Boundary conditions for the Boltzmann equation from gas-surface interaction kinetic models

Boundary conditions for the Boltzmann equation are investigated on the basis of a kinetic model for gas-surface interactions. The model takes into account gas and physisorbed molecules interacting with a surface potential and colliding with phonons. The potential field is generated by fixed crystal molecules, and the interaction with phonons represents the fluctuating part of the surface. The interaction layer is assumed to be thinner than the mean free path of the gas and physisorbed molecules, and the phonons are assumed to be at equilibrium. The asymptotic kinetic equation for the inner physisorbate layer is derived and used to investigate gas distribution boundary conditions. To be more specific, a model of the boundary condition for the Boltzmann equation is derived on the basis of an approximate iterative solution of the kinetic equation for the physisorbate layer, and the quality of the model is assessed by detailed numerical simulations, which also clarify the behavior of the molecules in the layer.

Figure 1

Typical surface interaction potential $\mathrm{W}$ as a function of $\zeta$.

Figure 2

Schematic view of the characteristic lines $(1/2){\widehat{c}}_z^2+\widehat{\mathrm{W}}\left(\zeta \right)=\text{const}$ of Eq. (34) in the $(\zeta ,{\widehat{c}}_z)$ plane. The direction of the lines is shown by the arrows.

Figure 3

$\widehat{\alpha }\left({\widehat{c}}_z^2\right)$ and ${\widehat{\alpha }}_{\text{E}}\left(\widehat{\mathit{c}}\right)$. (a) $\widehat{\alpha }\left({\widehat{c}}_z^2\right)$ vs ${\widehat{c}}_z$ for the Lennard-Jones (9, 3) potential (88) and the algebraic-type relaxation time (91) in the case of $\kappa =1, n=4$, and $({\kappa }_{\tau },\sigma )=(1,0.5), (1,1), (1,2), (0.5,1)$, and $(2,1)$. (b) ${\widehat{\alpha }}_{\text{E}}\left(\widehat{\mathit{c}}\right)$ vs ${\widehat{c}}_z$ for $({\upsilon }_1,{\upsilon }_2,B)=(6,6,0.2), (12,3,0.2)$, and $(3,12,0.2)$ at ${\widehat{c}}_x={\widehat{c}}_y=0$ and for $(6,6,0.2)$ at ${\widehat{c}}_x={\widehat{c}}_y=1$.

Figure 4

Lennard-Jones potentials: (a) the (9, 3) potential with $\kappa =1$, 5, and 10; (b) the (12, 6) and (9, 3) potentials with $\kappa =5$; and (c) characteristic lines $(1/2){\widehat{c}}_z^2+\widehat{\mathrm{W}}\left(\zeta \right)=\varepsilon$ with $\varepsilon =-0.99,-0.9,-0.8,...,-0.1,0,0.1,...,0.9$, and 1 for the (9, 3) potential with $\kappa =1$.

Figure 5

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (12, 6) potential (85) and the relaxation time of algebraic type (91) ($n=7$) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(1,-0.5,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). The function ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ with ${\widehat{c}}_z<0$ indicates the input, i.e., Eq. (104), and ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ with ${\widehat{c}}_z>0$ indicates the output.

Figure 6

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (12, 6) potential (85) and the relaxation time of exponential type (96) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(1,-0.5,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). See the caption of Fig. 5.

Figure 7

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (9, 3) potential (88) and the relaxation time of algebraic type (91) ($n=4$) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(1,-0.5,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). See the caption of Fig. 5.

Figure 8

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (9, 3) potential (88) and the relaxation time of exponential type (96) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(1,-0.5,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). See the caption of Fig. 5.

Figure 9

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 5: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. The green (light gray in grayscale) lines indicate the input distribution $h_{\infty }$ (${\widehat{c}}_z<0$), the (thick) red lines indicate the output distribution $h_{\infty }$ (${\widehat{c}}_z>0$) based on the numerical solution, and the (thin) black lines indicate that based on the approximate formula (102).

Figure 10

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 6: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. See the caption of Fig. 9 about the color and thickness of the lines.

Figure 11

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 7: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. See the caption of Fig. 9 about the color and thickness of the lines.

Figure 12

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 8: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. See the caption of Fig. 9 about the color and thickness of the lines.

Figure 13

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (9, 3) potential (88) and the relaxation time of algebraic type (91) ($n=4$) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(1,0.5,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). See the caption of Fig. 5.

Figure 14

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (9, 3) potential (88) and the relaxation time of algebraic type (91) ($n=4$) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(0.6,0,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). See the caption of Fig. 5.

Figure 15

Contour lines of ${\mathsf{h}}_{\infty }({\widehat{c}}_x,{\widehat{c}}_z)$ for the Lennard-Jones (9, 3) potential (88) and the relaxation time of algebraic type (91) ($n=4$) in the $({\widehat{c}}_x,{\widehat{c}}_z)$ plane in the case of $({\widehat{T}}_{\infty },{\widehat{v}}_{z\infty },{\widehat{v}}_{x\infty })=(0.6,-0.5,0.5)$ and $(\kappa ,{\kappa }_{\tau },\sigma )=(1,1,1)$. (a) Numerical solution. (b) Approximate formula (102). See the caption of Fig. 5.

Figure 16

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 13: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. See the caption of Fig. 9 about the color and thickness of the lines.

Figure 17

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 14: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. See the caption of Fig. 9 about the color and thickness of the lines.

Figure 18

Profiles of ${\mathsf{h}}_{\infty }$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at ${\widehat{c}}_z=\text{const}$ and ${\widehat{c}}_x=\text{const}$ in the case of Fig. 15: (a) profiles at ${\widehat{c}}_z=\text{const}$, (b) those at ${\widehat{c}}_x=\text{const}>0$, and (c) those at ${\widehat{c}}_x=\text{const}<0$. See the caption of Fig. 9 about the color and thickness of the lines.

Figure 19

Dimensionless number density $\widehat{\mathsf{n}}$ vs the scaled normal coordinate $\zeta$ ($=z/\delta$) in the case of Figs. 7 and 13, 14, 15 obtained by the numerical solution (see the captions of Figs. 7 and 13, 14, 15).

Figure 20

Profiles of $\varphi (\zeta ,{\widehat{c}}_z)$ (${\widehat{c}}_z<0$ and ${\widehat{c}}_z>0$) at various positions $\zeta$ in the case of Fig. 8: (a) profiles for $2.293\le \zeta \le \infty$, (b) for $1.201\le \zeta \le 2.293$, (c) for $1\le \zeta \le 1.201$, and (d) for $0.901\le \zeta \le 1$. The dashed line indicates the jump between the values at $\varepsilon =0-$ and $0+$.

Figure 21

Magnified figures of some of the profiles in Fig. 20 around the discontinuities. See the caption of Fig. 20.

Figure 22

Grid points $({\zeta }^{\left(i\right)},{\varepsilon }^{\left(j\right)})$ in the $(\zeta ,\varepsilon )$ plane.