How does a gel become mechanically stable?

Gels are strange materials. They are mostly made of a liquid, however they are solid. This solidity comes from the structure of the gel: a dilute solid network.

Transmission electron micrograph of a casein gel.

See the network in the above picture? Here the strands of the network are made of aggregated proteins. The holes, or pores, are permeated by water.

A gel network can be made of various building blocks. For example if you take very long, linear molecules that you crosslink from time to time, you get a polymer network. If this network is swelled by a solvent, then it is a gel. That is the case of panna cotta, where long collagen molecules from the gelatin associate into a network to imprison the creamy milk.

In the case of a yogurt, you start from globular proteins called caseins (think of little balls) dispersed in water. If you make them attract each other, the proteins will spontaneously aggregate into a network.

From a mechanical point of view, these two cases look quite different. Long molecules are floppy, adopt many configurations. If you pull on such a long molecule, you are decreasing the number of configurations it can adopt. This is costly in terms of (free) energy, so it is hard to pull, as if the chain was a little spring. Now is you assemble such springs into a network, it is quite obvious that you will get a solid that is easy to deform.

Using the path in black, it is possible to go from one end of the sample to the other. This sample is percolated. The black path is a percolation path.

Now, if you assemble sticky balls into a network, and if you pull on it, will it deform elastically or immediately break down into pieces? What are the conditions to make a particle gel stable? The answer to these question is not obvious and there have been many different answers in the literature.

1. In polymer gels, having a percolating network is enough to have mechanical stability. Percolation means that it is possible to find a path along the network that goes from one end of the sample to the other. But many observation concluded that this was not enough to have mechanical stability in the case of particle gels.
2. A glass is mechanically stable because its particles are not able to rearrange, trapped by the cage formed by their neighbours. This concept was imported into gels by saying that particles were trapped by the attractive bonds they formed with their neighbours. This theory allows arrest, but does not explain the network structure of the gel. For a long time it was thought that a combination of percolation and glassy arrest could explain gel stability.
3. About a decade ago the paradigm shifted. There was this class of phase separation called "spinodal decomposition" that spontaneously formed a bicontinous pattern: a phase rich in particles intertwined with a phase poor in particles. Usually the pattern coarsen with time until complete separation of the phases. But it was found that if the particle-rich phase is dense enough to undergo a glass transition, then it becomes arrested and the structure stop evolving. So we get a network structure and thus a gel. This is called the arrested spinodal theory. However at very low volume fractions it does not describes well some observations. More importantly, it does not predicts the mechanical behaviour of the gel.
4. But there are other ideas in the literature that come more from a mechanical point of view. Back in the 19th century, Maxwell stated that mechanical stability was ensured if the number of mechanical constraints is equal to the number of degrees of freedom. From example, if you live in 3D space (I do), you have 6 degrees of freedom : you can move east, west, north, south, up and down and any combination of theses. If you are blocked so that you can't move north, can't move south, can't move east, can't move west, can't move up and can't move down, then you cannot move anywhere. So you are stable. This concept of "isostaticity" is very useful at the macroscopic scale, when you want to build a bridge or when you want to understand sand piles. But is this concept relevant when each particle is jiggling around due to thermal motion? In 2012, a paper found that when you look at a gel before and after breaking it, you find a decrease of the particles that have 6 or more bonds. These are actually the particles that should be stable, since each bond is a constrains, so higher than 6 you shouldn't be able to move.
5. In 2016, percolation stroke back. A paper found that not all paths along the network were important. If you considered only paths that never turned back, that were "directed", then you are interested in "directed percolation", not "isotropic percolation". The authors found that all their samples where directed percolation took place were mechanically stable, and that no other sample was. So, is directed percolation the cause of mechanical stability? For the moment, just remember that their samples were very dilute (less than 10% of particles).

The path in black is a percolation path that never turn back. This is a directed percolation path.

Here we arrive on the scene with our method to follow experimentally each individual particle during the process of gelation. All previous experiments were done by observing the state of the already formed gel. So the only influence that had been studied was the fraction of solid particles and the strength of the interactions. Here we add another axis: time.

We asked this question: can we observe when a gel become mechanically stable, and how is the microstructure changing at that moment?

First, we had to find a way of checking whether the suspension was mechanically stable or not. Usually you do that by putting the suspension in a rheometer, you oscillate at very low amplitude to measure the mechanical response. This response has two parts: elastic and viscous. If the viscous part is larger than the elastic part, the suspension is still liquid. If the elastic part is larger than the viscous part, the suspension is solid, so the gel has become mechanically stable. But in our case we need large (3 micron) particles to be able to observe everything in minute details. Large particles makes extremely soft gels. So soft that the rheometer does not have enough sensitivity.

Therefore, we ditch the idea of a rheometer altogether and only look at the response of the system to thermal agitation. The sample is just under the microscope and we record the trajectories of all the particles due to thermal agitation. If particles are able to diffuse away freely, it means that the suspension is liquid. If the particles are stuck and vibrate around a mean position, it means that the suspension is solid. Actually, a clever method call microrheology manages to transform the displacement of the particles into a measure of the mechanical response, elastic or viscous. In this way, we know exactly when our suspension becomes a mechanically stable gel.

In parallel, we can follow the progress of percolation. The usual isotropic percolation occurs always way before mechanical stability. Explanation (1) is out of the game.

We can also follow directed percolation. It occurs later than isotropic percolation, and actually for dilute gels (below 10-12% particle fraction), the time of directed percolation matches very well the time of mechanical stability. But, it does not work at all a higher volume fraction. Mechanical stability occurs sometimes 5-15 times later than the directed percolation. So, obviously explanation (5) is not general.

Now, we can look at isostatic particles, these particles that have 6 or more neighbours. Are these particles percolating? Yes, they are, and always at the exact time where mechanical stability sets in (see error bars in the paper, because science has error bars). So explanation (4) seems to be the correct one.

But then, why does explanation (4) give the right time at low particle fraction?

Look at the cartoons below. This is in 2 dimensions, so isostaticity means having 4 neighbours, not 6 like in 3D. Isostatic particles are shown in purple.

Path to gelation in the dilute case

In the dilute case (less than 10-12%)

• When particles begin to aggregate, they form very loose clusters, very open and far from each other.
• These clusters compact. Particles make more bonds within the same cluster, but clusters are too far away from each other to meet now. Because of this compaction, the heart of each cluster is now isostatic.
• Clusters finally begin to meet and form larger and larger clusters, until you get a percolating path. In this situation, you have groups of isostatic particles that are bridged by floppy non-isostatic bridges.
• In order to make more bonds, floppy bridges have to straighten. So at the time isostaticity percolates through the system, the network is mostly made of straight paths. So directed percolation is easy and occur at the same time.

Path to gelation in the dense case

In the dense case (more than 10-12%)

• When particles begin to aggregate, they are so close together that there is no space between clusters. Very quickly a percolating network is formed.
• That is only after the formation of this initial network that rearrangements occur. More bonds are created locally and this is enough to obtain a directed percolating path.
• Compaction continues to proceed until isostaticity percolates. That only then that the gel is mechanically stable.

That explains why directed percolation looks like a good explanation if you look only at dilute samples.

To conclude, we can view a gel as a network that is sculpted and arrested by mechanical forces. Our mechanical explanation does not really contradict the arrested spinodal theory that is more thermodynamics. But we hope that explaining gels by mechanics will help understand the mechanics of gels. For example, we still do not understand why some particle gels break into well-defined and irreversible fractures (think yogurt) whereas some others just flow plastically and reversibly (think toothpaste or fresh cement).

If you want to dig more in depth, our paper is scheduled to be published the 31th of May 2019 in Science Advances, and a preprint is already available on ArXiv:
H. Tsurusawa, M. Leocmach, J. Russo, H. Tanaka, Direct link between mechanical stability in gels and percolation of isostatic particles. Sci. Adv. 5, eaav6090 (2019). arXiv:1804.04370

Want to study how flocs, swarms or crowds flow? I'm looking for prospective PhD student

Have you ever seen the patterns made by birds, sheeps or insects when they come together? There is no boss in the mass of living bodies, however they self-organize.



The same kind of patterns can be seen at the microscopic scale in colonies of bacteria or in living tissues.



And recently, I've been working on non-living microscopic particles that self-propel, displaying the same kind of patterns.

Long exposure picture of self-propelled particles

A lot of beautiful physics has been done in the last decade about this class of system. Now it's known as "Active Matter".

Patterns are mesmerizing, physics is intellectually pleasing, but what is the purpose? What do you make out of this "Active Matter"? Can you build a bridge? Power your home? Cure cancer ?

Well, maybe.

It becomes clearer by the day that cancer metastasis is a swarm of cancer cells migrating through the body, very close to the above patterns. In which conditions the solid tumour becomes a liquid-like swarm? How to influence this transition? These are actual questions physicists and biophysicists are asking right now.

Maybe you can't power a house with active matter, but maybe you can make it do the worm of pumping the water of your shower. It has been found that a suspension of swimming micro-organisms flow more readily (has a lower viscosity) than the suspension of dead micro-organisms. Furthermore, in some situations, the viscosity is even lower than the liquid alone, without micro-organisms! They swim, converting their food into movement, and this moment helps the water flow up the pipe.

And can we build something? Maybe not a bridge, but a muscle that would be able to contract from the action of trillions of micro-motors. Or some kind of new material that would react to a push to the left by deforming to the right?

To understand this last point, I am looking for a PhD student interested in physics, mechanics, materials and afraid neither to code nor to perform experiments. This is an experimental investigation. See the announcement (2 pages, FR then EN) on my lab's website.