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14.2.2 Theory

Consider the $r$th wall surface reaction written in general form as follows:


 \sum_{i=1}^{N_g} g'_{i,r} G_i + \sum_{i=1}^{N_b} b'_{i,r} B_... ...sum_{i=1}^{N_b} b''_{i,r} B_i + \sum_{i=1}^{N_s} s''_{i,r} S_i (14.2-1)

where $G_i$, $B_i$, and $S_i$ represent the gas phase species, the bulk (or solid) species, and the surface-adsorbed (or site) species, respectively. $N_g$, $N_b$, and $N_s$ are the total numbers of these species. $g'_{i,r}$, $b'_{i,r}$, and $s'_{i,r}$ are the stoichiometric coefficients for each reactant species $i$, and $g''_{i,r}$, $b''_{i,r}$, and $s''_{i,r}$ are the stoichiometric coefficients for each product species $i$. $K_r$ is the overall reaction rate constant.

The summations in Equation  14.2-1 are for all chemical species in the system, but only species involved as reactants or products will have non-zero stoichiometric coefficients. Hence, species that are not involved will drop out of the equation.

The rate of the $r$th reaction is


 {\cal R}_r = k_{f,r} \prod_{i=1}^{N_g} \left[G_i\right]_{\rm wall}^{g'_{i,r}} \left[S_i\right]_{\rm wall}^{s'_{i,r}} (14.2-2)

where $\left[\;\;\right]_{\rm wall}$ represents molar concentrations on the wall. It is assumed that the reaction rate does not depend on concentrations of the bulk (solid) species. From this, the net molar rate of production or consumption of each species $i$ is given by


$\displaystyle \hat{R}_{i,{\rm gas}}$ $\textstyle =$ $\displaystyle \sum_{r=1}^{N_{\rm rxn}} (g''_{i,r} - g'_{i,r}) {\cal R}_r \; \; \; \; \; i = 1, 2, 3, \dots, N_g$ (14.2-3)
$\displaystyle \hat{R}_{i,{\rm bulk}}$ $\textstyle =$ $\displaystyle \sum_{r=1}^{N_{\rm rxn}} (b''_{i,r} - b'_{i,r}) {\cal R}_r \; \; \; \; \; i = 1, 2, 3, \dots, N_b$ (14.2-4)
$\displaystyle \hat{R}_{i,{\rm site}}$ $\textstyle =$ $\displaystyle \sum_{r=1}^{N_{\rm rxn}} (s''_{i,r} - s'_{i,r}) {\cal R}_r \; \; \; \; \; i = 1, 2, 3, \dots, N_s$ (14.2-5)

The forward rate constant for reaction $r$ ( $k_{f,r}$) is computed using the Arrhenius expression. For example,


 k_{f,r}= A_r T^{\beta_r} e^{-E_r/RT} (14.2-6)


where $A_r$ = pre-exponential factor (consistent units)
  $\beta_r$ = temperature exponent (dimensionless)
  $E_r$ = activation energy for the reaction (J/kgmol)
  $R$ = universal gas constant (J/kgmol-K)

You (or the database) will provide values for $g'_{i,r}$, $g''_{i,r}$, $b'_{i,r}$, $b''_{i,r}$, $s'_{i,r}$, $s''_{i,r}$, $\beta_{r}$, $A_{r}$, and $E_r$.



Wall Surface Reaction Boundary Conditions


As shown in Equations  14.2-2- 14.2-5, the goal of surface reaction modeling is to compute concentrations of gas species and site species at the wall; i.e., $\left[G_i\right]_{\rm wall} $ and $\left[S_i\right]_{\rm wall}$. Assuming that, on a reacting surface , the mass flux of each gas species is balanced with its rate of production/consumption, then


$\displaystyle \rho_{\rm wall} D_i \frac{\partial Y_{i, \rm wall}}{\partial n} - \dot{m}_{\rm dep} Y_{i,\rm wall}$ $\textstyle =$ $\displaystyle M_{w,i} \hat{R}_{i,{\rm gas}} \; \; \; \; \; i = 1, 2, 3, \dots, N_g$ (14.2-7)
$\displaystyle \frac{\partial \left[S_i\right]_{\rm wall}}{\partial t}$ $\textstyle =$ $\displaystyle \hat{R}_{i,{\rm site}} \phantom{M_{w,i}} \; \; \; \; \; i = 1, 2, 3, \dots, N_s$ (14.2-8)

The mass fraction $Y_{i,\rm wall}$ is related to concentration by


 \left[G_i\right]_{\rm wall} = \frac{\rho_{\rm wall} Y_{i, {\rm wall}}}{M_{w,i}} (14.2-9)

$\dot{m}_{\rm dep}$ is the net rate of mass deposition or etching as a result of surface reaction; i.e.,


 \dot{m}_{\rm dep} = \sum_{i=1}^{N_b} M_{w,i} \hat{R}_{i, {\rm bulk}} (14.2-10)

$\left[S_i\right]_{\rm wall}$ is the site species concentration at the wall, and is defined as


 \left[S_i\right]_{\rm wall} = \rho_{\rm site} z_i (14.2-11)

where $\rho_{\rm site}$ is the site density and $z_i$ is the site coverage of species $i$.

Using Equations  14.2-7 and 14.2-8, expressions can be derived for the mass fraction of species $i$ at the wall and for the net rate of creation of species $i$ per unit area. These expressions are used in FLUENT to compute gas phase species concentrations, and if applicable, site coverages, at reacting surfaces using a point-by-point coupled stiff solver.



Including Mass Transfer To Surfaces in Continuity


In the surface reaction boundary condition described above, the effects of the wall normal velocity or bulk mass transfer to the wall are not included in the computation of species transport. The momentum of the net surface mass flux from the surface is also ignored because the momentum flux through the surface is usually small in comparison with the momentum of the flow in the cells adjacent to the surface. However, you can include the effect of surface mass transfer in the continuity equation by activating the Mass Deposition Source option in the Species Model panel.



Wall Surface Mass Transfer Effects in the Energy Equation


Species diffusion effects in the energy equation due to wall surface reactions are included in the normal species diffusion term described in Section  14.1.1.

If you are using the pressure-based solver, you can neglect this term by turning off the Diffusion Energy Source option in the Species Model panel. (For the density-based solvers, this term is always included; you cannot turn it off.) Neglecting the species diffusion term implies that errors may be introduced to the prediction of temperature in problems involving mixing of species with significantly different heat capacities, especially for components with a Lewis number far from unity. While the effect of species diffusion should go to zero at Le = 1, you may see subtle effects due to differences in the numerical integration in the species and energy equations.



Modeling the Heat Release Due to Wall Surface Reactions


The heat release due to a wall surface reaction is, by default, ignored by FLUENT. You can, however, choose to include the heat of surface reaction by activating the Heat of Surface Reactions option in the Species Model panel and setting appropriate formation enthalpies in the Materials panel.



Slip Boundary Formulation for Low-Pressure Gas Systems


Most semiconductor fabrication devices operate far below atmospheric pressure, typically only a few millitorrs. At such low pressures, the fluid flow is in the slip regime and the normally used no-slip boundary conditions for velocity and temperature are no longer valid.

Knudsen number , Kn, which is defined as the ratio of mean free path to a characteristic length scale of the system, is used to describe various flow regimes. Since the mean free path increases as the pressure is lowered, the high end of Kn values represents free molecular flow and the low end the continuum regime. The range in between these two extremes is called the slip regime ( $0.001 < {\rm Kn} < 0.1$).

In the slip regime, the gas-phase velocity at a solid surface differs from the velocity at which the wall moves, and the gas temperature at the surface differs from the wall temperature. Maxwell's models are adopted for these physical phenomena in FLUENT for their simplicity and effectiveness.

figure   

The low-pressure slip boundary formulation is available only with the pressure-based solver.


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