A New Science-based Tool to Develop Particles-Polymer Dispersions Faster

TAGS:  Science-based Formulation    

A typical formulation challenge with particles is to add some sort of dispersant or stabilizer so the particles can be put at high concentration into an ink, a battery electrode or into a polymer to confer desirable properties.

Unfortunately, our formulations are complex, so we get unexpected gelling, high viscosities or particles crashing out for no obvious reason.

• What we need is a theory that lets us play with how strongly the various components interact with the particles, the solvent and with each other.
• What we have are a few theories that are rarely usable in practice due to complex calculations.

Here, we discover that a new app-based implementation of a well-established theory provides a science-based understanding of what is going wrong and how to fix it. This, for the first time, gives the ordinary formulator, like yourself, the ability to optimally rationalize complex particle systems for success.

Let’s first begin by understanding complexity of particle stabilization in real-world applications.

Particle Stabilization – What Continues to Remain the Main Challenge?

It is surprising that after decades of “particle science” and “colloid science”, there are only few science-based tools to help us in formulation where particles, polymers and solvents (including water) interact in complex ways.

Putting to one side classic DLVO theory of charged particles in water, we mostly have the idea that we need to add “dispersants” or “stabilizers” to our particles where these can be:

• As simple as surfactant molecules or
• As complex as a brush-comb hyperdispersant.

1. Trial and error dominates. We generally have to try a bunch of dispersants from the suppliers who are reluctant to tell us anything about them other than that they are “exceptional” or “high performance”, omitting to tell us that this applies only to specific circumstances about which even they might not be clear.
2. Too much or too little of anything is a bad thing. It means that too little of a dispersant might cause failure but also adding too much makes things worse, for reasons that are not at all obvious.

The Complexity of Real-world Formulations

Even if our job was just to have a single type of particle of uniform size, a single dispersant and a single solvent, life would still be difficult. But we almost never have such a luxury, so things are even tougher.

• We often have multiple particles in the formulation and even more often we have to add extra polymers for formulation reasons that are not connected to stabilization.
• And, we often have solvent blends which change over time, e.g. when the formulation is drying after processing.

A formulation that is perfectly happy with lots of the solvent blend can crash out when the more volatile solvent disappears.

Part of the problem for the poverty of formulation tools is the insistence that we are dealing with “dispersions” which, somehow, follow special but rarely defined rules. I have not yet found a consistent definition of what a “dispersion” is, nor is it clear what special rules they should follow.

HSP Power – A Rational Approach Towards Dispersions

Fortunately, all that changed this year when my colleague Dr. Seishi Shimizu published a paper¹ showing that the term “dispersion” as usually defined (a separate phase within the formulation) is wrong at a fundamental level and that we are perfectly able to use “solubility” science to describe interactions with particles in formulations.

This paper fortunately coincides with a pragmatic movement within the particle community of measuring the Hansen Solubility Parameters (HSP) of particles via techniques that follow the same methodology as measuring the HSP of polymers.

• Find the “happiness” of the particles in a range of different solvents (with their δD, δP and δH parameters).
• Fit them to a sphere inside which are the good solvents, and outside which are the bad ones.
• The center of the sphere is the HSP of the particles.

We have a few ways to assess the stability of particles. For example, we can visually inspect them; we can put them in an analytical centrifuge; or we can measure the NMR solvent relaxation time. The important thing is to get a good set of measurements across a wide range of solvents with the least effort.

For some time, there was a debate about whether these could be called “solubility parameters” but the Shimizu paper closes that debate: they can.

With the measured particle HSP values it becomes routine to formulate rationally with different polymers and solvent blends.

This is all a familiar story to those who have encountered my views elsewhere.

 »  Learn More About the Fundamentals of Hansen Solubility Parameters (HSP)!

Making Complex Theories Accessible

The reason for this article is my “discovery” (about 30 years too late!) of a theory that we can all use routinely to understand the complexities of particle/polymer/solvent interactions. It is firmly based in solubility theory and has been validated extensively by the academic community. Unfortunately, even if the formulation community had encountered the theory (which we mostly haven’t) there was little we could do about it because it involved computational tools far beyond the capabilities of most of us.

I’m pleased to say that that’s all changed. Let’s begin with understanding the theory first.

Scheutjens-Fleer/ Self-Consistent Field (SF-SCF) Theory for Polymer Colloids

The theory started development in the 1970s and has been continually improving ever since. It is highly respected and, within the limits of its own assumptions, has been extensively validated.

How Does it Work?

SF theory puts the particle on the wall of a theoretical lattice then places the chosen polymer (which can be a single A polymer or a di-block or comb A-B or even [as per normal formulation] two separate polymers A and B) onto the lattice in some reasonably guessed configuration, with the solvent filling the rest of the lattice.

As with all lattice theories (such as Flory-Huggins), it is possible to calculate the energy of the system by summing all the good and bad neighbor-to-neighbor interactions.

Now, move a few polymer units around and recalculate the energy. If it drops, carry on doing what you were doing, if it’s worse, find something else to do. With algorithms that have become increasingly refined and robust over the decades, within a fraction of a second, an optimal configuration is found which tells us how much of the polymer is present as:

• “Trains” (Scheutjens’ fanciful name for polymer chains in contact with the particle surface),
• “Loops” (the bulk of the polymer) and
• “Tails” sticking out, along with free polymer doing nothing useful and potentially making things worse.

Now we need to add a few elementary facts, known to all theoreticians and mostly unknown to the rest of us:

1. First, although we like to draw polymers on particles according to the diagram on the left, most of the time they are like those on the right.
   
   
   
2. Second, although we like to think of steric stabilization as shown in the picture below, but the fact is that most of the time we have very few tails sticking out and this makes it less effective than we might imagine.
   
   
   
3. And, although we imagine bridging flocculation to take place like shown below (T), with the bridging chain spanning the gap, the image is completely wrong. It takes place via loops (B):
   
   
   
   

There is an amusing story about Scheutjens and the Nobel Prize winner de Gennes. Scheutjens proudly showed de Gennes the results of his first SF calculations. de Gennes merely glanced at them then said, “They’re wrong”. Indeed, it turned out that Scheutjens had made a programing error. The reason de Gennes knew instantly that the results were wrong was because they violated a key symmetry argument which, translated into simple terms, states: “Tails repel, loops attract”.

Science-based App to Simplify Complex Calculations of SF Theory

That digression means that we should now all be very keen to know whether we have tails or loops. That is where SF theory comes in – if we can find a way to do the complex calculations. I took one look at SF theory and knew that I would never be able to write a program or app to do the calculations.

Fortunately, I was introduced to Prof. Frans Leermakers at Wageningen U where SF theory was created. He had written a general-purpose SF theory solver “black box” which he called SFBox. Even this is too difficult for most of us to use, so with his generous help I wrote two versions of SFBox-FE, the front end to SFBox.

Here’s one of them.

SFBox-FE – A New Science-based Formulation Tool

User-Friendly App Determines Particle-Resin-Solvent Interaction Faster

Most of us want polymers as stabilizers/dispersants for particles. So when polymers are added in the formulation, we want most of polymer attached to the particle with plenty of tails sticking out (it's tails that provide steric stabilization), and with very little polymer "wasted" within the bulk of the solution.


We have a 338 “lattice unit” polymer, which is approximately 100nm extended length. The polymer somewhat likes the solvent because its χ (χA→Solvent) is 0.25 and it very much likes the particle because the χ_S parameter which describes the relative attraction of the polymer and the solvent for the particle is (for obscure historical reasons) -1.5 compared to a neutral value of 0 (for those unfamiliar with χ parameters, they are discussed below). We have a volume fraction φ of particle and polymer of 0.1 each.

So, what do the calculations tell us?

The top curve is φ All, the volume fraction of polymer in any form at a number of lattice units Z. At the surface of the particle it’s a large value, 0.8 then, as we can see, this quickly falls to ~0.06 (in fact, the value is in the φ Bulk box, but we can read it with the mouse over the graph). The φ T+L is the total of trains and loops and by 7 units this is down to a small value. Indeed, the radius of gyration of the polymer, R_g is shown as 7.6 units or ~2nm. So, most of the “100nm” polymer chain is within 2nm of the surface. The φ Tails rises to a small maximum at ~6 units and approaches zero near 20 units (this view is limited to 8 units, but that is adjustable). In other words, tails do not play much of a role. By the time we are at 20 units we are in the bulk concentration of polymer, φ Free.

Just pause for a moment. By loading a simple app and sliding a few sliders, you now know how a typical polymer behaves around a particle. Because the calculations have been done by SFBox (not by me!) and because SF theory is so well-established, you can be confident that the results are meaningful.


It is just as easy to explore what happens with a di-block (Figure above), a comb and a brush (not shown). For the di-block the A polymer is mostly within the first 4 units, with the B extending out beyond 8 units, even though it is only have the size of A. It is extraordinary that an app that can run on a phone can tell you so much about a di-block protected particle.

Identifying (In)Compatibility Issues in the App

In real life, it is unlikely that particle will have such an absolute preference for the polymer, especially because the polymer and solvent are fairly compatible. So, reduce χ_S and see what happens. Or help the distinction by making the solvent less compatible with the polymer. When you do the latter, things seem to get much better – less residual polymer in the solvent and a bigger build-up of polymer on the particle. From basic DLVO theory and from advanced SF calculations discussed below, we know that this is heading into troubled territory.

So now try some di-blocks and combs. Try varying the 5 (!) χ parameters that control the interactions – solvent to polymers A & B then the particle to the two polymers and finally the polymer-polymer interaction χ_A-B. It is far better to explore the issues of (in)compatibility in an app than through a bunch of failed formulations where most of the time we could only rationalize the results through hand waving. Then try with two separate polymers, the sort of situation that’s common in real-world formulations. You very quickly build up a powerful sense for what might happen when you formulate for real.

The χ Values – Measure of Stability Based on HSP Values

It is only science-based formulation if you can provide plausible inputs to the model.

So how do we get up to 5 χ values? First, what is a χ value?

χ value is a measure of similarity between, say, polymer and solvent where by convention 0 implies perfect compatibility and 0.5 is borderline (theta solvent). It is well known that χ values are related to the measure of similarity based on HSP values, the HSP Distance.

• For polymers and solvents, a Distance of 0 is perfect compatibility, equal to χ=0
• For an idealized polymer, a Distance of 8 is borderline solubility, equivalent to χ=0.5

So, we have 2 of our 5 parameters if we know (from datasets or manufacturers) or have measured the HSP of both A and B polymers. We can estimate χ_A-B from the distance between the two polymers. This leaves us with the χ_S values. At present I propose that we calculate χ values between particle and polymer (P-P) and particle and solvent (P-S) using the conventional formula. Then we can calculate χ_S = 3(χ_P-P- χ_P-S) because this naturally gives the -1.5 for when P-P and P-S are very different and 0 when they are the same. There are some subtleties to these calculations, and it will take time and effort to refine them to sufficient accuracy, but right from the start we have a rational, science-based approach to complex particle/polymer/solvent formulations.

This is one of the many reasons that SpecialChem has placed 10s of 1000s of HSP values on its site so that formulators can better understand compatibilities and incompatibilities based on Distance and χ.

Particle-particle Stability – Full Power of SFBox-FE

Amazingly, SFBox can calculate the system energy as the particle-particle distance decreases from far apart (defined as 0) to close together.

• If the energy goes negative, then we have an unstable system.
• If there is a large barrier then the system is stable.

Reluctantly, I decided that these calculations were not suitable for the app and those who want to carry them out will find my second implementation, a full-power SFBox-FE which is part of the HSPiP software package. I don’t like writing articles which imply “Go and buy my software”, but there are limits to apps, and in any case the large HSPiP user community always get free updates so this capability is now available at no charge to that community – all they have to do is download the new version.

I will include one pair of screenshots from that version because even I was surprised by it.


The setup is a Polymer A that likes the particle and B that does not, though they equally like the solvent so they have some similarities. This is not too far from real life. The graph on the left shows the free energy, ΔG in thermal units, kT. As the particles come to within 6 lattice units there is a huge barrier, typical of steric stabilization. All very straightforward. But let’s make the stability even greater by making A even bigger – after all, more stabilizing polymer means more stabilization. To my surprise, the system became totally unstable. What’s going on?

This takes us back to the law that for repulsion, tails are good, loops are bad. With the shorter polymer A I had a modest amount of loops but plenty of tail, after all, polymers must end in tails. When I increase the polymer length, the amount of tail cannot change much, yet the amount of loop has increased. With more loops, the interactions tip towards attraction and instability.

Again, the exact details are not important, and the calculations are a simplification. But the phenomenon that it shows is all too typical. With our folk physics of particles and polymers, increasing the polymer length can only be a good thing. So, when things fail, we have no idea how to formulate ourselves out of the problem. But seen through the lens of the real physics of interactions, we can dare to be different.

We can say “OK, let’s try a shorter polymer” something that few formulators would have the courage to try because “more” is always seen as “better”.

The Next Steps – Improve Your Formulation Capabilities

The academic SF community will find this all rather quaint. They have known this stuff for decades, so what is the fuss? After all, they have published papers describing what I’ve described in this article. But we tend not to read academic papers that quote obscure theories for which we have no hope of doing the calculations. I’ve known about it for a few months and I’m still excited about it, though also relieved that the hard work of assembling the HSPiP program and the app are now behind me.

For most readers this is a lot to take in, and many readers will be rightly skeptical that an app describing a theory they have never heard of is just what they need to improve their formulation capabilities in this complex area.

All I can do is encourage the community to give it a go. And do not take my word for it. Reach out to the academic community and ask their views.

Adopt SF Theory via a Science-based App

A surprising number will be familiar with SF theory and will be glad that a couple of user-friendly versions exist because even for academics, running SF theory has been challenging. Maybe they will welcome the chance to discuss the theory now they know that it can be explored hands-on.

And those readers who know me know that I welcome criticism and correction. I’m just a chemist who got lucky and via a friend of a friend got to know Prof. Leermakers, so in no way am I an expert in SF theory. Let’s get things right together; let me know where I’m wrong and I’ll fix things as best I can.

But above all, look around at the state of particle formulation science.

Do you like what you see? I don’t.
Does SF theory look like something that can change things for the better? That’s my view. What’s yours?

Coming soon!

In the next article on SF theory, we will explore how it helps us to understand the micellar behavior of surfactants and the di-block polymers used in many formulations and how di-blocks can be increase the solubility of hard-to-dissolve materials.


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