Stress dependence, diffusion coefficient

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

For the turbulent motion in a tube, the mass transfer coefficient k is proportional to the diffusion coefficient at the power of 2/3. It is easy to realize by inspection that this value of the exponent is a result of the linear dependence of the tangential velocity component on the distance y from the wall. For the turbulent motion in a tube, the shear stress t r0 = const near the wall, whereas for turbulent separated flows, the shear stress is small at the wall near the separation point (becoming zero at this point) and depends on the distance to the wall. Thus, the tangential velocity component has, in the latter case, no longer a linear dependence on y and a different exponent for the diffusion coefficient is expected. For separated flows, it is possible to write under certain conditions that [90]... 

Another distinctive feature of strong tunnelling recombination could be seen after a step-like (sudden) increase (decrease) of temperature (or diffusion coefficient - see equation (4.2.20)) when the steady-state profile has already been reached. Such mobility stimulation leads to the prolonged transient stage from one steady-state y(r,T ) to another y(r,T2), corresponding to the diffusion coefficients D(T ) and >(72) respectively. This process is shown schematicaly in Fig. 4.2 by a broken curve. It should be stressed that if tunnelling recombination is not involved, there is no transient stage at all since the relevant steady state profile y(r) — 1 - R/r, equation (4.1.62), doesn t depend on >( ). 

Water uptake in plasticized polyvinylchloride based ion selective membranes is found to be a two stage process. In the first stage water is dissolved in the polymer matrix and moves rapidly, with a diffusion coefficient of around 10 6 cm2/s. During the second stage a phase transformation occurs that is probably water droplet formation. Transport at this stage shows an apparent diffusion coefficient of 2 x 10 8 cm2/s at short times, but this value changes with time and membrane addititives in a complex fashion. The results show clear evidence of stress in the membranes due to water uptake, and that a water rich surface region develops whose thickness depends on the additives. Hydrophilic additives are found to increase the equilibrium water content, but decrease the rate at which uptake occurs. 

In all expressions the Einstein repeated index summation convention is used. Xi, x2 and x3 will be taken to be synonymous with x, y and z so that o-n = axx etc. The parameter B will be temperature-dependent through an activation energy expression and can be related to microstructural parameters such as grain size, diffusion coefficients, etc., on a case-by-case basis depending on the mechanism of creep involved.1 In addition, the index will depend on the mechanism which is active. In the linear case, n = 1 and B is equal to 1/3t/ where 17 is the linear shear viscosity of the material. Stresses, strains, and material parameters for the fibers will be denoted with a subscript or superscript/, and those for the matrix with a subscript or superscript m. 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

In this formulation, neither stress state, nor temperature (and thus, temperature dependent material characteristics, such as D or Q) are required to be stationary, but can be time dependent. To simplify this preliminary study, diffusion coefficient, D, as well as temperature is considered to be spatially uniform, i.e., their gradients are zero, although this is not an essential restriction. This leads to the equation of stress-assisted hydrogen diffusion in terms of concentration ... 

Of course, eventually, particles will reappear, through the equation = Po + RTln the existence of R depends on the existence of particles. But a theory of stress-driven deformation of a continuum does not require particles, even with stress-driven self-diffusion coefficients for viscosity and self-diffusion are the only things required. 

Here Qt is absorbed weight of water at a time t, Qs is the weight of water at saturation, d is gel thickness, and D is the diffusion constant. Figure 2 shows diffusion constants and their temperature dependence. Many data are plotted on this graph. Fluorinert and freon immersion samples were significantly swelled by their liquids. Swelling behavior would have an effect on the diffusion coefficients due to induced stresses on the gel networks. These test values contained relatively large errors. The immersion tests at 44°C for freon and at 140°C for fluorinert were also done. 

In crystals, impurities can take simple configurations. But depending on their concentration, diffusion coefficient, or chemical properties and also on the presence of different kind of impurities or of lattice defects, more complex situations can be found. Apart from indirect information like electrical measurements or X-ray diffraction, methods such as optical spectroscopy under uniaxial stress, electron spin resonance, channelling, positron annihilation or Extended X-ray Absorption Fine Structure (EXAFS) can provide more detailed results on the location and atomic structure of impurities and defects in crystals. Here, we describe the simplest atomic structures more complicated structures are discussed in other chapters. To explain the locations of the impurities and defects whose optical properties are discussed in this book, an account of the most common crystal structures mentioned is given in Appendix B. 

The last expression, which follows directly from the Doi-Edwards expression for stress, provides a link between mechanical behavior and the center-of-gravity diffusion coefficient. The tube parameter N can be eliminated between Eqs. 21 and 37, and D can be estimated from independent experiments as discussed earlier. Confirmation of this relationship11, the observed separability of strain and time dependences in step strain relaxation experiments23 as required by Eq. 33 ... 

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