D.Frolov, S.Bagdonas, R.Rotomskis
Laser Research Centre, Vilnius University, Sauletekio 9, 2040 Vilnius Lithuania

• Introduction
• Materials and Methods
• Results and Discusion
• Conclusions
• References

The absorption spectrum of TPPS₄ in neutral solution has five absorption bands with peaks at 414, 515, 552, 580 and 635 nm (Fig.2). In slightly acidic solutions at pH <5 TPPS₄ exists as dianion and its absorption spectrum displays three bands with maxima at 432, 594 and 645 nm (Fig.2). At lower pH values or in the presence of KCl significant changes in the absorption spectrum of TPPS₄ occur [5]: the new absorption bands with the maxima at 490 and 708 nm appear with simultaneous disappearance of all other bands (Fig.2). Such spectral changes are attributed to the formation of J-aggregates of TPPS₄ [5,6].

Fig.2 Absorption spectra of different ionic forms of TPPS₄ (black line represents non-protonated form (pH 7), red – dianion (pH<5), blue – J-aggregates (pH<4)). Concentration 5× 10^-5 M, l = 10 mm.

The absorption difference spectra of J-aggregates (Fig.3) of TPPS₄ registered with different delay times (0 ps, 10ps and 100 ps) have the same shape (only the intensity of signal becomes lower with longer delay times). At all delay times the induced absorption is observed throughout the whole absorption difference spectrum except the regions of the ground state absorption. This indicates that the absorption of the excited state is more intensive than that of the ground state (but it does not reflect any structural changes in the J-aggregate since no dependence of the relaxation kinetics shape on the wavelength of probe light is observed).

Fig.3 Absorption difference spectra of TPPS₄ obtained under various delay times. Filled circles – 0 ps, empty circles – 10 ps, squares – 100 ps. Concentration 2.5× 10^-5 M, l = 1 mm, l _ex =527 nm.

The typical kinetic curve of the excited state relaxation in J-aggregates of TPPS₄ is presented in Fig.4. In the region of 450-750 nm the shape of the curves does not depend on the wavelength of probing light. The kinetic curve consists of the fast initial part which gradually turns into slower relaxation. The kinetic curve can be approximated biexponentially (t ₁ »20 ps and t ₂ »1 ns) or monoexponentially (t » 100 ps+plato). The fast part of the kinetics is presumably consistent with exciton annihilation.

Fig.4 Relaxation kinetics of the excited state. Concentration 2.5× 10^-5 M, l = 1 mm, l _mon =490 nm, l _ex =527 nm. In the presence of annihilation the excited state relaxation is described as:

where n - exciton density, G - exciton generation speed, t - real (excluding exciton annihilation) relaxation time of the excited state, g (t ) - annihilation constant [7].

Theory of a diffusion-restricted movement as well as the case of long range Förster dipole-dipole interaction leads to the dependence of exciton annihilation constant on time, which could be approximately presented as g =g ₀t^-h [8-10], where h takes values in the range from 1/6 to 1/2. Parameter h is related to a spectral dimension d_s, 1-h=d_s/2, and reflects probability to revert to a ground state within a period of time t after random excitation P~t^-d_s^/2 [11].

When linear relaxation is slow, the member n/t is negligible and equation (1) becomes:

where n₀ -exciton density at initial time moment [10].

The ratio of excited molecules with non-excited (n/N) is related to stationary (A) and difference (D A) absorption and to extinction coefficients of steady state (e ) and excited state (e * ) by the following formulae [12]:

It is evident that when e >>e * (e.g. in the region where absorption bands are bleached), it can be assumed that n/N» D A/A. Consequently, equation (2) can be transformed into equation (4), which includes parameters obtained experimentally:

Experimental results - the dependence of D A of J-aggregate band (at 490nm) on time - could be described on the basis of presented exciton annihilation model. The best coincidence of the dependence of A/D A - A/D A₀ » N/n - N/n₀ on time with a model curve is at h=1/2 (Fig.5). This corresponds to the diffusion limited one-dimensional annihilation and therefore it might be concluded that J-aggregates are of linear structure.

Fig.5 Dependence of A/DA - A/DA₀ »  N/n - N/n₀ on t^1/2. Concentration 2.5× 10^-5 M, l = 10 mm, l _mon=490 nm, l _ex=527 nm. Dots represent experimental data, line – theoretical approach.

Investigation of the excitonic annihilation allows one to evaluate some parameters of exciton transitions in the matrix of J-aggregates of TPPS₄. It is known [10,13] that exciton transitions in linear structures are well described by Smoluchowski equation for one-dimensional diffusion limited transitions. The precise solution of the equation gives value of g ₀ (static annihilation coefficient) which could be used for the calculation of exciton diffusion coefficient:

where D - diffusion coefficient, d - the distance between molecules. Incorporation of (5) into (4) yields :

Assuming the distance between molecules equal 5Å, the exciton diffusion coefficient can be calculated from the slope of the curve representing dependence A/D A - A/D A₀ » N/n - N/n₀ on time t^1/2 (Fig.5).

The calculated exciton diffusion coefficient (D) in J-aggregates of TPPS₄ is equal (4± 2) 10^-3 cm²/s^-1. Using the obtained value of the diffusion coefficient, the characteristic exciton transition time [10] was calculated: t =d²/2D= (1.75± 0.75) ps.

The size of aggregate could be evaluated under assumption that after annihilation in every separate domain only one exciton is left, which then relaxes linearly [14]. Thus, taking into account that A/D A » N/n in the region where absorption bands are bleached in difference spectra (as it was indicated above), number of molecules in the aggregate could be calculated from a relation N/n=f(t). However, as it is seen in Fig.5, the dependence of A/D A - A/D A₀ » N/n - N/n₀ on time t^1/2 is linear up to the maximal delay time (about 1.5 ns), therefore we can only determine the minimal size of J-aggregates of TPPS₄. According to our evaluation such an aggregate contains at least forty porphyrin molecules.

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