Solids error function

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

Fig. 10 Volume fractions of the FE phase state for x = 0.20, x = 0.25 and x = 0.30 versus temperature. The solid lines are guide lines for the eye, whereas the dashed line is a fit of an error function to the data...

Figure 3-16 H2O diffusion profile in (a) a dehydration experiment (Zhang et al., 1991a) and (h) a hydration experiment with data read from a figure in Roberts and Roberts (1964). Two outlier points are excluded in (a). The solid curves are fits to the data assuming D is proportional to C, and the dashed curves are fits assuming constant D (error function).

Figure 3-28 H2O diffusion profile for a diffusion-couple experiment. Points are data, and the solid curve is fit of data by (a) error function (i.e., constant D) with 167 /irn ls, which does not fit the data well and (b) assuming D = Do(C/Cmax) with Do = 409 /im ls, which fits the data well, meaning that D ranges from 1 /rm /s at minimum H2O content (0.015 wt%) to 409 firn ls at maximum H2O content (6.2 wt%). Interface position has been adjusted to optimize the fit. Data are adapted from Behrens et al. (2004), sample DacDC3.

Figure 3-29 A half-space diffusion profile of Ar. Ca, = 0.00147 wt% is obtained by averaging 45 points at 346 to 766 /an. Points are data, and the solid curve is a fit of (a) all data by the error function with D = 0.207 /im /s and Co = 0.272 wt%, and (b) data at v < 230 fim (solid dots) by the inverse error function. In (b), for larger x, evaluation of erfc [(C —Cot)/(Co —Coa)] becomes increasingly unreliable and even impossible as (C - Cot)/(Cq — Coo) becomes negative. Data are adapted from Behrens and Zhang (2001), sample AbDArl.

The solution to this partial differential equation depends upon geometry, which imposes certain boundary conditions. Look np the solution to this equation for a semi-infinite solid in which the surface concentration is held constant, and the diffusion coefficient is assumed to be constant. The solution should contain the error function. Report the following the bonndary conditions, the resulting equation, and a table of the error function. 

The surface heat flux is determined by evaluating the temperature gradient at x = 0 from Eq. (4-11). A plot of the temperature distribution for the semiinfinite solid is given in Fig. 4-4. Values of the error function are tabulated in Ref. 3, and an abbreviated tabulation is given in Appendix A. 

Fig. 26 Representative ellipsometric spectra (A as a function of the incidence angle F is omitted for clarity) of PMAA brushes on LaSFN9 prisms swollen in aqueous solutions of NaN03 and Ca(N03)2. The concentrations are 10 5, 10 4, 10-3, 10 2, 10 1 and 10° mol L-1 from the top downwards and the offset in A by which the curves are shifted for clarity is 20. The solid lines represent model calculations using a complementary error function to describe the segment density profile...

Notice the tailing at depths greater than about 0.1 micron. Maximum error in data is indicated, (b) Error-function fit (solid line) to the tail portion of the profile, (c) Error-function fit to the residual profile. Symbols are residual-concentration data calculated by subtracting the error-function fit in b from data in a, (d) Fit of data is by the linear combination of the error functions shown in b and c. Taken from the work of Moore et al. (1998). 

Fig. 9. Number of attractors correctly discovered from the same random set by a HEDA with a standard linear learning rule (square markers and solid error bars) and a HEDA trained with the log-Hebbian rule. In both cases, the target function contained n(2 Inn) attractors and the graph shows how many of them were found.

The diffusion and solid solubility of Cr were investigated by using radiochemical and electrical methods. At a diffusion temperature of 1250C, the doping profile could be approximated by an error function. Assuming that the error function also held at lower temperatures, the diffusion coefficient was determined at 1100 to 1250C by using the pn-junction method. The expression,... 

Figure 5.1 Composition profiles of Co determined normal to the Al-Co interface in the initially prepared state and after 5 min of annealing at 300 °C [6j. The solid line represents the approximated error function.

Fig. 8.16 (a) Segment density profiles for three thicknesses of adsorbed n-hexadecane films at 350 K, plotted vs. distance for the solid surface, with the original at the center of atoms in the top layer of the solid substrate. An error-function fit for the tail region (marked by arrows) of the thickest film is shown in the inset, (b) Diffusion constants in the directions parallel and perpendicular to the surface plotted vs. distance. Circles and triangles correspond to the thinnest (about 0.1 nm) and thickest (about 4.0 nm) films, respectively. SoUd symbols denote parallel diffusion and empty ones perpendicular diffusion. (Redrawn from Ref. 30.)... 

FIG. 16 Phase diagram of fluid vesicles as a function of pressure increment p and bending rigidity A. Solid lines denote first-order transitions, dotted lines compressibility maxima. The transition between the prolate vesicles and the stomatocytes shows strong hysteresis efifects, as indicated by the error bars. Dashed line (squares) indicates a transition from metastable prolate to metastable disk-shaped vesicles. (From Gompper and KroU 1995 [243]. Copyright 1995 APS.)... 

Successful recrystallization of an impure solid is usually a function of solvent selection. The ideal solvent, of course, dissolves a large amount of the compound at the boiling point but very little at a lower temperature. Such a solvent or solvent mixture must exist (one feels) for the compound at hand, but its identification may necessitate a laborious trial and error search. Solvent polarity and boiling point are probably the most important factors in selection. Benzhydrol, for example, is only slightly soluble in 30-60 petroleum ether at the boiling point but readily dissolves in 60-90° petroleum ether at the boiling point. 

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

The specific conductivities of molten salts are frequently represented, as a function of temperature by an Arrhenius equation, but it is unlikely that the unit step in diffusion has a constant magnitude, as in the corresponding solids and the results for NaCl may be expressed, within experimental error, by the alternative equations... 

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