Other Interparticle Forces

In this analysis, we will disregard other interparticle forces, such as Van der Waals attraction and electrostatic repulsion, although, in principle, such contributions could be included. 

The higher order dependence on for spheres, in eq. 10.2.15, is shown in Figure 10.2.3. This departure from the Einstein equation is due to hydrodynamic interactions between spheres and to other interparticle forces. We will examine these effects in Section 10.4, but first we look at the influence of particle shape on the rheology of dilute suspensions. 

New experimental techniqnes for the direct measurement of interparticle forces are now available and can be nsed to nnderstand the physicochemical factors that control adhesion, coating phenomena, tribology, and others. 

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. 

Once nanoparticles have been formed, whether in an early state of growth or in a more or less final size, their fate depends on the forces between the individual particles and between particles and solid surfaces in the solution. While particles initially approach each other by transport in solution due to Brownian motion, convection, or sedimentation, when close enough, interparticle forces will determine their final state. If the dominant forces are repulsive, the particles will remain separate in colloidal form. If attractive, they will aggregate and eventually precipitate. In addition, they may adsorb onto a solid surface (the substrate or the walls of the vessel in which the reaction is carried out). For CD, both attractive particle-sur-... 

Thus, within the context of the Newtonian force atom and the caloric theory of heat, solids, liqitids, and gases were all viewed as organized arrays of particles produced by a static equilibrium between the attractive interparticle forces, on the one hand, and the repulsive intercaloric forces, on the other. The sole difference was that the position of eqitilibriitm became greater as one passed from the solid to the liqitid to the gas, due to the increasing size of the caloric envelopes siuToittrding the component atoms (Figures 5 and 6). 

Let us now consider coagulation of particles in the absence of any repulsive barrier. In addition, we assume that, although there are no interparticle forces that contribute to the transport of particles toward each other, there is sufficient attraction between the particles on contact for them to form a permanent bond. As early as 1917, Smoluchowski formulated the equations for the collision rate for particles transported by diffusion alone (Smoluchowski 1917), and we develop the same idea here. 

Such small particles usually are generated by air-jet micronization and less frequently by controlled precipitation or spray drying. As bulk powder, they usually tend to be very cohesive and exhibit poor flow and insufficient dispersion because of large interparticle forces such as van der Waals and electrostatic forces (Zeng et al. 2001 Podczeck 1998 Hickey et al. 1994). The control of sufficient powder flow and deaggregation (dispersion) is thus of utmost importance to ensure efficient therapy with a dry-powder aerosol. Two different formulation approaches are used currently in marketed DPI preparations to fulfill the requirements. Most often, coarse particles of a pharmacologically inactive excipient, usually a-lactose monohydrate, are added that act as a carrier and provide sufficient powder flow to the mixture. Other carbohydrates, amino acids, and phospholipids have been suggested frequently (Crowder et al. 2001). 

Double layers in non-polar media recur in colloid stability (Volume IV). The slow decay dv /dr (or dy//dx) means that the field strength is low. and so is the interparticle force. On the other hand, the range of the Interaction is very high, so that even in dilute sols the particles feel each other s presence. Absence of screening means that the pair interaction between particles is completely described by Coulomb s law. In emulsions and at oil-water Interfaces a "double diffuse double layer may be formed, which is more extended in the oil phase K... 

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)... 

On the other hand, solids are characterized by a very ordered structure in which each ion or molecule is surrounded by a fixed number of neighbors whose nature and orientation are determined by the interparticle forces in the crystal. These may be chiefly ion-ion interactions, as in an ionic crystal, or intermolecular forces, as in a molecular crystal. Because of the high state of order in crystals it is a reasonably straightforward problem to calculate their thermodynamic properties on the basis of quite simple statistical mechanical models. 

Just as a gliding air-hockey puck models a reboimding gas particle, Figure 10.7 shows that marbles in a beaker model some behaviors of liquids. When liquids form puddles, interparticle forces maintain their volmne but not their shape. The particles of a liquid can slide past each other, but they are so close together that they don t move as straight or as smoothly as an air-hockey puck. When you try to walk across a crowded sidewalk, you can t move quickly in a straight line, either. 

Problem 7-24. Sedimentation of a Colloidal Aggregate. Colloidal particles often aggregate because of London-van der Waals or other attractive interparticle forces unless measures are taken to stabilize them. The aggregation kinetics are such that the aggregate formed has a fractal dimension Df, which is often less than the spatial dimension. The fractal dimension measures the amount of mass in a sphere of radius R, i.e., mass R D<. For a fractal aggregate composed of Aprimary particles of radius Op with mass mp, estimate the sedimentation velocity of the aggregate when the Reynolds number for the motion is small. What is the appropriate Reynolds number ... 

Tadros (1986) describes four types of interparticle forces hard sphere, soft (electrostatic), van der Waals, and steric. Hard-sphere interactions, which are repulsive, become significant only when particles approach each other at distances slightly less than twice the hard-sphere radius. They are not commonly encountered. 

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