Nonequilibrium kinetic boundary condition at the vapor-liquid interface of argon

A boundary condition for the Boltzmann equation (kinetic boundary condition, KBC) at the vapor-liquid interface of argon is constructed with the help of molecular dynamics (MD) simulations. The KBC is examined at a constant liquid temperature of 85 K in a wide range of nonequilibrium states of vapor. The present investigation is an extension of a previous one by Ishiyama, Yano, and Fujikawa [Phys. Rev. Lett. 95, 084504 (2005)] and provides a more complete form of the KBC. The present KBC includes a thermal accommodation coefficient in addition to evaporation and condensation coefficients, and these coefficients are determined in MD simulations uniquely. The thermal accommodation coefficient shows an anisotropic behavior at the interface for molecular velocities normal versus tangential to the interface. It is also found that the evaporation and condensation coefficients are almost constant in a fairly wide range of nonequilibrium states. The thermal accommodation coefficient of the normal velocity component is almost unity, while that of the tangential component shows a decreasing function of the density of vapor incident on the interface, indicating that the tangential velocity distribution of molecules leaving the interface into the vapor phase may deviate from the tangential parts of the Maxwell velocity distribution at the liquid temperature. A mechanism for the deviation of the KBC from the isotropic Maxwell KBC at the liquid temperature is discussed in terms of anisotropic energy relaxation at the interface. The liquid-temperature dependence of the present KBC is also discussed.

Figure 1

Definitions of velocity distribution functions $f$ and molecular fluxes $J$ at a vapor-liquid interface.

Figure 2

(a) The molecular dynamics system. (b) Schematic of the control variables in the moving coordinate system $z^{*}$.

Figure 3

Density, velocity, and temperature distributions for some cases of $(\beta ,\gamma )$. The dashed line in (a) denotes the saturated vapor density ${\rho }_v$ for $T_{\ell }=85\mathrm{K}$: ${\rho }_v=4.59\times {10}^{-3}\mathrm{g}/{\mathrm{cm}}^3$.

Figure 4

Normalized velocity distribution function of molecules incident onto the interface, ${\widehat{f}}^{\text{coll}}$ (red squares), and outgoing from the interface, ${\widehat{f}}^{\text{out}}$ (blue circles), calculated in MD simulations at $T_{\ell }=85$K, where ${\zeta }_{\iota }={\xi }_{\iota }/\sqrt{2R{T}_{\ell }}(\iota =x,z)$. The solid curves in the left panels denote one-dimensional, normalized reduced half-Maxwell velocity distributions with $T_{\ell }$ ($=85$ K): $(2/\sqrt{\pi })\exp (-{\zeta }_z^2)$ $({\zeta }_z>0)$; those in the right panels denote one-dimensional, normalized reduced Maxwell velocity distributions with $T_{\ell }$: $(1/\sqrt{\pi })\exp (-{\zeta }_x^2)$. The dotted curves in the right panels are one-dimensional, normalized reduced Maxwell velocity distributions with different temperatures $T_t$: $\sqrt{T_{\ell }/\left(T_t\pi \right)}\exp (-{\zeta }_x^2{T}_{\ell }/T_t)$.

Figure 5

The condensation coefficient ${\alpha }_c$ in nonequilibrium states at $T_{\ell }=85\mathrm{K}$. The open triangle denotes the evaporation coefficient estimated in the previous study [37].

Figure 6

The thermal accommodation coefficient ${\alpha }_t$ in nonequilibrium states at $T_{\ell }=85\mathrm{K}$. Note that data for $\gamma =1.0$ are omitted since ${\alpha }_t$ is indefinite (see the text). (a) The three-dimensional plot for ${\alpha }_t$ as a function of $(\beta ,\gamma )$ sets, and (b) its projection onto the $(\beta ,{\alpha }_t)$ plane. The dotted line in (b) is the fitting function Eq. (36).

Figure 7

The time correlation function of molecular velocity tangential (thick red lines) and normal (thin blue lines) to the interface for the initial molecular positions $z^{*}=\pm 1.75$.

Figure 8

The relaxation time of molecular velocities tangential (red circles) and normal (blue squares) to the interface. The solid line indicates the density profile of the equilibrium state at 85 K.

Figure 9

The space-dependent friction coefficient of argon near the vapor-liquid interface in the equilibrium state at 85 K. Red and blue lines denote ${\gamma }_x^{\prime }$ and ${\gamma }_z^{\prime }$ defined in Eqs. (43), respectively. The dashed line is the friction coefficient of bulk liquid estimated from the experimental diffusion coefficient and the Einstein relation (see text). The thick black line is the potential of the mean force $G$ calculated from the thermodynamic integration method [50, 51] (see also the Supplemental Materials [54]): $G\left(z\right)=-{\int }_{z_{\text{liq}}}^{z_{\text{vap}}}{\langle F_z\left(z\right)\rangle }_{z}dz$.

Figure 10

Temperature profile of molecules incident onto and outgoing from the interface. The upper, middle, and lower graphs show the results of simulations for $(\beta =0.5,\gamma =2.0)$, $(\beta =0.5,\gamma =4.0)$, and $(\beta =4.0,\gamma =2.0)$, respectively. The left and the right panels are the normal and the tangential components, respectively. The open (filled) circles are the temperatures of molecules incident onto (outgoing from) the interface $T_{\iota }^{\text{in}}$ ($T_{\iota }^{\text{out}}$), calculated from ${\int }_{{\xi }_z<(>)0}{\xi }_{\iota }^{2}fd\mathbf{\xi }/\left(\rho R\right)(\iota =x\text{or}z)$. The dashed lines indicate the temperatures estimated from $f^{\text{coll}}(\beta ,\gamma )$ [Eq. (31)], which become $\gamma T_{\ell }$ for $\iota =x$ and $\gamma (1-2/\pi )T_{\ell }$ for $\iota =z$, respectively. The solid lines indicate the temperatures estimated from the reduced Maxwell velocity distribution at $T_{\ell }$, which become $T_{\ell }$ for $\iota =x$ and $(1-2/\pi )T_{\ell }$ for $\iota =z$, respectively (see the text for details).

Figure 11

Normalized velocity distribution function of molecules incident onto the interface, ${\widehat{f}}^{\text{coll}}$ (red squares), and outgoing from the interface, ${\widehat{f}}^{\text{out}}$ (blue circles), calculated in MD simulations at $T_{\ell }=90\mathrm{K}$, where ${\zeta }_{\iota }={\xi }_{\iota }/\sqrt{2R{T}_{\ell }}(\iota =x,z)$. The solid curves in the left panels denote one-dimensional, normalized reduced half-Maxwell velocity distributions with $T_{\ell }$ ($=90$ K): $(2/\sqrt{\pi })\exp (-{\zeta }_z^2)$ $({\zeta }_z>0)$; those in the right panels denote one-dimensional, normalized reduced Maxwell velocity distributions with $T_{\ell }$: $(1/\sqrt{\pi })\exp (-{\zeta }_x^2)$. The dotted curves in the right panels are one-dimensional, normalized reduced Maxwell velocity distributions with different temperatures $T_t$: $\sqrt{T_{\ell }/\left(T_t\pi \right)}\exp (-{\zeta }_x^2{T}_{\ell }/T_t)$.

Figure 12

Normalized velocity distribution function of molecules incident onto the interface, ${\widehat{f}}^{\text{coll}}$ (red squares), and outgoing from the interface, ${\widehat{f}}^{\text{out}}$ (blue circles), calculated in MD simulations at $T_{\ell }=100\mathrm{K}$, where ${\zeta }_{\iota }={\xi }_{\iota }/\sqrt{2R{T}_{\ell }}(\iota =x,z)$. The solid curves in the left panels denote one-dimensional, normalized reduced half-Maxwell velocity distributions with $T_{\ell }$ ($=100$ K): $(2/\sqrt{\pi })\exp (-{\zeta }_z^2)$ $({\zeta }_z>0)$; those in the right panels denote one-dimensional, normalized reduced Maxwell velocity distributions with $T_{\ell }$: $(1/\sqrt{\pi })\exp (-{\zeta }_x^2)$. The dotted curves in the right panels are one-dimensional, normalized reduced Maxwell velocity distributions with different temperatures $T_t$: $\sqrt{T_{\ell }/\left(T_t\pi \right)}\exp (-{\zeta }_x^2{T}_{\ell }/T_t)$.

Figure 13

The condensation coefficient ${\alpha }_c$ in nonequilibrium states at $T_{\ell }=90\mathrm{K}$. The open triangle denotes the evaporation coefficient estimated in the previous study [37].

Figure 14

The condensation coefficient ${\alpha }_c$ in nonequilibrium states at $T_{\ell }=100\mathrm{K}$. The open triangle denotes the evaporation coefficient estimated in the previous study [37].

Figure 15

The thermal accommodation coefficient ${\alpha }_t$ in nonequilibrium states at $T_{\ell }=90\mathrm{K}$. Note that data for $\gamma =1.0$ are omitted since ${\alpha }_t$ is indefinite (see the text). (a) The three-dimensional plot for ${\alpha }_t$ as a function of $(\beta ,\gamma )$ sets, and (b) its projection onto the $(\beta ,{\alpha }_t)$ plane. The dotted line in (b) is the fitting function Eq. (36).

Figure 16

The thermal accommodation coefficient ${\alpha }_t$ in nonequilibrium states at $T_{\ell }=100\mathrm{K}$. Note that data for $\gamma =1.0$ are omitted since ${\alpha }_t$ is indefinite (see the text). (a) The three-dimensional plot for ${\alpha }_t$ as a function of $(\beta ,\gamma )$ sets, and (b) its projection onto the $(\beta ,{\alpha }_t)$ plane. The dotted line in (b) is the fitting function Eq. (36).

Figure 17

The fitting parameter $k_t$ in Eq. (36) for the thermal accommodation coefficient ${\alpha }_t$ as a function of the temperature $T_{\ell }$. $T^{*}$ is the reduced temperature defined by Eq. (A1), and the dotted line is the fitting function Eq. (A3).

Figure 18

The thermal accommodation coefficient for the normal velocity component at $T_{\ell }=85,90$, and $100\mathrm{K}$.

Figure 19

The evaporation coefficient ${\alpha }_e$, which equals the condensation coefficient in the equilibrium state, for argon [37], water [43], and methanol [43], as a function of the reduced temperature $T^{*}$. The dotted line is the fitting function Eq. (A2).