Diffusion circle

Benthic oxygen fluxes as a function of longitude and depth on the northwest US continental margin. The fluxes were determined using either benthic lander measurements (boxes) or calculated from micro-electrode oxygen gradients in the top few centimeters of porewaters, assuming molecular diffusion (circles). Error bars are the standard deviation of replicate measurements. Redrawn from Archer and Devol (1992). 

A low value for Kp means a big precursor effect since k 4 is low, that is the probability of desorption is low and the lifetime in the precursor state is high allowing a wide area of diffusion on the surface and thus a high probability for adsorption into the final chemisorbed state. Approximate diffusion circles are shown in fig. 13 based on the following simple Frenkel relationships,... 

Fig. 13. A model of the extent of molecular diffusion on surfaces, so-called diffusion circles, showing the strong temperature dependence of the number of diffusion events. Circles are shown here superimposed on a (100) lattice for surface temperatures of 400 K and 950 K at lower temperatures the diffusion circles arc much more extensive (see text).

The size of the diffusion circle is then approximated by the following, assuming completely random directional diffusion. 

Fig. 9 Transport diffusivities (squares) and corrected diffusivities (circles) obtained for CF4 in silicalite at 200 K, by QENS (filled symbols) and simulations (open symbols)...

FIGURE 4.7 Simulated c spectral profile for constant I p (dashed line) and parabolic i>p(x) (solid line) with equal Fp and no axial diffusion, circles are for paraboUc t (x) with axial diffusion. (Adapted from Shvartsburg, A.A., Tang, K., Smith, R.D., J. Am. Soc. Mass... 

Filament with diffuse circle near complex LI-LMC 1249 (DEM 227). 

Finally, astigmatism is a lens error similar to coma this type of lens error is found at the outer portions of the field of view (far off-centre object points) in uncorrected lenses. It causes the image of such an object point, which in an ideal system would be a circular point image, to blur into a diffuse circle, elliptical patch, or line, depending upon the location of the focal plane. 

Figure A3.6.13. Density dependence of die photolytic cage effect of iodine in compressed liquid n-pentane (circles), n-hexane (triangles), and n-heptane (squares) [38], The solid curves represent calculations using the diffusion model [37], the dotted and dashed curves are from static caging models using Camahan-Starling packing fractions and calculated radial distribution fiinctions, respectively [38],...

The proof that these expressions are equivalent to Eq. (1.35) under suitable conditions is found in statistics textbooks. We shall have occasion to use the Poisson approximation to the binomial in discussing crystallization of polymers in Chap. 4, and the distribution of molecular weights of certain polymers in Chap. 6. The normal distribution is the familiar bell-shaped distribution that is known in academic circles as the curve. We shall use it in discussing diffusion in Chap. 9. 

Fig. 4. (a) Multistage diffusion pump (b) insert A, jet spray, where closed circles indicate gas molecules and open circles indicate vapor-jet molecules P. ... 

FIG. 16 36 Dimensionless time-distance plot for the displacement chromatography of a binary mixture. The darker lines indicate self-sharpening boundaries and the thinner lines diffuse boundaries. Circled numerals indicate the root number. Concentration profiles are shown at intermediate dimensionless column lengths = 0.43 and = 0.765. The profiles remain unchanged for longer column lengths. 

The ripple experiment works as follows In Fig. 6, HDH and DHD are depicted by open and filled circles where the filled circles represent the deuterium labeled portions of the molecule and the open circles are the normal (protonated) portions of the chains. Initially, the average concentration vs. depth of the labeled portions of the molecules is 0.5, as seen along the normal to the interface, unless chain-end segregation exists at / = 0. If the chains reptate, the chain ends diffuse across the interface before the chain centers. This will lead to a ripple or an excess of deuterium on the HDH side and a depletion on the DHD side of the interface as indicated in the concentration profile shown at the right in Fig. 6. However, when the molecules have diffused distances comparable to Rg, the ripple will vanish and a constant concentration profile at 0.5 will again be found. 

Fig. 6. The ripple experiment at the interface between a bilayer of HDH- and DHD-labeled polystyrene, showing the interdifussion behavior of matching chains. The protonated sections of the chain are marked by filled circles. The D concentration profiles are shown on the right. Top the initial interface at / = 0. The D concentration profile is flat, since there is 50% deuteration on each side of the interface. Middle the interface after the chain ends have diffused across (x < / g). The deuterated chains from Que side enrich the deuterated centers on the other side, vice ver.sa for the protonated sections, and the ripple in the depth profile of D results. A ripple of opposite sign occurs for the H profile. Bottom the interface when the molecules have fully diffused across. The D profile becomes flat [20,56].

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

Fig. 13. Simulated diffraction space of a 10-layer monochiral MWCNT with Hamada indices (40+8/ , 5+k) with / =0,...,9. In (a), (a ) and (02) the initial stacking at ( q was ABAB. whereas in (b), (b[) and (b2) the initial stacking was random, (a) The normal incidence pattern has a centre of symmetry only. (3 )(a2) The cusps are of two different types. The arc length separating the cusps is c (b) The normal incidence pattern now exhibits 2mm symmetry. (b )(b2) The cusps are distributed at random along the generating circles of the evolutes. These sections represent the diffuse coronae referred to in the "disordered stacking model" [17].

Fig. 5. Tentative mixed potential model for the sodium-potassium pump in biological membranes the vertical lines symbolyze the surface of the ATP-ase and at the same time the ordinate of the virtual current-voltage curves on either side resulting in different Evans-diagrams. The scale of the absolute potential difference between the ATP-ase and the solution phase is indicated in the upper left comer of the figure. On each side of the enzyme a mixed potential (= circle) between Na+, K+ and also other ions (i.e. Ca2+ ) is established, resulting in a transmembrane potential of around — 60 mV. This number is not essential it is also possible that this value is established by a passive diffusion of mainly K+-ions out of the cell at a different location. This would mean that the electric field across the cell-membranes is not uniformly distributed.

FIG. 17 Diffusion coefficients of the counterions and coions of a 1 1 RPM model electrolyte in a cylindrical nanopore of i = lOd. The circles and triangles represent the results of coions and counterions, respectively. 

After a drop of ink is added to a beaker of water (left), the ink diffuses slowly through the liquid (center) until eventually the ink is distributed uniformly (right). The molecular views indicate that the motion of ink molecules and water molecules is responsible for this diffusion. Ink molecules (red-violet circles) and water molecules (blue circles) move about continually, even after they are well mixed. 

In the commonest form of microbiological assay used today, samples to be assayed are applied in some form of reservoir (porcelain cup, paper dise or well) to a thin layer of agar seeded with indicator organism. The drug diffuses into the medium and after incubation a zone of growth inhibition forms, in this case as a circle around the reservoir. All other factors being constant, the diameter of the zone of inhibition is, within limits, related to the concentration of antibiotic in the reservoir. 

Figure 5. Cartoon models of the reaction of methanol with oxygen on Cu(llO). 1 A methanol molecule arrives from the gas phase onto the surface with islands of p(2xl) CuO (the open circles represent oxygen, cross-hatched are Cu). 2,3 Methanol diffuses on the surface in a weakly bound molecular state and reacts with a terminal oxygen atom, which deprotonates the molecule in 4 to form a terminal hydroxy group and a methoxy group. Another molecule can react with this to produce water, which desorbs (5-7). Panel 8 shows decomposition of the methoxy to produce a hydrogen atom (small filled circle) and formaldehyde (large filled circle), which desorbs in panel 9. The active site lost in panel 6 is proposed to be regenerated by the diffusion of the terminal Cu atom away from the island in panel 7.

Another example is dendritic crystal growth under diffusion-limited conditions accompanied by potential or current oscillations. Wang et al. reported that electrodeposition of Cu and Zn in ultra-thin electrolyte showed electrochemical oscillation, giving beautiful nanostmctured filaments of the deposits [27,28]. Saliba et al. found a potential oscillation in the electrodeposition of Au at a liquid/air interface, in which the Au electrodeposition proceeds specifically along the liquid/air interface, producing thin films with concentric-circle patterns at the interface [29, 30]. Although only two-dimensional ordered structures are formed in these examples because of the quasi-two-dimensional field for electrodeposition, very recently, we found that... 

Figure 13.9 Reaction scheme for Ci molecule oxidation on a Pt/C catalyst electrode, including reversible diffusion from the bulk electrolyte into the catalyst layer, (reversible) adsorption/ desorption of the reactants/products, and the actual surface reactions. The different original reactants (educts) and products are circled. For removal/addition of H, we do not distinguish between species adsorbed on the Pt surface and species transferred directly to neighboring water molecule (H d, H ) therefore, no charges are included (H, e ). For a description of the individual reaction steps, see the text.

Fig. 5.18. Self-diffusion constants for a bidisperse (i.e. two different chain lengths) PE melt with Mn = 20 coarse-grained monomers. Open triangles are for d = 2, filled diamonds for d = 4, open squares for d = 6 and filled circles for d = 8. There are always two symbols of the same kind shown in the figure, since the bidisperse melt contains two species of different chain length. The numbers quoted in the figure correspond to these chain lengths for a given polydisparsity d. For instance, d = 8 corresponds to Mi = 12 and M2 = 52. From [184].

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

Fig. 3.1.7 The surface diffusion coefficient / surface of cyclohexane (squares) and acetone (circles) in porous silicon with 3.6-nm mean...

Fig.3.1.9 (a) The adsorption-desorption isotherm (circles, right axis) and the self-diffusion coefficients D (triangles, left axis) for cyclohexane in porous silicon with 3.6-nm pore diameter as a function of the relative vapor pressure z = P/PS1 where Ps is the saturated vapor pressure, (b) The self-diffusion coefficients D for acetone (squares) and cyclohexane (triangles) as a function of the concentration 0 of molecules in pores measured on the adsorption (open symbols) and the desorption (filled symbols) branches. 

Fig. 3.1.11 Relative effective intracrystalline diffusivities D(t)/D0 as function of JDot for n-hexane under single-component adsorption (circles) and for n-hexane (triangles) and tetrafluoromethane (rectangles) under two-...

Fig. 14. Integral diffusivities of Cu2 +, corrected for migration effect, as a function of true ionic strength, /r. Circles and triangles indicate values reported by Selman (S8) and Hsueh (H7), squares indicate results of Arvia el al. (A5). Diaphragm cell diffusivities according to Fenech (F3) and capillary cell diffusivities according to Hsueh (H7) are also shown. [From Selman (S8).]...

Fig. 1. Schematic of an FCS experiment. For simplicity we consider an FCS measurement on a chemical reaction system confined to a plane, e.g., a membrane. The reaction is a two-state isomerization A (circles) B (squares). In the region of the plane illuminated by a laser beam (dark gray), A and B molecules appear white and light gray, respectively. Fluorescence fluctuations arise from interconversion of A and B and by A and B molecules diffusing into or out of the illuminated region. Molecules outside the illuminated region (black) are not detected.

Fig. 19 Our hybrid microrelaxation model. The solid circles are occupied by a polymer chain. The dashed lines show the new bond positions produced by a move consisting of kink generation and partial sliding diffusion along the chain. The arrows indicate the directions of monomer jumping [134]... 

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