Vacuum Fundamentals

Pressure units and regimes

Vacuum work uses several pressure units, often on the same drawing. The SI unit is the pascal; the millibar dominates European equipment and most pump datasheets; torr survives in North American labs and older literature.

Pressure spans more than fifteen orders of magnitude between atmosphere and the best laboratory vacuum, so it is divided into regimes. The boundaries below are the ones most pump and gauge vendors use; standards differ slightly, and what matters is that each regime brings different physics, pumps, and gauges.

The pressure unit converter handles the arithmetic and labels the regime for any value.

Gas kinetics: mean free path and flow regimes

The single most useful number in vacuum physics is the mean free path, the average distance a molecule travels between collisions with other molecules:

For air at 20 °C (effective molecular diameter \(d \approx 0.372\) nm), this works out to \(\lambda \approx 6.6\ \text{mm}\) at 1 Pa, or equivalently \(\lambda\,[\text{mm}] \approx 0.066 / P\,[\text{mbar}]\). The classic rule \(\lambda \approx 5\ \text{cm}\) at 1 mTorr is the same number in other units.

What matters is how λ compares with the size of the system, expressed as the Knudsen number \(\mathrm{Kn} = \lambda / D\), where D is a characteristic dimension such as a tube diameter:

• Viscous (continuum) flow, Kn < 0.01. Molecules collide mostly with each other, the gas behaves as a fluid, and conductance increases with pressure. For air this means \(P\cdot D \gtrsim 0.66\ \text{Pa}\cdot\text{m}\) (≈ 0.5 Torr·cm).
• Molecular flow, Kn ≳ 0.5–1. Molecules travel wall to wall without meeting each other. Each one moves independently, and conductance depends only on geometry and gas species, not pressure. For air, Kn > 1 means \(P\cdot D \lesssim 6.6\times10^{-3}\ \text{Pa}\cdot\text{m}\).
• Transitional flow lies in between and requires interpolation formulas.

For a 25 mm line, flow is viscous above roughly 26 Pa (0.26 mbar) and molecular below about 0.26 Pa (2.6 × 10⁻³ mbar). Turbopumps, gauges, and process chambers spend most of their working life in the molecular regime, where intuition about "suction" fails: a pump cannot pull gas toward it. It can only capture molecules that happen to wander into its inlet.

Two more kinetic quantities explain why surfaces dominate high vacuum. The number density \(n = P/k_BT\) is 2.5 × 10²⁵ m⁻³ at atmosphere and still 2.5 × 10¹⁶ m⁻³ at 10⁻⁶ mbar. The wall impingement rate

means that at 10⁻⁶ Torr a clean surface receives enough molecules to form a monolayer in a few seconds if every molecule sticks. That rate is the origin of the langmuir exposure unit (10⁻⁶ Torr·s). The mean free path calculator computes all of these for common gases.

Throughput, pumping speed, and conductance

Vacuum systems are analyzed with an electrical analogy. Throughput \(Q\) is the gas flow, expressed as pressure × volume per time (mbar·L/s or Pa·m³/s, with 1 Pa·m³/s = 10 mbar·L/s). A pump of speed \(S\) at inlet pressure \(P\) removes \(Q = PS\). A tube, valve, or port has a conductance \(C\) defined by \(Q = C\,(P_1 - P_2)\). Conductances combine like electrical conductances: in parallel they add, and in series their reciprocals add.

The result that drives most vacuum design is the effective pumping speed at the chamber:

Whichever of S and C is smaller dominates. In molecular flow, for air at 20 °C, the standard results are:

• Thin aperture: \(C = 11.6\,A\) L/s, with A in cm².
• Long round tube: \(C = 12.1\,d^3/L\) L/s, with d and L in cm.
• Short tube: combine the aperture and long-tube conductances in series (Dushman's approximation, typically within 10–15 % of exact transmission-probability results).

The \(d^3\) dependence is unforgiving: halving a tube's bore cuts its molecular conductance by a factor of eight. Molecular conductance also scales as \(\sqrt{T/M}\), so helium flows about 2.7× faster than air through the same tube. In viscous flow, by contrast, \(C \approx 136\, d^4 \bar P / L\) L/s (d and L in cm, mean pressure \(\bar P\) in mbar), which is large at roughing pressures. That is why roughing lines rarely limit pump-down, while high-vacuum plumbing almost always does.

Worked example. A 300 L/s turbopump connected to a chamber through a 40 mm bore tube 30 cm long. The aperture conductance is 11.6 × π(2 cm)² ≈ 145 L/s, the long-tube term is 12.1 × 4³/30 ≈ 26 L/s, and together they give C ≈ 22 L/s. The effective speed at the chamber is about 20 L/s, roughly 7 % of the pump's rating. The same pump on a 100 mm bore, 20 cm long port has C ≈ 364 L/s and S_eff ≈ 164 L/s. A useful design target is C ≥ 2–3 S, so that S_eff is at least 67–75 % of S. Elbows, valves, baffles, and liquid-nitrogen traps each subtract further. The conductance and pump-down calculator runs these combinations quickly.

Pump-down: four regimes

A chamber pumped down from atmosphere passes through distinct regimes. Each one has its own time dependence and its own lever for speeding it up.

1. Volume gas. At first the gas in the chamber volume dominates, and with constant speed \(P(t) = P_0\,e^{-t/\tau}\) with time constant \(\tau = V/S_\text{eff}\). The time to reach pressure P is \(t = (V/S_\text{eff})\ln(P_0/P)\). A 100 L chamber on a 10 L/s (36 m³/h) roughing pump has τ = 10 s and reaches 1 mbar in about 69 s, assuming the pump holds its speed. Real pumps lose speed as they approach their ultimate pressure, so allow margin.
2. Surface desorption. Below roughly 10⁻²–10⁻³ mbar (the crossover is system-dependent), molecules leaving the walls, mainly water, outnumber those left in the volume. The pressure now falls roughly as \(1/t\). A bigger pump lowers the curve but does not change its slope: every additional decade of pressure takes about ten times longer.
3. Diffusion from the bulk. Gas dissolved in materials, especially hydrogen in metals and absorbed gases in polymers, diffuses to the surface and desorbs, giving a slower \(t^{-1/2}\) decline.
4. Permeation. Gas passing through the walls themselves, such as helium through glass and elastomer O-rings or atmospheric gases through polymer seals, sets a constant floor that no amount of pumping removes.

The ultimate pressure is set by gas load, not pump size: \(P_\text{ult} \approx Q_\text{total}/S_\text{eff}\). Unbaked stainless steel typically outgasses on the order of 10⁻⁸–10⁻⁷ mbar·L/(s·cm²) after an hour of pumping, falling roughly as 1/t. One square meter (10⁴ cm²) of internal surface at 10⁻⁸ mbar·L/(s·cm²) produces 10⁻⁴ mbar·L/s. With 200 L/s effective speed, that holds the chamber at about 5 × 10⁻⁷ mbar. Elastomers, adhesives, plastics, and porous coatings outgas orders of magnitude more per unit area than clean metal, so a small amount of the wrong material can dominate the gas load.

Outgassing and bake-out

Water is the dominant residual gas in an unbaked system in the high-vacuum range. It adsorbs in multiple layers on every surface exposed to room air, and its desorption rate depends steeply on temperature. Heating is therefore the most effective way to shorten the desorption phase.

• Bake-out. Heating a stainless-steel system to roughly 150–250 °C for one to two days removes most adsorbed water. After cooling, hydrogen diffusing out of the steel becomes the main gas, and outgassing rates of order 10⁻¹² mbar·L/(s·cm²) are typical. Bake temperatures are limited by the weakest component: fluoroelastomer seals, viewports, feedthroughs, sensors, and the product itself.
• Partial heating still helps. In production, where a full bake is impractical, even moderate heating of the part or chamber walls during pump-down shortens cycle time noticeably, because desorption rates rise roughly exponentially with temperature.
• Vent with dry nitrogen. Venting with dry gas and minimizing the time a chamber stands open to humid air limits water re-adsorption. The next pump-down is faster and more repeatable.
• Keep it clean. Fingerprints, cutting fluids, and hydrocarbon films outgas slowly and persistently. Clean, degrease, and handle vacuum-side parts with gloves.

Leaks, virtual leaks, and rate-of-rise

Leak rates are throughputs, quoted in mbar·L/s (or Pa·m³/s). A quick system-level diagnostic is the rate-of-rise test: isolate the chamber from its pump and record pressure versus time. The total gas load is

A 50 L chamber rising from 1 × 10⁻³ to 2 × 10⁻³ mbar in 100 s has Q = 5 × 10⁻⁴ mbar·L/s. The shape of the curve indicates the source:

• Real leak: a straight-line rise that continues indefinitely, because the external source is effectively infinite.
• Outgassing: a rise that bends over and levels off as surface coverage approaches equilibrium.
• Virtual leak: gas trapped in an internal pocket (a blind tapped hole under a screw, a double weld enclosing a gap, an O-ring groove, stacked washers, a porous casting) slowly bleeds into the chamber. It behaves like a leak with a long time constant, and spraying helium on the outside finds nothing. The fixes are design fixes: vented screws, single continuous welds on the vacuum side, and no trapped volumes.

Locating a real leak requires a helium leak detector, a mass spectrometer tuned to mass 4, used either by spraying helium outside a pumped part or by sniffing outside a helium-filled part. Good detectors resolve leaks well below 10⁻¹⁰ mbar·L/s. The response time scales with the volume-to-helium-speed ratio of the test setup, so spray slowly, starting at the top, and wait.

Leak specifications should come from the application. A sealed device of internal volume V that may gain no more than ΔP over its service life t needs a total leak-plus-permeation rate \(Q < V\,\Delta P / t\). For 1 L, 10⁻³ mbar over 10 years, that is about 3 × 10⁻¹² mbar·L/s. Rate-of-rise tests cannot verify a number that small, which is why sealed vacuum products rely on helium leak testing and internal getters.

Choosing gauges

No single gauge covers the full range, and most read correctly only for the gas they were calibrated in (usually N₂).

Two practical rules follow. First, never control a backfill or vent of a non-air gas with a thermal-conductivity gauge. Above roughly 1 mbar its reading can be off by large factors, and chambers have been over-pressurized that way. Use a capacitance manometer. Second, put the gauge where the pressure matters. In molecular flow there are real pressure gradients, and a gauge mounted next to the pump reads lower than the pressure at the work.

An RGA is the fastest way to diagnose a slow system. A spectrum dominated by mass 18 means water, which needs time or heat. Peaks at 28 and 32 in roughly a 4:1 ratio, with argon at 40, point to an air leak. A series of peaks spaced 14 mass units apart indicates hydrocarbon contamination.

Design rules for production evacuation

Evacuating a product in a production cycle (a sealed device, a package, or a part processed in a load-locked chamber) differs from running a laboratory system. Cycle time, repeatability, and diagnosability matter as much as ultimate pressure.

1. Find the rate-limiting phase. In most production cycles it is desorption from internal surfaces, not removal of volume gas. Size the conductance, heating, and cycle around that phase.
2. Put conductance where it counts. When a sealed part is evacuated through a small port, the port dominates everything downstream. A 2 mm bore, 20 mm long port has a molecular-flow conductance of only about 0.043 L/s for air. With a 300 L/s pump behind it, the effective speed at the part is still 0.043 L/s, and a 1 L internal volume has a 23 s time constant. Because conductance scales as d³, a modest increase in port diameter is worth more than any pump upgrade.
3. Heat the part. Raising the temperature of internal surfaces speeds desorption far more than adding pumping speed. Respect the product's thermal limits and allow for the effect of cooling on residual pressure.
4. Measure at the part where you can. If you can't, characterize the pressure difference between the gauge and the part once, and keep the plumbing identical between stations.
5. Treat the pump-down curve as process data. Time-to-pressure, pressure at a fixed time, and rate-of-rise for every cycle are excellent statistical-process-control variables. They flag leaks, contamination, wet parts, and pump degradation long before yield drops. See SPC and process capability.
6. Control exposure and venting. Use dry-gas venting, short open times, and consistent part storage. Water picked up before the cycle is water you pump during it.
7. Specify leak tightness from service life. Use \(Q < V\,\Delta P/t\), then choose a test method (helium spray, accumulation, or bombing) able to resolve that rate with margin.
8. Design out virtual leaks and trapped volumes in fixtures and tooling as well as in the product.
9. Plan maintenance from trends. Rising base pressure or a slower time-to-pressure points to pump wear, seal wear, or contamination. Re-zero capacitance gauges on a schedule.

Quick reference

• λ (air, 20 °C) ≈ 6.6 mm × (1 Pa / P), or about 5 cm at 1 mTorr.
• Viscous flow when P·D > 0.66 Pa·m; molecular flow when P·D < 6.6 × 10⁻³ Pa·m (air).
• Molecular-flow conductance (air, 20 °C): aperture 11.6 A L/s (cm²); long tube 12.1 d³/L L/s (cm).
• S_eff = S·C/(S + C); aim for C ≥ 2–3 S.
• Volume pump-down: t = (V/S) ln(P₀/P). Ultimate pressure: P ≈ Q/S_eff. Rate-of-rise: Q = V dP/dt.

References

1. J. F. O'Hanlon, A User's Guide to Vacuum Technology, 3rd ed., Wiley (2003).
2. K. Jousten (ed.), Handbook of Vacuum Technology, 2nd ed., Wiley-VCH (2016).
3. J. M. Lafferty (ed.), Foundations of Vacuum Science and Technology, Wiley (1998).