Self-diffusion coefficient defined

We return now to the important question introduced in Section III. B dealing with p, V, and T dependence of transport properties in the vicinity of the glass transition. The most accurate data so far seem to be those obtained for the soft-sphere model. These have in part been seen already in Fig. 4 in normal units based on argonlike soft spheres. It is more natural, for internal comparisons, to use reduced soft-sphere ( = 12), self-diffusion coefficients defined by (2). Fig. 12 plots results for several authors against the reduced volume, which is defined as follows ... 

Pulsed field gradient NMR (PFG-NMR) spectroscopy has been successfully used for probing interactions in several research fields.44-53 The method was developed by Stejskal and Tanner more than 40 years ago54 and allows the measurement of self-diffusion coefficient, D, which is defined as the diffusion coefficient in absence of chemical potential gradient. 

The first and the second diffusion coefficients, Ddif and Diep are defined by relations (5.11) and (5.20), respectively. The two competing mechanisms have a different length dependence of the self-diffusion coefficient... 

The Kirkendall effect arises from the different values of the self-diffusion coefficients of the components of a substitutional solid solution, determined by Matano s method. Matano s interface is defined by the condition that as much of the diffusing atoms have migrated away from the one side as have entered the other. If DA = DB, its position coincides with the initial interface between phases A and B. If I)A f DB, it displaces into the side of a faster diffusant (see Fig. 1.22c). Note that KirkendalFs discovery only relates to disordered phases. It was indeed a discovery since at that time most reseachers considered the relation l)A = DB to hold for any solid solution of substitutional type. KirkendalFs experiments showed that in fact this is not always the case. 

Fick s first law, in the form given by Equation 5.12, allows us to define the tracer- and the self-diffusion coefficients. Diffusion of a tracer isotope is the case when a diffusing atom, which is marked by their radioactivity of by their isotopic mass (see Figure 5.1) [7], is introduced in an extremely dilute concentration in an otherwise homogeneous crystal with no driving force [4], In this case, the tracer gradient of concentration will give rise to a net flow of tracer atoms. Consequently,... 

The delocalized state can be considered to be a transition state, and transition state theory [105], a well-known methodology for the calculation of the kinetics of events, [12,88,106-108] can be applied. In the present model description of diffusion in a zeolite, the transition state methodology for the calculation of the self-diffusion coefficient of molecules in zeolites with linear channels and different dimensionalities of the channel system is applied [88], The transition state, defined by the delocalized state of movement of molecules adsorbed in zeolites, is established during the solution of the equation of motion of molecules whose adsorption is described by a model Hamiltonian, which describes the zeolite as a three-dimensional array of N identical cells, each containing N0 identical sites [104], This result is very interesting, since adsorption and diffusion states in zeolites have been noticed [88],... 

In order to develop a consistent free-volume diffusion model, there are some issues which must be addressed, namely i) how the currently available free-volume for the diffusion process is defined, ii) how this free-volume is distributed among the polymer segments and the penetrant molecules and iii) how much energy is required for the redistribution of the free-volume. Any valid free-volume diffusion model addresses these issues both from the phenomenologic and quantitative points of view such that the diffusion process is described adequately down to the microscopic level. Vrentas and Duda stated that their free-volume model addresses these three issues in a more detailed form than previous diffusion models of the same type. Moreover, it was stated that the model allows the calculation of the absolute value of the diffusion coefficient and the activation energy of diffusion mainly from parameters which have physical significance, i.e. so-called first principles . In the framework of this model the derivation of the relation for the calculation of the self-diffusion coefficient of the sol-... 

The aim of this Chapter is the development of an uniform model for predicting diffusion coefficients in gases and condensed phases, including plastic materials. The starting point is a macroscopic system of identical particles (molecules or atoms) in the critical state. At and above the critical temperature, Tc, the system has a single phase which is, by definition, a gas or supercritical fluid. The critical temperature is a measure of the intensity of interactions between the particles of the system and consequently is a function of the mass and structure of a particle. The derivation of equations for self-diffusion coefficients begins with the gaseous state at pressures p below the critical pressure pc. A reference state of a hypothetical gas will be defined, for which the unit value D = 1 m2/s is obtained at p = 1 Pa and a reference temperature, Tr. Only two specific parameters, Tc, and the critical molar volume, VL, of the mono-... 

Self-diffusion under equilibrium conditions may also be monitored in multicomponent systems. Again, with both eqs 2 and 3 a self-diffusion coefficient (of a particular component) may be defined. This coefficient depends on the nature and the concentration of all molecular species involved as well as on the nature of the catalyst particle. 

Local motions which occur in macromolecular systems can be probed from the diffusion process of small molecules in concentrated polymeric solutions. The translational diffusion is detected from NMR over a time scale which may vary from about 1 to 100 ms. Such a time interval corresponds to a very large number of elementary collisions and a long random path consequently, details about mechanisms of molecular jump are not disclosed from this NMR approach. However, the dynamical behaviour of small solvent molecules, immersed in a polymer melt and observed over a long time interval, permits the determination of characteristic parameters of the diffusion process. Applying the Langevin s equation, the self-diffusion coefficient Ds is defined as... 

Flo. 37. NMR intracrystalline self-diffusion coefficient Dm, (a) and effective self-diffusivity Dcir ( ) of methane in HZSM-5 crystals that were coked for different times by n-hexane cracking (131-133). Before loading with methane (9.2 CHa per u.c.), the coked ZSM-5 crystals were carefully outgassed at 623 K and 10 Pa. The remaining carbonaceous residues were defined as coke. Amounts of coke after different times on stream 1 h, 0.8 wt% C 2 h, 1.3 wt% C 6 h, 3.2 wt% C 16 h, 4.8 wt% C. The starting self-diffusion coefficient is 8.1 x 10" m s . ... 

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]... 

Consider the difficult case of self-diffusion in a hard-sphere fluid. It is useful to define the caging factor x as < / p(0) >/8/iBMkT and a shielding factor S by = (S/2)(Jm- Then Eq. (3.7) can be rearranged to give the self-diffusion coefficient... 

Very recently, a novel Fourier transform NMR method was employed by Lindman, et al. (21) to obtain multicomponent self-diffusion data for some single phase microenulsion systems. Because of the large values obtained for the self-diffusion coefficients of water, hydrocarbon, and alcohol, over a wide range of concentrations, the authors concluded that there are no extended, well-defined structures in these systems. In other words, the Interfaces which separate the hydrophobic from the hydrophilic regions appear to open up and reform at a short time scale. 

Reverting to the unprimed coordinates defined in (5.4.21) the echo amplitude is given for unrestricted Brownian motion with the self-diffusion coefficient D by... 

The diffusion coefficient of the particles in suspension depends on concentration of particles due to the interparticle interactions. Furthermore, we should distinguish the self-diffusion (or tracer diffusion) coefficient, D, from the collective diffusion (or mutual diffusion) coefficient, The self-diffusion coefficient accounts for the motion of a given particle and can be formally defined as an autocorrelation function of the particle velocity ... 

Diffusion in zirconia is closely linked to ionic conductivity. Consequently, some diffusion data has already been presented in Sect. 5. This section will include additional results particularly for monoclinic zirconia. Oxygen self-diffusion at a pressure of 300 Torr, as determined by testing zirconia spheres of diameters between 75 and 105 jam, behaves as shown in Fig. 17 [57], where D is the diffusion coefficient, t is time, and a is the sphere radius. At a pressure of 700 Torr, the behavior changes to that shown in Fig. 18 [58]. In this case D is the self-diffusion coefficient and the rest of the terms are as defined before, with a = 100-150 jam. Both of these experiments were performed in an oxygen atmosphere of 180-160. The self-diffusion coefficients calculated from the diffusion data obey Arrhenius expressions as illustrated in Fig. 19 [57, 58]. The linear fits describing the diffusion coefficient at 300 and 700 Torr, are given by ... 

Diffusion coefficients are important for mass-transfer operations (see Chapter 8, Mass Transfer ). There are several differently defined diffusion coefficients (self-diffusion coefficient, interdiffusion [or mutual diffusion] coefficient, intradiffusion [or tracer diffusion] coefficient) this can be a source of confusion. These are delineated in standard references [15, 68, 69]. 

Since the solvent mobility will generally increase with concentration and the thermodynamic factor decreases with concentration, D12 will usually display a maximum when data are obtained over a sufficiently large concentration range. The thermodynamic factor was obtained analytically from the derivative of the Flory-Rehner relation. For consistency, the value was calculated at the adjusted concentration defined in equation 4 above. The resulting values of D2, shown in Figure 6, increase rapidly at low concentration then reach a maximum at a value of 1.0 x 10-6 cm2/sec. This is a factor of about twenty lower than th estimated self diffusion coefficient for toluene, 1.75xl0 cm /sec (14). 

As noted above, in the case of isotopic exchange, we generally measure the self-diffusion coefficient which is defined as the rate of movement of a given isotopic species through a host with a bulk composition that is composed, in large part, of that species... 

These values are to be considered tentative since there is some drift in the values with concentration, and it is likely that the above equilibria do not completely define the system. Self-diffusion coefficients of the zirconium-EDTA complex in slightly acidic solution also indicate a considerable degree of polymerization [417) as do the hydrogen ion dependence data of Ermakov [172). In the direct analytical titration of zirconium with EDTA, polymeric species must be depolymerized. This may be accomplished by boiling the solution with 5 N sulfuric acid [430). 

Incoherent flow is often referred to as pseudo-diffusion. An apparent diffusion coefficient which can be significantly bigger than the self-diffusion coefficient is then defined. Pulse sequences to measure coherent flow (figure B 1.14.9) can also be used for (spatially) incoherent motion although the theory has to be reconsidered at this point [M, M and M] ... 

The diffusion coefficients used to describe multi-component diffusion are mutual diffusion coefficients. In the multi-component system, mutual diffusion coefficients are defined by Equation 4-13 the matrix of diffusion coefficients depends on the concentration of individual components. The diffusion coefficients used in the earlier sections of the chapter, however, describe solute molecules diffusing in a medium at infinite dilution. The isolated molecule is called a tracer these tracer diffusion coefficients are defined by the physics of random walk processes, as described in Chapter 3. The self-diffusion coefficient, used in Equation 4-11, is a tracer diffusion coefficient in the situation where all of the molecules in the system are identical. The self-diffusion coefficient, T>aa is defined by (recall Equation 3-12) [62] ... 

Figure 5 The variation of the obstruction factor A defined as the reduced self-diffusion coefficient D/Do of small solvent molecules in solutions as a function of the volume fraction of obstructing particles of different geometries. Curve a denotes spheres b, long prolates c, d, and e. oblates of axial ratios 1 5. 1 10, and 1 100, respectively and f, large (i.e., infinite) oblates. (Redrawn from...

This "phenomenological / should not be confused with the "molecular friction coefficient which is defined as a force needed to have a molecule translate steadily at unit velocity. The latter appears below when the self-diffusion coefficient is introduced. 

Equation 7-2.3 defining the self-diffusion coefficient Dg may be written in terms of rG(r) as... 

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