1.3.3 Desorption, diffusion, permeation and leaks
In addition to water, other substances such as vacuum pump operating fluids can be adsorbed on surfaces. Substances can also diffuse out of the metal walls, which can be evidenced in the residual gas. In the case of particularly rigorous requirements, stainless steel vessels can be baked out under vacuum, thus driving the majority of the volatile components out of the metal walls.
Desorption
Gas molecules, (primarily water) are bound to the interior surfaces of the vacuum chamber through adsorption and absorption, and gradually desorb again under vacuum. The desorption rate of the metal and glass surfaces in the vacuum system produces a gas yield that declines over time as the coverage rate decreases. A good approximation can be obtained by assuming that after a given point in time $t \gt t_0$ the reduction will occur on a linear basis over time. $t_0$ is typically assumed to be one hour.
The gas yield can thus be described as:
\[Q_\mathrm{des}=q_\mathrm{des}\cdot A\cdot\frac{t_0}t\]
Formula 1-32: Desorption rate
| $Q_\mathrm{des}$ | Desorption rate | [Pa m3 s-1] |
| $q_\mathrm{des}$ | Desorption flow density (area-specific) | [Pa m3 s-1 m-2] |
| $A$ | Area | [m2] |
| $t$ | Time | [s] |
Diffusion with desorption
At operation below 10-6 hPa desorption of plastic surfaces, particularly the seals, assumes greater significance. Plastics mainly give off the gases that are dissolved in these plastics, which first must diffuse on the surface. Following extended pump downtimes, desorption from plastics can therefore dominate desorption from metal surfaces. Although the surface areas of the seals are relatively small; the decrease in the desorption rate over time occurs more slowly than in the case of metal surfaces. As an approximation it can be assumed that the reduction over time will occur at the square root of the time.
The gas produced from plastic surfaces can thus be described as:
\[Q_\mathrm{diff}=q_\mathrm{diff}\cdot A_d\cdot\sqrt\frac{t_0}t\]
Formula 1-33: Desorption rate from plastics
| $Q_\mathrm{diff}$ | Diffusion rate | [Pa m3 s-1] |
| $q_\mathrm{diff}$ | Diffusion flow density (area-specific) | [Pa m3 s-1 m-2] |
| $A_d$ | Surface of plastic material in the vessel | [m2] |
| $t$ | Time | [s] |
Similar effects also occur at even lower pressures in metals, where hydrogen and carbon escape in the form of CO and CO2, and can be seen in the residual gas spectrum. Formula 1-33 also applies in this regard.
Permeation and leaks
Seals, and even metal walls, can be penetrated by small gas molecules, such as helium,through diffusion. Since this process is not a function of time, it results in a sustained increase in the desired ultimate pressure. The permeation gas flow is proportional to the pressure gradient across the wall thickness and a material-dependent permeation constant.
\[Q_\mathrm{perm}=k_\mathrm{perm}\cdot A\cdot\frac{p_a}d\]
Formula 1-34: Permeation
| $Q_\mathrm{perm}$ | Diffusion rate | [Pa m3 s-1] |
| $p_a$ | Pressure outside the vessel | [Pa] |
| $d$ | Wall thickness | [m] |
| $A$ | Surface of the vessel | [m2] |
| $k_\mathrm{perm}$ | Permeation constant | [m2 s-1 ] |
Permeation first manifests itself at pressures below 10-8 hPa.
$Q_L$ describes the leak rate, i.e. a gas flow, which enters the vacuum system through leaks. The leakage rate is defined as the pressure rise over time in a given volume:
\[Q_L=\frac{\Delta p\cdot V}{\Delta t}\]
Formula 1-35: Leak rate
| $Q_L$ | Leak rate | [Pa m3 s-1] |
| $\Delta p$ | Pressure change during measurement period | [Pa] |
| $V$ | Volume | [m3] |
| $\Delta t$ | Measurement period | [s] |
If a vessel is continuously pumped out at a volume flow rate $S$, an equilibrium pressure $p_\mathrm{eq}$ will be produced if the throughput (Formula 1-16) is equal to the leakage rate $Q_L = S\cdot p_\mathrm{eq}$.
A system is considered to be adequately tight if the equilibrium pressure $p_\mathrm{eq}$ is approximately 10 % of the working pressure. If, for example, a working pressure of 10-6 hPa is to be attained and the vacuum pump that is being used has a pumping speed of 100 l s-1, the leakage rate should not be more than s 10-6 Pa m3 s-1.
Leakage rates $Q_L$ < 10-9 Pa m3 s-1 can usually be easily attained in clean stainless steel vessels.
The ultimate pressure achievable after a given period of time $t$ primarily depends upon all of the effects described above and upon the pumping speed of the vacuum pump. The prerequisite is naturally that the ultimate pressure will be high relative to the base pressure of the vacuum pump.
\[Q_\mathrm{des}(t) + Q_\mathrm{diff}(t) + Q_\mathrm{perm} + Q_L = p(t)\cdot S\]
Formula 1-36: Ultimate pressure as a function of time
The various gas flows and the resulting pressures can be calculated for a given pumping time $t$ by using Formula 1-36 and by solving the equations in relation to the time. The achievable ultimate pressure is the sum of these pressures.
