Modeling of the mean photoluminescence intensity ratio during blinking events

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=11/2(e(Xx0)/ld+ex0/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 segmentint(ld,X)=0Xp(x0,ld,X)dx0=X+ld(eX/ld1)(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 byint(L,ld,NSM,φ)=n=01/(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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