STAND Surface Tension for Aggregation Number Determination

The CMC and CMC Saturation apps both address key issues of what happens when you add surfactant to water up to somewhat above the CMC value. In this app we find that it is possible to do a very full analysis of the surface tension versus concentration plot and derive an important extra value, the micelle aggregation number, N.

The starting point is the Langmuir isotherm, but expressed in terms of the free surfactant concentration, c_free rather than the total concentration c_tot. The isotherm requires θ, the degree of saturation of the surface which, in turn, depends on the absorption coefficient, K. Note that to be consistent with the nomenclature in the other apps on this site I am changing nomenclature from the authors' paper; for example, they use β for the absorption coefficient, not K.. The key equation used in their analysis of the surface tension σ is:

`σ_"water"-σ = RT.Γ_mln(1+K.c-"free")`

We have already encountered Γ_m as the maximum surface concentration. At the same time there is another equilibrium, between monomers and micelles. Where the concentration of micelles is M, k is the (very, very large) equilibrium constant given by:

`k=M/c_"free"^N`

Using this it is easy to see that:

`c_"tot"=c_"free"+kNc_"free"^N`

All that is needed to characterise the whole system is to find the parameters K, Γ_m, k and N via fitting of the surface tension versus concentration curve. Unfortunately this is not a simple task. In the real app a sophisticated optimisation is used. Here we use human fitting, by sliding the four property sliders to get an adequate fit. Seven datasets from the original paper (inaccurate because I have converted them from graphs in the paper) are provided for you to explore. The reason for the row containing a CMC input and some "suggested value" outputs is discussed below.

Outputs

The free energy of micelle formation, Δμ⁰ is derived from as -RT/Nln(k) and is in kJ/mole. K (from your input ln(K)) is shown for convenience. The area per surfactant molecule is calculated as A=1/(Avogadro.Γ_m). Although the STAND paper shows that CMC can be derived via their methodology, CMC is not provided as an output as it is fairly obvious by inspection to a good-enough precision.

Further comments

Although fitting my slider is not perfect, it is very instructive. It builds up an intuition about which parameters affect which part of the curve. Clearly K and Γ_m dominate one part of the curve and k and N dominate the other, so you can quickly get to an approximate solution. Then, within the limitations of sliders, you can use your eyes and/or the RMS Error value to find a decent fit.

The absorption coefficient, K, is input via ln(K) because it covers a wide range. The equilibrium coefficient is input as ln(k). If you take a typical value of 300 this means that k is e³⁰⁰ = 10¹³⁰, a truly enormous number, but necessary because the other part of the equation is c_free^N which, for 10^-6M and N=50 = 10^-300.

The surface tension for water is taken to be 71 mN/m as that is the value used/fitted by the original authors. For simplicity, the temperature is assumed to be 25°C. The original paper has data at other temperatures. Errors from fitting data taken at other temperatures will be small because RT changes by only 3% going from 20° to 30°.

Your own data

The authors' app lets you load your own data which can include 4 columns of data for their more sophisticated fitting, with a warning that just 2 columns will not be so accurate. For this app the file format just uses 2 data columns but with a format that is convenient for generalised reading. The first row simply says Item,Conc,gamma and subsequent rows say Data, followed by the concentration in mM and the surface tension in mN/m. In both cases you can use comma or tab separated variables. The app handles up to 20 pairs of datapoints. If you want to see my versions of the datasets, you can download them as STANDdata.zip.

Alternative fitting

Dr Théophile Gaudin examined the STAND paper and realised that if CMC was taken as an objective input (something that STAND explicitly rejects because of the ill-defined nature of CMC) it was possible to produce a simple fit to the Langmuir curve below CMC (assuming that c_free = c_tot) to access Γ_m and K, using:

σ_water-σ = RTΓ_mln(1+K.c_tot)

Note that the assumption that c_free = c_tot is explicitly not part of the STAND approach and is introduced here as a useful approximation to arrive at plausible values to feed to the STAND model.

Above the chosen CMC, the concentration of micellised surfactant c_m is calculated by solving the equation:

σ_water-σ - RTΓ_mln(1+K.(c_tot-c_m)) = 0

Finally, by calculating c_free=c_tot-c_m it is possible to obtain N and k by fitting to the equation:

ln(c_m) = N.ln(c_free) + ln(k.N)

Some simple checks are made to ensure that the entered CMC gives the fitting procedures a chance to create meaningful values, i.e. with a reasonable number of pre- and post-CMC data values, otherwise a "No solution" message appears in the relevant outputs. If you would like to enter the calculated values directly into the main slider inputs, click the Set Inputs button.