8.3 Thermodynamics of Surfaces and Interfaces

Surfaces and interfaces have unique thermodynamic properties because molecules there experience an imbalanced force environment compared to the bulk. This section covers surface free energy, surface tension, and surface excess quantities, then builds toward key equations (Gibbs adsorption, Laplace, Kelvin, Young) and their applications in adsorption and wetting.

Thermodynamics of Surfaces

Unique Properties of Surfaces and Interfaces

Molecules in the bulk of a material are surrounded on all sides by neighbors, so intermolecular forces roughly cancel. At a surface or interface, that symmetry breaks: molecules have fewer neighbors on one side, leading to a net inward pull. This reduced coordination is the origin of every surface thermodynamic quantity discussed here.

• Surface free energy ($\gamma$) is the excess Gibbs free energy per unit area created by the presence of a surface. It quantifies the energetic cost of having molecules in that under-coordinated environment rather than in the bulk.
• Surface tension is numerically equal to the surface free energy for a pure liquid, but conceptually it's a force per unit length acting parallel to the surface. The system "wants" to contract its surface area to minimize $\gamma$, and surface tension is the mechanical manifestation of that tendency.
• Surface excess quantities capture the difference between the actual amount of a substance (or any extensive property) at the interface and the amount you'd predict by assuming bulk properties extend uniformly right up to a mathematical dividing surface. Common examples:
  • Surface excess concentration ($\Gamma_i$): the extra moles of component $i$ per unit area accumulated at the interface.
  • Surface excess entropy: the extra entropy per unit area associated with the interfacial region.

Derivation of Surface Free Energy and Surface Tension Expressions

Starting from the Gibbs free energy of a system that can change its surface area at constant $T$, $P$, and composition:

1. Write the differential of $G$ including a surface work term:

$dG = -S\,dT + V\,dP + \sum_i \mu_i\,dn_i + \gamma\,dA$

1. At constant $T$, $P$, and $n_i$, this reduces to:

$dG = \gamma\,dA$

1. Surface tension is therefore identified as the partial derivative:

$\gamma = \left(\frac{\partial G}{\partial A}\right)_{T,P,n_i}$

This tells you $\gamma$ is the reversible work per unit area needed to create new surface.

Gibbs adsorption equation. For a multicomponent system at constant $T$, the variation of surface tension with composition is:

$d\gamma = -\sum_i \Gamma_i\,d\mu_i$

• $\Gamma_i$ is the surface excess concentration of component $i$.
• $d\mu_i$ is the change in chemical potential of component $i$.

For a two-component system (solvent 1, solute 2) using the Gibbs dividing surface convention where $\Gamma_1 = 0$, this simplifies to:

$d\gamma = -\Gamma_2\,d\mu_2$

If the solute is dilute and ideal, $d\mu_2 = RT\,d\ln c_2$, so:

$\Gamma_2 = -\frac{1}{RT}\left(\frac{\partial \gamma}{\partial \ln c_2}\right)_T$

This is extremely useful: you can extract the surface excess of a solute just by measuring how surface tension changes with concentration. A surfactant that lowers $\gamma$ as concentration rises gives a positive $\Gamma_2$, confirming it accumulates at the interface.

Surface Free Energy and Tension

Curvature Effects on Surface Thermodynamics

A flat interface is a special case. When the interface is curved, the pressure on the concave side must exceed the pressure on the convex side to balance the inward pull of surface tension.

Laplace equation. The pressure difference across a curved interface with principal radii of curvature $R_1$ and $R_2$ is:

$\Delta P = \gamma\left(\frac{1}{R_1} + \frac{1}{R_2}\right)$

For a spherical surface (e.g., a small droplet or bubble of radius $R$), both radii are equal, so:

$\Delta P = \frac{2\gamma}{R}$

Notice that as $R$ shrinks, $\Delta P$ grows. A nanometer-scale droplet can have an internal pressure many atmospheres above ambient.

Kelvin equation. Curvature also shifts the equilibrium vapor pressure. The vapor pressure $P$ over a curved liquid surface of radius $r$ relative to the flat-surface vapor pressure $P^0$ is:

$\ln\!\left(\frac{P}{P^0}\right) = \frac{2\gamma V_m}{rRT}$

• $V_m$ is the molar volume of the liquid.
• $R$ is the gas constant; $T$ is temperature.

For a convex surface (e.g., a small droplet, $r > 0$), the vapor pressure is higher than over a flat surface. This means small droplets evaporate more readily than large ones, which drives Ostwald ripening: large droplets grow at the expense of small ones.

For a concave meniscus ($r < 0$), the vapor pressure is lower, which explains capillary condensation: vapor condenses in narrow pores at pressures below the normal saturation pressure.

Surface Thermodynamics Applications

Adsorption

Adsorption is the accumulation of molecules (adsorbates) at a surface, driven by the reduction of surface free energy. There are two broad categories:

• Physisorption: the adsorbate binds through weak van der Waals forces. Binding energies are typically on the order of 5–40 kJ/mol, and the process is readily reversible.
• Chemisorption: the adsorbate forms actual chemical bonds with the surface. Binding energies are much larger (40–400 kJ/mol), and the process may be irreversible or require significant activation energy.

Adsorption isotherms describe how much adsorbate sits on the surface as a function of bulk-phase concentration (or pressure for gases) at constant $T$:

• Langmuir isotherm: assumes monolayer coverage on equivalent, independent sites. For gas-phase adsorption with pressure $P$:

$\theta = \frac{KP}{1 + KP}$

where $\theta$ is fractional surface coverage and $K$ is the equilibrium adsorption constant. At low $P$, coverage is linear in pressure; at high $P$, the surface saturates ($\theta \to 1$).

• Freundlich isotherm: an empirical relation, $\theta \propto P^{1/n}$ (with $n > 1$), that works well for heterogeneous surfaces where binding sites have a distribution of energies.

Wetting

Wetting describes how well a liquid spreads on a solid surface. It's governed by the competition between adhesive forces (liquid–solid) and cohesive forces (liquid–liquid).

Young's equation gives the equilibrium contact angle $\theta$ of a droplet on a flat, ideal solid:

$\gamma_{sv} = \gamma_{sl} + \gamma_{lv}\cos\theta$

• $\gamma_{sv}$, $\gamma_{sl}$, $\gamma_{lv}$ are the solid–vapor, solid–liquid, and liquid–vapor surface tensions, respectively.
• $\theta < 90°$: the surface is hydrophilic (wetting). Adhesive forces dominate.
• $\theta > 90°$: the surface is hydrophobic (non-wetting). Cohesive forces dominate.
• $\theta \approx 0°$: complete wetting (the liquid spreads into a thin film).

Dupré equation defines the thermodynamic work of adhesion, the reversible work per unit area to separate the liquid from the solid:

$W_{\text{adh}} = \gamma_{sv} + \gamma_{lv} - \gamma_{sl}$

Combining Dupré with Young's equation gives the Young–Dupré relation:

$W_{\text{adh}} = \gamma_{lv}(1 + \cos\theta)$

This is practical because you can determine the work of adhesion from just two measurable quantities: the liquid's surface tension and the contact angle.