Modeling of the mean photoluminescence intensity ratio during blinking events

AG Antoine G. Godin AS Antonio Setaro MG Morgane Gandil RH Rainer Haag MA Mohsen Adeli SR Stephanie Reich LC Laurent Cognet

This protocol is extracted from research article:

Photoswitchable single-walled carbon nanotubes for super-resolution microscopy in the near-infrared

**
Sci Adv**,
Sep 27, 2019;
DOI:
10.1126/sciadv.aax1166

Photoswitchable single-walled carbon nanotubes for super-resolution microscopy in the near-infrared

Procedure

To model the mean photoluminescence intensity probability during the blinking processes (normalized to the luminescence without defects), we derived an analytical expression for the mean intensity of a segment of arbitrary length *X*. It corresponds to the probability of an exciton recombining before reaching the end of the nanotube and gives $p({x}_{0},{l}_{\mathrm{d}},X)={\int}_{0}^{X}c(\mathrm{x},{x}_{0},{l}_{\mathrm{d}})\mathit{dx}/{\int}_{-\infty}^{+\infty}c(\mathrm{x},{x}_{0},{l}_{\mathrm{d}})\mathit{dx}=1-1/2({e}^{-(X-{x}_{0})/{l}_{d}}+{e}^{-{x}_{0}/{l}_{d}})$. Assuming that excitons are generated uniformly along the nanotube segment allows to estimate the average intensity of a nanotube segment of arbitrary length *X* by integrating *p*(*x*_{0}, *l*_{d}, *X*) over the nanotube segment$$\text{int}({l}_{\mathrm{d}},X)={\displaystyle {\int}_{0}^{X}}p({x}_{0},{l}_{\mathrm{d}},X)d{x}_{0}=\mathrm{X}+{l}_{\mathrm{d}}\bullet ({e}^{-X/{l}_{\mathrm{d}}}-1)$$(1)

The integrated intensity from a nanotube with *n* quenching sites is given by the sum of the intensity of each (*n* + 1) nanotubes segments. The intensity of each segment is given by Eq. 1 and fig. S3A. The mean normalized intensity for a nanotube of length *L* = 300 nm can then be calculated (fig. S3B). This curve was obtained by numerically generating 50,000 random configurations having *n* quenchers. The relation is well approximated by int(*L*, *l*_{d}, n) = 1/(1 + *a*(*l*_{d}/*L*) ∙ *n*), where *a*(*l*_{d}/*L*) depends only on the length ratio *l*_{d}/*L*, and relates the probability of an exciton to encounter a quencher [*a*(*l*_{d}/*L*) = *c*_{1} ∙ (*l*_{d}/*L*)^{c2}/(1 + *c*_{3} ∙ (*l*_{d}/*L*)^{c2})], where *c*_{1}, *c*_{2}, and *c*_{3} are constants (fig. S3C). The average luminescence intensity of a nanotube of length *L* with diffusion length *l*_{d} having *N*_{SM} SP-MC molecules is then obtained by weighing each int(*L*, *l*_{d}, *n*) by the probability of having exactly *n* SP-MC molecules in the MC state on the nanotube at a given time: Poi(*N*_{SM}/φ, *n*) = (*N*_{SM}/φ)* ^{n}* ∙

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