Interparticle distances

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... 

The three internal coordinates aie expressed as combinations of squares of the interparticle distances ... 

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. 

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... 

Figure 7 Notched impact strength vs. interparticle distance for PBT-maleic anhydride grafted EOR blends. Source Ref. 58.

The use of DNA as a template to fabricate mesoscale structures was also demonstrated in a recent work of Torimoto and coworkers. They used preformed, positively charged 3-nm CdS nanoparticles with a thiocholine-modified surface to be assembled into chains by using the electrostatic interaction between positively charged nanoparticle snr-faces and the phosphate groups of DNA. As determined by TEM analysis, the CdS nanoparticles were arranged in a qnasi-one-dimensional dense packing. This revealed interparticle distances of about 3.5 nm, which is almost equal to the height of one helical tnm of the DNA double strand [98]. 

At short interparticle distances, the van der Walls forces show that two metallic particles will be mutually attracted. In the absence of repulsive forces opposed to the van der Walls forces the colloidal metal particles will aggregate. Consequently, the use of a protective agent able to induce a repulsive force opposed to the van der Walls forces is necessary to provide stable nanoparticles in solution. The general stabihzation mechanisms of colloidal materials have been described in Derjaguin-Landau-Verway-Overbeck (DLVO) theory. [40,41] Stabilization of colloids is usually discussed... 

Interpretation of pubhshed data is often comphcated by the fact that rather complex catalytic materials are utilized, namely, poly disperse nonuniform metal particles, highly porous supports, etc., where various secondary effects may influence or even submerge PSEs. These include mass transport and discrete particle distribution effects in porous layers, as confirmed by Gloaguen, Antoine, and co-workers [Gloaguen et al., 1994, 1998 Antoine et al., 1998], and diffusion-readsorption effects, as shown by Jusys and co-workers for the MOR and by Chen and Kucemak for the ORR [Jusys et al., 2003 Chen and Kucemak, 2004a, b]. Novel approaches to the design of ordered nanoparticle arrays where nanoparticle size and interparticle distances can be varied independently are expected to shed hght on PSEs in complex multistep multielectron processes such as the MOR and the ORR. 

In general, moments of inertia are determined relative to an axis of rotation. In this case the axis is perpendicular to the interparticle distance R and passes through the center of mass. Thus, we have... 

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).

In the IRT model, reactions of products can be incorporated indirectly and approximately by one of the following procedures (Green et al, 1987) (1) the diffusion approach, (2) the time approach, or (3) the position approach. The diffusion approach is conceptually the simplest. In it, the fundamental entity is the interparticle distance, which evolves by diffusion independently of other such distances along with IRT. Thus, if the interparticle distance was at t = 0, that at time t is simulated as f = r + R3, where R3 is a three-dimensional normally distributed random number of zero mean and variance 2D t. When reaction occurs at t, the product inherits the position of one of the parents taken at random. The procedure is then repeated with new interparticle distances so obtained. 

Small metal particles are unstable with respect to agglomeration to the bulk. At short interparticle distances, two particles would be attracted to each other by van der Waals forces and, in the absence of repulsive forces to counteract this attraction, an unprotected sol would coagulate. To counteract this, stabilization can be achieved in two ways electrostatic stabilization and steric stabilization. 

Figure 10.9. (a) Schematic structure of a silicon quantum dot crystal and (b) its calculated electronic structure as a function of interparticle distance H. The size of the nanoparticles is L = 6.5 nm. At small H, a splitting of the quantized energy levels of single dots results in the formation of three-dimensional minibands. Reproduced from Ref. 64, Copyright 2001, with permission from the American Institute of Physics. 

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. 

Every gas consists of particles, whether as atoms (such as neon) or as molecules (such as methane). To a relatively good first approximation, any atom can be regarded as a small, incompressible sphere. The reason why we can compress a gas relates to the large separation between the gas particles. The first effect of compressing a gas is to decrease these interparticle distances. 

In order to obtain a finely sized dispersed phase in the PET matrix, the use of reactive compatibilization has been found to be important. Small dispersed rubber particles and a small interparticle distance are necessary to induce high toughness. For effective rubber toughening of PET, it is important that the rubber domains be less than 3 im in diameter (and preferably less than 1 xm) and that the interparticle distance be between 50-300 nm. 

The following equation relates the interparticle distance (ID) to the volume fraction of the impact modifier (4)) and the weight-average particle size (dlT) [28] ... 

Figure 14.12 shows that the impact strength increases sharply as the interparticle distance is reduced. The toughness increases as the interparticle distance decreases to a critical size, but becomes lower again as the distance becomes too small. It can be seen that the critical interparticle distance for PET is 50 nm. 

Pecorini and Calvert [28] attribute the role of small particles and a small interparticle distance to inducing high toughness in PET by promoting massive shear yielding in the matrix. Their study showed that the non-reactive impact modifier gives a system in which the rubber phase is not well dispersed. It was shown that this is not effective in toughening PET at levels of either 10 or 20%. The... 

Figure 14.12 Notched Izod impact strength data (on crystallized PET) for samples of toughened polymer as a function of the ratio of interparticle distance O, amorphous x, crystalline [28]. Reprinted with permission from Pecorini, T. J. and Calvert, D., in Toughening of Plastics - Advances in Modelling and Experiments, Pearson, R. A., Sue, H.-J. and Yee, A. F. (Eds), ACS Symposium Series, 759, American Chemical Society, Washington, DC, 2000, Ch. 9, pp. 141-158. Copyright (2000) American Chemical Society...

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