# Interfacial Thermal Resistance

In very thin films, as noted above, the attractive force from the solid surface to the liquid produces a pressure difference (disjoining pressure) across the liquid-vapor interface, in addition to the capillary effect. These two effects reduce the saturated vapor pressure over a thin film with curvature in comparison with the normal saturated condition. Consider a thin liquid film with liquid thickness δ over a substrate with liquid interface temperature Tδ and normal saturation vapor pressure corresponding to Tδ of ${{p}_{sat}}\left( {{T}_{\delta }} \right)$. At equilibrium, the chemical potential in the two phases must be equal:

 ${{\mu }_{\ell }}={{\mu }_{v}}$ (1)

Integrating the Gibbs-Duhem equation,

 dμ = − sdT + vdp (2)

at constant temperature from the normal saturated pressure ${{p}_{sat}}\left( {{T}_{\delta }} \right)$ to an arbitrary pressure gives

 $\mu -{{\mu }_{sat}}=\int_{psat(T\delta )}^{p}{vdp}$ (3)

Using the ideal gas law $\left( {{v}_{v}}={{R}_{g}}{{T}_{\delta }}/{{P}_{v}} \right)$ for the vapor phase, and assuming the liquid phase is incompressible $(v={{v}_{\ell }})$, one obtains the following relations upon integration of eq. (3) for the vapor and liquid chemical potentials, respectively:

 ${{\mu }_{v,\delta }}={{\mu }_{sat,v}}+{{R}_{g}}{{T}_{\delta }}\ln \frac{{{p}_{v,\delta }}}{{{p}_{sat}}({{T}_{\delta }})}$ (4)
 ${{\mu }_{\ell ,\delta }}={{\mu }_{sat,\ell }}+{{v}_{\ell }}[{{p}_{\ell }}-{{p}_{sat}}({{T}_{\delta }})]$ (5)

Since ${{\mu }_{sat,\ell }}={{\mu }_{sat,v}}$, substituting eqs. (4) and (5) into eq. (2) yields

 ${{p}_{v,\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left\{ \frac{{{v}_{\ell }}[{{p}_{\ell }}-{{p}_{sat}}({{T}_{\delta }})]}{{{R}_{g}}{{T}_{\delta }}} \right\}$ (6)

The pressure difference in the vapor phase pv and the liquid phase ${{p}_{\ell }}$ due to capillary and disjoining effects are related as follows:

 ${{p}_{v,\delta }}-{{p}_{\ell }}={{p}_{cap}}-{{p}_{d}}$ (7)

where pcap is capillary pressure. Equation (7) can be used to eliminate ${{p}_{\ell }}$ in eq. (6).

 ${{p}_{v,\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left\{ \frac{[{{p}_{v,\delta }}-{{p}_{sat}}({{T}_{\delta }})-{{p}_{cap}}+{{p}_{d}}]}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right\}$ (8)

When the interface is flat and pd = 0, ${{p}_{v,\delta }}={{p}_{sat}}\left( {{T}_{\delta }} \right)$. For a curved interface and pd = 0, eq. (8) is reduced to the following Kelvin equation:

 ${{p}_{v,\delta }}={{p}_{sat}}({{T}_{\delta }})\exp \left\{ \frac{[{{p}_{v,\delta }}-{{p}_{sat}}({{T}_{\delta }})-{{p}_{cap}}]}{{{\rho }_{\ell }}{{R}_{g}}{{T}_{\delta }}} \right\}$ (9)

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.