5. Electrical double layer and adsorption

5.1. Gibbs adsorption isotherm

Adsorption on electrode surfaces can take place with or without simultaneous charge transfer between the adsorbing species and the electrode. When charge transfer is involved, it can be either partial or complete charge transfer. In this chapter, we look at adsorption, which takes place without charge transfer. The best tool for this is thermodynamics.

Consider an interface of surface area A separating two phases, \alpha and \beta . These two phases can be, for example, a metal electrode and an electrolyte solution, or two immiscible liquids in contact with one another. In a system with several components, the (electro) chemical potential of each component has to be equal in both phases and at the interface. Experience has shown that real systems tend to either minimize or maximize the interfacial area. Because of this, the free energy of the system has to depend on the area:

\displaystyle dG=\left(\frac{\partial G}{\partial T}\right)dT+\left(\frac{\partial G}{\partial P}\right)dP+\left(\frac{\partial G}{\partial A}\right)dA+\sum\limits_{i}\left(\frac{\partial G}{\partial n_i}\right)_idn_i (5.1)

We can give a name to the partial derivative (\partial G/\partial A) : it is the surface tension \gamma . The forces that bind particles together are weaker at the interface because the particles at the surface have fewer neighboring atoms compared to the ones in the bulk. Instead, the particles at the surface interact with particles in the neighboring phase. This changes the equilibrium of forces and can lead to a new force called surface tension. Surface tension is therefore a force at the interface that tends to increase or decrease the growth of the surface. It is significant only in a very thin zone, a few molecules thick, near the interface.

Gibbs energy differentials \alpha and  \beta are

\displaystyle dG^{\alpha}=\left(\frac{\partial G^{\alpha}}{\partial T}\right)dT+\left(\frac{\partial G^{\alpha}}{\partial P}\right)dP+\sum\limits_{i}\left(\frac{\partial G^{\alpha}}{\partial n_i^{\alpha}}\right)_idn_i^{\alpha} (5.2)
\displaystyle dG^{\beta}=\left(\frac{\partial G^{\beta}}{\partial T}\right)dT+\left(\frac{\partial G^{\beta}}{\partial P}\right)dP+\sum\limits_{i}\left(\frac{\partial G^{\beta}}{\partial n_i^{\beta}}\right)_idn_i^{\beta} (5.3)

We will only look at experiments conducted at constant temperature and pressure, and thus the two first terms of the Gibbs equation can be neglected. Since the phases are in equilibrium, the chemical potentials of each species are equal in the whole system:

\displaystyle\left(\frac{\partial G}{\partial n_i}\right)=\left(\frac{\partial G^{\alpha}}{\partial n_i^{\alpha}}\right)=\left(\frac{\partial G^{\beta}}{\partial n_i^{\beta}}\right)=\mu_i (5.4)

Subtracting the Gibbs energies of phases \alpha and \beta , the Gibbs energy of the surface is:

\displaystyle dG^{\sigma}=\gamma dA+\sum\limits_i\mu_id(n_i-n_i^{\alpha}-n_i^{\beta}) (5.5)

There is either an excess or a deficit of components on the surface compared to the bulk of the solution. The difference in the amounts of substance on the surface and in the bulk is called the surface concentration, or surface excess, n_i^{\sigma} :

\displaystyle n_i^{\sigma}=n_i-n_i^{\alpha}-n_i^{\beta} (5.6)

Consider a situation in which a surface is formed slowly between two phases from zero to a finite value A, while pressure, temperature, and the composition of the solution are constant. In this case, the surface concentration increases from zero to the value n_i^{\sigma} , but \gamma and \mu_i  are constant. Equation (5.6) is substituted into Equation (5.5) and integrated

\displaystyle dG^{\sigma}=\gamma dA+\sum\limits_i\mu_idn_i^{\sigma}

(5.7)
\displaystyle\int_{0}^{G^{\sigma}}{dG^{\sigma}}=\gamma \int_{0}^{A}{dA} +\sum\limits_i\mu_i \int_{0}^{n_i^{\sigma}}{dn_i^{\sigma}} (5.8)

And we obtain

\displaystyle G^{\sigma}=\gamma dA+\sum\limits_i\mu_in_i^{\sigma} (5.9)

The total differential of Gibbs energy is

\displaystyle dG^{\sigma}=\gamma dA+\sum\limits_i\mu_in_i^{\sigma}+Ad\gamma+\sum\limits_in_i^{\sigma}d\mu_i (5.10)

Equations (5.7) and (5.10) have to be equivalent, and thus the sum of the last two terms of Equation (5.10) is zero.

\displaystyle Ad\gamma+\sum\limits_in_i^{\sigma}d\mu_i=0 (5.11)

It is often convenient to talk about surface excess per unit area. We therefore define the surface concentration  \Gamma_i=n_i^{\sigma}/A  , and the previous equation becomes


\displaystyle -d\mu_1\=\sum\limits_i\Gamma_id\mu_i

(5.12)

This equation is called the Gibbs adsorption isotherm. The equation implies that measurements of surface tension will give us information on the structure of the interface.

It is useful to describe the adsorption of other components with respect to one predominant component, e.g. a solvent, in dilute solutions in particular. Here, the main component is given the subscript 1. In the bulk of the solution, the Gibbs-Duhem equation holds. According to it, for any phase at constant temperature and pressure, the following is true:*

\displaystyle\sum\limits_in_id\mu_i=0 (5.13)

The change in the electrochemical potential of the main component is therefore

\displaystyle -d\mu_1=\sum\limits_{i \neq1} \frac{n_i}{n_1}d\mu_i (5.14)

Using this, the Gibbs adsorption isotherm gives

\displaystyle -d\gamma=\sum\limits_{i\neq 1} \Gamma_id\mu_i+\Gamma_1d\mu_1=\sum\limits_{i\neq 1}\left(\Gamma_i-\frac{n_i}{n_1}\Gamma_1\right)d\mu_i (5.15)

The bracketed term is called the relative surface excess \Gamma^{\sigma}

\displaystyle\Gamma_i^{\sigma}=\Gamma_i-\frac{n_i}{n_1}\Gamma_1 (5.16)


In very dilute solutions n1 » ni, and \Gamma_i^{\sigma} \approx \Gamma_i .


*See also Equation (2.5) and Chapter 2.2.