Interparticle Potentials

Several potential interparticle forces can be distinguished. Dispersion (i.e., London-van der Waals) forces always exist because of interactions between induced dipoles in the molecules of neighboring particles (Mahanty and Ninham, 1976). Integration of the interactions between all induced dipoles in two bodies results in an expression for the total attraction force. It is characterized by the Hamaker constant A, which depends on the material composition of the particle. In principle, it can be calculated ffom the ffequency-depenrtent polarizabUity of the material, and it determines the attraction forces in vacuum. For particles with Hamaker constant Ap in a. medium with Hamaker constant A/, an effective Am can be calculated from... 

Marlow and Rowell discuss the deviation from Eq. V-47 when electrostatic and hydrodynamic interactions between the particles must be considered [78]. In a suspension of glass spheres, beyond a volume fraction of 0.018, these interparticle forces cause nonlinearities in Eq. V-47, diminishing the induced potential E. 

The fimctiong(ri is central to the modem theory of liquids, since it can be measured experimentally using neutron or x-ray diffraction and can be related to the interparticle potential energy. Experimental data [1] for two liquids, water and argon (iso-electronic with water) are shown in figure A2.4.1 plotted as a fiinction ofR = R /a, where a is the effective diameter of the species, and is roughly the position of the first maximum in g (R). For water, a = 2.82 A,... 

The relationship between g(r) and the interparticle potential energy is most easily seen if we assume that the interparticle energy can be factorized into pairwise additive potentials as... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

There is a very convenient way of writing the Hamiltonian operator for atomic and molecular systems. One simply writes a kinetic energy part — for each election and a Coulombic potential Z/r for each interparticle electrostatic interaction. In the Coulombic potential Z is the charge and r is the interparticle distance. The temi Z/r is also an operator signifying multiply by Z r . The sign is - - for repulsion and — for atPaction. 

Chemical Grafting. Polymer chains which are soluble in the suspending Hquid may be grafted to the particle surface to provide steric stabilization. The most common technique is the reaction of an organic silyl chloride or an organic titanate with surface hydroxyl groups in a nonaqueous solvent. For typical interparticle potentials and a particle diameter of 10 p.m, steric stabilization can be provided by a soluble polymer layer having a thickness of - 10 nm. This can be provided by a polymer tail with a molar mass of 10 kg/mol (25) (see Dispersants). 

The density functional approach has also been used to study capillary condensation in slit-like pores [148,149]. As in the previous section, a simple model of the Lennard-Jones associating fluid with a single associative site is considered. All the parameters of the interparticle potentials are chosen the same as in the previous section. Our attention has been focused on the influence of association on capillary condensation and the evaluation of the phase diagram [42]. 

Let us consider a simple model of a quenched-annealed system which consists of particles belonging to two species species 0 is quenched (matrix) and species 1 is annealed, i.e., the particles are allowed to equlibrate between themselves in the presence of 0 particles. We assume that the subsystem composed of 0 particles has been a usual fluid before quenching. One can characterize it either by the density or by the value of the chemical potential The interparticle interaction Woo(r) does not need to be specified for the moment. It is just assumed that the fluid with interaction woo(r) has reached an equlibrium at certain temperature Tq, and then the fluid has been quenched at this temperature without structural relaxation. Thus, the distribution of species 0 is any one from a set of equihbrium configurations corresponding to canonical or grand canonical ensemble. We denote the interactions between annealed particles by Un r), and the cross fluid-matrix interactions by Wio(r). 

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... 

In the simulations the maxima and minima of n y are shifted to slightly smaller porewldths compared to predictions of the theory. This trend Is consistent with the fact that the 6-12 Lennard-Jones potential Is not Infinitely repulsive at an Interparticle separation of (7, whereas the 6-oo potential Is Infinitely repulsive at a. 

Fig. 1 Illustration of the DLVO theory interaction of two charged particles as a function of the interparticle distance (attractive energy curve, VA, repulsive energy curve, VR and net or total potential energy curve, Vj).

Many investigators of steric stabilization have measured colloidal stability without taking the effort to find out whether the stability actually resulted from electrostatic stabilization. In many published articles it has been concluded that steric stabilization had been attained and further study showed this was not the case. One such example is a recent paper on "steric" stabilization by an additive of the same type used in this work. (12) The published photograph shows the silica particles in oil stabilized at interparticle separations several times the distances provided by the adsorbed films no electrical measurements had been made, but it they had, this particular dispersant would have provided about -200 mV of zeta-potential and given excellent electrostatic repulsion. The reader should be wary of any claims of steric stabilization unless the electrostatic contribution has been measured. 

As the temperature is increased, the interparticle potential kx1 2/2 becomes more and more important until a critical value Tcr V/(2tt)2 is reached (we take the Boltzman constant equal to unity), when the kinetic energy is large enough to overcome the on-site potential barrier. At this point low frequency appears and this happens at the critical temperatures Tcr = 0.13 for V = 5 (left), and Tcr = 0.025 for V = 1 (right). This is in quite good agreement with the data of Fig.8. 

Various anionic compounds such as halides, carboxylates or polyoxoanions, generally dissolved in aqueous solution, can establish electrostatic stabilization. Adsorption of these compounds onto the metallic surface and the associated countercations necessary for charge balance produces an electrical double-layer around the particles (Scheme 9.1). The result is a coulombic repulsion between the particles. At short interparticle distances, if the electric potential associated with the double layer is sufficiently high, repulsive forces opposed to the van der Waals forces will be significant to prevent particle aggregation. 

All matter seeks to minimize its energy and entropy see Chapter 4. This concept explains, for example, why a ball rolls down a hill, and only stops when it reaches its position of lowest potential energy. These interparticle interactions form for a similar reason. 

There are other approximations in the literature that may be more appropriate for particular conditions.12,27,29,35 In many colloidal systems the value of <5 is 2 nm < <5 < 10 nm and until h < 2S the pair potential will be dominated by the other types of interparticle interactions, i.e. it is of short range, albeit very steep. 

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