Ward-Tordai equation

The time, f,-, for the induction period (region I) to end is an important factor in determining the surface tension as a function of time, since only when that period ends does the surface tension start to fall rapidly. The value of f,- has been shown (Gao, 1995 Rosen, 1996) to be related to the surface coverage of the air-aqueous solution interface and to the apparent diffusion coefficient, Dap, of the surfactant, calculated by use of the short-time approximation of the Ward-Tordai equation (Ward, 1946) for diffusion-controlled adsorption (equation 5.6) ... 

This adsorption model can be solved analytically using Laplace Transforms (Hansen 1961, Miller 1983) but the result is a non-linear Volterra integral equation similar to the Ward Tordai equation (4.1) ... 

The classical Ward Tordai equation results when Eq. (4E.13) is rearranged to... 

The Ward Tordai equation (4.1) is comparatively complex and large numerical efforts are necessary for its application. Therefore, simple asymptotic and approximate solutions are very... 

WardTordai, which implements the solution of the Ward-Tordai equation for Langmuir, Frumkin, Two-State Reorientation Quasiequilibrium, Two-State Reorientation Kinetics and Aggregation models. Using this module, the user can compare (both visually and numerically) his experimental data with the kinetic curves calculated from any of these models. 

The equation which is implemented in the software is somewhat more general than the ordinary Ward-Tordai equation (4.3) first proposed in [6], and accounts also for the existence of an adjacent second liquid phase. 

As mentioned above, a complete set of equation involves and equation of the type of Eq. (7.35), otherwise a numerical solution of the Ward-Tordai equation is not available. The software package includes all adsorption models described in Chapter 3, i.e. the classical Langmuir and Frumkin model as well as the reorientation and 2D-aggregation models. 

In mathematical terms, the adsorption being diffusion-limited means that the variation of the free energy with respect to 0o can be taken to zero at all times whereas the variation with respect to (j> x> 0) cannot. This has two consequences. The first is that the relation between 0o and (j)i is given at all times by the equilibrium adsorption isotherm [(3) in our model]. The solution of the adsorption problem in the non-ionic, diffusion-limited case amounts, therefore, to the simultaneous solution of the Ward-Tordai equation (8) and the adsorption isotherm. Exact analytical solution exists only for the simplest, linear isotherm, °c 0i [19]. For more realistic isotherms such as (3), one has to resort to numerical techniques (useful numerical schemes can be found in refs. [2, 8]). The second consequence of the vanishing of 5Ay/5 o is that the dynamic surface tension, Ay t), approximately obeys the equilibrium equation of state (4). These two consequences show that the validity of the schemes employed by previous theories is essentially restricted to diffusion-limited cases. 

Fig. 2B, one such curve published by Lin et al. [8]. The theoretical solid curve was obtained by these authors using a scheme similar to the one just described - solution of the Ward-Tordai equation together with the Frumkin isotherm and substitution in the equation of state to calculate the surface tension. Note that the parameters a, P and a can be fitted from independent equilibrium measurements, so the dynamic surface tension curve has only one fitting parameter, namely the diffusion coefficient, D. As can be seen, the agreement with experiment is quite satisfactory. However, when the adsorption is not diffusion-limited, such a theoretical approach is no longer applicable, as will be demonstrated in the next section. 

In 1907, Milner [8] first suggested that the variation of surface tension of a surfactant solution could be mediated by molecules diffusing to the interface. Some considerable time later, Langmuir and Schaeffer [9] made a significant advance when they looked at the diffusion of ions into monolayers and proposed a mathematical model of the diffusion process. However, it was not until the seminal 1946 paper of Ward and Tordai [10] that the first complete model for diffusion-based kinetics emerged. The Ward-Tordai model accounts for three variables the bulk concentration, the subsurface concentration, and the surface tension. This led to the celebrated Ward-Tordai equation ... 

As the Ward-Tordai equation contains two independent variable functions (surface excess and subsurface concentration), its application requires a further equation relating the two functions. The first attempt at this was by Sutherland [11], who incorporated a linear adsorption isotherm. This, however, proved to be quite limiting, and so various other isotherms were employed [12, 13]. Even so, these extended theorems accurately matched experimental results only in the case of some nonionic surfactants. 

A classic model for accounting for adsorption/desorption dynamics is the Ward and Tordai equation [2]. This model predicts the presence of a subsurface layer, a few molecular diameters in size. Molecules present in the subsurface immediately adsorb to the surface at early times, while at later times they might linger there or possibly diffuse back into the bulk. Therefore, transport from the bulk to the subsurface is purely diffusive, while the mechanism between the subsurface and the surface depends on time in a more complex manner. The Ward-Tordai equation may be written as... 

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