Boundary conditions

Interaction between gas atoms and inner surface of the channel is a very important issue that significantly influences on flow behaviors [34, 68-70]. In description of interaction between gas molecules and solid surface, there are two simple models proposed by Maxwell [19] in 1879: the specular reflection model and the diffuse reflection model. The word 'specular' originated from specularis in Latin which means 'mirror'. The word 'diffuse' originated from diffusus which in Latin means 'spread abroad', 'scatter'.

It should be noted, that both models mentioned above are limiting simplifications of the real gas-solid boundary dealing with atomic structure of solid.

This is so-called realist boundary condition or wall with atomic structure. The last approach allows us to carry out precise investigation of gas flow, but in the same time it supposes use of huge amount of computational power. Consequently it would be beneficial to couple merit of the first group of BC (gas atoms’ parameters after collision with solids can be found analytically) with high precision of the realistic BC. Such

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combination allowed us to create New statistical Boundary Conditions for argon-tungsten interactions, which will be discussed below.

3.3.1 Specular model

The specular reflection model (Figure 3.3-1a) assumes that the incident molecules reflect on the body surface as the elastic spheres reflect on the entirely elastic surface, i.e., the normal to the surface component of the relative velocity reverses its direction while the parallel to the surface components remain unchanged. Thus the normal pressure originated from the reflected molecules equals to that originated from the incident molecules; the shear stress subjected by the surface from the reflected molecules has the opposite sign to that from the incident molecules and the net shear stress is zero; the total energy exchange with the surface is zero. Implementation of specular reflection is simple. When particle reaches surface, normal velocity component changes sign.

3.3.2 Diffuse model

The diffuse reflection model (Figure 3.3-1b) assumes that molecules leaving surface scatter with an equilibrium Maxwell distribution. The condition of equilibrium is the equality of the surface temperature, the temperature in the Maxwell distribution and the static temperature of the incident flow. In diffuse reflection, velocity of each molecule after reflection is independent of its incident velocity. However, velocities of reflected molecules are randomly distributed within half-range Maxwell distribution.

Equilibrium diffuse reflection requires that both the surface temperature and the temperature associated with the reflected gas according to Maxwellian distribution must be equal to the gas temperature [14].

In current studying diffuse reflection model was used to make sure that created computational model is able to reproduce the theoretical results based on diffusive

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boundary conditions. Implementation of such model is more sophisticated than in case of specular reflection. Maxwellian cumulative velocity distribution of scattered molecules corresponding to the surface’s temperature Tsurf (Eq. 3.2-2) is used for this reflection. According to the sampling method based on inversion of the cumulative distribution [71], by letting F(V^s) equal to a random function ranf uniformly distributed between 0 and 1, solving the equality, the velocity of scattered atom is obtained. The next step is velocity vector direction definition, for 3D case two angles must be defined (Figure 3.3-2 shows local and global 3D coordinate systems for a scattered particle), first angle α^s (in surface plane) is uniformly distributed between -π and π:

(^ranf) ^{/ 2}

αs =π −π (3.3-1)

The second and angle β^s (in plane perpendicular to the surface) is uniformly distributed between-π and π:

(^ranf) ^{/ 2}

βs =π −π (3.3-2)

The described above reflection model is called Maxwellian type boundary condition. Diffusive model of BC has been extensively used in numerical and analytical studies of rarefied gas flow through micro- channels [15, 47, 48, 72]. Unfortunately, those simple models are fail in case of temperature driven flow [2], moreover, recently it has been shown, that interactions of gas with clean metal surface cannot be precisely described by using diffuse BC[61].

It should be noted that there is a special specular-diffusive type of boundary conditions where the degree of diffusivity or secularity is measured in terms of accommodation coefficient [73-75]. This coefficient shows how much energy and/or momentum gas atom loses while collision with solids. Implementation of

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accommodation coefficients partially fixes demerits of diffusive model, but this approach requires determination of accommodation coefficients for all flow regimes because their values are found to be in strong dependence with thermodynamic parameters of gas and surface [76, 77].

3.3.3 Wall constructed with atomic structures

The third model of BC that has been used in current study is “Real channel”, i.e.

the channel with walls represented by a set of tungsten atoms. “Real channel” was created by removing of excess atoms of the tungsten bar (atoms of the bar are arranged according to BCC structure of tungsten). This process is illustrated in Figure 3.3-3a.

Each wall of the channel consists of six atomic layers and atoms that comprised the outer layers are held fixed in their equilibrium lattice positions, while the atoms in other layers were permitted to move according to the appropriate classical equations of motion. In order to demonstrate the influence of channel’s irregularities on the flow we have considered two kinds of real channel: with smooth and wavy walls, see left and right pictures, respectively, in the Figure 3.3-3b. Initial velocities of the tungsten atoms were determined from the Maxwell distribution function corresponding to the desired temperature of the channel’s surface Tsurf.

This model of BC is the most accurate and comprehensive one, nevertheless it requires a lot of computational resources and increases the number of simulated molecules significantly. Consequently the time of computations rises as well.

3.3.4 New statistical Boundary Conditions

The energy transfer and other processes accompanying the scattering of rarefied gases from solid surfaces have been the subject of a series of studies. Weinberg and Merrill [78] determined angular distributions for gas atoms scattered by a single-crystal W(110) surface. The experimental results of Janda, Hurst and co-workers [102] allowed

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the researchers to relate the average kinetic energy of scattered argon atoms to the surface temperature, as well as to the incident kinetic energy. The theoretical explanations for argon atoms scattering from a self-assembled monolayer on Ag(111) have been proposed recently by Fan and Manson [79]. Furthermore, Gibson et al. [80]

conducted a detailed study of Ar scattering from an ordered 1-decanethiol–Au(111) monolayer. Recently, Chase et al. [81] have conducted experimental and molecular dynamics studies of argon scattering from liquid indium. They have shown how the angular and energy distributions of scattered atoms depend on incident energy.

Inapplicability of the simple hard-sphere model (this model assumes that atoms don’t have any potential energy on interaction, i.e. atoms influence on each other while collision only) for the description of gas-surface interactions is also presented.

While there are many publications related to gas-surface interactions, their results are still insufficient to define boundary conditions that can describe the gas flow in micro/nano systems. The best way to study natural processes is to conduct an experiment, but one should realize that accurate measurements of gas-surface interactions on a microscopic level are very expensive and time consuming. Fortunately, the recent achievements in computer science and numerical methods made it possible to investigate such processes using MD simulation method.

In this thesis, MD method was applied to study the argon gas scattering processes on a W(110) surface. This approach made it possible to precisely describe the interaction between argon gas and tungsten. The aims of this part of the work were to study effects of argon scattering on the tungsten surface and to propose boundary conditions describing correlations between the parameters of incident and scattered atoms. The method applied in the present thesis can be simply expressed as the bombardment of a tungsten surface with argon atoms, where further analysis of the

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scattered atoms’ trajectories was conducted. Analysis of both angular distributions and distributions of velocities of scattered atoms were performed using mean values and root mean square deviations (RMSDs). The combinations of these parameters provide complete information about process of gas atoms scattering process. It is shown that results of current study are in line with experimental and theoretical results obtained by the other researches. The information obtained in simulations was statistically analyzed and represented by polynomial functions of incident energy and angle of incidence. All the functions that state relationship between parameters of impinging gas atoms and scattered atoms have been obtained using the Least Squares Method (LSqM). As a conclusive step of the work, an algorithm describing an implementation of the relations mentioned above will be applied to a real gas flow with tungsten boundary in this study.

These relations can be used to specify boundary conditions for argon-tungsten interactions. This model of boundary condition as well as results of its implementation is discussed in next chapter.