Error function concentration profile

Fig. 16. Deuterium concentration profiles obtained by SIMS from which diffusion coefficients are obtained. The initial deuterated layer is indicated by the vertical lines the broken lines show error-function fits to the concentration profiles. The samples structure is indicated in the inset (Street et al., 1987b). 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

If D is constant, an experimental diffusion profile can be fit to the analytical solution (such as an error function) to obtain D. If it depends on concentration and the functional dependence is known. Equation 3-9 can be solved numerically, and the numerical solution may be fit to obtain D (e.g., Zhang et al., 1991a Zhang and Behrens, 2000). However, if D depends on concentration but the functional dependence is not known a priori, other methods do not work, and Boltzmann transformation provides a powerful way (and the only way) to obtain D at every concentration along the diffusion profile if the diffusion medium is infinite or semi-infinite. Starting from Equation 3-58a, integrate the above from Po to 00, leading to... 

The experimental duration and the length of the diffusion couple are designed such that the length of the diffusion profile is short compared to the total length of the diffusion couple. Hence, the diffusion medium may be treated as infinite, meaning that at the ends of the two halves, the compositions are still the initial compositions. If D does not vary with concentration or distance, the concentration profile (C versus x) would be an error function. Hence, the first step to try to understand the profile is to fit an error function to the profile (Equation 3-38) ... 

Calculated concentration profiles in the melt and in the crystal are shown in an example below. Equation 4-103 means that the solution is an error function with respect to the lab-fixed boundary (y = y + 2a VD t). In the interface-fixed reference frame, the solution appears like an error function, and its shape is often error function shape, but the diffusion distance is not simply For a... 

Another example for treating concentration profiles during mineral dissolution can be found in Figure 3-32b, which shows a Zr concentration profile during zircon dissolution. In this case, the dissolution distance is very small compared to the diffusion profile length. Hence, the diffusion profile is basically an error function. 

Figure 5-25 (a) Diffusion profile across a diffusion couple for a given cooling history. This profile is an error function even if temperature is variable as long as D is not composition dependent, (b) Diffusion profile across a miscibility gap for a given cooling history. Because the interface concentration changes with time, each half of the profile is not necessarily an error function. 

Next we turn to the inference of cooling history. The length of the concentration profile in each phase is a rough indication of (jDdf) = (Dot), where Do is calculated using Tq estimated from the thermometry calculation. If can be estimated, then x, Xc and cooling rate q may be estimated. However, because the interface concentration varies with time (due to the dependence of the equilibrium constants between the two phases, and a, on temperature), the concentration profile in each phase is not a simple error function, and often may not have an analytical solution. Suppose the surface concentration is a linear function of time, the diffusion profile would be an integrated error function i erfc[x/(4/Ddf) ] (Appendix A3.2.3b). Then the mid-concentration distance would occur at... 

Recall that for one-dimensional diffusion in infinite medium, the concentration profile is an error function and the mid-concentration point is the interface. The above profile is also roughly an error function (e.g., fitting the profile by an error function would give D accurate to within 0.1% if (4Df) la < 0.5), but the mid-concentration point is not fixed at Tq = a rather it moves toward the center as To = a 2Dtla. The evolution of concentration profile is shown in Figure A3.3.4. 

When Fick s law applies, the concentration profile generally contains information about the concentration dependence of the diffusivity. For constant D, step-function initial conditions have the error function (Eq. 4.31) as a solution to dc/dt = Dd2c/dx2. When the diffusivity is a function of concentration,... 

For identical initial conditions, the difference between a measured profile and the error-function solution is related to the last (nonlinear) term in Eq. 4.43. When diffusivity is a function of local concentration, the concentration profile tends to be relatively flat at a concentration where D(c) is large and relatively steep where D(c) is small (this is demonstrated in Exercise 4.2). Asymmetry of the diffusion profile in a diffusion couple is an indicator of a concentration-dependent diffusivity. 

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary case. 

Solution, The method of diagonalization described in Section 6.2.1 is employed. The initial conditions for all components are step-function concentration profiles and hence error-function profiles having the form... 

In fact, there is no reason why a detector should not have a logarithmic, exponential or any other functional output as long as the function can be explicitly defined. However, chromatograms obtained from such detectors would be unfamiliar and difficult to interpret and calculations for quantitative analysis would become very involved. The concentration profile of an eluted peak closely resembles that of the Gaussian or Error function where the independent variable is the volume of mobile phase passed through the column and the dependent variable solute concentration in the mobile phase. If the flow rate through the chromatographic system is constant, then the independent... 

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...

The resolution p Rg(N) already allows us to obtain an explicit measure of the brush layer thickness L (see Fig. 36). In this case the simplest step-function profile ( )(z) with constant composition in the brush layer region is assumed (see Fig. 33). While the de Gennes-Leibler model assumes all end-attached chains to stretch at the same distance z=L from the interface, the situation with a lower free energy is conceivable [226,228] characterized by the non-uniform stretching and the total brush concentration decreasing with z. Measurements performed with higher resolution reveal [242,243,261,264] the profiles ( )(z) of the stretched brushes which might be approximated by an error function [266] ... 

The solution of the ordinary differential equation can be formulated as an error function. The concentrations profile is thus given as ... 

The final result in equation (6.5.28) shows that the concentration falls from its initial value of Cq at the wall in a manner determined by the error function and the diffusion coefficient D. Typical concentration profiles estimated at various values of t are shown in fig. 6.3. Obviously, the concentration increases at a given distance from the wall as time increases. 

Figure 16.1 (a) Change of concentration in the interior of a sample where the concentration at the boundary increases with time if the increase is exponential with time, the profiles can be exponential through space, (b) Change of concentration where the concentration at the boundary is jumped to a new value and then kept constant the profiles are of error-function type. 

For analysis, let us assume that the concentration profile migrates inward as shown in Figure 16.1b with an error-function profile whose width increases in proportion to (time), and let us assume for a start that the stress state has no effect on the migration of potassium. Let us assume that the compressive stress at any point tends to rise because of this potassium concentration effect but tends to fall because of creep or relaxation of the glassy host. And let us assume that at small distances (less than 1 pm) from the surface, stress is relieved also by self-diffusion of the glass, so that right... 

At the other extreme, when the diffusion layer grows much larger than ro (as at a UME), the concentration profile near the surface becomes independent of time and linear with 1/r. One can see this effect in (5.2.22), where error function complement approaches unity for (r - ro) << 2(DoO - In that case. 

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. 12.4 Effects of the depth resolution in pore water concentration profiles on calculating the rates of diffusive transport. Three samples drawn from surface sediments are shown to possess different resolutions (intervals 0.5 cm - dots, 1.0 cm diamonds, 2.0 cm - squares). All values are sufficient to plot the idealized concentration profile within the hounds of analytical error, yet very different flux rates are calculated in dependence on the depth resolution values. In the demonstrated example, the smallest sample distance indicates the highest diffusion (2.98 mmol cmA f ). As soon as the vertical distance between single values increases, or, when the sediment segments under study grows in thickness, the calculated export across the sediment-water boundary diminishes (2.34-t.64mmol cm yr ). In our example, this error which is due to the coarse depth resolution can be reduced by applying a mathematical Fit-function. A truncation of 0.05 cm yields a flux rate of 2.84 mmol cm yr. (The indicated values were calculated under the assumption of the presented porosity profile according to Pick s first law of diffusion - see Chapter 3. A diffusion coefficient of 1 cmA f was assumed. Adaptation to the resolution interval of 2.0 cm was accomplished by using a simple exponential equation).

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