## DST Choice

### Quick Start

The DST app showed the general principles. Here we look at how to best select a surfactant for a rapid reduction of surface tension.

It turns out that the secret is to have a bad surfactant with a high CMC value. Hopefully this will intrigue you enough to explore what is going on.It is especially nice that the theory is simple and leads to clear conclusions.

### DST Choice

Log
Linear
Tablet
Laptop
Γm 10-6 mol/m²
D 10-6 cm²/s
Tmax S
CMC (µM)
a 10-6 M
C (µM)
K l/mol
γc
h μm
τD S

The DST app based on the Rosen equation helps understand many things, but doesn't answer the key question: "Which surfactant should I choose to get the best reduction in surface tension for my specific purpose?" This page solves the problem of DST choice.

The starting point is a key paper1 by Prof Stebe and colleagues, with a title Which surfactants reduce surface tension faster? that gets right to the point. The only number that matters is τD which is a characteristic diffusion time which tells you when most of the reduction in surface tension is going to take place. With academic modesty, Prof Stebe says that τD is accurate within a factor of 10, but mostly it's good enough within a factor of 2 which, for DST, is amazing. Unlike many concepts in surfactancy, τD is simplicity itself:

τ_D = h^2/D

h is an absorption depth which says how deep you have to go below the interface to be able to find the number of surfactant molecules needed to fill the surface sufficiently to reduce the ST to the required level. D is the diffusion coefficient. Both factors make sense. If you have a large D and a small h it is very easy for molecules to reach the interface.

h itself is also easy to calculate when the surfactant is at concentration C:

h=Γ_m/(a(1+C/a))

To see what Γm is, go to the CMC app, but briefly it is the limiting surface concentration of surfactant at saturation of the interface. a is a parameter from the Langmuir isotherm (a = 1/K for those who use K-based Langmuirs. K is shown below for reference). Briefly, a small a value means that the surfactant likes being at the surface. Importantly, both Γm and a are easily obtained from a standard plot of surface tension versus surfactant concentration, the one that plateaus at CMC. The limiting surface tension, γc is shown below. It is a useful sanity check for your input parameters - if it falls below, say, 20mN/m a is too small or CMC is too high.

The surfactant concentration C is the other part of h. A high concentration implies a small h. That's obvious. What is not obvious is that the one thing which destroys our hopes of a fast DST performance is the CMC. C cannot be (effectively) higher than CMC. Which means that "good" surfactants with low CMC values are truly awful in terms of DST. A super-low CMC means that h is always large which means that τD is always large.

This leads to the golden rule of DST: always choose a "bad" surfactant with a large CMC but which is also "good" in that it also has a low γc which is the equilibrium surface tension at its high CMC.

There is obvious resistance to the idea that bad surfactants are needed for good DST performance. So in addition to the two factors from Prof Stebe it's necessary (as done in her papers) to calculate the DST behaviour from the key parameters. This is done by using the standard Ward and Tordai DST formula. This is, unfortunately, a complicated differential equation and the solutions to it are not trivial. Fortunately Prof Stevenson's group provided2 an explicit algorithm and code for implementing Ward and Tordai. So a reasonably fast-response app became possible.

The groups of Prof Stebe and Stevenson are well aware that a simple Langmuir isotherm and planar geometry are not sufficient for all cases, and they explain all the extra complexities. Here things are kept deliberately simple; those who require Frumkin isotherms and spherical geometries are free to implement their own apps.

The inputs are the basic data from a Langmuir fit to surface tension data. If you only know CMC and γc then use the CMC app to find Γm and a. The diffusion coefficient, D, varies rather little between ordinary surfactants and 5.10-6 is a good-enough estimate. Those who use protein surfactants are in a different league - there 5.10-7 is more common, with a big effect on DST behaviour. The CMC is required in order to limit the maximum usable concentration in the Stebe and Stevenson calculations. See the comments below. Finally, you specify the surfactant concentration you intend to use in your formulation. Units are a nightmare in this whole area; I've tried to be as consistent as possible with other apps. You will regularly find yourself with factors of 103 and 104 to convert from whatever units your data arrive in.

Numerical solvers can sometimes "explode". They can always be made more robust at the expense of speed. Use the default "Tablet" mode which is a reasonable balance of speed and robustness. If you hit a problem (the time graph will show a strange spike) then select Laptop mode and be aware that response will be rather slow on a tablet.

Once you start to experiment you will quickly find that the only way to get small h, small τD and, therefore, fast DST behaviour is via large C, which, because of the limitations of "good" low CMC surfactants, absolutely requires high CMC. I'm happy to be contradicted, but like everyone I could never understand why the "acetylenic" surfactants were often recommended for good DST performance. They don't have super-high diffusion coefficients (at this size range of surfactants it's not possible to do much better the 8.10-6cm²/s) and they don't have super-low γc. They just have high CMC so you can add plenty (e.g. 0.5% which is "small" for a formulator) and get their full benefit in reducing h and τD. I believe it's as simple as that!

These comments on CMC aren't quite right. Other data suggest that CMC has no effect on the effective concentration - that if you go to 2*CMC, although you have more micelles you still have 2*concentration to reduce h. Unfortunately I don't have the theory to handle this because all the relevant calculations use the Langmuir isotherm which knows nothing about CMC. So in the app you get zero benefit from being above CMC. If/when I have access to better theory I will improve the app.

I've adopted a few tricks in the numerics of the Stevenson solution to make the app workable on iPads etc. If the DST graph "explodes" just choose a shorter timescale. For many users the only interest is in events that happen over a few seconds and the app is OK even on an iPad. For those interested in events over 100s or 1000s of seconds it might be advisable to use a fast laptop. The combination of fast processes (over in seconds) and long timescales is the hardest for the numerics to cope with.

1James K. Ferri, Kathleen J. Stebe,Which surfactants reduce surface tension faster? A scaling argument for diffusion-controlled adsorption, Advances in Colloid and Interface Science 85, 2000, 61-97

2Xueliang Li, Ryan Shaw, Geoffrey M. Evans, Paul Stevenson, A simple numerical solution to the Ward–Tordai equation for the adsorption of non-ionic surfactants, Computers and Chemical Engineering 34 (2010) 146–153