Fluctuation concentration

The relationship can be obtained from the statistical mechanics for the open system (Problem 2.25). From the virial expansion of II given in Eq. 2.20, 

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We start with a simple example the decay of concentration fluctuations in a binary mixture which is in equilibrium. Let >C(r,f)=C(r,f) - be the concentration fluctuation field in the system where is the mean concentration. C is a conserved variable and thus satisfies a conthuiity equation ... 

Therefore, the locus of the values ( ) with a vanishing second derivative of A delimits the region of the miscibility gap in which spinodal decomposition occurs. This locus is referred to as the spinodal (figure C2.1.10 (bl). The length scale of the concentration fluctuations at the beginning of the separation process is controlled by... 

Dimensionless pumping number and blend time are independent of Reynolds number under fully turbulent conditions. The magnitude of concentration fluctuations from the final well-mixed value in batch mixing decays exponentially with time. 

Turbulent eddies larger than the cloud size, as such, tend to move the cloud as a whole and do not influence the internal concentration distribution. The mean concentration distribution is largely determined by turbulent motion of a scale comparable to the cloud size. These eddies tend to break up the cloud into smaller and smaller parts, so as to render turbulent motion on smaller and smaller scales effective in generating fluctuations of ever smaller scales, and so on. On the small-scale side of the spectrum, concentration fluctuations are homogenized by molecular diffusion. 

Wilson, D. J., J. E. Fackrell, and A. C. Robins. 1982a. Concentration fluctuations in an elevated plume A diffusion-dissipation approximation. Atmospheric Environ. 16(ll) 2581-2589. 

Often the most important effective cluster interaction of the concentration fluctuation expansion of the configurational free energy is the EPI which is defined as... 

For a more complete discussion of critical phenomena, we consider the scattering intensity S(q) from concentration fluctuations (q is the wavenumber of the scattering) which can be derived when one supplements Eq. (1) by a gradient-square term " ... 

Figure 22. Schematic depiction of asymmetrical concentration fluctuation. In the case of dissolution, it takes a positive value from the electrostatic equilibrium state.78...

Nonequilibrium fluctuation can be described as spatial waves with various wave numbers, which are two-dimensional plane waves composed of x and y components. The concentration fluctuation and surface-form... 

With the boundary condition that the concentration fluctuation disappears at the bulk solution, i.e.,... 

Moreover, as the reaction progresses, the mass flux of the asymmetrical concentration fluctuation forms a diffusion layer outside the double layer, and nonequilibrium fluctuations of another type occur in the diffusion layer these come from the mass transfer of dissolved metal ions perturbed by the thermal motion of solution particles. A fluctuation of this type was analyzed by Aogaki et a/.78 As shown in Fig. 31, the mass flux fluctuates around its average value, i.e., the fluctuations are both positive and negative, and in this sense have symmetry. This type of fluctuation is... 

Figure 31. Schematic depiction of symmetrical concentration fluctuation, the fluctation takes positive and negative values around the average value.78...

The concentration fluctuation is solved under the following boundary condition... 

The concentration fluctuations arising from the diffusion process in a static solution with a large amount of supporting electrolyte are followed by... 

The average value of the asymmetrical fluctuation is defined as the root mean square (rms) value. Using the Rayleigh theorem,94 the average value of the surface concentration fluctuation, for example, can be written as... 

In general, with the normalization of the critical fluctuation, the initial spatial spectrum of the concentration fluctuation corresponding to the wave number components, A and Ay, is given as... 

As the unstable growth of the fluctuations proceeds, the minimum current starts to increase. According to Section III.l, the unstable component of the asymmetrical concentration fluctuation is provided by the amplitude equation... 

Using the data of lJO), Eq. (116) allows us to reckon the value of the average critical concentration fluctuation on the completely active surface,... 

Since the nonequilibrium concentration fluctuation arises from the dissolution of substrate metal, as shown in Fig. 43, the value of is independent of the metallic ion concentration in the bulk solution. 

Figure 43. Average critical concentration fluctuation Pa on a completely active surface against Ni2+ ionic concentration.91 T = 300 K. (Reprinted from M. Asanuma and R. Aogaki, Nonequilibrium fluctuation theory on pitting dissolution. II. Determination of surface coverage of nickel passive film, J. Chem. Phys. 106,9938,1997, Fig. 13. Copyright 1997, American Institute of Physics.)...

First, the amplitude cj,(0,Of, is calculated by Eq. (91). Then, taking the Fourier inversion of the amplitude, the space-variable component of the surface concentration fluctuation is obtained,... 

As shown by the flow chart in Fig. 49, at first a wave number plane is divided into a mesh with 128 x 128 rectangular mesh points. Then the amplitude of the concentration fluctuation at t = 0 corresponding to each... 

Concentration fluctuation, in the double layer, 267 of metal ions in the double layer, 274 Concentration variation of diffusion layer, 270... 

Double layer capacitance of solid at rough electrodes, 52 Double layer parameters, for aluminum, tabulated, 129 and concentration fluctuations, 268, 269 for copper in aqueous solutions, 91 on iron, tabulated, 124... 

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