Gibbs Treatment of the Interfacial Energy

Note the atoms at the surface were treated as being somehow different than those of the bulk. This idea can be extended rigorously to treat the interface as a relatively thin layer. This thin layer will be treated as a separate ``quasi-two-dimensional phase.''

The original Gibbs idea was as follows: Suppose the composition is different between to phases (but dictated by $\mu^{\alpha}_i = \mu^{\beta}_i$, of course). There is no requirement that the concentration should be uniform in the vicinity of the interface:

Figure 35-2: Illustration of the composition in the vicinity of a real interface.

The Gibbs treatment extends each of the homogeneous phases up to a mathematical surface:

Figure 35-3: Gibbs idealization of the interface as a mathematical dividing surface.

Subtract the real system in Figure 35-2 from its idealization in Figure 35-3 to define an excess quantity associated with the mathematical surface ``The Gibbs Surface'' which has no volume associated with it, but excess extensive quantities.

If the following illustration represents a thermodynamic system:

Figure 35-4: Illustration of an interface separating two phases. The extra surface represents the motion normal to itself and is used in the derivation of Equation 35-13.

For the surface phase:

$$\input{equations/surf-dU}$$ (35-7)

where $A$ is the surface area of the interface.

What is $\gamma$?

Obviously,

$$\input{equations/28-6A}$$ (35-8)

But, it also represents the work done to increase the surface area

$$\input{equations/28-6B}$$ (35-9)

where a force $F$ is applied to a surface of width $w$ to extend its length $l$.

Figure 35-5: The mechanical implications of surface tension.

This is the surface tension, it has units of force per unit length or, equivalently, surface energy per area. It is the energy associated with creating surface.¹

Consider the entire system:

$$\input{equations/28-7A}$$ (35-10)

Because the surface represents an object that can resist pressure, it can no longer assumed that the pressures on the inside and the outside phase are equal.

For equilibrium $dU = 0$, along with the temperature and chemical potentials being uniform:

$$\input{equations/28-7B}$$ (35-11)

Imagine that the surface moves normal to itself (such as in Figure 35-4)

Then the following relations hold:

$$\input{equations/28-8A}$$ (35-12)

$$\input{equations/28-8B}$$ (35-13)

where $K_1$ and $K_2$ are the ``curvatures'' of the surface in each of two perpendicular planes, where the axis of intersection is normal to the surface.

$$\input{equations/28-8C}$$ (35-14)

is called the ``mean curvature'' of the surface ${1}/{R_1}=K_1$, and ${1}/{R_2}=K_2$ where $R_1$ and $R_2$ are the radii of curvature.

Putting equations 35-12, 35-13, and 35-14 together, the following important relation holds:

$$\input{equations/gibbs-thompson}$$ (35-15)

which is known as the Gibbs-Thompson equation. It relates the difference in pressure at an interface to its surface tension and curvature.