Boltzmann sigmoid and Pyrene fluorescence ratio
XiaO / 20160220
Figure 1. Fluorescence emission spectra of pyrene.
Pyrene 373/384 ratio
 As a microenvironmental polaritysensitive probe, the pyrene fluorescent ratio of 373/384 will be much smaller than one while sensing a more hydrophobic environment. However, if in a more hydrophilic microenvironment, such as water and alcohol, the ratio would increase dramatically, basically much bigger than one. The value trend corresponds to the total concentration of amphiphilic materials in solution.
 The plots of the Pyrene 373/384 ratio as a function of the material concentration can be fitted using Boltzmann sigmoidal curve.^{1}
Boltzmann sigmoidal & CMC calculation
Assumption: pyrene 373/384 ratio plots can be adequately described by a decreasing sigmoid of the Boltzmann type, which is given by
$$ y = \frac {A_1  A_2}{ 1+ e^{( xx_0 ) / \Delta x}} + A_2 \tag{1}$$
where the variable y corresponds to the pyrene 373/384 ratio, the independent variable $(x)$
is the total concentration of amphiphilic material, $A_1$
and $A_2$
are the upper and lower limits of the sigmoid, respectively. $x_0$
is the centre of the sigmoid, and $\Delta x$
is directly related to the independent variable range where the abrupt change of the dependent variable occurs, as shown in Figure 2.
Figure 2. Decreasing sigmoid of the Boltzmann type.
The $CMC_1$
value is the centre of the sigmoid $x_0$
. Basically, $A_1$
, $A_2$
, $x_0$
and $slop (x_0) $
are all given as fit parameters of the experimental data.
$$CMC_1 = x_0 \tag{2}$$
The $CMC_2$
must be analytically determined. After calculating and simplifying some terms, finally:
$$ {\left( {d_y} \over {d_x} \right)} _{x_0}= \frac {A_2  A_1}{4\Delta x} \tag{3}$$
$$CMC_2 = x_0 + 2\Delta x \tag{4}$$
The goodness of the fitting curve can be evaluated according to the $R^2$
and $Sy. x$
values, basically the $R^2$
would be above 0.95 and the $Sy. x$
value would be very small. Therefore the plots of the Pyrene 373/384 ratio as a function of the material concentration can be fitted using Boltzmann sigmoidal curve.
CMC determination
For single material micellar systems, experienceably:
 Ionic surfactants:
$CMC_2$
 Nonionic surfactants:
$CMC_1$

Aguiar, J., et al., On the determination of the critical micelle concentration by the pyrene 1 : 3 ratio method. Journal of Colloid and Interface Science, 2003. 258(1): p. 116122. ↩︎