Inter-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... 

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 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. 

In Fig. 42, the full-width at half maximum of the (narrower) exchange propagator provides an estimate of the effective diffusion coefficient of water molecules moving between the pore space of the catalyst and the inter-particle space of the bed. In this example, the value is 2 x lO- m s which gives a lower limit to the value for the mass transfer coefficient of 4x 10 ms This value was obtained by defining a mass transfer coefficient as Djd where d is a typical distance traveled to the surface of the catalyst that we estimate as half a typical bead dimension (approximately 500 pm). This value of the mass transfer coefficient is consistent with the reaction occurring under conditions of kinetic as opposed to mass transfer control. 

Data analysis methods depend upon the level of order in the sample. The degree of order, in turn, depends upon the scale of distance on which the sample is viewed. For example, casein micelles show great variation in size (20 to 300 nm diameter) and so must be treated as a polydisperse system. However, the density variations ( submicelles ) within the whole micelle are much more uniform in size. They can be treated as a quasi-monodisperse system (Stothart and Cebula, 1982) and analyzed in terms of inter-particle interference (Stothart, 1989). 

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]. 

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. 

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. 

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 ... 

MECC separations are conducted in open capillaries, hence eddy diffusion is not problematic. However, the columns behave in many ways like packed columns, with the micelles functioning as uniformly sized and evenly dispersed packing particles. In packed columns, resistance to mass transfer in the mobile phase is reduced (i.e., efficiency improved) when smaller particles are used because the "diffusion distance" between particles is decreased. Average inter-micellar" distance is the analogous parameter in MECC. This distance can be decreased by increasing surfactant concentration. 

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]... 

The inter-particle distance used in the simulation is calculated using the minimum image convention. It dictates that the distance between two particles m and k is the smallest of all the possible distances between particle m and k including all the replica images of particle k. 

DLVO [2,3] theory estimates the repulsive and attractive force due to the overlap of electric double layers and London-van der Waals force in terms of inter particle distance, respectively. The summation of them gives the total interaction force and can be used for the interpretation of colloid stability in terms of the nature of interaction force-distance curve. If a small interparticle separation (H) is assumed, van der Waals forces for a sphere and substrate can be expressed to... 

For collisions of antiprotons with atomic hydrogen, a quasidipole is formed during the collisions. The dipolar antiproton-proton system does not support bound states for inter-particle distances below 0.64 a.u. [52]. For finite velocities and larger impact parameters b (in the figure, Z = 1) there is still a significant ionization contribution. As can be observed in the figure at... 

The inter-particle force F can be computed as / = A]idol l2Hl), where Ah is the Hamaker parameter for the liquid-particle system and is the distance between two primary particles. The coordination number is based on experimental observation and can be calculated as kc 150p, where 0p is the volume fraction of solid within the aggregates. In the case of compact (or solid) particles 0p is close to unity, whereas in the case of fractal aggregates 0p can be determined once the fractal dimension T)f of the aggregates is known 0p = (0.414T)f - 0.21 l)(r/p/(io) , where dp is the size of the particle and do is the size of the primary particle (Vanni, 2000b). 

The linear approximation for the velocity differences in (2.67) is only valid while <5(t) Lyapunov exponents can be defined as the growth rate of the inter-particle distance as... 

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