Surface thermodynamics

The keys of experimental thermodynamics were explained in other two documents. If we want to apply those to the surface, some certain modifications must be taken. Below, we talk about the special treatment on surfaces.

Definition of surface

To study a surface, one has to have a well defined concept of surface. As shown in the Fig. below, surface is normally defined by the two boundaries that separate surface from bulk (can be solid or liquid) and gas. The only requirement for the boundaries are the bulk and gas has to include phases that are truly bulk and gas.

There are two important issues one has to keep in mind:

1) A surface never stands alone. In fact, when one talks about a surface, the implication is always that it is between a bulk and a gas phases. Because of that, the number of independent variables for a surface system is often lower than a stand-alone gas system. When there are coexisting phases, all the intensive state functions (, and ) must be the same. These constraints take away a certain independent variables.

Following example is very revealing on this issue.

Consider a single substance system, (e.g. Cu surface without strain),

i) one begins with 2 independent variables: e.g. and .

ii) on a surface, there is an additional variable: surface tension .

iii) because of the equilibrium with the bulk and gas, one has:

which eliminates two independent variables.

Hence, there is only one independent variables left for such a system.

2) Obviously, there are infinite number of ways to draw the lines, which suggests ambiguity. Fortunately, by introducing the concept “surface excess”, the ambiguity problem can be resolved.

Surface excess

The concept of “surface excess” can never be overemphasized because it lies on the most fundamental part of the surface thermodynamics. The formal definition of surface excess can be exemplified as the following.

Consider the extensive quantities ,, and , their corresponding values in the bulk, surface and gas phase are indicated by the subscript , and. Suppose that the surface is composed of two components of bulk and gas, one can solve the equations:

to get and .

Using this composition, we can try to guess the other extensive value

This value is going to be different from the true .

The different is called the surface excess.

It is not hard to prove that this value does not depend on the boundary we choose, as long as the surface is complete.

Law of thermodynamics on surface

Once we understand the discussion above, we are ready to take on the law of surface thermodynamics.

In principle, the Gibbs-Duhem equation for a surface is very similar to the one in single phase case:

.

Basically we only added the surface tension terms here. As we discussed, this is not the whole story, because there is still the constraint that the surface is in equilibrium with the bulk and the gas. The constrain can be taken into account by considering the Gibbs-Duhem equations of the bulk and the gas:

We can eliminate, for example,  and . The result is:

,

This is the most fundamental (although surprisingly simple) law of surface thermodynamics.

The symbolrepresents the solution above, which can be reached using simple math. The detail can be found in [Chung2001]. Interestingly, it is exactly equal to the surface excess discussed above.

Application

1. Surface excess of particle numbers

When organic molecules are dissolved in water, it is often the case that the concentration of on the surface is very different from the concentration in the solution. In this problem, we are talking about two substances. Hence:

,

where the subscript  and  correspond to organic molecules and water respectively. Therefore,

is the surface excess of the organic molecule.

Since ,

where  is the concentration in the solution, one can determine the surface excess by measuring the relation between  and  and use the formula

.

2. Surface adsorption

One can follow almost exactly the same steps for gas adsorption on surface to reach:

,

where the only change is to use  instead of .

Unfortunately, unlike the liquid surfaces, the tension of a solid surface is very hard to measure directly. Therefore, the formula we found here is not that useful for surface adsorption problem. Instead, the Langmuir isotherm approach is a better way to deal with this problem.

[Chung2001] Yip-Wah Chung Practical Guide to Surface Science and Spectroscopy, Academic press 2001.