Diffusion hydrodynamic theory

Extension of the hydrodynamic theory to explain the variation of detonation velocity with cartridge diameter takes place in two stages. First, the structure of the reaction zone is studied to allow for the fact that the chemical reaction takes place in a finite time secondly, the effect of lateral losses on these reactions is studied. A simplified case neglecting the effects of heat conduction or diffusion and of viscosity is shown in Fig. 2.5. The Rankine-Hugoniot curves for the unreacted explosive and for the detonation products are shown, together with the Raleigh line. In the reaction zone the explosive is suddenly compressed from its initial state at... 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

Monteiro, C. Herve du Penhoat, C. Translational Diffusion of Dilute Aqueous Solutions of Sugars as Probed by NMR and Hydrodynamic Theory. J. Phys. Chan. A 2001, 105, 9827-9833. 

Callaghan and Pinder48,49 used the PGSE method in a detailed examination of the diffusion of linear polystyrene molecules dissolved in CC14. They applied standard dilute hydrodynamic theory to self-diffusion (as distinct from mutual diffusion) and identified the lowest-order concentration dependence of D with the coefficient kF, writing... 

The decrease in the alpha factor to values below a = 1 can be due to a decrease in either kL or a or both. Two theories are commonly used to explain the reduction in kp. the barrier effect and the hydrodynamic effect. In the barrier theory, the presence of the surfactants at the phase interface creates an additional resistance to mass transfer due to diffusion through the surfactant layer. In the hydrodynamic theory, the layer of surfactant molecules at the gas-liquid interface depresses the hydrodynamic activity (Gurol and Nekouinaini, 1985). 

Other than dynamical correlations, transport properties have also been derived using hydrodynamic theory. In hydrodynamics the diffusion of a tagged particle is defined by the Stoke-Einstein relation that is given by the following well-known expression ... 

As seen from Eq. (8), Im II(kw) is finite for k — 0, whereas cjk vanishes in this limit. Therefore the spin Green s function has a purely imaginary, diffusive pole near the T point, in compliance with the result of the hydrodynamic theory [12]. 

In the hydrodynamic theory, the diffusion coefficient of a solute molecule A or single particle through a stationary medium B, DAB, is given by the Nemst-Einstein equation ... 

The characteristics of pore structure in polymers is a key parameter in the study of diffusion in polymers. Pore sizes ranging from 0.1 to 1.0 pm (macroporous) are much larger than the pore sizes of diffusing solute molecules, and thus the diffusant molecules do not face a significant hurdle to diffuse through polymers comprising the solvent-filled pores. Thus, a minor modification of the values determined by the hydrodynamic theory or its empirical equations can be made to take into account the fraction of void volume in polymers (i.e., porosity, e), the crookedness of pores (i.e., tortuosity, x), and the affinity of solutes to polymers (i.e., partition coefficient, K). The effective diffusion coefficient, De, in the solvent-filled polymer pores is expressed by ... 

Rotational Rdaxation.- As far as small molecules having dimensions of a few angstroms are concerned, the rotational relaxation time t, under ordinary conditions (particularly in aqueous solution at room temperature) has the order of magnitude of 10 —10 s. This corresponds to dielectric dispersion in the microwave range of 1 —10 GHz. Macromolecular particles display dielectric relaxation far below these frequencies. Sudi behaviour is readily e>q>lained by the strong dependence of r, on the lei h of the dipolar axis of the molecule. By calculation of the rotational diffusion coefiident Dx according to hydrodynamic theory and inserting the result in (19) one obtains for spherically shaped dipoles ... 

The theoretical description of translational diffusion in a lipid bilayer depends on the size of the diffusing particle. Theoretical descriptions based on fluid hydrodynamic theory (51, 52) have been shown to be applicable to particles whose radius in the plane of the bilayer is significantly larger than the radius of the lipid molecules that constitute the bilayer, in which case the diffusion coefficient may be given by ... 

The influence of temperature on diffusion coefficients of solutes in liquids has been studied in less detail. Diffusion coefficients often can be estimated from viscosity measurements. Hydrodynamic theory relates self-diffusion coefficients to the viscosity by... 

Hydrodynamic theory [67], based on Stokes-Einstein equation, postulates that solute is represented by a very large sphere in comparison with the surrounding small liquid phase molecules. Solute mobility, and thus its diffusion coefficient, depends on the frictional drag exerted by liquid phase molecules. For heterogeneous gels (rigid polymeric chains), Cukier [85] suggests... 

To evaluate the solute diffusion coefficient in the stationary phase, and the solute partition coefficient, fCq, a model for the pore is required. A simple model where the pore is considered as an infinitely long cylinder and the solute is a rigid sphere adequately describes the elution process [58]. Using this model, Dg, the solute diffusivity within the porous particles, can be estimated from the hydrodynamic theory of hindered diffusion [59] ... 

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