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 $T δ$ and normal saturation vapor pressure corresponding to $T δ$ 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,

$d μ = - s d T + v d p$	(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 $p v,δ$ 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 $p c a p$ 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.

External Links

Retrieved from "http://www.thermalfluidscentral.org/encyclopedia/index.php/Interfacial_Thermal_Resistance"

Interfacial Thermal Resistance

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