Self-diffusion mass dependence

In binary ion-exchange, intraparticle mass transfer is described by Eq. (16-75) and is dependent on the ionic self diffusivities of the exchanging counterions. A numerical solution of the corresponding conseiwation equation for spherical particles with an infinite fluid volume is given by Helfferich and Plesset [J. Chem. Phy.s., 66, 28, 418... 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

The primary difference between D and D is a thermodynamic factor involving the concentration dependence of the activity coefficient of component 1. The thermodynamic factor arises because mass diffusion has a chemical potential gradient as a driving force, but the diffusivity is measured proportional to a concentration gradient and is thus influenced by the nonideality of the solution. This effect is absent in self-diffusion. 

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

Figure 7. Plot of In ( >1 /D2) versus In (M/m) at p = 0.844 and T = 0.728, where D is the self-diffusion of the solvent and D2 is that of the solute. M and m are the masses of the solute and the solvent, respectively. The squares are the calculated values, and the solid line is the linear fit. The slope of the straight line is 0.099. The plot shows a power law mass dependence of the solute diffusion. The slope of the plot suggests that this mass dependence is weak.

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

Relation (5.4) is used to describe the dependence in the region of lengths below 2Me, whereas in the region above 2Me, the two mechanisms of the displacement of the centre of mass of the macromolecule are optional, so that the resulting coefficient of self-diffusion has to be defined as... 

Here, Ds and Dd are the coefficients representing the Soret and Dufour effects, respectively, Du is the self-diffusion coefficient, and Dik is the diffusion coefficient between components / and k. Equations (7.149) and (7.150) may be nonlinear because of, for example, reference frame differences, an anisotropic medium for heat and mass transfer, and temperature- and concentration-dependent thermal conductivity and diffusion coefficients. 

The diffusion mode of macroradicals is predominantly determined by the molecular mass distribution both of the radicals and of the inactive polymer. The self-diffusion coefficient of a discrete species depends, of course, on the molecular mass of that particle itself [24],... 

In many structured products, water management includes several mass transport mechanisms such as hydrodynamic flow, capillary flow and molecular self-diffusion depending on the length scale. Hydrodynamic flow is active in large and open structures and it is driven by external forces such as gravity or by differences in the chemical potential, that is, differences in concentrations at different locations in the structure. Capillary flow also depends on surface tension and occurs in channels and pores on shorter length scales than in hydrodynamic flow. A capillary gel structure will hold water, and external pressures equivalent to the capillary pressure will be needed to remove the water. 

Once an appropriate frame of reference is chosen, a two components (A, B) system may be described in terms of the mutual diffusion coefficient (diffusivity of A in B and vice versa). Unfortunately, however, unless A and B molecules are identical in mass and size, mobility of A molecules is different with respect to that of B molecules. Accordingly, the hydrostatic pressure generated by this fact will be compensated by a bulk flow (convective contribution to species transport) of A and B together, i.e., of the whole solution. Consequently, the mutual diffusion coefficient is the combined result of the bulk flow and the molecules random motion. For this reason, an intrinsic diffusion coefficient (Da and Db), accounting only for molecules random motion has been defined. Finally, by using radioactively labeled molecules it is possible to observe the rate of diffusion of one component (let s say A) in a two component system, of uniform chemical composition, comprised of labeled and not labeled A molecules. In this manner, the self-diffusion coefficient (Da) can be defined [54]. Interestingly, it can be demonstrated that both Da and Da are concentration dependent. Indeed, the force/acting on A molecule at point X is [1]... 

Quantum Effects in Collisional Friction The Tlansition Regime of Aerodynamics Hydrodynamic and Viscoelastic Effects Mass Dependence of Self-Diffusion Pair Dissociation—A Stochastic Approach A. Diatomic Dissociation on a J-Averaged Potential Adiabatic Effects in Diatomic Dissociation... 

Figure 10. Mass dependence of self-diffusion coefficient of hard-sphere test particle of mass mg in hard-sphere fluid of particles of mass m for Rg = R at density pR, = 0.9 for Rodger-Sceats friction (-----) and Smoluchowski friction (--------). 0—simulation results of Herman and Alder.

SOLVENT MOBILITIES. One check on the physical significance and the reliability of the data representing the concentration dependence of the diffusion coefficient is to convert these results to solvent mobilities. The values should increase rapidly with increasing concentration and extrapolate to the self-diffusion coefficient for toluene. The procedure for carrying out the calculations was outlined in previous publication (11) and is repeated here in a brief form for convenient reference. The diffusion coefficient obtained directly in the vapor sorption experiment is a polymer, mass-fixed, mean diffusion coefficient, D, in the sorption interval. Duda et. al. (12) have shown that, if the concentration interval is small, the true diffusion coefficient, D, is simply related to the mean diffusion coefficient at a prescribed intermediate concentration in the interval ... 

The variety of diffusion mechanisms involved in intracrystalline molecular mass transfer is most vividly reflected in the different patterns of the concentration dependence of intracrystalline self-diffusion. A classification of the various concentration dependences so far observed by PFG NMR is presented in Fig. 10. 

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