Sliding diffusion

The whole problem of computing pressure distributions in particulate packings is one of great complexity. In addition to the fact that we are unable to deal with a material whose apparent density is not uniform, we must consider added difficulties such as diffusion, sliding friction, deformation of individual particles, cohesive forces, and perhaps others. The quantitative relationships of these factors to particle size must remain empirical for the time being. In the paragraphs to follow we shall be concerned only with a limited theory of the problem of particles under pressure. 

It is known that even condensed films must have surface diffusional mobility Rideal and Tadayon [64] found that stearic acid films transferred from one surface to another by a process that seemed to involve surface diffusion to the occasional points of contact between the solids. Such transfer, of course, is observed in actual friction experiments in that an uncoated rider quickly acquires a layer of boundary lubricant from the surface over which it is passed [46]. However, there is little quantitative information available about actual surface diffusion coefficients. One value that may be relevant is that of Ross and Good [65] for butane on Spheron 6, which, for a monolayer, was about 5 x 10 cm /sec. If the average junction is about 10 cm in size, this would also be about the average distance that a film molecule would have to migrate, and the time required would be about 10 sec. This rate of Junctions passing each other corresponds to a sliding speed of 100 cm/sec so that the usual speeds of 0.01 cm/sec should not be too fast for pressurized film formation. See Ref. 62 for a study of another mechanism for surface mobility, that of evaporative hopping. 

The exponential term appears for the same reason as it does in diffusion it describes the rate at which molecules can slide past each other, permitting flow. The molecules have a lumpy shape (see Fig. 5.9) and the lumps key the molecules together. The activation energy, Q, is the energy it takes to push one lump of a molecule past that of a neighbouring molecule. If we compare the last equation with that defining the viscosity (for the tensile deformation of a viscous material)... 

Whenever the polymer crystal assumes a loosely packed hexagonal structure at high pressure, the ECC structure is found to be realized. Hikosaka [165] then proposed the sliding diffusion of a polymer chain as dominant transport process. Molecular dynamics simulations will be helpful for the understanding of this shding diffusion. Folding phenomena of chains are also studied intensively by Monte Carlo methods and generalizations [166,167]. 

The recent theory of sliding diffusion (see Sect. 3.8.2) considers the possibility of growth in all dimensions. 

First we consider the theory initiated by Wunderlich and Mehta [145] which they termed molecular nucleation . The we turn to recent work by Hikosaka [146, 147] who introduced the idea of sliding diffusion . 

Fig. 3.20. A two-dimensional nucleus which may grow either by the addition of a new unit onto the lateral surface or by sliding diffusion within the nucleus (from [146] by permission of the publishers, Butter-worth-Heinemann Ltd. )...

The relative magnitude of these two activation free energies determines the size and shape of the critical nucleus, and hence of the resulting crystal. If sliding diffusion is easy then extended chain crystals may form if it is hard then the thickness will be determined kinetically and will be close to lmin. The work so far has concentrated on obtaining a measure for this nucleus for different input parameters and on plotting the most likely path for its formation. The SI catastrophe does not occur because there is always a barrier against the formation of thick crystals which increases with /. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

FIGURE 26.18 Theoretical temperature rise in the contact area of a pad sliding over a semi-infinite solid for different depths from the surface. Width 2b 2 mm, speed 3 m/s, pressure 2 Mp, p—l, heat conductivity 0.15 W/m/K, heat diffusivity 10 " m /s. 

The diffusion of one liquid into another also demonstrates molecular motion. Figure 2 shows that if a drop of ink is added to a beaker of still water, the color slowly but surely spreads throughout the water. The water molecules and the molecules that give ink its color move continuously. As they slide by one another, the ink molecules eventually become distributed uniformly throughout the volume of liquid. 

An analogy exists between mass transfer (which depends on the diffusion coefficient) and momentum transfer between the sliding hquid layers (which depends on the kinematic viscosity). Calculations show that the ratio of thicknesses of the diffnsion and boundary layer can be written as... 

Fig. 19 Our hybrid microrelaxation model. The solid circles are occupied by a polymer chain. The dashed lines show the new bond positions produced by a move consisting of kink generation and partial sliding diffusion along the chain. The arrows indicate the directions of monomer jumping [134]... 

Sliding diffusion Small angle X-ray scattering (SAXS) Topology... 

Fig-1 Schematic illustration of the crystallization and melting processes of polymers. The crystallization process corresponds to processes of disentanglement and chain sliding diffusion. The melting process is the reverse of the crystallization process. Between equilibrium melt and ideal crystal, there exists metastable melt and crystal. Cross marks indicate entanglement... 

Hikosaka presented a chain sliding diffusion theory and formulated the topological nature in nucleation theory [14,15]. We will define chain sliding diffusion as self-diffusion of a polymer chain molecule along its chain axis in some anisotropic potential field as seen within a nucleus, a crystal or the interface between the crystalline and the isotropic phases . The terminology of diffusion derives from the effect of chain sliding diffusion, which could be successfully formulated as a diffusion coefficient in our kinetic theory. 

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