Particle limiting diffusion coefficient

In the limit of infinite dilution the friction coefficient can be related to the single particle translational diffusion coefficient... 

Brownian motion of a single noninteracting particle can be described in terms of self-diffusion characterized by Do, the particle self-diffusion coefficient in the infinite dilution limit. The probability / (Ar. r) of a particle displacement Ar in time r satisfies the diffusion equation... 

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

Difoggio and Corner [1982] and Wang and Comer [1985] have discovered tunneling diffusion of H and D atoms on the (110) face of tungsten. They saw that the Arrhenius dependence of the diffusion coefficient D sharpy levels-off to the low-temperature limit (D = D ) at 130-140 K (fig. 47) the values of depend but slightly on the mass of the tunneling particle for the D and... 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

As diffusion to the surface of a polymer is one of the limiting steps in extraction, the particle size or film thickness of a sample is also important [278,333,337-340]. With the typical diffusion coefficients of additives in polymers a particle diameter of about 0.3 mm is required for an extraction time of about 1000 s at 40 °C. An exception to this is the extraction of thin films and foams, for which the shortest dimension is small. It is not surprising that no more than 50 % of antioxidants could be extracted from PP pellets as opposed to 90 % recoveries from the same polymer extruded into film [341]. Grinding of the polymer is usually an essential step before extraction. Care should be taken to avoid loss of volatile additives owing to the heat generated in such processes. Therefore, cryogrind-ing is preferred. 

In any case, exceptions to the FIAM have been pointed out [2,11,38,44,74,76,78]. For example, the uptake has been shown to depend on the Cj M or rMI (e.g. in the case of siderophores [11] or hydrophobic complexes [43,50]), rather than on the free c M. Several authors [11,12,15] showed that a scheme taking into account the kinetics of parallel transfer of M from several solution complexes to the internalisation transporter ( ligand exchange ) can lead to exceptions to the FIAM, even if there is no diffusion limitation. Adsorption equilibrium has been assumed in all the models discussed so far in this chapter, and the consideration of adsorption kinetics is kept for Section 4. Within the framework of the usual hypotheses in this Section 3, we would expect that the FIAM is less likely to apply for larger radii and smaller diffusion coefficients (perhaps arising from D due to the labile complexation of M with a large macromolecule or a colloid particle, see Section 3.3). 

Moreover, the influence of the motions of the particles on each other (i.e., when the motion of a particle affects those of the others because of communication of stress through the suspending fluid) can also influence the measured diffusion coefficients. Such effects are called hydrodynamic interactions and must be accounted for in dispersions deviating from the dilute limit. Corrections need to be applied to the above expressions for D and Dm when particles interact hydrodynamically. These are beyond the scope of this book, but are discussed in Pecora (1985), Schmitz (1990), and Brown (1993). 

In Fig. 42, the full-width at half maximum of the (narrower) exchange propagator provides an estimate of the effective diffusion coefficient of water molecules moving between the pore space of the catalyst and the inter-particle space of the bed. In this example, the value is 2 x lO- m s which gives a lower limit to the value for the mass transfer coefficient of 4x 10 ms This value was obtained by defining a mass transfer coefficient as Djd where d is a typical distance traveled to the surface of the catalyst that we estimate as half a typical bead dimension (approximately 500 pm). This value of the mass transfer coefficient is consistent with the reaction occurring under conditions of kinetic as opposed to mass transfer control. 

In Figure 7.4 the effectiveness factor is plotted against the Thiele modulus for spherical catalyst particles. For low values of 0, Ef is almost equal to unity, with reactant transfer within the catalyst particles having little effect on the apparent reaction rate. On the other hand, Ef decreases in inverse proportion to 0 for higher values of 0, with reactant diffusion rates limiting the apparent reaction rate. Thus, decreases with increasing reaction rates and the radius of catalyst spheres, and with decreasing effective diffusion coefficients of reactants within the catalyst spheres. 

The rotational effect on the correction term for diffusion-limited rate coefficients is shown as the intercepts on the ordinate of Fig. 17. Here, B is a small particle and only A can be re-oriented. As the size of A decreases, it can re-orient more quickly, but the mutual diffusion coefficient decreases till rA = rB. Hence, the very fact that the correction term increases with increase of rA shows that rotational diffusion is very important. 

It shows the clear link between the change of motion of the particle and its diffusion coefficient. In Fig. 50, the velocity autocorrelation function of liquid argon at 90 K (calculated by computer simulation) is shown [451], The velocity becomes effectively randomised within a time less than lps. Further comments on the velocity autocorrelation functions obtained by computer simulation are reserved until the next sub-section. Because the velocity autocorrelation function of molecular liquids is small for times of a picosecond or more, the diffusion coefficient defined in the limit above is effectively established and constant. Consequently, the diffusion equation becomes a reasonable description of molecular motion over times comparable with or greater than the time over which the velocity autocorrelation function had decayed effectively to zero. Under... 

The experimental diffusion parameters, D /r., at 30°C. are presented in Table II for all the coals. Clearly, no correlation exists between diffusion parameter and rank. If r<> is taken as the average particle radius for the 200 X 325 mesh samples, an upper limit to the values of diffusion coefficient, D, is obtained. The diffusion coefficient ranges from 1.92 X 10 9 sq. cm./sec. for Kelley coal to 1.41 X 10"8 sk. cm./sec. for the Dorrance anthracite. Our previous studies on the change of D /n with particle size suggested that n is not necessarily the particle radius (7) but is a smaller distance related to the average length of the micropores in the particles. That is, the calculated... 

It is obvious that the mass-transfer coefficient, ks, can be influenced by the diffusion coefficient, Di2, which is also included in the Sherwood number, Sh, as in the Schmidt number, Sc. Diffusion can be increased by shortening the diffusion length. For solid materials this is achieved by smaller particle sizes, which further leads to a higher specific interfacial area, as. However, there is a limit for reducing the particle size because if the particles are too fine, the problem of channelling arises, so an optimum has to be found. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

This equation reproduces correctly the limiting cases mentioned above and has been derived in [53] for the particular case of a single B particle surrounded by two A particles. Careful checking has shown that this exponent a depends indeed only on the ratio, but not separately on the individual values of the diffusion coefficients, Da and Dq. 

The kinetics of the A + B - 0 bimolecular reaction between charged particles (reactants) is treated traditionally in terms of the law of mass action, Section 2.2. In the transient period the reaction rate K(t) depends on the initial particle distribution, but as f -> oo, it reaches the steady-state limit K(oo) = K() = 47rD/ieff, where D — Da + >b is a sum of diffusion coefficients, and /4fr is an effective reaction radius. In terms of the black sphere approximation (when AB pairs approaching to within certain critical distance ro instantly recombine) this radius is [74]... 

Here, D is the diffusion coefficient of the reacting particles and n(t) and N(t) are the current concentrations of the two reagents. The solution of the system of equations (46) is written in the form of a complex series (see, for example, ref. 27). However, it is substantially simplified in two practically important limits t td and t > td, where td = a2/D is the time of diffusion travel of the reagents at a distance of the order of a. For the sake of simplicity we shall consider only the case of random spatial distribution of the reagents and assume that n(0) N. If t tu, then the solution of eqns. (46) is given by expression (35), i.e. it coincides with the equation for the kinetics of electron tunneling reactions for immobile reagents. At t > tu, from eqn. (46) it is possible to obtain... 

Gartek and Goldman relate the expansion of the concentration limit in upward propagation in lean hydrogen mixtures to the fact that due to the large coefficient of diffusion in the mixture, high temperatures may be achieved on the surfaces of particles of the catalyst. Comparison of hydrogen and deuterium confirms this point of view. In addition, a number of facts show that it is not the absolute value of the diffusion coefficient, but its... 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

---