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Colloids Interfaces 2019, 3(1), 23; https://doi.org/10.3390/colloids3010023

Article
Practical Formulation Science for Particle-Based Inks
1
Steven Abbott TCNF Ltd., Ipswich IP1 3SZ, UK
2
School of Mechanical Engineering, University of Leeds, Leeds LS6 1AN, UK
Received: 4 November 2018 / Accepted: 29 January 2019 / Published: 1 February 2019

Abstract

:
There is a big gap between idealized colloid science and the needs of a printing ink formulator. This often leads to formulations based on intuition and experience rather than good science. By bringing together the most relevant colloid and interface science with the capabilities of modern apps, it is possible to bridge the gap between theory and reality to the benefit of both the colloid science community and those who need the science to improve their formulations. The process of making current science usable also exposes the limitations of available theories. This suggests a methodology for highlighting the grand challenges to the colloid science research community, including the challenge of making any new science more usable.
Keywords:
particle formulation; inks; rheology; printing; apps

1. Introduction

A harassed formulator of a particle-based ink might wish to consult the standard literature on colloids and interfaces in order to seek guidance. Any such text would paint a picture of idealized particles at idealized interfaces. In terms of particles, there would be much discussion of van der Waals and London forces between particles, and tables of Hamaker constants that are unnervingly similar across a large number of particles, which the formulator knows to behave very differently. There will be discussions, with a few graphs, of DLVO (Derjaguin & Landau, Verwey & Overbeek) theory [1,2] and much emphasis on zeta potentials, though these turn out to be largely irrelevant in all solvents except water, which has a high dielectric constant. More advanced DLVO-style treatments [3] will include steric stabilization via polymer chains. Advice will follow that too little polymer is bad but, unfortunately, too much will also be bad thanks to depletion flocculation. Surfactants, similarly, will be presented as providing conflicting capabilities with both too little and too much being a bad thing.
In terms of interfaces, the focus will be on surface energies and surface tensions. The smart formulator might be puzzled by this because most of the surfaces encountered have very similar surface energies, within a factor of two, yet orders of magnitude difference in performance. Furthermore, although different solvents behave dramatically differently in formulations, their surface tensions vary, similarly, by only a factor of two, once water is excluded. The formulator may also note that viscosities and speeds/timescales of processes vary by many orders of magnitude, making surface energy an unlikely focus of interest. In particular, printing (excluding the relatively gentle inkjet process) involves rather dramatic processes of film splitting where surface tension forces are overwhelmed [4].
Attempts to gain insights via rheology into the key steps of printing, such as splitting or spreading of inks, will also prove discouraging. There is a confusing and disjointed array of rheological tests, many of which have little obvious purchase on the key processes in printing, which cover a wider range of stresses, strains, and strain-rates. Often it seems that measurements are made because they are possible rather than because there is a clear logical link between the measurement and a key part of a printing process [4].
In the standard texts, a number of formulae, of greater or lesser complexity, are presented with precious little guidance as to what to do with them and, at best, a few graphs illustrating one or two examples.
The net result is that most formulators shrug their shoulders and carry on formulating via instinct and experience. Although there are multiple problems with this approach, the biggest is that formulators tend to remain trapped in non-optimal zones because attempts to break out of them (unless backed up by some smart, hypothesis-driven statistical experimental design) take the formulations into unstable domains, with no signals that better domains lie further away.
To put things succinctly, the academic colloids and interfaces world, despite protestations that the work is important for many aspects of real-world formulations, has produced surprisingly few tools usable by those the work is supposed to support. This is both unfortunate and unnecessary.
The aim of this article is to indicate how this community can better foster a culture of science-based formulation by providing usable scientific tools, based on best-available science, that address the common problems faced by printers and ink formulators.
The role of blue-sky research is not the issue here. Most useful theories started off as blue-sky ideas. The issue is the majority of work which is claimed to be of practical relevance but which, all too often, adds no usable value to the community.
Most of the science discussed here is “well known,” in that the science has been around a long time and is encoded in a few key papers. The next section claims that the relatively few useful papers are swamped by a large number of unusable ones. Those who think that this pessimistic assessment is wrong may wish to recall the famous Lancet review [5] stating that 85% of investment in biomedical research is wasted or the Ionnidis paper [6] entitled Why most published research findings are false. If, generously, we say that of the 85% wasted, 45% of this is due to the fact that science is hard and unpredictable, this still leaves 40% which is simply poorly conceived or executed science.

2. Appification

The overwhelming number of papers in the fields that interest the present author are of zero use to the formulation community, even though most claim, at the start, to be concerned with an issue of practical importance. Many papers are “phenomenological” in that they report observed effects, without providing a meaningful theoretical framework or a refutable hypothesis, so outside their specific system there is no generalized learning that can add to a corpus of principles. A paper that tells us that “viscosity” (however they choose to measure it) and print speed are key parameters in controlling some aspect of print quality is both stating the obvious and telling us nothing usable. There is not a single viscosity relevant to the process, and both rheology and speed impact multiple aspects of the process. Many other papers contain fascinating theoretical analyses that have no possibility of being linked to a plausible set of formulation conditions. A rare few contain actionable science, with inputs and outputs that a formulator could, in principle, use. However, most formulators lack the time and ability to, for example, solve a differential equation via a fourth-order Runge–Kutta algorithm. Given that everyone has access to a powerful browser, and given that the modern HTML5-CSS3-Javascript infrastructure contains a vast array of open-source tools (mathematics libraries, graphing, data handling), those who do the good science should finish off the job by “appifying” their research, i.e., providing an on-line version accessible, as much as possible, via phones, tablets, and laptops. For models too complex to run on low-powered machines, it is still possible to appify models via systems such as Code Ocean [7,8], which run models in the user’s chosen language on high-performance platforms, while allowing users to view the graphs, change inputs, etc. A straightforward Javascript app for a theory with an analytical formula and a bunch of inputs, outputs, and graphs can be written and debugged in a few hours. A more sophisticated numerical simulation might require a couple of days if it already exists in the researcher’s lab in the form of code such as MATLAB or can be uploaded to a system like Code Ocean with little extra effort.
In the current general absence of a research and publication culture that promotes appification, there seem to be rather few examples of bringing science to life in this way [7,8]. Therefore, the present author has chosen to appify a wide range (≈300) of theories that formulators can apply to their every-day concerns, and which the author himself uses for troubleshooting and education. The majority work equally well on a phone as on a laptop, so usable knowledge is always available. The remainder of this article shows how the combination of relevant theory with usable apps can create a science-based formulation ethos. The theories are not the author’s and no claim is made that the apps themselves are optimal. Because the apps are all open source and the code freely downloadable, the author would positively welcome readers to take the code and improve upon it. Even better would be if readers became inspired (perhaps by funding bodies insisting on this as part of research impact) to appify their own excellent science for the formulation community. A rough-and-ready criterion for a good paper of science intended for the formulation community should be: “Is it appable?” A discussion of this question is provided in the conclusion.

3. Examples

The following examples (taken from ≈300 apps in areas of interest to the present author) cover a range of typical colloid- and surface-related topics, chosen to illustrate the many opportunities within the community to bring good science to life and to serve those who, in funding applications, are claimed to be the potential beneficiaries of the work.

3.1. DLVO

The general idea behind DLVO [1,2] is simple enough: all particles are attracted by van der Waals forces that are generally very similar between particles in general formulations, so there is nothing we can do about it other than provide some repulsions that are either charge-based in water or steric-based in water or other solvents. The problem is that the equations behind each effect involve multiple parameters, most of which are unknown (or even unknowable) to the formulator. Therefore, the pure theory is unusable. Yet with an app (Figure 1) that handles most of the difficulties with adequate approximations, the key issues for the formulator become clear:
As long as the formulator knows their salt concentration (and type), their zeta potential, their particle size, and knows that the dielectric constant (ε) of water is ≈80, then largely irrespective of the (generally unknown) Hamaker constant A12, they can find whether their system is likely to be stable. For those puzzled by the zeta potential, a convenient app, www.stevenabbott.co.uk/practical-solubility/zeta.php (not shown), demonstrates why it is such an tricky concept (the app illustrates how the “true” charge on the particle interacts with the ionic environment and falls away to a dividing line between “particle + ion cloud” and ionic solution), and why minor changes to a formulation can produce major changes to the measured potential. The practical advice is: “never measure a single zeta potential—make some judicious formulation changes to see if the potential is in a stable zone or near a knife edge”.
In the DLVO app, if users slide the value of ε to that of almost any other solvent, they will quickly find (Figure 2) that charge stabilization is not a viable possibility for anything other than water.
If charge stabilization is impossible then the only choice is to use steric stabilization, which appears very easy, until the Flory–Huggins χ parameter exceeds 0.5, at which point the system becomes unstable. The colloid science explanation is that at 0.5, the stabilizing polymer chain has reached the theta solvent point, and beyond that, it prefers to self-associate rather than provide a barrier. Without a way of knowing what χ might be, such an explanation is unhelpful.
A practical way forward (discussed in more detail later) is to say that the particle-plus-stabilizer has a set of Hansen solubility parameters (HSP) [9] or, as they are increasingly known, Hansen similarity parameters. These can be measured by assessing the particle stability in a modest range of test solvents and determining a sphere that encompasses all the good solvents and excludes the bad ones. The center is the HSP and the radius in terms of HSP distance discussed later, defines when χ = 0.5. This equivalence allows us to convert from χ-based theories where, in general, we have no idea what the χ values are, to an HSP-based theory where estimation of χ is easy. Crucially, the HSP of a solvent blend is the volume-weighted average of the values of the different solvents. This means that when (as frequently happens) a formulator has to add some co-solvents for other reasons, their effect on the overall HSP can be calculated and the risk of approaching the χ = 0.5 zone assessed.
The HSP-based approach, like DLVO itself, is not precise. However, it has routinely been shown to be an adequate guide. The alternative is trial and error or a 3-year PhD to more accurately measure the relevant DLVO parameters. In general, both alternatives are far worse than the good-enough HSP approach. The idea that particles might be helpfully discussed in solubility terms is the topic of a later section.
Discussions of depletion flocculation tend to invoke Asakura and Oosawa [10]. An app implementing their formula (https://www.stevenabbott.co.uk/practical-solubility/depletion.php, not shown) is of modest scientific interest but it is hard to see how anyone can apply it to a real formulation. This means that after decades of experiments and theoretical refinements, formulators are basically on their own when it comes to depletion. If an app is of little value because it is badly implemented or is using the wrong theory, the problem is readily rectified via a better implementation. If, as seems to be the case here, we still have no really usable formulation approach, then this is a problem and an opportunity. The opportunity opened up by a re-imagination of an old (1989) theory of depletion flocculation is discussed in the larger section on HSP.

3.2. Coating Defects

An overemphasis on contact angle measurements in terms of wetting has addressed a problem that is of relatively little concern because most formulators can produce coatings that wet adequately. This has taken attention away from topics that are far more relevant to high-quality coatings and where apps provide much-needed insight.

3.2.1. Pinholes

If a coating already contains a small defect, e.g., from an air bubble, the defect might self-heal or it might expand further to create a highly-visible pinhole defect. Of course, the contact angle θ affects this, but it is little known that the ratio of coating thickness, h, to defect diameter, d, is also vital. In simple terms, as shown by Sharma and Ruckenstein [11], if h/d < 2(1 – cosθ) then the defect will tend to form a large pinhole.
It might be the case, as in Figure 3, that a coating that is pinhole-free with a certain level of process defects will become full of pinholes not because surface energy has changed but simply because someone asked for the coating to be a little thinner.
A more complex app (https://www.stevenabbott.co.uk/practical-coatings/Dewetting.php, not shown) based on recent work [12] seems to indicate that most features printed with contact angles greater than, say, 20° would tend to dewet with astonishing speed. A key feature of printing (again, excluding inkjet) is that irrespective of contact angle, the substrate in the printed areas is fully wetted by the printed plate system, so what happens during and after the separation process (i.e., with potentially receding contact angles) is far more important than processes tested by gently applying drops to a free surface. The fact that in general we do not see such dewetting tells us something very interesting, perhaps that receding contact angle effects are dominant. If this is the case, then decades of work on advancing contact angles would be largely irrelevant and the focus should have been on the more difficult but perhaps more important receding angle.

3.2.2. Levelling

Anyone who looks at what comes out of a printing/coating head might see a range of defects and wonder why some of them disappear in the short time before drying/curing, and others remain highly visible. Additives are often included “to change the surface tension” in the hope that the defects will go away. Again, it seems to be little-known that levelling theory (usually attributed to Orchard [13]) shows that such additives are largely irrelevant (and work in the opposite direction to what is commonly supposed), and that other factors are far more important. If, as in Figure 4, we can assign a wavelength λ to defects (the classic example is the spacing of paint brush marks) in a coating of thickness h, levelling theory tells us that the time to level depends on λ4 and 1/h3 with only a linear dependence on viscosity η and an inverse linear dependence on surface tension σ (contrary to intuition).
The change of emphasis from viscosity and surface tension to parameters that are far more important, and generally overlooked, shifts the efforts to the factors that have the largest chance of solving the problem.

3.2.3. Drop Spreading

Although classic Tanner theory [14] of drop spreading (the velocity of spreading, v, depends on contact angle θ, surface tension σ, and viscosity η via v = cosθ3.σ/η) can be applied directly to the spread of an inkjet drop, a more sophisticated approach is required to explore the spread of, say, a screen-printed line. Building on the work of McHale [15], it was possible to create an app that explains much that is confusing to screen printers. A typical screen ink is highly viscous and examination of a printed line after printing but before drying shows no obvious sign of line spread, yet the measured line width is often much larger than expected.
What the app in Figure 5 shows is that even for a viscous ink, the initial rate of spreading is very high because it depends on the cube of the contact angle. As the line spreads, the contact angle decreases such that the rate of increase falls off rapidly. As can be found in the app, the only way to stop drop spreading is to arrange for a relatively high equilibrium contact angle. In the example, if that angle is 40° then the drop spreads only to 70 μm. Therefore, just as has been found with inkjet, the contact angle challenge for the formulator is not the classic one of getting “wetting” but of getting a high-enough angle to stop drop spreading. Because there is a totally erroneous belief that adhesion depends strongly on contact angle (see the Practical Adhesion website [16] for a long explanation of why this belief is wrong), formulators find it difficult to know how to restrict drop spreading while obtaining good adhesion. Using the correct approach to adhesion science, there is no inherent contradiction, so formulation is more straightforward.

3.2.4. Marangoni Defects

A common type of coating defect can be described as mottle or orange peel and various causes and cures can be suggested for such vague symptoms. One probable cause for such defects (which, at an extreme, show as hexagonal patterns) is the Marangoni effect (terminology is vague, and such defects can be named, with various degrees of precision, as Bénard cells, or Gibbs or Rayleigh instabilities). Again, the standard approach for those who are unaware of Marangoni is to worry about “surface tension” and, for aqueous coatings, to add surfactants to fix the surface tension. With the help of an app, it becomes clear that “surface tension” is not a cause; rather it is “change of surface tension with temperature” (δσ/δT), which is altogether different and can be caused by thermal gradients and/or concentration gradients caused by the wrong mixture of solvents or a bad choice of surfactant. The key, in a coating of thickness h, viscosity η, thermal diffusivity κ, with a temperature difference across the coating of ΔT, is the Marangoni number, Ma = −δσ/δT.h.ΔT/(η.κ). If this exceeds a critical value, generally assumed to be ≈80 [17], then instabilities are likely.
Like so much of practical interfacial formulation science, it is unlikely that the formulator will know many of the key values such as the rate of change of surface tension with change in concentration, δσ/δC. However, the app (Figure 6) can show that for a borderline problem, a higher viscosity or thinner coating might be appropriate. For the concentration-driven process, there are two formulation strategies, depending on whether the system is based on a single solvent or a solvent blend. For the single solvent, given that the polymer/particle system is likely to be of higher surface energy, increasing the surface tension of the solvent will reduce δσ/δC. For the mixed-solvent system, if the more volatile solvent is of a higher surface tension, then δσ/δC will be negative, making the system self-correcting. For a mixed-solvent system, this means that the faster-evaporating solvent should be of a higher surface tension. An alternative strategy (adopted under the erroneous assumption that things get better by “reducing surface tension”) is to swamp the surface with a surfactant so that δσ/δC stays at zero.

3.3. Rheology

Rheology should be at the heart of much of coating and printing science. Sadly, it is often replaced by a single “viscosity” measurement in a flow cup. A set of app images generated from the author’s Practical Rheology website help to explain why many formulators find rheology confused and confusing, discouraging them from trying to use it. Rather than burden this article with the many, rather complex formulae used within the various rheology apps, interested readers will find the relevant formulae and/or academic references. For example, the app images in Figure 7 are each different views of the same data calculated from the well-known Cross model:
The first set in Figure 7 shows, at the top left, the familiar viscosity versus log(shear rate) graph that most formulators know about. The other three are plots that rheologists might choose to show and which mean nothing to most formulators. Each can be invoked from the app by selecting an option and show (top to bottom) log and linear shear rates and (left to right) viscosity and shear stress. The apparently arbitrary choices of X versus Y plots made by rheologists add serious confusion and frustration to a difficult field.
The next example in Figure 8 shows that although formulators might be encouraged to measure yield stress for those situations where it is important to initiate or to inhibit flow, there are at least six methods for measuring a value [18], though the different measurements might produce different values. A formulator who might wish to better understand yield stress will rapidly get discouraged when discussions of similar yield phenomena are based on different measurement techniques with no acknowledgement of the existence of, or reasons for rejecting, other techniques.
The third example in Figure 9 shows a profound truth about rheology: that a measurement of one effect, such as creep, provides, in principle, measurements of five other effects. The six effects are: relaxation modulus, G’ & G’’, relaxation spectrum, creep compliance, J’ & J’’, and the retardation spectrum. A formulator could learn a great deal about their material if they could see it in all six different ways and different aspects of the printing process can be best described via different viewpoints. Classically, G’ & G’’ are used to describe an ink. However, the idea of relaxation times is especially valuable for printing processes that cover a very large range of timescales, so in principle we should all be familiar with the relaxation spectrum of our formulations. If we can get the whole picture from a single set of measurements or, more likely, can construct a whole picture by piecing together measurements that work comfortably only in a narrow area, then we can far better understand our process.
Yet this profound truth about rheology is largely absent from the formulation world where measuring shear-dependent viscosities and, perhaps, G’ & G’’ over the range of a few Hz is seen as normal. It might be, for example, that a relatively easy creep method could supply vital relaxation data, but formulators do not, in general, know that this could be a viable option. There is no shortage of academic papers on the difficult problem of interconversion, and suppliers of rheometers have found methodologies that work for them. To create an app that would help formulators grasp the possibilities was a challenge because academic rheologists do not generally regard the ability to implement their published methodologies to be of much importance, and rheometer suppliers like to keep their methods as proprietary. The app uses basic techniques from Ferry [19] and from Tschoegl [20], combined with a relatively comprehensible methodology of Park and Schapery [21]. By adding datasets from Ferry’s comprehensive guide plus some examples from the Park paper, it is possible to see a wide range of phenomena. The user can also provide their own dataset of relaxation moduli from which the other five effects are calculated. A more powerful and general app is beyond the capabilities of the present author. Arguably, bringing this important science to the user community in a more general and usable manner should be the task of those who receive public funds to do the research. This idea is discussed further in the Conclusion.
A further, related truth about rheology was a deep shock to this author. It is well known that we can measure effects via rotational, oscillatory, and extensional rheology, with each having distinctive capabilities of relevance to some parts of printing and coating. The shock is that there is no way to interconvert between them and, even worse, that there are even wild differences between the “same” measurements within the “same” technique. In addition to the six ways of measuring “the” yield stress mentioned earlier, the opening statement of Petrie’s critical discussion on extensional viscometry [22] shows the extent of the problem: “The issue of whether extensional viscosity is a concept that causes more confusion than enlightenment is addressed.” The conclusion is that confusion is more likely than enlightenment. This means that even if a formulator purchased and mastered state-of-the-art rotational, oscillatory, and extensional rheometers, there is little hope of building a grand picture of how to improve some key aspect of a formulation. Faced with this, why not just buy some simple flow cups and use them as a quick-and-easy quality control tool? Again, the Conclusion section of this article has some suggestions for avoiding this negative viewpoint.

3.4. Particles

Rheology is especially useful for the sorts of filled systems that are common in printing and coating, and there are large numbers of papers from rheologists with variations on the theme that viscosity of a dispersion with a carrier fluid of viscosity η0 depends on the volume fraction φ and some critical volume fraction φc according to some power law of something like η0/(1 − φ/φc)n. What is generally lacking is a way for the formulator to explore what this means, both at low shear rates (where the formula impacts on yield stress), and at the high shear rates of the printing processes. Appification is not itself so hard; the problem is to find a set of formulae that can relate to things that formulators might know about their system. It is especially frustrating that the core formulae commonly found are independent of particle size, even though everyone knows that the higher surface area and (for a given mass) greater number of smaller particles can lead to problems. A few key references (see Reference [23] for an excellent guide through the complexities) handle the distinction between “hydrodynamic” and “Brownian” modes (pure flow versus particle–particle interactions), which in turn are size and volume-fraction dependent. As an attempt to broaden appreciation of these size effects, an app based on mode coupling theory is available (https://www.stevenabbott.co.uk/practical-rheology/Particle-Viscosity-MCT.php, not shown), though it merely illustrates the ideas rather than provides a methodology for the formulator to adopt.
The app (not shown) for low shear behavior, www.stevenabbott.co.uk/practical-rheology/Low-Shear-Particles.php, shows the differences near φc between three typical contenders for best theory: Dougherty–Krieger [24]; Yaron, Gal-Or [25]; and Pal [26]. Because of uncertainties about φc, the values of the large viscosities at high volume fractions are unlikely to be accurate. However, less appreciated is that the η0 term means that a trivial change from 1 to 2 cP in the carrier fluid will take a 100 cP dispersion up to 200 cP. A formulator’s intuition says that it would go from 100 to 101 cP. Although the formula is trivial, having an app to show what happens is genuinely useful. Things get more interesting when we look at the high-shear behavior as in Figure 10.
The app (based on References [27,28,29]) shows that things get tricky for a formulator even with an assembly of theories that are as simple as possible whilst still capturing most of the key effects (the absence of a radius effect on the main shear effects is clearly a problem). The theories are not especially difficult, but for most of us, it is hard to grasp their implications. In an ideal world, users would ask the theoreticians for education and the theoreticians would ask the users how they could make the theories more usable. Somehow the mindsets of the interested parties have not come together to make this sensible approach the norm.
As explained on the app page, the fractal dimension input helps to cope with incipient flocculation, i.e., takes into account particle–particle interactions, which are proportionately more significant at smaller particle sizes, though there is no guidance on how to do this. The yield stress part addresses another aspect of smaller particles.
If a real expert created a rheology app that allowed the formulator to better grasp the effects of size and particle–particle interactions, a question would still remain. How does a formulator, in a complex system far removed from the world of DLVO, go about making the formulation more tractable by reducing the tendency for particles to self-associate? One possible answer is to think of the particles in terms of solubility.

Particle Solubility

The idea that particles can be “soluble” rather than “dispersed” strikes most people as bizarre, but the present author knows of no law of physics that provides a demarcation line between “dispersed” and “soluble” and the practical upsides of dealing with solubility rather than dispersibility are so significant that formulators who adopt the idea can make rapid progress both in optimizing a formula within a comfortable domain and in taking a formulation across a plateau of instability into a valley of higher stability. This argument is made at greater length in the present author’s free eBook [30] which is linked to the relevant apps.
The trick is to find a solubility tool that is simple enough to be useable yet complex enough to capture real science. There are many possible approaches based, for example, on hydrogen bonding parameters, donor/acceptor numbers, hydrophilic–lipophilic balances, and dielectric constant matching. Furthermore, there is the original Hildebrand solubility parameter approach. As argued in the same eBook, none of these is adequate for the real world because each emphasizes just one or two parameters while chemistry demands at least three. Therefore, this balance is currently best obtained via Hansen solubility parameters (HSP) that describe solvents, polymers, and particles in terms of three components that make sense to most formulators: δD, the van der Waals term; δP, a polar term; and δH, a hydrogen bonding term. If we know the three values for any pair of materials, e.g., a particle and a solvent, we can calculate how “like” they are from their distance, D, given by D2 = 4(δD1 − δD2)2 + (δP1 − δP2)2 + (δH1 − δH2)2. When D is small then the components are alike and will formulate well together, when D is large the components are not alike and will be difficult to bring together.
Why not four parameters, for example splitting hydrogen bonding into donor and acceptor? Because none of the many such schemes (again see the discussion in the eBook) has so far proved to be viable in practice.
This simple idea has worked well for 50+ years and in its early years was especially popular in the paint industry because Charles Hansen, who developed the theory, happened to work in that industry. As industrialists, they were concerned with what worked, so a debate about whether paint pigment particles were “soluble” was irrelevant given that the idea allowed them to formulate sophisticated paint systems by co-adjusting the HSP of the pigments, dispersants, polymers, and solvent blends. Only recently has the nanoparticle world started to catch up with the idea and use it to formulate effectively [31]. Because the term “solubility” is contentious when applied to particles, it is becoming increasingly popular to think of the S as “similarity”; therefore, Hansen similarity parameters can describe how similar a particle and a solvent might be. How are the HSP of particulate systems measured? In the same way that materials such as polymers are measured: test the compatibility with a range of solvents with known HSP values and construct a sphere containing all the good solvents and excluding all the bad ones [32]. The center of the sphere provides the HSP. For (nano)particles, the tests can be done using sedimentation times, with particles in good solvents falling slower than those in bad ones (where the particles quickly come together). Judgement can be via the human eye looking at test tubes or via automated centrifugal techniques [31].
One more thing needs to be known about HSP. The HSP of a mix of solvents is the volume-weighted average of their individual HSP [9]. Imagine two solvents each outside the solubility sphere (so they are bad solvents) and on opposite sides. A 50:50 mix brings them to the center of the sphere. This says, against intuition, that two bad solvents can create a good solvent. The fact that this is found in practice is liberating for the formulator. A bad solvent (in solubility terms) with other desirable attributes (low cost, low VOC, green, etc.) can be used when carefully paired with another solvent with complementary attributes.
The app (Figure 11) shows how a formulator can use these attributes of HSP.
We have two spheres that might represent a polymer and a nanoparticle. We also have a collection of solvents that we might consider using, scattered throughout the 3D Hansen space. The golden sphere shows the ideal point between the two spheres with the minimum distance. There happen to be no usable solvents at that point, so one is created (light blue) by mixing two of them and optimizing the HSP of the mix along the line joining the two solvents.
This sort of sophisticated method for solving difficult formulation issues is not possible via standard colloid science. The combination of a good-enough theory (HSP) and some basic apps is a powerful enabler of innovative formulations. If someone is stuck in formulation space in a comfort zone, by seeing things in 3D and by thinking of the possibilities of combining bad solvents, it is possible to move rationally to a different part of formulation space. Similarly, to swap to a different polymer system might involve changes of both polymer and solvent; this is straightforward with a 3D solubility tool, yet very hard without it.
The challenge for colloid scientists is not to write yet another paper about DLVO nor to point out the many flaws in HSP. The challenge is to come up with a better combination of theory and practice to allow the formulator to navigate more reliably to the optimum part of a complicated formulation space.
A specific example of such an approach arises from Vincent’s delightfully named “pragmatic theory” of depletion flocculation [33]. Rather than the generally unrealistic case of pure spheres and pure, non-interacting polymer discussed by Asakura and Oosaka [10], Vincent has three interactions described by χ parameters: χ12, interaction between free polymer and the solvent; χ13, interaction between steric stabilization polymer on the particle and the solvent; and χ23, interaction between the two polymers. This takes us beyond mere depletion flocculation because, via χ13, it also includes steric repulsion and the relative interactions of the protective and added polymers (which might be the same) with themselves and the solvent. Because, as noted earlier, it is possible to translate from χ values to HSP distances, we now have a working model with which the adventurous formulator can explore a complex part of formulation space. At the time of writing, a working prototype has been added to the Depletion Flocculation app mentioned earlier.

4. Conclusions

The problems faced by formulators are complex and are often at the limits of known science. While academic research should not solely be defined by “utility,” it is appropriate that parts of the academic community should receive funding to help push those limits to enable superior formulations. However, the majority of the literature purporting to be of relevance to such a community is simply unusable by the community (recall the Lancet study on biomedical research [5]) and the cumulative effect of the recent decades of such research has not, as far as this author can determine, led to significant advances in usable ideas. The apps used here, and the many other formulation-related apps on the author’s website, rarely depend on significant ideas from beyond 2000, though there are some welcome tweaks to older theories that make it easier to write a clearer app.
This rather negative view can be turned into two positives. First, those embarking on such research can pre-filter potential projects by asking whether any hoped-for outcomes would be appable and usable by the formulation community. By re-framing the research in a way that could have usable benefits, and, perhaps, planning in advance how to publish the research along with an app, the impact of a successful outcome would be larger, to everyone’s benefit. This links to the second positive point. The idea of pre-filtering has the danger of encouraging safe projects with worthy but unexciting outcomes. In the long term, this is in no one’s interest. What we really need are some high risk, high reward projects that try to answer questions, which if answered, would have a dramatic impact on the community. What might such projects be? To the extent that the author’s collection of apps provides a snapshot of the state of the art, it provides a starting point for exploring the problems and limitations of the current generally-accepted knowledge.
One example is that even when complex rheology is made as friendly as possible, it is still relatively unhelpful at answering many of the big formulation questions. The most-used techniques of rotational and low-strain oscillatory rheology are currently not bridged by any useful theory. So the high shear rates that rotational techniques can easily manage do not provide useful information to help extend the relatively pedestrian few-Hz rates of oscillatory (though in well-behaved systems this range can be extended by time–temperature superposition, as illustrated in the app https://www.stevenabbott.co.uk/practical-rheology/WLF.php not shown here). A new technique or a new theory that could cover large ranges of stresses, strains, and relaxation rates would have a dramatic effect on our ability to understand printable systems. Such a theory would automatically address the need for ready interconversion from one type of measurement, such as relaxation moduli, to other forms, such as retardation spectra, if such a conversion aided understanding of a key printing phenomenon.
Another example is that our ability to grasp particle–particle interactions (and their interactions with solvents and polymers) is shockingly rudimentary. An over-emphasis on the niceties of DLVO and an under-emphasis on robust “solubility-style” theories has left the formulation world with very few tools for rational navigation through a complex formulation space. HSP seem to be a technique with much to offer in the short term. The present author happens to believe that the recent upgrading of Kirkwood–Buff’s 1950s theory (see the appropriate tutorials in the Practical Solubility website [34]) offers one way beyond HSP. Other possibilities would be welcome.
Both these examples represent grand challenges, with the further challenge that anyone who embarks on them should include a usability criterion in their approach. However wonderful the new theory, if it is not translatable into parameters that formulators can adjust rationally, it will sit on the virtual shelves of academic literature unloved. Above all, this represents a community challenge. Those with minds that can devise brilliant new theories are often not those who have a grasp of the difficulties of implementing them; they need to interact with minds who can help translate the theory into something usable. A theory so abstruse as not to be testable can benefit from being re-configured into a testable format. This is a tough intellectual challenge. However, that is what academia is supposed to be about.
Finally, this paper comes with a double guarantee. First, if expert readers find errors in, or ways to improve, the current apps, the present author will be happy to implement the changes with full acknowledgement of the source of new information. Second, if an expert reader notifies the present author about their magnificent, useful theory which has not been appified, then provided there is some reasonable set of inputs and outputs connected by some reasonable algebra or a manageable numerical technique, then the app will be added, in virtual reality [35] if required, to the collection with full acknowledgement.

Funding

This research received no external funding and all the apps mentioned are open source, free to use, and free from advertising.

Acknowledgments

The author’s collection of apps has often been aided by the generous support of those behind the theories used. Any such support is acknowledged on the individual apps.

Conflicts of Interest

The author declares no conflict of interest.

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Figure 1. A stable DLVO setup. The formulation is in water (ε = 80), there is a reasonable zeta potential of 40 mV, and the salt concentration of monoionic species is not too high. The Vh (Hamaker) attraction eventually overwhelms the Vd (Debye) charge repulsion, but there is a satisfactory 25 kT barrier at ≈0.5 nm. The app is at www.stevenabbott.co.uk/practical-solubility/dlvo.php.
Figure 1. A stable DLVO setup. The formulation is in water (ε = 80), there is a reasonable zeta potential of 40 mV, and the salt concentration of monoionic species is not too high. The Vh (Hamaker) attraction eventually overwhelms the Vd (Debye) charge repulsion, but there is a satisfactory 25 kT barrier at ≈0.5 nm. The app is at www.stevenabbott.co.uk/practical-solubility/dlvo.php.
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Figure 2. With charge stabilization turned off (φ ≈ 0), a 1 nm layer thickness δ is enough to provide a strong barrier. Playing with most other parameters makes no difference, including the Flory–Huggins χ parameter, until this exceeds 0.5 and the system suddenly collapses.
Figure 2. With charge stabilization turned off (φ ≈ 0), a 1 nm layer thickness δ is enough to provide a strong barrier. Playing with most other parameters makes no difference, including the Flory–Huggins χ parameter, until this exceeds 0.5 and the system suddenly collapses.
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Figure 3. In this example, the 100 μm defect spontaneously closes. However, the unfortunate formulator might be unaware that a modest reduction of thickness to 30μm will allow a large pinhole to form. The app is at www.stevenabbott.co.uk/practical-coatings/pinholes.php.
Figure 3. In this example, the 100 μm defect spontaneously closes. However, the unfortunate formulator might be unaware that a modest reduction of thickness to 30μm will allow a large pinhole to form. The app is at www.stevenabbott.co.uk/practical-coatings/pinholes.php.
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Figure 4. A 200 μm feature in a 20 µm coating levels in ≈1 ms. A well-intentioned reduction in thickness to 10 μm would increase the time to 8-fold while defects of 240 μm wavelength would double the time. The app is at www.stevenabbott.co.uk/practical-coatings/levelling.php.
Figure 4. A 200 μm feature in a 20 µm coating levels in ≈1 ms. A well-intentioned reduction in thickness to 10 μm would increase the time to 8-fold while defects of 240 μm wavelength would double the time. The app is at www.stevenabbott.co.uk/practical-coatings/levelling.php.
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Figure 5. What should be a 50 μm line becomes 118 μm after only 0.2 s, with an increase to 165 μm over 2 s. The rapid initial spreading in a 1 Pa·s ink is surprising to most of us. The app is at www.stevenabbott.co.uk/practical-coatings/drop-spread.php.
Figure 5. What should be a 50 μm line becomes 118 μm after only 0.2 s, with an increase to 165 μm over 2 s. The rapid initial spreading in a 1 Pa·s ink is surprising to most of us. The app is at www.stevenabbott.co.uk/practical-coatings/drop-spread.php.
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Figure 6. If a coating exceeds the critical Marangoni number (≈80), it is likely to show orange-peel defects. In this example, the thermal Marangoni number (Ma T) is only 24, so there is no problem. However, the concentration number, Ma C is very large, so defects are likely. The app is at www.stevenabbott.co.uk/practical-coatings/Marangoni.php.
Figure 6. If a coating exceeds the critical Marangoni number (≈80), it is likely to show orange-peel defects. In this example, the thermal Marangoni number (Ma T) is only 24, so there is no problem. However, the concentration number, Ma C is very large, so defects are likely. The app is at www.stevenabbott.co.uk/practical-coatings/Marangoni.php.
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Figure 7. Only an expert could work out that these are the same data plotted with four different options. The x-axis is always shear rate, as log along the top and linear along the bottom. The left-hand plots are of viscosity, the right-hand are of shear stress. The app allows the user to view the same data with a total of six plot options and is at www.stevenabbott.co.uk/practical-rheology/Shear-Viscosity.php.
Figure 7. Only an expert could work out that these are the same data plotted with four different options. The x-axis is always shear rate, as log along the top and linear along the bottom. The left-hand plots are of viscosity, the right-hand are of shear stress. The app allows the user to view the same data with a total of six plot options and is at www.stevenabbott.co.uk/practical-rheology/Shear-Viscosity.php.
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Figure 8. Yield stress is important in many aspects of printing. However, which method should be used to measure it? These six plots show the same yield stress being measured in six different ways. Arguably the top-left and top-middle are the most common, though the others are relatively common. The app is at www.stevenabbott.co.uk/practical-rheology/Yield-Measurement.php.
Figure 8. Yield stress is important in many aspects of printing. However, which method should be used to measure it? These six plots show the same yield stress being measured in six different ways. Arguably the top-left and top-middle are the most common, though the others are relatively common. The app is at www.stevenabbott.co.uk/practical-rheology/Yield-Measurement.php.
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Figure 9. Relaxation modulus, G’ & G’’, relaxation spectrum, creep compliance, J’ & J’’, and the retardation spectrum are all fundamentally interconvertible. The app is at www.stevenabbott.co.uk/practical-rheology/Interconversions.php.
Figure 9. Relaxation modulus, G’ & G’’, relaxation spectrum, creep compliance, J’ & J’’, and the retardation spectrum are all fundamentally interconvertible. The app is at www.stevenabbott.co.uk/practical-rheology/Interconversions.php.
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Figure 10. High shear-rate properties of relevance to printing. There are two critical phase volumes (percolation and packing), plus the effects of aspect ratio, plus the effect of fractal dimension—a surrogate for flocculation effects. The effects of particle radius on yield stress are included. The equations are too complex to show here but are fully described in the app, which is at www.stevenabbott.co.uk/practical-rheology/High-Shear-Particles.php.
Figure 10. High shear-rate properties of relevance to printing. There are two critical phase volumes (percolation and packing), plus the effects of aspect ratio, plus the effect of fractal dimension—a surrogate for flocculation effects. The effects of particle radius on yield stress are included. The equations are too complex to show here but are fully described in the app, which is at www.stevenabbott.co.uk/practical-rheology/High-Shear-Particles.php.
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Figure 11. The screenshot tries to show how two rather incompatible materials (e.g., a polymer and nanoparticle) can be brought together via a mix of two rather bad solvents. The reader is urged to explore the app to get a better feel for what is going on. The app is at www.stevenabbott.co.uk/practical-solubility/HSP-3DO.php.
Figure 11. The screenshot tries to show how two rather incompatible materials (e.g., a polymer and nanoparticle) can be brought together via a mix of two rather bad solvents. The reader is urged to explore the app to get a better feel for what is going on. The app is at www.stevenabbott.co.uk/practical-solubility/HSP-3DO.php.
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© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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