Diffusion coefficient from surface

A width (compared to about 5 A for the free water surface). The authors observed a continuous change of the self-diffusion coefficient from liquidlike to solid-like values over this interval. The results have been extensively reviewed by Laird and Haymet [197]. 

The analogy for transport processes is readily interpreted from Stokes theory if we consider the generalization that forces or fluxes of a property are proportional to a diffusion coefficient, the surface area of the body, and a gradient in property being transported. In the case of momentum, the transfer rate is related to the frictional and pressure forces on the body. The diffusion coefficient in this case is the kinematic viscosity of the gas (vg = p-g/pg, where pg is the gas density). The momentum gradient is Pjg Uoo/B. 

In the third simulation example, we carried out an analysis of some of the aspects that characterize the case of the mass transfer of species through a membrane which is composed of two layers (the separative and the support layers) with the same thickness but with different diffusion coefficients of each entity or species. To answer this new problem the early model has been modified as follows (i) the term corresponding to the source has been eliminated (u) different conditions for bottom and top surfaces have been used for example, at the bottom surface, the dimensionless concentration of species is considered to present a unitary value while it is zero at the top surface (iii) a new initial condition is used in accordance with this case of mass transport through a two-layer membrane (iv) the values of the four thermal diffusion coefficients from the original model are replaced by the mass diffusion coefficients of each entity for both membrane layers (v) the model is extended in order to respond correctly to the high value of the geometric parameter 1/L. 

The binary K j may be calculated as a function of the appropriate Maxwell-Stefan diffusion coefficient from a suitable correlation or physical model (e.g., the surface renewal models of Chapter 10). These binary must also be used directly in the calculation of the rate factor matrix [ ] (cf. Eqs. 8.3.28 and 8.3.29). 

Figure 12.7 shows values of xy calculated in two different ways as a function of temperature for panicles of dilTerent. size. One set of curves (solid lines) was calculated front the phenomenological relationships (12.5) and (12.6). For the solid particles (low temperature). MD simulations were used to obtain the diffusion coefficient and surface tension that appear in (12.6). For the liquid-like particles (high temperature), data for the viscosity and surface tension that appear in (12.5) were obtained from experimental results reported in the literature. The dashed lines in Fig. 12.7 were calculated directly from MD simulations of the decrease in the moment of inertia of two coalescing spheres. 

Erbil, H.Y. and Avci, Y. (2002). Simultaneous Determination of Vapor Diffusion Coefficient from Thin Tube Evaporation and Sessile Drop Evaporation on Solid Surfaces. Langmuir, 18,... 

A stochastic dynamics simulation requires a value to be assigned to the collision frequency friction coefficient 7. For simple particles such as spheres this can be related to the diffusion constant in the fluid. For the simulation of a rigid molecule it may be possible to derive 7 via the diffusion coefficient from a standard molecular dynamics situation. In the more general case we require the friction coefficient of each atom. For simple molecules such as butane the friction coefficient can be considered to be the same for all atoms. The optimal value for 7 can be determined by trial and error, performing a stochastic dynamics simulation for different values of 7 and comparing the results with those from experiment (where available) or from standard molecular dynamics simulations. For large molecules the atomic friction coefficient is considered to depend upon the degree to which each atom is in contact with the solvent and is usually taken to be proportional to the accessible surface area of the atom (as defined in Section 1.5). 

In order to derive self-diffusion coefficients from the TEX-PEP experiments, a mathematical model is needed to describe the re-exchange process in the zeolite reactor bed. A common way to describe diffusion in packed beds is to use a set of diffusion equations, describing the mass transport in the zeolite bed and inside the crystals [3,38,39]. The model to analyze the TEX-PEP experiment study is basically a modification of the equations by Noordhoek et al. [25]. The process is thought to consist of the transport of molecules via convection and axial diffusion in the space between the crystals, adsorption and desorption at the zeolite crystal surface and diffusion inside the pores... 

The ability of surface acoustic wave (SAW) devices to monitor adsorption of Nj onto the surfaces of porous films and diffusion of species Into polymer films has been demonstrated. Calculations based on the N2 adsorption Isotherms Illustrate how sol-gel solution chemistry can be used to tailor the surface area and pore size distribution of thin films. BET surface areas from unity to over 30 cm /cm of film have been obtained on various samples with median pore diameters from less than 0.4 nm to greater than 6 nm. SAW frequency transients occurring during the diffusion of small molecular species Into polymer films have been used to determine diffusion coefficients from 10 to 10 cm /sec. 

Johnson DL (1969) New method of obtaining volume grain-boundary and surface diffusion coefficients from sintering data. J Appl Phys 40 192-200... 

The first term on the left-hand side occurs on account of diffusion of CF from compartment A to compartment B and kj is a constant, which would be related to diffusion coefficient and surface area. The second term is related to interaction of CF and NH4 ion, k2 is a constant which depends on the extent of NHj (amine) adsorption. It may be noted that Cl would be depleted with increase in (NH4) and consequently AC would increase. 

The pure gas (99.99% or higher purity) transport properties were tested with an instrument (GKSS, Geesthacht, Germany) with eonstant permeate volume described elsewhere [14], The short response time of the instrument allows one to record transient permeation behaviours of less than 1 second. The pure gas permeability is the amount of gas permeating in the unit time, multiplied by the thickness of the membrane and normalized for the membrane surface and the pressure gradient. The recorded pressure vs. time plots were used to derive diffusion coefficients from the initial transient permeation, and permeability at the steady state. The time-lag 9 is the intercept of the linear part of the pressure vs. time curve with the axis of time. In a homogeneous membrane in which the solubility of a gas obeys Henry s law, its diffusion coefficient D can be calculated from the ratio ... 

Rg. 11 Evolution of the self-diffusion coefficient and surface tension of aqueous solutions of n-pentyl a-glyceryl ether (nCsGly). The dotted curve showing the solubilisation of a hydrophobic model dye (Disperse Red 13) as a function of nCsGly concentration is superimposed for comparison. Reproduced from Ref. 77. 

FIGURE 4.7 Comparison of water diffusion coefficients along surface normals for different alkali chlorides. (Zero distance corresponds to the position where the nnmber density of water molecules is zero.) (From Du, H. and Miller, J. D., J. Phys. Chem. C, 111 10013, 2007c. With permission.)... 

The second part of ISO 17497 specifies a method of measuring the directional diffusion coefficient of surfaces in a free field (ISO, 17497-2, 2012). The diffusion coefficient characterizes the sound reflected from a surface in terms of the uniformity of the reflected polar distribution. The directional diffusion coefficient is a frequency dependent value derived from the polar distribution of the scattered sound (Cox and D Antonio, 2009). First the scattering from a surface is measured in terms of a polar distribution. Then the diffusion coefficient is evaluated at 1/3 octave bandwidth intervals, which has the advantage of smoothing out some of the local variations in the polar responses. 

Many statistical parameters can be used to describe a diffusion coefficient from measurements of polar sound distributions. In the standard, an autocorrelation function is used to measure the scattered energy s spatial similarity with receiver angles. A surface which scatters sound uniformly to all receivers produces high values in a spatial autocorrelation function conversely a surface which concentrates scattered energy in one direction has a low value (Cox and D Antonio, 2009). By using the directional diffusion coefficient calculated by Eq. (6.10), the normalized directional diffnsion coefficient can be obtained by ... 

The release of steroids from silicone implants follows Ficks law of diffusion with the flux of diffusion being affected by a diffusion coefficient, membrane surface area, thickness of the membrane and the concentration gradient across the diffusion path in the membrane. 

Most recently we have devised a method of taking surface diffusion into account explicitly and obtaining the surface diffusion coefficient from the data. Thus we will no longer need to rely on the plots used here to infer the presence or absence of significant surface diffusion. 

D. L. Johnson, New Method of Determining Volume, Grain Boundary, and Surface Diffusion Coefficients from Sintering Data, /, Appl Phys., Jan., 1969. 

A linear plot of Jd vs. co, the Levich plot, implies that the electrocatalytic reaction is faster than the rate of substrate delivery to the electrode, so the current is limited only by mass transport to the catalyst surface. Also, it allows the determination of diffusion coefficient from the slope if other variables are kept constant. 

On the other hand, the MOR electrochemical oxidation on glassy carbon electrodes in 0.2 M phosphate buffer solutions (pH 7.00) at high concentrations of reagent showed a mixed control diffusion/adsorption. Therefore, plots of 7pv vs. v (eq. 32, Conway Kannangara, 1987)) were linear. Values of 7.3 x 10 cm s, and 1.5 x 10 " mol cm were determined for MOR diffusion coefficient, and surface concentration from the slope, and the intercept of those plots, respectively (Tesio, 2013). 

It is known that even condensed films must have surface diffusional mobility Rideal and Tadayon [64] found that stearic acid films transferred from one surface to another by a process that seemed to involve surface diffusion to the occasional points of contact between the solids. Such transfer, of course, is observed in actual friction experiments in that an uncoated rider quickly acquires a layer of boundary lubricant from the surface over which it is passed [46]. However, there is little quantitative information available about actual surface diffusion coefficients. One value that may be relevant is that of Ross and Good [65] for butane on Spheron 6, which, for a monolayer, was about 5 x 10 cm /sec. If the average junction is about 10 cm in size, this would also be about the average distance that a film molecule would have to migrate, and the time required would be about 10 sec. This rate of Junctions passing each other corresponds to a sliding speed of 100 cm/sec so that the usual speeds of 0.01 cm/sec should not be too fast for pressurized film formation. See Ref. 62 for a study of another mechanism for surface mobility, that of evaporative hopping. 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

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