration of the large protein molecules or nanoparticles is far less than that of the other constituents. 

Structure factor of the sticky hard sphere model. The structure factor of a system cf interacting spheres is determined by the inter-particle interaction potential u(r). We consider a system of hard spheres with adhesive surfaces. The pair-wise inter-particle interaction potential is written as ... 

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

Another closely related constraint is that of Galileian invariance. Suppose that a many-body wavefunction (r, t2, r ) satisfies the time-independent interacting N-particle Schrddinger equation with an external one-particle potential r(r). Then, provided that the inter-particle interaction depends on coordinate differences only, it is readily verified that a boosted wavefunction of the form... 

Here pj = pj +pjy + pj7 and U is the potential associated with the inter-particle interaction. The function/(/ , p ) is an example of a joint probability density function (see below). The staicture of the Hamiltonian (1.184) implies that f can be factorized into a term that depends only on the particles positions and terms that depend only on their momenta. This implies, as explained below, that at equilibrium positions and momenta are statistically independent. In fact, Eqs (1.183) and (1.184) imply that individual particle momenta are also statistically independent and so are the different cartesian components of the momenta of each particle. 

Remember that ideal gases should have no inter-particle interactions. This gives the first virial coefficient, R, and the ideal gas law. Unfortunately, the atoms or molecules of real gases do interact. The energy E of one mole of a real gas is the sum of the kinetic and intermolecular potential energies of all its molecules, E = K + v. 

The second term in eqs. 1-3 describes the dissipative forces between the particles and the plates and is proportional to their relative velocities and to the X -dependent damping T1j, x(X ) = r j exp(—X /A), accounting for dissipation that arises from interaction with phonons and other excitations. The interactions between the particles and the plates are represented by the periodic potential with respect to x, 4>(x,X ) = —0() exp(—X /A) cos 2nx/h). Concerning the inter-particle interaction, we assume here nearest neighbor harmonic interaction (jc/— jc/ i) = k/2)[xi - Xj af- and free boundary conditions, although the Lennard-Jones interaction and periodic boundary conditions have been investigated as well [45]. The two plates do not interact directly. These equations are integrated numerically [50]. 

The basic frequency in the problem is chosen as the frequency of the top plate oscillation in the periodic potential = 2k/h). The other frequencies in the model are the frequency of the particle oscillation in the potential co = (27c// )y5o/m, the characteristic frequency of the inter-particle interaction d) = y/kJTn, and the frequencies of the free lateral and normal oscillations of the top plate Qy x =... 

In order Eq. (1.184) may be further worked out the form of the interaction function should be specified it act as an inter-particle interaction energy, being thus related with the inter-particle bilocal potential itself in fact it was found having the functional form (Vetter, 1997 Putz, 2011b-c, 2012a,c) ... 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

The average dissociation time (7") for doublets of equal-size particles, in air at 1 atm and 298 K, has been calculated using Eq. [38]. The overall interaction potential between the constituent particles has been obtained by the integration of the Lennard-Jones 6-12 inter-molecular potential, under the assumption of pairwise additivity. The expression for this overall interaction potential in terms of the Hamaker constant can be found elsewhere (1). All the calculations have been performed for a Hamaker constant of 10 11 erg. The time scale of Brownian motion f-1 of the constituent particles was calculated from... 

The main results are that a reduction in particle size at one position of the array increases the potential at this point which may lead, at least, to localization, i. e. the single excess electron in the array might be trapped. At a packing defect, which affects the inter-particle capacitance at one point and acts like an inhomogenity, the soliton will interact with its mirror-image soliton (or anti-soliton) and will therefore be attracted. Concerning the practical use of this method, it was emphasized that the total reflection amplitude obtained from these calculations is directly related to the Landauer resistance,and reflects the electrical characteristics of such multijunction arrays. 

As aheady said, apart from the initial conditions, the only input information in a computer simulation are the details of the inter-particle potential, almost always assumed to be pair-wise additive. Usually in practical simulations, in order to economize the computing time, the interaction potential is truncated at a separations r (the cut-off radius), typically of the order of three molecular diameters. Obviously, the use of a cut-off sphere of small radius is not acceptable when the inter-particle forces are very long ranged. 

It should be noted that Eq. (39) is derived from DFT, and it presents the local expression similar to RHNC in Eq. (35). By noting that at bulk (r) = p r)/p recovers to g i) and the external potential recovers to the inter-molecular interaction (in that case the solute is virtually a solvent particle, and the system is locally inhomogeneous but represents a homogeneous system), both equations are essentially equivalent. This suggests that DFT and integral equation theory are closely related and moreover the closure relation for the integral equation can be derived from DFT. 

The DLVO model (named after its principal creators, Deqaguin, Landau, Verwey and Overbeek) is the most widely used to describe inter-particle surface force potential (1,2). It assumes that the total inter-particle potential is the sum of an attractive van der Waals force and a repulsive double-layer force. The repulsive force due to the double-layer coulombic interaction between equal spheres separated by a distance D generates a positive potential energy Vr. If the radius r of the spheres is large compared to the double-layer thickness 1/k (Kr l, with K the Debye-Hiickel parameter), Fr is described approximately by ... 

Note that stays for the Hohenberg-Kohn functional, V(r) is the external applied potential, p is the chemical potential, W die inter-particle pair interaction, the kinetic energy of the real system, and J(r) the external... 

Figure D.2. Effective interaction potential for a pair of bosons or fermions the inter-particle distance r is measured in units of the thermal wavelength A,.

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