Martin Knudsen made a tremendous contribution to vacuum science and especially in the the understanding of flow in different parts of the vacuum spectrum. He was born in the Fyn region of Denmark in 1871 and in 1896, after 6 years of study, gained a Masters degree in science majoring in (the relatively new established discipline of) physics. This was at the the University of Copenhagen during which time he worked as an assistnat to Christiansen who soon later was to guide Neils Bohr.

Knudsen was highly interested in kintec theory of gases and was the first to apply it to  rarefied gases so as to become the ‘father’ of modern vacuum science. His supreme experimental skills allowed the verification of the prediction from the Maxwell-Boltzmann distribution of the flow of gases through an aperture. From this is  the concept of a Knusden cell which is the basic element for molecular beam epitaxy.

His analysis of the thermal effects at surfaces led him to develop the Knusden gauge and introduce the thermal accommodation coefficients. He later looked at a viscosity gauge by analysing the movement of gas molecules between moving plates.

Knudsen is probably best known and remembered as giving his name to the Knudsen number Kn = λ/d where λ is the mean free path of a gas molecule in the system and d is a characteristic dimension (usually the pipe or chamber diameter or transverse section length).

The continuum or viscous flow regime ii where Kn < 0.01 and molecule-molecule dominate gas behaviour which behaves as a fluid. In molecular or Knudsen flow where Kn > 1 (or for some authors >0.5 or > 3) molecule-surface collisions dominate and the interaction of a gas molecule with for example a chamber wall is crucial to understanding this flow regime. The transitional flow regime is where 1 > Kn > 0.01 this is a particularly difficult regime to analyse.

For a box of length l the number of molecule-surface/molecule-molecule collisions is 3 λ/l. hence the range 1 < Kn < 10 is known as an ‘almost-free’ passage of molecules down a tube. An application of this fact is particularly important in calculations of molecular transmission probabilities.

Knudsen’s analysis of the behavior of the molecules on a surface was also seminal. A molecule impinging on a surface accommodates to the surface and after a residence time (which can vary in a huge range). After leaving the surface (a process known as desorption) the molecule has no memory of the direction it took (or speed it had) to travel to the surface. Since molecule-wall collisions dominate in molecular flow this action at the molecule-wall interface dictates the molecular flow behavior.

This situation is that described by the Knudsen Cosine Law which states that the relative probability W of molecules leaving a surface into a solid angle dω forming an angle θ; with the normal to the surface is proportional to cosω  i.e. W = (dω/π)cosθ or the flux per unit solid angle is

                                                  
where J(0) is the flux (per unit solid angle) normal to the surface (θ = 0) which is the most probable direction. On average molecules depart at an angle normal to the surface.

Knudsen cosine distribution The polar diagram represents the locus of the flux (number density) of molecules emitted from a flat (average) surface element. The magnitude of each vector is proportional to cosine θ.

Interestingly this understanding if specific nature of gas desorption corrected Wolfgang Gaede’s erroneous propositions.

Knudsen had also a keen interest in hydrography and developed methods of defining properties of seawater (he was editor of Hydrological Tables in 1901), however it is his great contributions to vacuum science makes him a true Hero of Vacuum.
References

An early classic paper is from 1910 (with Willard Fisher): The Molecular and the Frictional Flow of Gases in Tubes:Physical Review (Series I, Volume 31, p586 (1910).

His ideas on Kinetic Theory are summarized in his book, The Kinetic Theory of Gases (London, 1934).

Walter Steckelmacher’s article: Knudsen flow 75 years on (Reports on Progress in Physics ,Volume 49, p1083 -1986) is an excellent summary.

Andrew D Chew;2008