Ward-Tordai model

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 ... 

The adsorption kinetics of interfacial active molecules at liquid interfaces, for example surfactants at the aqueous solution/air or solution/organic solvent interface, can be described by quantitative models. The first physically founded model for interfaces with time invariant area was derived by Ward Tordai (1946). It is based on the assumption that the time dependence of interfacial tension, which is directly correlated to the interfacial concentration T of the adsorbing molecules, is caused by a transport of molecules to the interface. In the absence of any external influences this transport is controlled by diffusion and the result, the so-called diffusion controlled adsorption kinetics model, has the following form... 

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) ... 

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. 

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. 

The present state of research allows describing the adsorption kinetics for most surfactants at liquid interfaces quantitatively. The first model derived by Ward Tordai (1946) was based on the assumption that the time dependence of interfacial tension, which is directly related to the interfacial concentration T of adsorbed molecules via an equation of state, is mainly caused by the diftusional transport of surfactant molecules to the interface. A schematic picture of this model is shown in Fig. 1. Transport in the solution bulk is controlled by surfactant diffusion. The transfer of molecules from the so-called subsurface, the liquid layer adjacent to the interface, to the interface itself is assumed to happen without transport. 

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. 

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... 

The first physically sound model for adsorption kinetics, which was derived by Ward and Tordai [18], is based on the assumption that the time dependence of a surface or interfacial tension (which is directly proportional to the surface excess F, in mol m ) is caused by diffusion and transport of surfactant molecules to the interface. This is referred to as diffusion-controlled adsorption kinetics model . The interfacial surfactant concentration at any time t, T(t), is given by the following expression,... 

A quantitative description of adsoiption kinetics processes is so far usually based on the model derived in 1946 by Ward and Tordai [3], The various models developed on this basis use mainly different boundary and initial conditions [2], as it becomes clear from the schematic in Fig. 4.1. The diffusion-controlled adsorption model of Ward and Tordai assumes that the step of transfer from the subsurface to the interface is fast compared to the transport from the bulk to the subsurface. It is based on the following general equation,... 

To derive an adsorption kinetics model the Ward and Tordai equation (4.1) is again the main relationship between the dynamic adsorption r(t) and the subsurface concentration c(0,t). As it was described in detail in paragraph 4.1.2, an adsorption isotherm as additional function r(c) is needed for a kinetic model. The isotherm equations (2.110) - (2.112) given in Chapter 2 represent such type of function, which accounts for a 2D-aggregation in the adsorption layer [48]. The set of equation is too complex to find an analytical solution. Only for the short time range and for low adsorption layer coverage, the following approximation is valid [65]... 

The penetration kinetics of a component 2 into an insoluble monolayer, can be monitored by measurements of the rate of the surface pressure change An(t). The above discussed integro-differential equation (4.1) derived by Ward and Tordai [3] is again the basis for a theoretical description of penetration processes. As shown in paragraph 2.9, a simple model for the diffusion mechanism of the penetration process can be obtained by using an equation of the following type interrelating the subsurface concentration and the adsorption 1-0, b C3(0,t)... 

For a modelling of adsorption processes the well-known integro-differential equation (4.1) derived by Ward and Tordai [3] is used. It is the most general relationship between the dynamic adsorption r(t) and the subsurface concentration e(0,t) for fresh non-deformed surfaces and is valid for kinetic-controlled, pure diffusion-controlled and mixed adsorption mechanisms. For a diffusion-controlled adsorption mechanism Eq. (4.1) predicts different F dependencies on t for different types of isotherms. For example, the Frumkin adsorption isotherm predicts a slower initial rate of surface tension decrease than the Langmuir isotherm does. In section 4.2.2. it was shown that reorientation processes in the adsorption layer can mimic adsorption processes faster than expected from diffusion. In this paragraph we will give experimental evidence, that changes in the molar area of adsorbed molecules can cause sueh effectively faster adsorption processes. 

The theoretical solution of models 1 and 2 is a generalized Ward and Tordai equation [Eq. (66)] and was first proposed by Hansen (112) ... 

As mentioned above reorientation processes in the adsorption layer can mimic adsorption processes faster than expected from diffusion. For modelling the adsorption process with molecular reorientation the Ward and Tordai equation (37) as the most general relationship between the dynamic adsorption F(t) and the subsurface concentration c(0,t) can be used. As additional relationship the reorientation isotherm Eqs. (18) to (20) are included into the model. 

Previous theoretical works have addressed these questions by adding appropriate assumptions to the theory. Sueh models can be roughly summarized by the following scheme (i) consider a diffusive transport of surfactant molecules from a semi-infinite bulk solution (following Ward and Tordai) (ii) introduce a certain adsorption equation as a boundary condition at the interface (iii) solve for the time-dependent surface coverage (iv) assume that the equilibrium equation of state is valid also out of equilibrium and calculate the dynamic surface tension [10]. 

Our formalism has led to a diffusive transport in the bulk [Eqs. (5) and (6)] coupled to an adsorption mechanism at the interface [Eq. (7)]. Yet unlike previous models, all of the equations have been derived from a single functional, and hence, various assumptions employed by previous works can be examined. Treating Eqs. (5) and (6) using the Laplace transform with respect to time, we obtain a relation similar to the Ward and Tordai result [1]. 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

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