(image) (image) [ Home ]

Phenomenological Single-Particle
Modeling of Reactive Transport
in Semiconductor Processing

3.4 Transitional Flow

The theory of Knudsen diffusion presented so far has focused on the molecular flow regime. However, as it has already been discussed for Eq. (3.40), the presence of molecule-molecule collisions affects the transport properties even for high \(\text {Kn}\). These issues become even more salient in the transitional flow regime (approximately \(10^{-1} < \text {Kn} < 10\)), as the molecule-molecule collisions become roughly as likely as molecule-geometry interactions.

A rigorous examination of the problem of diffusion in long cylindrical tubes was performed by Pollard and Present [116]. They perform an integration procedure similar in concept to that presented in Eq. (3.22), but including an additional integration over the \(s\) between the cross-sectional area and the surface element (c.f. Fig. 3.2). This additional integral captures an attenuation of the number of incoming molecules due to molecule-molecule collisions through an \(e^{-s/\lambda }ds\) term. They obtain a general expression for the combined diffusion coefficient

\begin{align} \label {eq::pollard_combined_diffusion} D_\mathrm {Kn+self-diffusion} = \frac {1}{3}\overline {v}\lambda \left [ 1 - \frac {6}{8}\frac {\lambda }{d} + \frac {12}{\pi }\frac {\lambda }{d}Q(d/\lambda ) \right ]\, . \end{align} In Eq. (3.42), \(Q(d/L)\) is an integral which must be calculated numerically. \(Q(d/L)\) has the key property of recovering both the self-diffusivity from Eq. (3.11) in the limit \(\lambda >> d\), as well as the standard Knudsen diffusivity from Eq. (3.24) after calculating the limit \(d >> \lambda \).

However, Eq. (3.42) is valid only for the combination of Knudsen with self-diffusion. For the issue of mutual diffusion, that is, the diffusion of two gases of different characteristics, mean free path approximations like Eq. (3.8) fail [117]. To address that issue, the Chapman-Enskog theory of diffusion was developed [120] which has as a fundamental result the following diffusivity \(D_{AB}\) between two "billiard ball" molecules \(A\) and \(B\)

\begin{align} \label {eq::chapman_enskog} D_{AB} = \frac {3}{8} \sqrt {\frac {\pi k_B T}{2m^*}}\frac {1}{n \pi d_{AB}^2}\, , \end{align} with \(m^* = m_Am_B/(m_A+m_B)\) being the reduced mass, \(d_{AB}\) the average molecular diameter \((d_A + d_B)/2\), and \(n\) the total concentration \(n_A + n_B\).

In order to capture a combined diffusivity \(D\) stemming from an arbitrary gas-phase diffusivity \(D_\mathrm {gas}\) and the Knudsen diffusivity \(D_\mathrm {Kn}\) in a more general manner, Bosanquet proposed an interpolation formula. This work is usually attributed to a British military technical report from 1944 [127], which is not easily available to the public. It was re-introduced to the scientific community at large by Pollard and Present [116]. The Bosanquet formula is not rigorously derived, instead, it is based on the intuition that the combined frequency of the total scattering events (i.e., both molecule-molecule and molecule-geometry collisions) is the sum of the individual frequencies of each type of collision. From that insight, each collision frequency is identified as approximately the inverse of the respective diffusion coefficient. Therefore, Bosanquet interpolation formula for the combined diffusivity \(D\) is

\begin{align} \label {eq::bosanquet} \frac {1}{D} \approx \frac {1}{D_\mathrm {gas}} + \frac {1}{D_\mathrm {Kn}}\, . \end{align} Equation (3.44) is an interpolation formula since it naturally recovers the limits \(D_\mathrm {gas} >> D_\mathrm {Kn}\) and vice-versa. Notably, Pollard and Present [116] achieve excellent agreement between their formula in Eq. (3.42) and Eq. (3.44) using the standard self-diffusion and Knudsen diffusivity formulas.