The Local Initial Adsorption Constant kx its Limit Expressions and Approximation

Full Solution and Limit Expressions

Let us recall that the general expression (Dejardin et al. 1994) for the constant k(x) at distance x from the channel entrance as a function of ka and kLev is not easily accessible for the determination of ka.

with

Lco where

with

1 rA

r'(n) is the usual gamma function (Abramovitz and Stegun 1972). The prime is added to avoid confusion with r used for the interfacial concentration.

It is preferable to have ka as a function of experimental k(x). It has been shown that rather good approximations of the general solution can be obtained at the two limits of the controls by transport and the interfacial reaction (Dejardin et al. 1994). These involve the inverse of the kinetic constants and, in fact, lead directly to the interpretation of the total resistance (time) of the adsorption process as being the sum of one resistance due to the transport and the other resistance due to the interfacial reaction. Close to the conditions of the transport-controlled process, k-1 = k-1 + 0.684k-1 , (6a)

while close to the conditions of the control by interfacial reaction k-1 = 0.827k-! + k-1 (6b)

Both expressions are similar to the simplest approximation, which considers no coupling between transport and the interfacial reaction:

Equation 7 can be also obtained by assuming that the thickness of the diffusion layer (when C(x,y = 0) = 0, Leveque model) is unaltered whatever the finite value of ka, say whatever the nonzero steady-state value of the volume concentration near the interface. Equation 7 corresponds to the work (Bowen and Epstein 1979) and matches Eq. 2.45 of Bowen et al. (1976).

Approximation of k(x)/ka as a function of k(x)/kLev(x) Ifthelinearapproximationsgiven in Eq. 6aandbare validattothe twolimits can be useful in practice, as it is easy to deduce ka from k(x)and kLev(x), they do not describe the entire domain as does the general expression (which writes k as a function of ka, not the reverse). Recently we have proposed (Déjardin and Vasina 2004) to approximate the general expression in Eq. 5a by the function y = f (u), where y = k/ka and u = k/kFev:

with a = 0.451707 and b = -0.624713, to satisfy the two limits of Eq. 6a,b. The greatest relative variation between Eq. 8 and the complete calculation

Fig. 3. Adsorption of a-chymotrypsin (V, dashed line, 10-2 M Tris; pH 8.6) onto mica. The normalization of the scales to the intercepts ka and D2/3 illustrates the magnitude of the depletion at the interface (Cy=0/Q,; right axis), where y is the distance to the interface, and the departure from the fully transport-limited process (k/kLev; top axis). For measurements performed at the same distance x, the experimental points corresponding to one given wall shear rate are positioned on a straight line passing through the origin. An example is provided with the points corresponding to experiments in 10-2 M (V), 0.2 M (♦) and 0.5 M (# dash-dotted line) Tris, pH 8.6

Fig. 3. Adsorption of a-chymotrypsin (V, dashed line, 10-2 M Tris; pH 8.6) onto mica. The normalization of the scales to the intercepts ka and D2/3 illustrates the magnitude of the depletion at the interface (Cy=0/Q,; right axis), where y is the distance to the interface, and the departure from the fully transport-limited process (k/kLev; top axis). For measurements performed at the same distance x, the experimental points corresponding to one given wall shear rate are positioned on a straight line passing through the origin. An example is provided with the points corresponding to experiments in 10-2 M (V), 0.2 M (♦) and 0.5 M (# dash-dotted line) Tris, pH 8.6

is 1% around u = 0.8. Let us note that k(x)/ka = C(x, 0)/Cb, so the ordinate illustrates the depletion in solution at the interface, while u = k/kLev estimates contribution of the transport. Figure 3 shows an example of these two items.

Equation 8 can be transformed into a two-parameter fit (D and ka) to experimental k(x, y) as a function of u' = k(x/y)1/3/0.538 = uD2/3, which does not require the knowledge of D.

In such a representation, the intercept of the fit with the ordinate axis gives ka, while the intercept with the abscissa axis gives D2/3 (see Fig. 3). Figure 4 illustrates the magnitude of error in the adsorption kinetic constant ka and diffusion coefficient D that can occur when using Eq. 7. Equation 7 corresponds there to the straight line k = ka(1 - u'/D2/3).

Fig. 4. Illustration of the maximum (Max.) error on adsorption constant ka and diffusion coefficient D when neglecting any coupling between transport and interfacial reaction (straight dash-dotted line; Eq. 7 in text), compared to exact solution (full curved line). The dashed lines demonstrate linear approximations at the two extreme limits

Fig. 4. Illustration of the maximum (Max.) error on adsorption constant ka and diffusion coefficient D when neglecting any coupling between transport and interfacial reaction (straight dash-dotted line; Eq. 7 in text), compared to exact solution (full curved line). The dashed lines demonstrate linear approximations at the two extreme limits

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