Why small particles can’t get through masks

Corona Wikivoyage bannerMasks are in the news at the moment. The basic idea of a mask is simple, it filters out some of the nasty particles from the air. The hope then is that if a droplet containing virus particles gets sucked into or blown out through the mask, it will be trapped by the mask and go no further. How masks filter out the bigger droplets is easy to understand. Masks are basically made from meshes of fibres that are very roughly around ten micrometres or so thick. So when the user breathes in or out, the air is forced through holes in this mesh of fibres. Some of these holes may be only around a micrometre or a few micrometres across, and of course any droplet bigger than the hole will get trapped and so not get through.

But naively I assumed that most droplets much smaller than the sizes of the holes in the mesh, would just zip through the mask. This is not so, in fact for small droplets, the smaller the droplet, the more likely it is to be caught in the mask. This looks a bit counter intuitive at first sight, but the reason is that small particles are very mobile, they diffuse rapidly. They do this even when they are in the air stream being sucked through the mask. And this mobility makes it likely that they will wander into one of the fibres and stick to it.

Simply speaking, the argument is as follows. Small particles diffuse in air, with some diffusion constant D. This means that in addition to being carried by the flow it also pin balls around in the air such that it undergoes random-walk-like motion over a distance about x ~ (Dt)½ metres, in a time seconds. Inside a mask, this particle is being carried by air flowing at u metres per second.

Inside the mask there are holes of various size but let us consider a narrow gap about l across and long between fibres. We want a rough estimate of the probability that the particle will diffuse into the surface of the fibres during the time it is carried through this gap, by the flow.

The time for the particle to be carried through the gap is just l/u, or in other words the rate of this process of being carried through the gap is u/l. Roughly speaking the probability that it diffuses into the fibres should be roughly the rate of diffusion*, D/l2, divided by rate it goes through the pore, u/l. This ratio, which is our estimate of the probability of being trapped, is then D/ul**.

This diffusion-and-capture probability is proportional to the diffusion constant D, so droplets with bigger diffusion constants are more likely to be filtered out. Now, the diffusion constant of a droplet is approximately inversely proportional to its size, so smaller droplets diffuse faster and are more likely to diffuse into a fibre and be filtered out.

I guess this is why the fancy masks, called N95 in the USA and FFP2 in Europe, are specified in terms of their ability to filter our droplets 0.3 micrometres across. This size may be the tough ones to filter out as they diffuse more slowly than smaller droplets.  And bigger ones are bigger than the gaps and so are easily caught. Note that you can’t just make the holes very small to filter out everything as the smaller the holes the harder it is to breath through them. If you halve the hole size you need to exert four times the force to breath through them.

So there may be some basic physics behind the difficulty of filtering out droplets around a third of a micrometre across. I don’t know if droplets these size are good or bad at spreading the corona virus, hopefully not too good.

* As I just want the scaling, a rate (dimensions of one over time) is just one over a timescale. The diffusive timescale is distance squared, divided by the diffusion constant.

** This is an inverse Péclet number, it compares flow and diffusion rates. Note that this argument is too simple, if you do the geometry better, the scaling is this inverse Péclet number to a power of two thirds. See Wang and Otani’s review of filtration.

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