Question 24.6: One step in the manufacture of optical fibers is the chemica...
One step in the manufacture of optical fibers is the chemical vapor deposition of silane \left(\mathrm{SiH}_{4}\right) on the inside surface of a hollow glass fiber to form a very thin cladding of solid silicon by the reaction
\mathrm{SiH}_{4}(\mathrm{~g}) \rightarrow \mathrm{Si}(\mathrm{s})+2 \mathrm{H}_{2}(\mathrm{~g})
as shown in Figure 24.4. Typically, the process is carried out at high temperature and very low total system pressure. Optical fibers for high bandwidth data transmission have very small inner pore diameters, typically less than 20 \mu \mathrm{m}\left(1 \mu \mathrm{m}=1 \times 10^{-6} \mathrm{~m}\right). If the inner diameter of the Si-coated hollow glass fiber is 10 \mu \mathrm{m}, assess the importance of Knudsen diffusion for \mathrm{SiH}_{4} inside the fiber lumen at 900 \mathrm{~K} and 100 \mathrm{~Pa}(0.1 \mathrm{kPa}) total system pressure. Silane is diluted to 1.0 \mathrm{~mol} \% in the inert carrier gas helium (He). The binary gas phase diffusivity of silane in helium at 25^{\circ} \mathrm{C} (298 \mathrm{~K}) and 1.0 \mathrm{~atm}(101.3 \mathrm{kPa}) total system pressure is 0.571 \mathrm{~cm}^{2} / \mathrm{s}, with \sigma_{\mathrm{SiH}_{4}}=4.08 Å and \varepsilon_{\mathrm{SiH}_{4}} / \kappa=207.6 \mathrm{~K}. The molecular weight of silane is 32 \mathrm{~g} / \mathrm{mol}.
The gas-phase molecular diffusivity of \mathrm{SiH}_{4}-\mathrm{He}, Knudsen diffusivity for \mathrm{SiH}_{4}, and effective diffusivity for \mathrm{SiH}_{4} at 900 \mathrm{~K} and 100 \mathrm{~Pa} total system pressure must be calculated. The gas-phase molecular diffusivity of silane in helium is scaled to process temperature and pressure using the Hirschfelder extrapolation, equation (24-41)
D_{A B_{T_{2},P_{1}}}=D_{A B_{T_{1},P_{1}}}\left(\frac{P_{1}}{P_{2}}\right)\left(\frac{T_{2}}{T_{1}}\right)^{3/2}\frac{\Omega_{D}|_{T_{1}}}{\Omega_{D}|_{T_{2}}} (24-41)
\left.D_{\mathrm{SiH}_{4}-\mathrm{He}}\right|_{0.1 \mathrm{kPa}} ^{900 \mathrm{~K}}=0.571 \frac{\mathrm{cm}^{2}}{\mathrm{~s}}\left(\frac{900 \mathrm{~K}}{298 \mathrm{~K}}\right)^{1.5}\left(\frac{101.3 \mathrm{kPa}}{0.1 \mathrm{kPa}}\right)\left(\frac{0.802}{0.668}\right)=3.32 \times 10^{3} \frac{\mathrm{cm}^{2}}{\mathrm{~s}}
It is left to the reader to show that the collision integral \Omega_{D} is equal to 0.802 at 298 \mathrm{~K} and 0.668 at 900 \mathrm{K} for gaseous \mathrm{SiH}_{4}-\mathrm{He} mixtures. Note that the gas phase molecular diffusivity is high due to high temperature and very low system pressure. The Knudsen diffusivity of \mathrm{SiH}_{4} inside the optical fiber is calculated using equation (24-58),
D_{K A}={\frac{d_{\mathrm{pore}}}{3}}u={\frac{d_{\mathrm{pore}}}{3}}{\sqrt{\frac{8\kappa N T}{\pi M_{A}}}} (24-58)
with d_{\text {pore }}=1 \times 10^{-3} \mathrm{~cm}(10 \mu \mathrm{m})
D_{K, \mathrm{SiH}_{4}}=4850 \mathrm{~d}_{\text {pore }} \sqrt{\frac{T}{M_{\mathrm{SiH}_{4}}}}=4850\left(1 \times 10^{-3}\right) \sqrt{\frac{900}{32}}=25.7 \frac{\mathrm{cm}^{2}}{\mathrm{~s}}
As the \mathrm{SiH}_{4} is significantly diluted in \mathrm{He}, the process is dilute with respect to \mathrm{SiH}_{4} and so equation (24-60)
{\frac{1}{D_{A e}}}={\frac{1}{D_{A B}}}+{\frac{1}{D_{K A}}} (24-60)
can be used to estimate the effective diffusivity
D_{\mathrm{SiH}_{4}, e}=\frac{1}{\frac{1}{D_{\mathrm{SiH}_{4}-\mathrm{He}}}+\frac{1}{D_{K, \mathrm{SiH}_{4}}}}=\frac{1}{\frac{1}{3.32 \times 10^{3}}+\frac{1}{25.7}}=25.5 \frac{\mathrm{cm}^{2}}{\mathrm{~s}}
The effective diffusivity for \mathrm{SiH}_{4} is smaller than its Knudsen diffusivity, reflecting the resistance in series approach. Finally, we calculate the Knudsen number for \mathrm{SiH}_{4}
\begin{aligned} \lambda & =\frac{\kappa T}{\sqrt{2} \pi \sigma_{A}^{2} P}=\frac{1.38 \times 10^{-16} \frac{\mathrm{erg}}{K} \frac{1 N \mathrm{~m}}{10^{7} \mathrm{erg}} 900 K}{\sqrt{2} \pi\left(0.408 \mathrm{~nm} \frac{1 \mathrm{~m}}{10^{9} \mathrm{~nm}}\right)^{2} 100 \frac{N}{\mathrm{~m}^{2}}}=1.68 \times 10^{-4} \mathrm{~m}=168 \mu \mathrm{m} \\ K n & =\frac{\lambda}{d_{\text {pore }}}=\frac{168 \mu \mathrm{m}}{10 \mu \mathrm{m}}=16.8 \end{aligned}
As K n \gg 1 and the effective diffusivity is close to the Knudsen diffusivity, then Knudsen diffusion controls the silane transport inside the optical fiber if no external bulk transport is supplied.