100 Nano-Stories: Capillary Pressure Equation!

Episode #11: Drying of Aerogels & Xerogels!

Carlos Manuel Jarquín Sánchez
7 min readJan 8, 2021
Xerogels! Photo by Dynasylan.


Hey, hey, aerogel enthusiasts! Now for the moment, we have all been waiting for in this series: The Production of our Silica Gel Products! Let’s dive right into our topic!

Uh, Carlos, I’m new here to your series on “100 Nano-Stories”, so can you please give me your previous article on silica gels?

Also, reader, I will be posting vocabulary terms and some extra resources if you want to learn more or if you don’t want me to leave you hanging in the middle of the article!

Intro to Drying

Drying is the most critical part of the sol-gel process when producing aerogel. For aerogel, there is supercritical and subcritical drying of the silica gels, and drying is critical because we can free the solid from the liquid of the gel, and that will prevent the silica structure/skeleton of the gel to collapse.

Wait, Carlos, isn’t the collapse of the gel called capillary action?

Photo from Reddit

I knew you were going to say that! You see, capillary action happens when we try and remove the liquid in the silica framework through evaporation, but it can't collapse by itself. That is why we have these wonderful terms such as capillary pressure and surface tension.

But, Carlos, how are those terms going to affect the silica framework?

Well, reader, this when some mathematics comes in! Don’t be scared, I’ll guide you through the equation! Are you ready?

Well… Sure! But you better tell every detail possible, Carlos! Oh yeah, and while you’re doing that, explain to me quickly…

What is Capillary Pressure?

To prevent a partial or total collapse of the silica gel structure, we control the drying process with capillary pressure. Capillary pressure is the pressure of two liquids in our nano-scale silica network, and this will result from the pressures of the liquid and the solid walls of the silica framework.

But what two liquids do we have, Carlos? I only see one, and that is our liquid in our silica framework!

Depending on if we do Low-temperature supercritical drying (LTSCD) or High-temperature supercritical drying (HTSCD), we will have our solvent, which is the liquid that was already in the gel, but the additional liquid could be either alcohol or liquid CO2!

Certain alcohols like ethanol, methanol, and pentanol are used for LTSCD Drying, while Liquid CO2 is used for HTSCD!

Woah! Can you talk about that for the next article?

Sure! But the reason I talked to you about the fluid is that it will relate to the Capillary Pressure Equation! But I will make that for your next article, I promise! 🤞🏽

Capillary Pressure Equation: Decoded!

Let’s bring up The Capillary Pressure Equation!

Here we go! Photo by Carlos Jarquin.

Okay, Carlos, you know what to do! Help me understand what all of this means!

P(c) is the variable for Capillary Pressure.

But wait, Carlos! If P(c) means Capillary Pressure, and there is an equal sign, then what are the rest of the components that equal to P(c)?

Good thinking reader! Love that you are paying close attention!

The top of the fraction (-γlv) is the surface tension of our pore liquid in the silica gel. (γ) is the symbol in math used to describe surface tension. (l) is used to describe the liquid in our silica network. (v) is the critical point. A critical point in aerogel is the pressure-temperature curve that designates conditions under which a liquid and its vapor can coexist. But why is surface tension negative (-γlv)? I will explain that after we complete breaking down the equation, reader!

Now, let’s look at the bottom half of the fraction: (2(Vp/Sp))!

(Vp) is defined as the pore volume of the silica network. (Sp) is defined as the surface area of the pores of the silica network. θ is defined as the thickness of a surface adsorbed layer. For those who don’t know what adsorbed means, adsorbed means the molecules are held loosely on the surface of the adsorbent (molecules of the solvent in the silica gel) and can be easily removed.

Wait, are we done, Carlos? I still have some questions!

Don’t worry, reader! I got you covered!

First of all, θ is usually defined as a plane angle in trigonometry and calculus, and not as the thickness. To clear this out, the usual sign for the equation is usually the lowercase Greek letter Delta (δ). “δ” is used to represent an angle in any geometric shape. With aerogel, we are trying to identify the geometry of the thickness of a surface adsorbed layer!

Carlos, you nearly confused us there with the θ and δ!

The next piece of information that I want to cover is the negative surface tension (-γlv) and why is it negative? To start, surface tension and capillary pressure are proportional to each other, which means both values have to equal the same. So if a solvent with high surface tension is introduced to the drying process, then the capillary pressure in the pores of the silica gel will have to increase as well.

But remember, we forgot about the liquid and critical point. We are extracting liquid from the gel and replacing it with either a certain type of alcohol or liquid CO2, so this means we are losing liquid, and that means we are losing surface tension (negative; -γ), but this will only happen once we introduce the additives of alcohol or liquid CO2.

Jesus! That’s crazy!

Finally, I want to talk about (Vp/Sp)! The pore volume (Vp) and surface area of the pores (Sp) are very critical in the mechanical damage of the drying in the silica gel. The increase of the capillary pressure within the pore of the gel could do that mechanical damage to the structure of the gel. If the capillary tension exceeds 200 MPa, the structure can shrink and crack, resulting in something that looks like a xerogel.

This is what could happen! Photo by Aerogel.org

Closing Thoughts

Well, reader, now you know a new math equation to know more about the drying process of aerogel!

But Carlos, why should I care? How is it going to affect me?

So you don’t want to wear the Kistler Aerogel Jacket to protect you from the cold anymore?

Kistler Aerogel Jacket! Photo by Indiegogo.

Oh well, reader! We’ll talk more about supercritical drying in the next article, but until then, see you later! 👋🏽

Wait, what!? An aerogel jacket!?

Excited? GIF by Giphy.

Vocabulary 📓

Pascal - The SI Unit of Pressure, and it is equal to one kilogram per meter per second squared.

Surface Tension - It’s what allows for the bulk of a liquid molecule to have less surface area. In elementary terms, a liquid surface will shrink to the minimum amount of surface area possible.

Capillary Pressure - The pressure of two liquids in our nano-scale silica network (unable to become a homogenous mixture) resulting from the pressures of the liquid and the solid walls of the silica framework.

Proportional - Two values/quantities have to equal the same.

Aerogel - The world’s lightest solid that is produced by the removal of the liquid component from a conventional gel.

Xerogel - A very light solid (similar to aerogel) caused by drying/evaporating with rapid shrinkage of the gel.

Solute - A substance (either solid or liquid or gas) that is dissolved in a solvent.

Solvent - The liquid in which a solute is dissolved to form a solution.

θ (Theta) - The symbol to write/denote an unknown measure of an angle; mostly used in trigonometry & calculus.

δ (Lowercase Delta) - The symbol used to denote small quantities in proofs (theorems) involving limits.

CO2 - Carbon Dioxide.

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© 2021 by Carlos Manuel Jarquin Sanchez. All Rights Reserved.