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Modeling of the mean photoluminescence intensity ratio during blinking events
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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 $p(x0,ld,X)=∫0Xc(x,x0,ld)dx/∫−∞+∞c(x,x0,ld)dx=1−1/2(e−(X−x0)/ld+e−x0/ld)$. 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(x0, ld, X) over the nanotube segment$int(ld,X)=∫0Xp(x0,ld,X)dx0=X+ld∙(e−X/ld−1)$(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, ld, n) = 1/(1 + a(ld/L) ∙ n), where a(ld/L) depends only on the length ratio ld/L, and relates the probability of an exciton to encounter a quencher [a(ld/L) = c1 ∙ (ld/L)c2/(1 + c3 ∙ (ld/L)c2)], where c1, c2, and c3 are constants (fig. S3C). The average luminescence intensity of a nanotube of length L with diffusion length ld having NSM SP-MC molecules is then obtained by weighing each int(L, ld, n) by the probability of having exactly n SP-MC molecules in the MC state on the nanotube at a given time: Poi(NSM/φ, n) = (NSM/φ)ne(−NSM/φ)/n!, where NSM/φ is the average number of n SP-MC molecules in the MC state and φ = tSP/tMC. The final normalized intensity is then given by$int(L,ld,NSM,φ)=∑n=0∞1/(1+a(ld/L)∙n)∙Poi(NSM/φ,n)$(2)and displayed on Fig. 3A for L = 300 ± 50 nm and varying φ. From Eq. 2, the ratio φ can thus be determined knowing L, ld, and NSM with an error ∣∆φ∣/φ < 12% for L = 300 ± 50 nm, error on ∣∆φ∣/φ < 13 % for NSM = 1 ± 0.1 per nm, and ∣∆φ∣/φ < 10 % for ld = 200 ± 50 nm (Fig. 3A and fig. S5).

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