Estimate of the Excess Energy Associated with Surfaces

In the treatment of the equilibrium of phases the effect of the surface that separates the various phases was neglected.

In other words, no distinction was made between systems that had an abundance of surface and those that do not--there was no distinction made between:

Figure 35-1: Including the effect of interfaces and surfaces. The treatment of equilibrium up until now treated these two systems as being alike even though one obviously has much more surface (and thus any energy associated with that surface) than the other.

Consider, as an example, that an atom on a surface as having a 50% higher energy than those in the bulk, then there will be an extra energy associated with the surface of a sphere.

To estimate how much energy is associated with the surface let:

surface area$$\equiv$$   surface area of $$\alpha{\mbox{-phase}} = 4\pi R_s^2$$

volume $$\equiv$$   volume of $$\alpha{\mbox{-phase}} = \frac{4}{3}\pi R_s^3$$

The energy of the system is:

$$\input{equations/28-2A}$$ (35-1)

Letting the energy of an atom on the surface be half again that of the bulk:

$$\input{equations/28-2B}$$ (35-2)

$$\input{equations/28-2C}$$ (35-3)

If $\Omega$ is the volume per atom and $R_A$ is the radius of an atom, then

$$\input{equations/28-2D}$$ (35-4)

Letting $R_s$ be the radius of the sphere:

$$\input{equations/28-3A}$$ (35-5)

How small does the sphere need to be in order that the excess is about 1%?

$$\input{equations/28-3B}$$ (35-6)

This is pretty small, but important important in many systems.