Diffusion coefficients hydrodynamical theories

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

A reciprocal proportionality exists between the square root of the characteristic flow rate, t/A, and the thickness of the effective hydrodynamic boundary layer, <5Hl- Moreover, f)HL depends on the diffusion coefficient D, characteristic length L, and kinematic viscosity v of the fluid. Based on Levich s convective diffusion theory the combination model ( Kombi-nations-Modell ) was derived to describe the dissolution of particles and solid formulations exposed to agitated systems [(10), Chapter 5.2]. In contrast to the rotating disc method, the combination model is intended to serve as an approximation describing the dissolution in hydrodynamic systems where the solid solvendum is not necessarily fixed but is likely to move within the dissolution medium. Introducing the term... 

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. 

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... 

Debye—Smolucholowski rate coefficient does indicate that these reaction rates are in broad agreement (see Table 2). If it is accepted that some hydrodynamic repulsion occurs, less than implied by the Deutch and Felderhof analysis [70], but as suggested by Wolynes and Deutch [71], then the reaction radii are as listed in Table 2. However, if allowance is also made for the larger size of some reactants than the hydrated electron, then the agreement between experiment and theory becomes satisfactory. Nevertheless, the uncertainty of diffusion coefficients and the crystallographic or true reaction radii, R, let alone the rate of reaction of encounter pairs, makes a comparison of these relatively small effects difficult. 

Fig. 14. Prediction of solute permeability for solutes of different hydrodynamic radii in swollen gels. Permeability is calculated as the product of the partition coefficient, K, using the size exclusion theory of Schnitzer and the ratio of the solute diffusion coefficient in the gel, D, to its value in solution, D , using the theory of Yasuda et al. [123, 159, 160]...

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 ... 

If the hydroxonium ions migrated only hydrodynam-ically A° 85 cm2 would be expected, which value can easily be derived by using the -> Stokes law and the known values of the -+ self-diffusion coefficient of water, the radius of the ion and the - viscosity. (See also proton, - Eigen complexes, - Zundel complexes, charge transfer, - Dahms-Ruff theory.)... 

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... 

For noninteracting particles D b is + D, but as the particles approach each other, the relative diffusion coefficient becomes dependent on their spatial separation. In liquids for large particles this arises from hydrodynamic interactions ( bow waves ), while in the gas phase the particles screen each other from the bath collisions. For small particles the viscoelastic projjerties of the fluid will become important near contact. The solution of Eq. (2.23) applies only for sufficiently large friction where the relative motion on all length scales is diffusive. In the other limit of very low friction, the general result obtained from molecular theory is of the form... 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

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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