Interfacial Thermal Resistance

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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}}


Integrating the Gibbs-Duhem equation,

dμ = − sdT + vdp


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}


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 }})}


{{\mu }_{\ell ,\delta }}={{\mu }_{sat,\ell }}+{{v}_{\ell }}[{{p}_{\ell }}-{{p}_{sat}}({{T}_{\delta }})]


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\}


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}}


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\}


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\}



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

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