Hansen Solubility Parameters in Practice (HSPiP) e-Book Contents
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Chapter 1               The Minimum Possible Theory (Simple Introduction)

Although we want HSP to be practical, we don’t want you to think that they are magic or “just a bunch of correlations”. At the same time, we don’t want to bog you down with unnecessary theory. So here is the minimum possible theory necessary for a practical user of HSP.

Kinetics versus Thermodynamics

Thermodynamics tells you if something is possible nor not. You can dissolve sodium chloride in water because solvated sodium and chloride ions are thermodynamically more stable (energy and entropy) than crystalline sodium chloride. Barium sulphate crystals are thermodynamically more stable than solvated barium and sulphate ions, so barium sulphate is essentially insoluble.

Kinetics tells you how fast something will happen if it is thermodynamically possible. So kinetics have nothing much useful to say about dissolving barium sulphate. But it’s entirely possible to have lots of salt and water in close proximity without much of the salt dissolving if you don’t get the kinetics right. One large lump of salt sitting in some very cold water will dissolve far less quickly than a well-stirred fine salt powder in warm water.

Thermodynamics and kinetics are both powerful. But ultimately it’s thermodynamics which is the more powerful. Kinetics might suggest that you should try harder to dissolve the barium sulphate, but thermodynamics tells you that you shouldn’t bother. The observation of a slow-dissolving lump of salt might suggest to you that it’s going to be impossible, but thermodynamics encourages you to try.

So let’s make it clear. The strength of HSP is that they are based on thermodynamics. They are all about whether something is fundamentally possible or not. We won’t hide from you the fact that kinetics can sometimes wreck even the best thermodynamic predictions of HSP. But the fact that HSP are essentially a way for you to reach profound thermodynamic conclusions is their prime strength.

It will become tedious to insert “thermodynamically” into every sentence which says “HSP show that thermodynamically A will dissolve in B”, so let’s take it that we now understand the difference between kinetics and thermodynamics.

Note to the sceptics: HSP really do come from deep thermodynamic insights. The fact that most HSP have been determined by correlation experiments reflects a limitation on our ability to do complex thermodynamic calculations rather than a limitation of HSP themselves. The recent work of Panayiotou has at last accurately derived HSP from first principles – with remarkable agreement with the experimentally derived values. Similarly the molecular dynamics work of Goddard’s group at CalTech has produced accurate numbers, showing that it is possible for anyone to obtain HSP from first principles.

Doing it the hard way

If you want to dissolve something in something else then you have to compare two energy losses with one energy gain. The first loss is the mutual interaction of the solvent with itself. You are effectively making a hole in the solvent and that takes energy. The more the solvent attracts itself, the more energy it takes. The second loss is from the mutual interaction of the solute with itself – for the same reason. And the gain is the interaction of the solvent with the solute. If this interaction is greater than the sum of the losses, then the solute will dissolve.

So if you want to know if A dissolves in B, “all” you have to do is to calculate the two losses and the one gain. For simple systems this can be done, but it becomes impossibly hard for more complex systems. And when you start trying to work out the best mixture of C, D and E in which to dissolve A it’s even more impossible.

The glory of HSP is that in 3 numbers, all those fussy thermodynamic calculations are done for you, with a high degree of accuracy.

1, 2, 3 (or more?) energies

If we are going to short-cut the hard way, we need to have numbers that characterise the internal energies (the energy required to create the hole in the solvent and break up the solute) and also the interactive energy.

You could imagine that if the chemicals were all of one general type then one energy value could be sufficient to enable the calculations. Hildebrand famously tried to do everything via just one energy, but although that one energy is fundamental, without partitioning it, its predictive value proved to be limited. Indeed we are astonished that Hildebrand parameters still continue to be used. There are many knock-out arguments against using Hildebrand (see the Hansen-Solubility website for a more detailed review) but one simple example says it all. Epoxies aren’t generally soluble in nitromethane or in butanol which, as it happens, have the same Hildebrand parameters. But a 50:50 mix of these two solvents is a good solvent for epoxies. As we will shortly see, this is easily explained by Hansen parameters and is inexplicable with Hildebrand.

As practical scientists we know that there are at least 4 fairly distinctive forms of energy:

• Dispersion forces (atomic). These are the general van der Waals interactions between just about everything. Put any molecule a few Angstrom from another molecule and you get a powerful attractive force between the atoms of the two molecules. Because they are everywhere, and because they are unglamorous we tend to ignore them, but they are the dominating force in most interactions! The famous gecko effect that allows a gecko to walk upside down on a ceiling is due almost entirely to the amazing strength of dispersion forces.
• Polar forces (molecular). These are the familiar “positive attracts negative” electrical attractions arising from dipole moments. They are important in just about every molecule except some hydrocarbons and special chemicals consisting of only carbon and fluorine.
• Hydrogen bond forces (molecular) are arguably a type of polar force. But their predictive value in many different aspects of science goes beyond simply thinking of them as polar forces so it seems worthwhile to make them distinct. More generally they can be considered as a form of electron exchange so that CO₂ shows strong “hydrogen bonding” forces that make it a good solvent for e.g. caffeine even though it contains no hydrogen atoms.
• Ionic forces. These are what keep inorganic crystals together.

If you are going to describe molecular interactions in simple numbers it’s clear that you would need at least 2 for every molecule: Dispersion and Polar. By including the third parameter, Hydrogen bonding, everything except strong ionic interactions became thermodynamically predictable. It turns out that even for organic salts the polar and hydrogen bonding contributions are sufficient. And as ionic interactions are mostly the domain of aqueous environments dominated by the extraordinary properties of water, it doesn’t seem to be useful to include a 4^th descriptive parameter when you are trying to understand interactions that don’t involve large amounts of water. There is a lot of progress being made, but the division of energy types in the aqueous domain is still not fully understood.

So it seems reasonable that three parameters could be used to describe solvent/solute interactions. But why should something as simple as 3 numbers be sufficient to describe a process which, by our own admission, is far too complex for the best computers to calculate?

Do 3 numbers give accurate predictions?

Yes. The data is overwhelming. We’ll come back to that in a moment.

Aren’t 4 numbers even better?

Yes, and no. In principle, dividing the Hydrogen Bonding parameter into Donar/Acceptor terms (as, for example, in MOSCED) should give even better results. But the practical problems of creating a large, self-consistent database with 4 parameters, and of visualising issues in 4D space mean that this has not proven to be a popular way forward.

Why (in principle) does it work?

The strength of HSP is that they are based on thermodynamics. And the key insight that led to the creation of thermodynamics is that the law of large numbers lets you calculate things that can’t be done by attending to individual details. It’s hard to calculate the force on the wall of a container containing 1 trillion gas molecules if you try to consider what’s happening to each of the trillion molecules, yet it’s easy, and accurate, to calculate via simple thermodynamic gas laws.

The same applies to HSP. The dispersion, polar and hydrogen bonding forces are impossibly hard to calculate via the interactions of trillions of individual molecules, yet are easily encoded in the HSP numbers.

We have to stress again, that if you can do the calculations (and it is becoming increasingly routine to do them), then the calculated results confirm the numbers you find listed in the tables of HSP.

Do 3 numbers give accurate predictions?

Let’s think of the most basic thermodynamic situation. We are trying to mix solvent A with solute B. The claim is that you will have to lose and gain energies. How can we calculate those?

A naïve approach would be to calculate the sum of the (absolute) differences of the three HSP. By definition, if B is so close to A that it’s the same molecule then these differences will be zero. So the definition of a perfect solvent is a difference of 0. If A and B are chemically fairly similar then you would expect their HSP to be similar, and the differences to be small. And if they are utterly different, the difference should be large.

So we might try:

Equ. 1‑1 Difference = [Dispersion_A-Dispersion_B] + [Polar_A-Polar_B] + [Hydrogen Bonding_A – Hydrogen bonding_B]

where the [square brackets] imply the absolute value.

As it happens, you can’t add and subtract energies quite like this. If we introduce δD, δP and δH for Dispersion, Polar and Hydrogen bonding parameters then the true difference is:

Equ. 1‑2 Difference² =4 (δD_A-δD_B)² + (δP_A-δP_B)² + (δH_A-δH_B)²

The squared terms mean that we don’t have to worry about absolute values as (δD_A-δD_B)² is the same as (δD_B-δD_A)². The units of these solubility parameters are (Joules/cm³)^½ or, equivalently, MPa^½. In older papers you will see the units expressed as (cal/cm³) ^½. If you ever need to convert between old units, simply multiply by a factor of 2 (or 2.046 if you want to be precise). Throughout this book and in the software, all quoted values are in (Joules/cm³)^½ or, if you prefer, MPa^½. As Molar Volume (MVol) is commonly used throughout the book it’s worth stating here that its units are cm³/mole. Note, too, that all quoted values are at the standard temperature of 25ºC.

You’ll have noticed that the famous factor of four in front of the δD term has crept into the formula. Many have questioned the justification for this factor. In the Handbook (pp30-31), Hansen provides some interesting possibilities based on Prigogine’s Corresponding States Theory. At the heart of the issue is whether the “geometric mean” is the best way to calculate the differential heat of mixing between components. For non-polar spherical molecules interacting via Lennard-Jones potential there’s a good case that this is a good approximation. But there is no reason to believe that the same should apply to polar and hydrogen-bonding interactions. Furthermore, there is universal agreement amongst diverse luminaries such as Prausnitz, Good, Beerbower and Gardon that the differential heat of mixing term should be less for polar and hydrogen bonding than for dispersive forces. How much less is a matter of debate, but values between 1/8 and 1/2 have received support in a wide range of experiments, with the value of 1/4 providing the best data fit for Hansens’s polymer/solvent data. So although we regret that we, like everyone else, cannot provide a compelling argument that the factor should be precisely four, we are confident that it should be at least a factor of 2. Because the factor of 4 gives spherical plots, fits well with the largest range of practical test correlations and has stood the test of time in such a wide variety of real-world uses we feel that its continued use is more than justified.

This famous difference equation is the core of HSP. For any problem you just calculate the difference. If it’s small then the thermodynamic chances are high that the two components will be mutually soluble (or compatible or, well, “happy” together if you have some interaction such as pigment dispersion where you know what “happy” means, even if it can’t be defined precisely). If the distance is large then the chances are small.

For those who are interested in the theory, for most cases, a distance greater than 0 means that mixing is enthalpically unfavourable. But of course mixing tends to increase entropy so the total is energetically favourable. The smaller the distance, the less you have to rely on entropy to help you. Large polymers have less entropy gain when they are dissolved so you need a smaller distance from the polymers’ HSP in order to dissolve them.

That would be fine, but rather limiting. The true power of HSP is that because they are based on the thermodynamic law of large numbers, a “solvent” can be a mixture of an arbitrary number of components and the “solvent’s” HSP are simply the average (weighted for % contribution) of the individual components.

Here are two examples to show the principle. In both cases there happens to be a 50:50 mixture (so you can check the answer by inspection). And in both cases you obtain effectively the same solvent, even though they are created from very different starting solvents.

	δD	δP	δH	%
Solvent X	16	8	2	50
Solvent Y	18	10	4	50
Mixture	17	9	3	100

 

	δD	δP	δH	%
Solvent X	14	0	0	50
Solvent Y	21	18	6	50
Mixture	17	9	3	100

Table 1‑1 Creating the same solvent properties from very different solvent blends

This is the real power of HSP. A striking example is when you want the HSP of a particular solvent but can’t use that solvent because it is toxic or too expensive. Simply mix together two (possibly widely different) safe/cheap solvents in proportions that give you the correct HSP and you have a fully functional solvent, indistinguishable (as far as the solute is concerned) from the original solvent.

Note that in this example we use the simple “linear mixing” rule that has been successfully applied to HSP for more than 40 years. In the 3^rd Edition a “squared mixing” option is available, as discussed in the chapter on optimisation.

As we will often be referring to sets of δD, δP, δH numbers always in that order we introduce the convention that [17, 9, 3] means “δD=17, δP=9 and δH=3”. This shorthand makes it easy for us to say that a 50:50 mix of [16, 8, 2] and [18, 10, 4] gives [17, 9, 3]. When, later on, we introduce a Radius, this will be the fourth element so [17, 9, 3, 8] means [17, 9, 3] with a Radius of 8.

Let’s go back to the question: Do 3 numbers give accurate predictions? The answer is overwhelming. Not only is it the case that the 3 numbers do work, but in fact they must work. This is thermodynamics and one of the great rules of life is never to argue with the laws of thermodynamics.

Do they work all the time? They can’t work outside their own thermodynamic area. So they cannot work for ionic solids and they are not much of a guide for anything to do with primarily aqueous solutions (though pioneers are doing good work in this area). And of course there will be times when the HSP will say that a given solvent blend will dissolve a certain polymer but experiments show that it merely swells. This is because its excessively high molecular weight means that it will take far too long to dissolve it and so only swells it. That’s the limit of kinetics versus thermodynamics.

Even here, the HSP can be deeply insightful. If the HSP for the polymer has been determined using a low molecular weight, then it is entropically probable that solvents which were just good enough for the low molecular weight version will be inadequate for the high molecular weight. The need for a solvent closer to the HSP of the polymer, or for a solvent with a lower molar volume, is therefore predictable.

For some reason, HSP have irritated many people over the years. There have been many attempts to overturn them, but there is an overwhelming amount of theory and practical success in support. One classic criticism was around “negative heats of mixing”. Both HSP and the earlier Hildebrand parameter seemingly allowed “positive” heats of mixing only. This situation was cleared up by some skilful thermodynamic calculations on solubility parameters which showed that both positive and negative heats of mixing were not only allowed, but were also required. Experiments by Patterson and Delmas (see, for example, Patterson D., Delmas G., New Aspects of Polymer Solution Thermodynamics, Off. Dig. Fed. Soc. Paint Technol., 34,677,1962) confirmed these calculations.

So for all practical purposes you can take it that HSP do work and must work.

So all you have to do is to get to grips with the Sphere in the next chapter.

Deeper theory

The Sticking, Flowing, Dissolving chapter contains some deeper theory about aspects of polymer/solvent solubility. Once you’ve become comfortable with HSP and HSPiP at this basic level of theory, you might want to dip into sections of that chapter to find out more.

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