100 Nano-Stories: Ostwald Ripening & Distribution!

Episode #09: The Kelvin Equation! (Section 2)

Carlos Manuel Jarquín Sánchez
6 min readJan 6, 2021



Welcome back, reader! Now that we talked about the fundamental concepts of the aging of silica gels, now we can talk about the mechanisms that affect the structure & properties of the aerogel!

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

Thank you, Carlos! You’re a real life-saver!

As a bonus, I will also be including vocabulary terms and extra resources at the end for anyone who got lost in the article or for anyone interested in diving more into the topic, reader!

Thanks, Carlos! But before you start talking about the mechanisms of silica gels, can you just briefly talk about where you ended in your last article?

Sure! Two different mechanisms might operate during aging that affects the structure and properties of a gel: the dissolution of small particles to larger particles and transportation of material to the neck regions of the silica gel. On some occasions during the aging process of the silica aerogel, some of these silica groups can dissolve back into the solvent and then reprecipitate into the neck regions of the gel. This allows for the overall structure to become thicker & stronger in the weak necks of the silica network. Now, it may sound wonderful, but the problem with his process is that it is time-inefficient, so it takes some time for this process to occur, and to get it right. This is known as Ostwald Ripening.

Can you at least show us a photo of this process?

Neck Regions of The Silica Groups/Network. Photo by ScienceDirect.

Very intriguing! But can you tell us more about Ostwald Ripening, Carlos?

Ostwald Ripening

To define Ostwald Ripening (disproportionation), it is the process of the disappearance of small particles or droplets by dissolution and deposition on the larger particles or droplets of a substance/mixture.

Ostwald Ripening mainly occurs in liquid sols such as our silica sol-gel solution, and what happens is that our smaller silica particles will dissolve and become deposited in larger molecules of the silica gel. The smaller particles (with a higher radius of curvature) are more soluble than the larger ones (with a lower radius of curvature), for more clarity on why the smaller particles dissolve and are deposited in the silica gel.

But what drives this weird phenomena of Ostwald Ripening, Carlos?

The main cause for Ostwald Ripening is the difference in solubility between the small and the large particles, reader. Solubility is a property of a substance (solute) to dissolve in a given solvent (look at definitions at the end of the article for more clarity).

Now, we will talk about how this material is being transported throughout the material/ silica gel, reader! Hold on to your seats!

What’s going to happen, Carlos? Am I going to blow up?

“De-Coding” The Kelvin Equation (Part 1)!

During the aging of the silica gel, the material is transported to the neck region between particles, giving a more rigid gel network. But what is the true drive behind the transportation of material and Ostwald Ripening?

The Kelvin Equation! Photo by Carlos Jarquin.

Oh… so that’s what you mean by hold on to your seats😅 … can you please explain what this means?

Okay, let’s do break it one by one.

S(r) is the solubility surrounding the particle of our radius (r). Are you following along, reader?

Yes, Carlos!

Next, S (∞) is the bulk solubility. Bulk is just a term for saying the majority of something; or in large amount; and in this case, the majority of our solubility!

Exp is an exponential function.

is the interfacial tension. The forces acting on interfacial tension are adhesive forces (tension) between the liquid phase of one substance and either a solid, liquid, or gas phase of another substance. In our case with the aging process, we are looking at a solid-liquid interfacial tension. Finally, the interaction occurs at the surfaces of the substances involved, otherwise known as their interfaces.

(Vm) is the molar volume of the solid. A molar volume is a volume occupied by one mole of a certain substance divided by a specified pressure and temperature. It is mostly used for gas

Molar volume is calculated as molar mass (M) divided by mass density (ρ): Vm = M / ρ

A Mole (mol) is the unit of measurement for the amount of a substance using the International System of Units (SI Units). However, moles do have a definite number of particles, and that number is 6.02214076×10²³ atoms/particles/electrons/molecules/ions.

The molar mass of a substance is the mass of 1 mole of that substance, but using multiples of the gram. The amount of substance is the number of moles given a certain sample. For example, the molar mass of canola oil is 876.6 g/mole. (More information on conversion in the resources below!)

So, you need 6.02214076×10²³ of particles to equal 1 mole? That’s a lot of particles, Carlos!😵

Yup! This number is also known as Avogadro's Number/Constant!

However, the SI Unit for molar volume is cubic meters per mole (m3/mol).

Can we go back to the actual equation, Carlos?

“De-Coding” The Kelvin Equation (Part 2)!

Now you have to divide by the next three variables.

r is just the term for “radius”.

R is the ideal/universal gas constant. This number can be precisely measured as 8.31446261815324 J⋅K^−1⋅mol^−1 (J- Joule) (K- Kelvin) (mol- Mole).

How do we find this number? Multiply the Avogadro Constant times the Boltzmann Constant:

38064852 × 10^–23 m^2 kg s^-2 K^-1 (Boltzmann Constant)

6.02214086 × 10^23 mol^-1 (Avogrado’s Constant)

T is the temperature of our substance/mixture.

The Kelvin Equation! Photo by Carlos Jarquin.

Closing Thoughts

So, now that we broke down the Kelvin Equation, what does it mean for the dissolution and distribution of the silica particles in the gel, Carlos?

Necks between particles of the silica gel network have negative curvature [radius] (r < 0), which means that they have low solubility. This is when the transportation of material comes into play.

The material of the gel will begin to pile up in the form of a convex surface. The convex surface will cause the liquid to cave downward, hence our curvature of the silica gel. But remember, the smaller particles of silica have larger solubility.

So this driving force will act on the smaller particles of silica to dissolve via precipitation onto the larger particles of silica!

The only downside? It takes so long for this process of Otswald Ripening on the Silica Gel Network!

Wait, are we done, pleaseee? This was probably one of the more complex articles on Aerogel, Carlos.

Yup, we are done for today! I promise the next articles will be shorter since we just completed the more complicated part of creating silica aerogel! Pinky promise! 🤞🏽

All right then… See you soon, Carlos!👋🏽

Vocabulary 📓

Solubility - The maximum amount of solute that can dissolve in a known quantity of solvent at a certain temperature is its solubility.

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.

Ostwald Ripening - The process of the disappearance of small particles or droplets by dissolution and deposition on the larger particles or droplets of a substance/mixture.

Dissolution - The act of dissolving; to become or caused to become included into a liquid to form a solution.

Precipitation - The process of converting a chemical substance into a solid from a solution, mostly a liquid solution.

Lord Kelvin Equation - equation of the effect of a droplet or particle size on the solubility of the disperse phase.

Dispersed Phase - The phase that is scattered or present in the form of colloidal particles (a solid dispersing in a liquid)

Colloid - A heterogeneous non-crystalline structure with nanoparticles from one single substance that is mixed with a second substance.

Resources 💻

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