Particle distance

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... 

A next step is to consider the surface-particle distance z and curvature (interfacial radius R) dependence of the interactions [1341, for which approximate expressions are ... 

For linear molecules, the coulombic potential is unchanged (because the set of all inter-particle distances are unchanged) by rotations about the molecular axis (the z axis) ... 

The tip-particle distance, using (Vbias = 1 V, /tmmei = 1 nA) as tunnel parameters does not correspond to that needed to observe the Coulomb staircase. The particle-substrate distance is fixed by the coating with dodecanethiol. Hence the two tunnel junctions are characterized by fixed parameters. Similar Coulomb blockade behavior has been observed [58,59]. 

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... 

Of course the cusp can be represented by including the interparticle distances in a trial wave functions, most simply by means of Jastrow factors, exponentials of the inter-particle distances. But the problems of integral evaluation with such fac-... 

X-ray scattering (SAXS) plots of nanoparticle assembly indicating systematic control of inter-particle distance by dendrimer generations. Reprinted with permission from Frankamp, Boal, et al. (2002). Copyright 2002 American Chemical Society. 

Fig. 3. Comparison the one- (Eq. 24) and the two dimensional model (Lin et al. [44]) results as a function of dispersed phase holdup at different values of particle distance from the interface...

Similar to zero-dimensional metal nanoparticles, most of the work on one-dimensional metal nanostructures focuses almost exclusively on gold nanorods. The high interest in anisometric gold nanoclusters arises from their unique optical and electronic properties that can be easily tuned through small changes in size, structure (e.g., the position, width, and intensity of the absorption band due to the longitudinal surface plasmon resonance is strongly influenced by the shell as well as the aspect ratio of the nanorods), shape (e.g., needle, round capped cylinder, or dog bone), and the inter-particle distance [157]. 

Figure 3.6 Radiation envelope for light scattered from small particles. Distances from the origin of the dotted, dashed and smooth lines represent the relative intensities of the horizontally polarised component, vertically polarised component and total scattered light, respectively...

Adhesion and cohesion forces in the immobile liquid films between individual primary powder particles. These forces may not contribute significantly to the final granule strength once the granules have been dried. These forces arise from the thin liquid layers around particles that may be generated due to moisture uptake by the powders. This immobile liquid film tends to decrease the inter-particle distances leading to an increase in the van der Waals forces between particles. 

Uhlemann and Mori [93] described the calculation of the average particle velocity wp within the fluidized bed. The impact time of a particle can be calculated with the help of particle distance ap and average particle velocity. Analogous to the previous derivations, the probability of destruction Pq> of the wetted particles surface - and thus of the liquid film - can be calculated by the ratio of the wetted surface to the total surface of the particle. 

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. 

For the above mentioned FePt particles, the particle diameter is clearly smaller than the critical particle size given by Eq. (8) for coherent rotation. Furthermore the strength of the magnetostatic interaction field acting on nearest neighbor particles is only about 2% of the anisotropy field for a particle distance of 2 nm. Thus the Stoner-Wohlfarth theory can be applied. 

Consider a one-dimensional (usually almost infinite) set of N atoms, molecules, or point masses, all equally spaced at inter-particle distances d along the real-space coordinate x, with Bom-von Karman periodic boundary conditions for the potential energy ... 

Eulerian equations for the dispersed phase may be derived by several means. A popular and simple way consists in volume filtering of the separate, local, instantaneous phase equations accounting for the inter-facial jump conditions [274]. Such an averaging approach may be restrictive, because particle sizes and particle distances have to be smaller than the smallest length scale of the turbulence. Besides, it does not account for the Random Uncorrelated Motion (RUM), which measures the deviation of particle velocities compared to the local mean velocity of the dispersed phase [280] (see section 10.1). In the present study, a statistical approach analogous to kinetic theory [265] is used to construct a probability density function (pdf) fp cp,Cp, which gives the local instantaneous probable num-... 

Fig. 5 a Nanoparticle assembly using PAMAM dendrimers. b Small-angle X-ray scattering plots, demonstrating increasing inter-particle distances with increasing dendrimer generation. Reprinted with permission from [76]... 

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