Boltzmann sigmoid and Pyrene fluorescence ratio

XiaO / 2016-02-20

Pyrene, a hydrophobic fluorescent dye, could be used as a microenvironmental polarity-sensitive probe for the determination of CMC values. Pyrene molecules could be excited by light at a wavelength of 339 nm and emitted fluoresces with maxima at wavelengths of 373 nm and 384 nm. Pyrene 373/384 ratio represents the ratio of fluorescent intensity at wavelength of 373 nm to 384 nm. [^1]

Pyrene fluorescence

Figure 1. Fluorescence emission spectra of pyrene.

Pyrene 373/384 ratio

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

$$ y = \frac {A_1 - A_2}{ 1+ e^{( x-x_0 ) / \Delta x}} + A_2 \tag{1}$$

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.

Boltzmann sigmoidal curve

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:

  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. 116-122. ↩︎