Interparticle forces potential

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. 

There is a wealth of microstructural models used for describing nonlinear viscoelastic responses. Many of these relate the rheological properties to the interparticle forces and the bulk of these consider the action of continuous shear rate or stress. We will begin with a consideration of the simplest form of potential, a hard rigid sphere. 

This is for a simple cubic lattice. As we include interparticle forces our ability to describe the system other than by numerical simulation becomes progressively more difficult to achieve. At present quasi-hard spheres at moderate to large volume fractions can only be modelled by analytical expressions that are empirical in origin. Simple models are available for other forms of interaction potentials. 

Stabilising a colloidal suspension implies that the total interparticle potential decreases with increasing inter particle distance. The different kinds of stabilisation all use some of the above-mentioned interparticle forces. 

The dimerization is easily understood considering the optical potential created by the trapping laser. Figure 18.2b shows the calculated optical potential experienced by a silver nanoparticle that is fi ee to move in a Gaussian laser focus at a wavelength of 830 nm. The particle is also affected by the optical interparticle force from an immobilized silver particle located at different separations from the laser focus. It is clear that a deep potential minimum is induced when the trapped particle approaches the immobilized one, giving rise to spontaneous optical dimerization and a SERS hot spot in the optical trap. Note that the two particles are expected to ahgn parallel to the laser polarization in this case, as has been demonstrated experimentally recently [88]. 

An example of the effective force (derivative of the pair potential with respect to the separation) experienced by the two approaching macroions is shown in Fig. 3b. The oscillatory decay part of the force profile reflects in an effective way the impact of the discrete nature of the solvent on interparticle forces. The... 

Solvent friction is measured by the Stokes friction coefficient = 6 r)is H- The interparticle forces = — d/dr, U ( rj ) derive from potential interactions of particle i with all other colloidal particles U is the total potential energy. The solvent shear-flow is given by v ° (r) = yyx, and the Gaussian white noise force satisfies (with a,j8 denoting directions)... 

Stable Systems. The viscoelastic response of a concentrated noncoagulating suspension is strong when the average distance between the suspended particles is of the same order as the distance at which the interparticle repulsive forces become important. Hence, the viscoelastic behavior originates from the interparticle repulsive potential. Several studies have been carried out on hard sphere systems (72, 204, 205), steric systems (88, 94, 203, 206), and electrostatic systems (163,... 

It is interesting to note that, if the formation of a double layer is regarded as the adsorption of counter-ions by the charged particle, then the principles concerning the effect of adsorption on interparticle forces set out in Chapter 5 ean be applied. Thus, as two double layers overlap at constant potential, counter-ions are rejected from the space between the particles, the adsorption of counter-ions decreases, and, according to the general principles discussed in Chapter 5, the particles repel one another. 

A theoretical derivation of the Schulze-Hardy rule can be developed on the basis of the interparticle forces described in Sec. 6.2. Each of the three forces is associated with a potential energy that contributes additively to the total potential energy between two planar particle surfaces a distance d apart. If (p(d) is the total potential energy per unit area of planar surface, then... 

The classical DLVO theory of interparticle forces considers the interaction between two charged particles in terms of the overlap of their electric double layers leading to a repulsive force which is combined with the attractive London-van der Waals term to give the total potential energy as a function of distance for the system. To calculate the potential energy of attraction Va between solid spherical particles we may use the Hamaker expression ... 

A third important advantage of colloidal systems over atomic ones is that the interparticle forces can be varied readily via the electrolyte concentration and surface charge density on the particles. In general, the interparticle potential used in ordering studies is not the DLVO potential, because the separation between particles is significantly larger than the range of van der Waals forces, and this term is usually dropped. Instead a screened Coulomb potential is used, usually referred to as the Yukawa potential,... 

The realistic modefing, simulating millions of atoms and molecules, involves more heuristic approach to define the interparticle forces. Instead of integrating the Gauss equation (which is very demanding computationally for large systems), the potential function is... 

Adopting a different point of view, we tried to fit the variation of D versus (p at low (j) with existing theories taking into account the role of hydrodynamic interactions and interparticle forces. This has been done for microemulsions A and B, using Felderhof theory (11) with an interaction potential sum of hard sphere repulsion and W = A(2R/r), A = B. The agreement with experimental a values is quite satisfactory. 

Similarly, the rate coefficient for a thermal reaction occurring with the influence of a spherically symmetric potential V(r) can be calculated from equation (63) by relating the cross-section to the potential. A useful relationship from classical scattering dynamics [16] is found in terms of the impact parameter, b. The impact parameter is the distance of closest approach between two particles in the absence of an interparticle force. At large separation, the collision trajectories of two particles will be parallel straight lines, and the impact parameter is the perpendicular distance between the trajectories. The cross-section is given by equation (64),... 

Consider first the effect of interparticle forces. We return to the view of relative motion of the particle of volume x from the center of the particle of volume x. The interactive force F is generally described as the gradient of an interactive potential, say F(r), where r is the position vector of the (center of the) particle of volume x relative to the particle of volume x. Thus, F = — VK Neglecting acceleration effects, we may relate the above force to a steady velocity of the particle of volume x by dividing the force by the friction coefficient / = 6nia 3x /4ny. Note in particular that this procedure continues to neglect the viscous interaction between the two drops. The stochastic differential equation for the relative displacement of particle of volume x should now be written as... 

Competition between two different local particle arrangements, arising from either directional or core-softened isotropic forces, is usually deemed to be responsible for anomalous thermodynamic behavior. However, our results show that the same behaviors may also occur for isotropic interactions characterized by a repulsion that is only marginally softened and yields a single structure at a local level. Such potentials can be relevant in the realm of soft matter, where engineering interparticle forces is possible, and also for hard matter under extreme conditions, where pressure-triggered rearrangements of the crystal structure induce a partial... 

Figure 5.2 Schematic representations of interparticle potential energy (V) and force (f) versus particle surface to surface separation distance (D). (a) Energy versus separation distance curve for an attractive interaction. The particles will reside at the separation distance where the minimum in energy occurs, (b) Force versus separation distance for the attractive potential shown in (a). (The convention used in this book is that positive interparticle forces are repulsive.) The particles feel no force if they are at the equilibrium separation distance. An applied force greater than a maximum is required to pull the particles apart, (c) Energy versus separation distance curve for a repulsive interaction. When the potential energy barrier is greater than the available thermal and kinetic energy the particles cannot come in contact and move away from each other to reduce their energy, (d) Force versus separation distance for the repulsive potential shown in (c). There is no force on the particles when they are very far apart. There is a maximum force that must be exceeded to push the particles into contact...

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