DSMC and PICMC documentation

1.1.1 Particle flow rate (sccm) and current (A)

In vacuum technology, particle flows are often measured in sccm, which stands for standard cubic centimeters per minute. This is actually proportional to the number of gas molecules per time unit and thus, it is a very convenient unit for kinetic simulation methods based on particles. The term standard conditions stands for atmospheric pressure on earth at a temperature of \(0\,^\circ\)C:

• Pressure \(p = 1.01325 \textrm{bar} = 1.01325\times 10^5\,\textrm{Pa}\)

• Temperature \(T = 0\,^\circ\textrm{C} = 273.15\,\textrm{K}\)

• Volume \(V = 1\,\textrm{cm}^3 = 10^{-6}\,\textrm{m}^3\)

The number \(N\) of particles within a given volume \(V\) can be obtained from the ideal gas equation (1.1).

\[N = \frac{pV}{k_B T}\](1.1)

With the Boltzmann constant \(k_B=1.38065\times 10^{-23}\,\textrm{J/K}\) and a volume of \(V = 1\,\textrm{cm}^3\) we obtain \(N=2.68675\times 10^{19}\). Since sccm is the flow rate given per minute but we would like to have the number of particles per second, we have to divide \(N\) by 60. This yields the number particles per second for 1 sccm as given in Eqn. (1.2).

\[1\,\textrm{sccm} \approx 4.47796\times 10^{17}\,\textrm{s}^{-1}\](1.2)

From the electron charge \(e = 1.60218\times 10^{-19}\,\textrm{(As)}\) follows, that one ampere corresponds to a number of \(6.24150\times 10^{18}\) particles (electrons or ions) per second. A comparison with the particle flow rate in Eqn. (1.2) yields the conversion factor between A and sccm.

\[1\,\textrm{A} \approx 13.9397\,\textrm{sccm}\](1.3)

1.1.2 Pumping speed

The volumetric pumping speed \(S_p\) is defined as pumped gas volume / time. In case of an orifice with area \(A\) and ideal pumping behavior, the maximum possible pumping speed \(S_{p.max}\) is

\[S_{p.max} = \frac{\overline{c}}{4}\times A\](1.4)

In Eqn. (1.4), \(A\) is the area of the pumping orifice, and \(\overline{c}\) is the mean thermal speed of a gas molecule, which follows from Eqn. (1.5).

\[\overline{c} = \sqrt{\frac{8 kT}{m\pi}} \simeq 145.5\textrm{ (m/s)} \times\sqrt{\frac{T\textrm{ (K)}}{m\textrm{ (u)}}}\](1.5)

At a temperature of 300 K, the following mean thermal speeds are obtained for a variety of gases:

1.1.3 How to specify the transmission factor for a surface with given pumping speed

Pumping surfaces in DSMC- or PIC-MC-simulation are characterized by an absorption factor \(f\in [0\ldots 1]\), while the volumetric pumping speed \(S_p\) is given as an external input (e.g. the nominal pumping speed of an attached turbomolecular pump). In such cases, the pump factor is selected as the ratio between given pumping speed \(S_p\) and maximum possible pumping speed \(S_{p.max}\) from Eqn. (1.4).

\[f = \frac{S_p}{S_{p.max}} = \frac{4 S_p}{A\overline{c}}\](1.6)

1.1.4 Relation between gas flow, pumping speed and pressure

Especially in DSMC simulation, it is helpful to estimate the mean equilibrium pressure, when the gas flow through the inlet and the pumping speed are given. This can be done based on the following consideration:

In equilibrium, the time derivative \(dN/dt\) of the total number of particles \(N\) in a chamber volume, which can be expressed as the balance between source and drain, is zero.

\[\frac{dN}{dt} = F_{in}f_{sccm} - S_p n = F_{in}f_{sccm} - S_p \frac{p}{kT} = 0\](1.7)

Here, \(F_{in}\) is the gas flow in sccm, \(f_{sccm}=4.47796\times 10^{17}\textrm{ (sccm}^{-1}\textrm{s}^{-1}\textrm{)}\) is the conversion factor from Eqn. (1.2), \(S_p\) is the nominal pumping speed, and \(n=N/V\) is the number density of gas molecules. According to the ideal gas equation (1.1), \(n\) can be written as \(p/kT\). Eqn. (1.7) can be written as

\[p = \Phi\frac{F_{in}}{S_p};\,\,\Phi = kTf_{sccm}\](1.8)

The factor \(\Phi\) is independent from the gas species, typical values as a function of temperature are listed in Table (1.2).