Adsorption isotherm generalised Langmuir

This isotherm does not consider any interaction between the adsorbing molecules. The generalised Langmuir isotherm regards for a competitive adsorption of all surface active component r at the interface. 

The kinetics of adsorption from solutions of surfactant mixtures are described on the basis of a generalised Langmuir isotherm. The simultaneous adsorption leads to the replacement of less surface active compounds by those of higher surface activity, which are usually present in the bulk at much lower concentration. More general descriptions of the process are possible on the basis of the Frumkin and Frumkin-Damaskin isotherms, which include specific interfacial properties of the individual surface active species. Quantitative studies of such very complex models can be performed only numerically. 

Only for a generalised linear adsorption isotherm the adsorption of the components i are independent and consequently the functions ai. Assuming a generalised Langmuir type adsorption isotherm Eq. (2.47) the following complex elasticity modulus results. 

There are different models developed in the past to describe the adsorption from mixed surfactant solutions, for example recently by Siddiqui and Franses (1997), Ariel et al. (1999), Mulqueen and Blankschtein (1999, 2000), Penfold et al. (2003). The simplest model is obviously a generalised Langmuir isotherm (for ideal behaviour in the bulk and at the interface) for mixtures of two surfactants 1 and 2 with similar partial molar surface area (O can be presented in the form (Fainerman et al. 2001)... 

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

We start this book with a chapter (Chapter 2) on the fundamentals of pure component equilibria. Results of this chapter are mainly applicable to ideal solids or surfaces, and rarely applied to real solids. Langmuir equation is the most celebrated equation, and therefore is the cornerstone of all theories of adsorption and is dealt with first. To generalise the fundamental theory for ideal solids, the Gibbs approach is introduced, and from which many fundamental isotherm equations, such as Volmer, Fowler-Guggenheim, Hill-de Boer, Jura-Harkins can be derived. A recent equation introduced by Nitta and co-workers is presented to allow for the multi-site adsorption. We finally close this chapter by presenting the vacancy solution theory of Danner and co-workers. The results of Chapter 2 are used as a basis for the... 

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