In a silicon oxidation model, if C Gand C Sare the oxidant concentration in the bulk of the gas and oxidant concentration adjacent to the oxide surface, respectively, then the gas phase flux are: (h Gis the gas phase mass transfer coefficient)
h G(C G - C S)
Understanding Gas Phase Flux in Silicon Oxidation
In the context of silicon oxidation, especially as described by models like the Deal-Grove model, the process of oxidizing the silicon surface involves the transport of oxidant species (like $\text{O}_2$ or $\text{H}_2\text{O}$) from the bulk of the gas phase to the silicon surface where it reacts to form silicon dioxide ($\text{SiO}_2$). This transport happens through different regions: the gas phase, the existing oxide layer, and the interface between the oxide and silicon.
The question focuses specifically on the gas phase transport of the oxidant. This transport is driven by the difference in the concentration of the oxidant in the main gas stream ($C_G$) and the concentration right next to the oxide surface ($C_S$). The rate of this transport, or flux, is proportional to this concentration difference.
Gas Phase Flux Calculation
The gas phase flux ($F_{gas}$) represents the number of oxidant molecules passing through a unit area per unit time from the bulk gas to the oxide surface. This flux is related to the concentration difference across the gas phase boundary layer and a coefficient that describes how easily the oxidant can move through this layer. This coefficient is known as the gas phase mass transfer coefficient, denoted as $h_G$.
Think of it like water flowing from a higher pressure area to a lower pressure area. Here, the "driving force" for the flow of oxidant molecules is the concentration difference ($C_G - C_S$). The higher the concentration in the bulk gas ($C_G$) compared to the concentration at the surface ($C_S$), the greater the driving force and thus the greater the flux. The coefficient $h_G$ determines the rate at which this transfer happens for a given concentration difference.
Based on transport phenomena principles, the gas phase flux is given by the product of the gas phase mass transfer coefficient and the concentration difference:
Here:
- $C_G$ is the oxidant concentration in the bulk of the gas phase.
- $C_S$ is the oxidant concentration adjacent to the oxide surface.
- $h_G$ is the gas phase mass transfer coefficient.
The concentration difference is taken as $(C_G - C_S)$ because the flux is from the higher concentration region (bulk gas, $C_G$) towards the lower concentration region (surface, $C_S$).
Analyzing the Options
Let's compare the derived formula with the given options:
- Option 1: $\frac{h_G}{C_G- C_S}$. This formula shows the coefficient divided by the concentration difference, which is incorrect for flux calculation in this context.
- Option 2: $\frac{h_G}{C_G+ C_S}$. This formula involves division and a sum of concentrations, neither of which represents the gas phase flux.
- Option 3: $h_G(C_G - C_S)$. This formula correctly represents the product of the gas phase mass transfer coefficient and the concentration difference between the bulk gas and the surface.
- Option 4: $h_G (C_G + C_S)$. This formula involves the sum of concentrations instead of the difference, which is not the driving force for diffusion/transport.
Therefore, the formula $h_G(C_G - C_S)$ accurately describes the gas phase flux in the silicon oxidation model.
Conclusion
The gas phase flux in a silicon oxidation model, given the oxidant concentration in the bulk gas ($C_G$), the oxidant concentration adjacent to the oxide surface ($C_S$), and the gas phase mass transfer coefficient ($h_G$), is determined by the product of the mass transfer coefficient and the concentration difference between the bulk gas and the surface. The correct expression is $h_G(C_G - C_S)$.
Revision Table: Silicon Oxidation Flux
| Term | Symbol | Description |
|---|---|---|
| Oxidant Concentration (Bulk Gas) | $C_G$ | Concentration of oxidant in the main gas stream. |
| Oxidant Concentration (Surface) | $C_S$ | Concentration of oxidant right at the oxide surface. |
| Gas Phase Mass Transfer Coefficient | $h_G$ | Represents how easily oxidant transfers through the gas layer. |
| Gas Phase Flux | $F_{gas}$ | Rate of oxidant molecules reaching the surface per unit area per unit time. |
Additional Information: Deal-Grove Model Overview
The silicon oxidation model mentioned is likely related to the Deal-Grove model, a fundamental model used to describe the thermal oxidation of silicon. This model considers three sequential steps for the oxidant to reach the silicon surface and react:
- Transport through the gas phase: Oxidant moves from the bulk gas to the oxide surface (flux $F_{gas}$). This step depends on $h_G$, $C_G$, and $C_S$.
- Transport through the oxide layer: Oxidant diffuses through the existing silicon dioxide layer to the oxide-silicon interface (flux $F_{oxide}$). This step depends on the diffusion coefficient of the oxidant in $\text{SiO}_2$ ($D$), the concentration of oxidant at the surface ($C_S$), the concentration at the oxide-silicon interface ($C_i$), and the oxide thickness ($t$).
- Reaction at the silicon surface: Oxidant reacts with silicon at the oxide-silicon interface (flux $F_{reaction}$). This step depends on a surface reaction rate constant ($k_s$) and the concentration of oxidant at the interface ($C_i$).
Under steady-state conditions during oxidation, all three fluxes are equal ($F_{gas} = F_{oxide} = F_{reaction}$). By equating these fluxes, the Deal-Grove model derives an equation that describes the oxide thickness as a function of time, temperature, and the concentrations/coefficients involved. The gas phase flux calculated here, $h_G(C_G - C_S)$, is the first term in this overall flux balance.
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