Theories of Self-diffusion

Altenberger, AR TirreU, M, On the Theory of Self-Diffusion in a Polymer Gel, Journal of Chemical Physics 80, 2208, 1984. 

This is precisely the result from kinetic theory of self-diffusion, neglecting the small Somine corrections, and also applies when the test particle is identical to the solvent particle. At intermediate densities the packing effects of the solvent around the test particle become important and Eq. (3.7) collapses to the Enskog result... 

Guenza M, Tang H and Schweizer K S (1998) Mode-coupling theory of self-diffusion in diblock copolymers I. General derivation and qualitative predictions, J Chem Phys 108 1257-1270. 

A number of important works have been devoted to the computation of selfdiffusion rates of small ions in polyelectrolyte solutions. Since this question extends beyond the present necessarily limited discussion we shall refer the reader for this question to the original memoirs [21-25]. Let us just mention that the effect of the electrostatic field of the macro-ion on the macroscopic self-diffusion constant of small ions has been computed analytically or numerically for polyelectrolytes of different shape and charge. The theory of self-diffusion starts usually with a cell model, the polyion being held fixed in space and the rate of motion of labelled ions in the spatially periodic electrostatic field set up by the polyion is obtained by solving the modified diffusion equation. 

Berne B. J. A self-consistent theory of rotational diffusion, J. Chem. Phys. 62, 1154-60 (1975). 

Further measurements of self-diffusion and diffusion in dense gas regions in order to establish the corresponding states behavior in diffusion and to test further the Enskog theory. 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

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 diffusion of small molecules in rubbers is of both theoretical and practical importance. The theories of diffusion based on consideration of free volume can be tested by measurement of self-diffusion using methods such as pulsed field gradient NMR. Selfdiffusion of small molecules must be understood for applications of rubbers as seals in contact with solvents, and for example for diffusion of plasticisers and other small molecules. 

The self-diffusion of benzene in PIB [36], cyclohexane in BR [37] and toluene in PIB [38-40] has been investigated by PFG NMR. In addition more recently Schlick and co-workers [41] have measured the self-diffusion of benzene and cyclohexane mixtures in polyisoprene. In the first reported study of this kind, Boss and co-workers [36] measured the self-diffusion coefficients of benzene in polyisoprene at 70.4 °C. The increase in Dself with increasing solvent volume fraction could be described by the Fujita-Doolittle theory which states that the rate of self-diffusion scales with the free volume which in turn increases linearly with temperature. At higher solvent volume fractions the rate of selfdiffusion deviates from the Fujita-Doolittle theory, as the entanglement density decreased below the critical value. 

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... 

Application to Polvmer-Solvent Systems. Fujita (231 was the first to use the free-volume theory of transport to derive a free-volume theory for self-diffusion in polymer-solvent systems. Berry and Fox (241 showed that, for the temperature intervals usually considered (smaller than 200°C), the theories that consider a redistribution energy for the voids gives results similar to those of the theories that assume a zero energy of redistribution for the free volume available for molecular transport. Vrentas and Duda (5.61 re-examined the free-volume theory of diffusion in polymer-solvent systems and proposed a more general version of the theory presented by Fujita. They concluded that the further restrictions needed for the theory of Fujita are responsible for the failures of the Fujita theory in describing the temperature and concentration dependence... 

It is important to note that diffusion is not a universally defined term. In foods, the processes of self-diffusion of the polymer matrix molecules, selfdiffusion of solutes, translational diffusion of solutes, and diffusion of moisture and other liquids have not been well discerned among theorists. All of these diffusion or mobility-based processes may occur in foods and pharmaceuticals. Yet recent theories do not clearly and consistently address which of these processes are of significance to chemical reactions, and how changes in water content or a as well as T or Tg affect each of these types of diffusion. 

Apparently, in the near future there will be developed (a) a detailed theory of surface excitons not only at the crystal boundary with vacuum but also at the interfaces of various condensed media, particularly of different symmetry (b) a theory of surface excitons including the exciton-phonon interaction and, in particular, the theory of self-trapping of surface excitons (c) the features of surface excitons for quasi-one-dimensional and quasi-two-dimensional crystals (d) the theory of kinetic parameters and, particularly, the theory of diffusion of surface excitons (e) the theory of surface excitons in disordered media (f) the features of Anderson localization on a surface (g) the theory of the interaction of surface excitons of various types with charged and neutral particles (h) the evaluation of the role of surface excitons in the process of photoelectron emission (i) the electronic and structural phase transitions on the surface with participation of surface excitons. We mention here also the theory of exciton-exciton interactions at the surface, the surface biexcitons, and the role of defects (see, as example, (53)). The above list of problems reflects mainly the interests of the author and thus is far from complete. Referring in one or another way to surface excitons we enter into a large, interesting, and yet insufficiently studied field of solid-state physics. 

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

It is interesting to attempt a simple theory of the parameter Tq. Relaxation of p occurs at the cluster surfaces via a diffusive jump across the interface. Using the free-volume model of self-diffusion (Section VII), we obtain the following for Tq,... 

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

FREE VOLUME THEORY FOR SELF-DIFFUSIVITY OF SIMPLE NON-POLAR LIQUIDS. 

Graessley owed that the above estimation of a is consistent with the results of diffuaon experiments. According to the reptation theory, the self diffusion constant Do is given by (see eqn (6.40))... 

To obtain a macroscopic theory for self-diffusion, we define a quantity F(r, t) that is the probability density for finding the tagged particle at the point r at time t. Since the number of tagged partides is conserved, P(r, t) satisfies a conservation law of the form... 

The microscopic theory based on the Liouville equation leads to diffusion equation (235) for t with the coefficient of self-diffusion D given by an equation identical to (236) except that in the microscopic theory the angular bracket is taken to be the average over an equilibrium ensemble. To be... 

Moreover, one can also show that the time integral of (248) for PD,o(t) leads to exactly the same values for the coefficient of self-diffusion D as given by the Boltzmann equation. Similarly the time integral of po e(0 Eq- (250), leads to an expression for D that is identical to the Enskog theory result. ... 

Further modeling steps were made in Ref [34]. First, the ad hoc function (5.4) that gives only a qualitative description of the gel effect [34] was replaced by a function that was derived from the expression [43] for the termination rate given in terms of self-diffusion rates of polymer radicals, which in turn are found by applying the free volume theory [44,45]. Secondly, the effect of adding an inhibitor to the mixture was considered [34]. 

For highly dilute and non-interacting solutions, this allows for the determination of self-diffusion coefficients Dq. Proceeding to the dilute regime where the polymer chains can interact with each other but still do not overlap, the dynamics is dominated by the hydrodynamic radius of the diffusing probe. According to the Kirkwood—Riseman theory [114], the diffusion coefficient can be calculated as follows ... 

Naghizadeh, J. and Rice, S. A., Kinetic theory of dense fluids. X. Measurement and interpretation of self-diffusion in liquid Ar, Kr, Xe, and CH4, /. Chem. Phys., 36, 2710,1962. 

This expression (for P) is derived from the WLF equation and h is Planck s constant, 7i is a constant, U is a constant (dimension J mol ), and is the temperature at which diffusion is stopped. Later versions of the LH theory use another expression derived from de Gennes s theory for self-diffusion (see Chapter 6) for the P parameter ... 

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