# How do I calculate capillary entry pressure for a given pore width and and a known surface tension?

This is not homework, it is research. I have drops of known mixtures that I drop onto dry samples and I measure the surface tension of the sample this way. I also have data about pore-size distributions on the sample. I can correlate surface tension to pore size but I am also interested in explaining why they correlate in a particular way.

I know that micropores will enhance hydrophobicity (drive up surface tension) because the pores are too small to admit water. I believe this is quantified by *capillary entry pressure* under a gravity-draining system at atmospheric pressure.

I want to find the pore size that *forbids* entry by water due to capillary pressure considerations.

Does anyone either:

1) Know how to calculate this?

2) Know what this theory might be called?

3) Have suggestions for another way to determine what size of pore is too small for water to enter it?

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## 8 Answers

I’m not a scientist but I did have a flash of intuition while reading your question. Must not these calculations have been made when the type of waterproof/breathable fabric that admits vapor but not water? The patents are certainly public.

Can I ask for a clarification? You know that under atmospheric pressure and gravity-conditions, some pore sizes are too small to admit water. If I’m correct, you are trying to figure out how this relationship works under different pressure conditions? Like, water sitting still on a material might not penetrate, but water getting shot at high speed at the material might? If this is correct, it’s a very interesting question, and I’ll think about it and see if I come up with anything…

@pdworkin Can you elucidate your response?

@hannahsugs Actually, I’m interested in solving this for atmospheric pressure. I know from experience that if a drop of water is dropped from a height it will penetrate the sample immediately. But if a drop is placed gently on the surface, it can take 5 hours of more for it to penetrate.

What I’m really trying to figure out is how to calculate a capillary pressure which is greater than atmospheric pressure plus gravity…I think.

You said it is a known fluid—from this I would trust that you know the viscosity of the fluid (I didn’t see that you said the fluid was water though). The diameter of the pore is part of it, but the depth of the pore would also seem relevant if calculating surface tension (i,e, the formation of a meniscus be important). The relative concavity or convexity of the meniscus would either draw the liquid up or down.

ok, let me see if I can help (no guarantee, especially because I imagine you already know most of this stuff):

when you let a drop of liquid fall from a height, it hits the material with a certain speed, let’s call it V. The drop of liquid comes to a stop because the material exerts a force on it. Because we know the initial speed, V, and the final speed can be assumed to be zero, you could calculate the impulse, which is the change in momentum of the drop, or the integral of the force over time. If you can time how long it takes for the drop to come to a complete stop (which would be hard, and probably require a high-speed camera), then you could know the force that’s being exerted between the material and the droplet. Since pressure=force/area, you could use the area of the pores, and figure out the pressure exerted on them by the falling droplet.

Of course, the above paragraph is based on basic freshman physics, and doesn’t account for a huge number of different variables that could affect this. On top of that, I don’t exactly understand what capillary pressure is, so I’m not even sure if the approach i described would be useful, in the event that you could put it into practice. I am very interested in learning about this though, if you want to try to explain it, I’d enjoy trying to work out the problem with you.

I’m sorry. I was saying that the type of fabric that allows vapor to escape, but does not allow water to enter is called breathable/waterproof fabric. There are several well known brand names. They must have done similar calculations which must have been published in the patents, so I was suggesting that you find the patents on line and see if the math has been done for you. They seem to me to have needed to get the same result you seek.

From what I know about capillary pressure: the smaller the radius—the larger the capillary pressure. This would mean that there is no radius where water is not admitted. This may change for extremely small pores when fluid-material (pore wall) and fluid-fluid interactions dominate.

Ive only ever encountered “capillary entrance pressure” in viscometers/rheomotry, As with all pressure drops its dependant on flowrate and capillary (“pipe”) dimensions. (so in your case the intake rate of water into the pores).

I googled it to see what other fields use capillary entrance pressure and there appears to be alot of information on this in geology, specifically the study of oil reserves. Where (from what I gather) its the pressure required before oil displaces water in porous media. Water being above the oil, and oil being the non-wetting phase. I dont think this comes into effect because your non-wetting phase (air) isnt displacing your wetting phase (water).

So – without wasting any more of your time as I dont actually have an equation only advice: I think with the scale of the pores your talking about, you should be searching for water absorption characteristics and mechanisms in micropores, not capillary entrance pressures. Though the entrance pressure will be good to calculate once you have pore intake rates.

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