Spectral diffusion in an amorphous polymer probed by single molecule spectroscopy

JOURNAL OF

LUMINESCENCE Journal

ELSEVIER

of Luminescence

64 (1995) 1-11

Spectral diffusion in an amorphous polymer probed by single molecule spectroscopy J. Tittel a, R. Kettner a, Th. Baschit a~*,C. Brtiuchle a, H. Quanteb, K. Mdlen b aInstitutfiir Physikalische Chemie, Universitiit Miinchen, Sophienstraje I I, 80333 Munich. Germany b~ax-Planck-Institut fir Polymerforschung, Ackermannweg IO. 55021 Mainz, Germany Received

16 September

1994; revised 29 November

1994; accepted

17 December

1994

Abstract

We investigated the spectra of single tetra-terc-butylterrylene (TBT) molecules in the amorphous matrix poly(isobutylene) (PIB). The distribution of line widths of TBT in PIB was measured and compared to that of TBT in poly(ethylene). The fluorescence intensity autocorrelation function as well as the two-point frequency autocorrelation function were determined for different single TBT molecules. Logarithmic-like decays of the fluorescence autocorrelation function could be reproduced by assuming a l/R fluctuation rate distribution for the two-level-tunnelling systems.

1. Introduction

One important feature of the optical spectroscopy of single chromophores in low temperature solids [l, 23 is its extreme sensitivity to the local environment of the chromophore on a length scale of nanometers. This characteristic of single molecule spectroscopy (SMS) became already evident in the very beginning of this new technique [3,4] when specific single pentacene molecules in a p-terphenyl crystal were observed to spectrally diffuse, i.e., changed their absorption frequency in a stochastic fashion with time. Recently, this behaviour, which was unexpected to occur in crystalline samples, was analyzed theoretically [S] and explained by a flip-flop motion of the inner phenylring of p-terphenyl molecules residing at domain walls. The potentials of the flip-flop motions were modelled as two-level systems, the (phonon*Correspondingauthor.

induced) transitions of which induce frequency fluctuations of the single molecule resonance line. In this specific case SMS has allowed a microscopic description of the dynamics of the nanoenvironment of a single molecule. The dynamics in polymers or glasses at low temperatures is known to be very rich and manifold because of the inherent disorder in these materials [6]. Detailed investigations of the dynamics in amorphous dye-matrix systems have already been performed using line-narrowing spectroscopies as spectral hole burning [7, S] and photon echoes [9]. From these measurements it is known that the dynamics at low temperatures is dominated by two-level-tunneling (TLS) systems which couple to the optical transition (TLSs). Phonon-induced transitions of these TLSs lead to dephasing and spectral diffusion. From the above it can be anticipated that SMS will be a valuable tool to investigate the nanoenvironment of single molecules in disordered solids.

0022-2313/95/%09.50 0 1995 - Elsevier Science B.V. All rights reserved SSDI 0022-2313(95)00002-X

2

J. Tittel et al. /Journal

of Luminescence

The first experiments were done on single perylene molecules in poly(ethylene) (PE) and a variety of effects as spectral hole-burning of a single molecule and spectral diffusion were reported [lo, 11). In this study the first indications show of the coupling of a single TLS to the chromophore were observed. Later on the system terrylene in PE was investigated in great detail and also here a variety of most interesting spectral diffusion effects were reported [ 12-141. Using fluorescence autocorrelation spectroscopy, Orrit and co-workers [ 12, 131 could unambiguously demonstrate that the frequency jumps (spectral diffusion) of a number of single molecules were induced by a single TLS probably in close proximity to the chromophore. It is clear that the ability to observe such phenomena is a mere result of SMS. In this paper we present a more complete description of recent experiments [lS] performed with single tetra-tert-butylterrylene (TBT) molecules in poly(isobutylene) (PIB). We chose this matrix because in contrast to semicrystalline PE it is completely amorphous and allows the preparation of thin spin-coating films with high optical quality. The main interest in this work was the study of spectral diffusion processes with different techniques. First, we examined the distribution of line widths for a large number of single chromophores. Additionally, we investigated spectral diffusion by recording the changes of a single molecules peak absorption frequency with time (peak frequency trend). The time resolution of such measurements was limited to 2.7 s. From these data the two-point-frequency autocorrelation function could be derived. By measuring a different autocorrelation function, namely the fluorescence intensity autocorrelation function, spectral diffusion processes on a time scale between ms and s could be studied. By analyzing such measurements at different laser intensities, information about the distribution function of TLS transition rates could be obtained.

2. Experimental Tetra-tert-butylterrylene (TBT) is a derivative of terrylene [ 161, the structure of which is displayed in

64 (1995) I-11

the inset of Fig. 2. The four tert-butyl-groups provide a high solubility in organic solvents. For the sample preparation TBT and PIB were dissolved in toluene. The solution was filtered with a 0.2 urn filter and spin-coated on a LiF-substrate. Under our experimental conditions this yields z 0.5 urn thin PIB-layers with excellent optical quality. By decreasing the concentration of PIB in toluene much thinner films down to % 50-100 nm can be prepared. The concentration of TBT in the PIB films ranged from low4 to 10m6 g/g. The samples were mounted at the joint focus of a parabolic mirror and a focusing lens inside a cryostat which was operated in helium-bath-mode at 1.4 K. The laser system was a R6G-dye laser (Coherent 699-21) pumped by an Ar+-ion laser. The bandwidth of the dye laser is typically l-2 MHz. The laser beam was first guided through a single mode fiber for conditioning the beam profile. The beam, after passing through the fiber, has a Gaussian intensity profile. Then the beam was stabilized by an electro-optic modulator and focused onto the sample. The parabolic mirror collects both fluorescence light and straylight whereby the latter contribution is filtered out by a red-pass filter (Schott RG 610). The fluorescence light which passes through the filter is focused onto a photomultiplier (PM) operated in photon-counting mode. With our setup three different kinds of measurements could be performed. First, by scanning the single mode R6G-dye laser over the molecular resonance line, fluorescence excitation spectra were recorded. To measure the fluorescence autocorrelation function, the laser was held at a constant frequency, the resonance frequency of the dye molecule investigated, and the output pulses of the PM were sent to a digital correlator (ALV-5000). Furthermore, trajectories of the spectral peak position of spectrally diffusing molecules were recorded. This was done by scanning the laser repeatedly in a fast mode, typically 2.7 s per scan, over the resonance line of a molecule. For each scan the spectral peak position was determined by a computer program and plotted versus time. The jitter of the spectral peak position due to experimental reasons was less than + 1% of the entire scan range. In the remainder of this paper this kind of measurement is called peak frequency trend. From these data

J. Tittel et al. J Journal of Luminescence 64 (1995) I-11

the two-point-frequency autocorrelation function (TPFCF) [S] can be calculated. To record the fluorescence excitation spectra and fluorescence line narrowing spectra (FLN) of bulk samples, we used a slightly different setup. For the measurement of fluorescence excitation spectra the light of a mercury vapour lamp was dispersed by a monochromator and focused onto the sample. The exciting light again was rejected by a red-pass filter and the total fluorescence light was detected as described above. To record a FLN spectrum, the sample was excited with the dye laser and the fluorescence light was dispersed by the monochromator.

3. Results and discussion 3.1. General aspects of the system TBT in PIB In Fig. 1 the fluorescence excitation spectrum and fluorescence line narrowing spectrum (FLN) of bulk samples of TBT in PIB are displayed. The spectra are not corrected for any wavelength dependence of excitation or detection. The circles in Fig. 1 describe the fluorescence excitation spectrum which shows two broad absorption bands in the wavelength region between 500 and 580 nm. This spectrum is very similar to the excitation spectrum of terrylene in polymer matrices. The FLN spectrum is given by the solid line in Fig. 1. The general

500

525

550

575

Wavelength [

nrny

625

650

Fig. 1. Fluorescence excitation spectrum (circles) and fluorescence line narrowing spectrum (solid line) of bulk samples of TBT in PIB. The wavelength of the exciting laser for the fluorescence line narrowing spectrum was /I = 561 nm.

3

features of the FLN spectra were already discussed in a recent publication [15] and we only want to state that the enhanced electron-phonon coupling as compared to terrylene in poly(ethylene) leads to slightly weaker single molecule signals. Fluorescence excitation spectra of single TBT molecules were recorded in thin spin-coated PIB films. Depending on the concentration of dye molecules either a very weak inhomogeneous band was observed (10m4 g/g) or the single molecule lines were distributed as isolated features over the investigated wavelength range between 570 and 578 nm. For selecting single molecules, the laser was at first scanned over 10 GHz. Using a sample with a concentration of lo- 5 g/g (TBT/PIB), in this scan range typically lo-14 molecules were detected. This is a smaller number than expected because assuming a laser spot diameter in the focal point of 5 urn, a film thickness of 0.5 urn and a given concentration of lo-’ g/g there should have been approximately 20 molecules in a 10 GHz scan range. We assume, that a certain percentage of the TBT molecules undergo fast spectral diffusion over a broad frequency range and cannot be detected as well resolved peaks above the background. In an earlier publication [lS], it was already mentioned that the background signal for the single molecule spectra of TBT in PIB was fairly large taking into account that thin spin-coating films with high optical quality were used. In the meantime, we performed experiments with a different PIB material which was prepared without the addition of any stabilizing agent. The background signal, however, did not decrease appreciably. We still assume that the background is due to a so far unknown impurity in the polymer. A small contribution to the background could also originate from TBT molecules that spectrally diffuse very fast over a broad frequency range. When the scan range was narrowed to l-2 GHz to zoom in strong features of the previous scan (10 GHz) a large number of molecules changed their absorption frequency irreversibly indicating spectral hole-burning. In these measurements the 10 GHz as well as the l-2 GHz frequency ranges were both always scanned in a time interval of 25 s. This translates into a higher fluence seen by the single molecules for the l-2 GHz laser scans and explains the increased probability of

J. Tittel et al. /Journal of Luminescence 64 (1995) I-l I

4

spectral hole-burning in this case. The following measurements could be performed only on stable molecules which survived the zoom-in procedure from 10 GHz to l-2 GHz. The fraction of unstable molecules was clearly higher for TBT in PIB when compared to similar measurements on single TBT molecules in PE. 3.2. Distribution of line widths Since the first single molecule experiments in polymers [lo] it is known that the line widths of different centers are distributed over a wide range. To get insight into this distribution for TBT/PIB and TBT/PE a large number of single molecule spectra were recorded. Lorentzians were fitted to the excitation line shapes to determine the line width. The distributions of line widths are displayed as histograms in Fig. 2. The data in Fig. 2 demonstrate that in both matrices a wide range of line widths was observed indicating spectral diffusion and dephasing processes which strongly depend on the specific single molecule, i.e., its local environment. It is obvious that measurements as presented in Fig. 2 strongly depend on the time scale of the experiment because all dynamical con-

010

,

,

,

(b)TBTiPIB

(a) TETIPE

’

1

10)

Optical Linewidth

m [MHz]

Optical Lmewidth

(MHz]

Fig. 2. Histograms for the distribution of line widths in (a) TBT in PE (113 molecules) and (b) TBT in PIB (173 molecules). The line widths are measured at T = 1.4 K in the wavelength range between 570 and 578 nm with an intensity between 3.5 and 7 mW/cm2. The probability density is defined as the probability of finding the linewidth of a molecule within a certain 15 MHz frequency interval. The inset shows the chemical structure of TBT (after Ref. [ 151).

tributions sampled during the time to scan over the molecular absorption line show up in the measured line width. Therefore, such distributions may look very different for different time scales of the experiment. In our experiment the data collected for both histograms were measured in the same time regime so that the histograms can be compared directly. The general features of both histograms in Fig. 2 were already discussed in a recent publication [ 151 where also the question was raised why the line width distributions in Fig. 2 look very similar, although the measurements were done in a semicrystalline (PE) and a completely amorphous (PIB) polymer, respectively. Before we try to answer this question let us briefly present some results from fluorescence spectroscopy of single terrylene molecules in PE [ 17, 181. These investigations revealed that when looking at the frequency of a totally symmetric Franck-Condon active in plane vibrational mode of terrylene two types of spectra were observed. Type I spectra showed a mode frequency of 243 cm- ’ and type II spectra a mode frequency of 212 cm- ‘. The authors assumed, supported by calculations, that the molecules giving rise to type I spectra are embedded in the crystalline region of PE while type II spectra originate from molecules in the amorphous regions of PE. In a recent investigation of terrylene in p-terphenyl single crystals [19] the respective mode frequency was found at 253 cm- ’ giving further support that type I spectra represent terrylene molecules in the crystalline environment of PE. Assuming that TBT molecules can also be embedded in the crystalline or amorphous parts of PE, respectively, we would expect different dynamical contributions to the line width depending on the surrounding. From this point of view the distributions of line widths as shown in Fig. 2 should be different for TBTjPE and TBTjPIB because in the latter case the environment for the TBT molecules can only be amorphous. The bulky tert-butyl substituents of TBT, however, may prohibit any crystalline ordering of the PE polymer chains or at least severely increase the disorder. Therefore all TBT molecules in PE are probably located in highly disordered or amorphous environments. As PE and PIB are both pure hydrocarbons, the principal type of interaction between the polymers and the chromophore should

J. Tittel et al. 1 Journal of Luminescence

be the same. Putting all the information together, the similarity of the line width histograms in Fig. 2 seems to be quite reasonable. 3.3. Investigations of spectral d@ision by measurements of autocorrelation functions We now want to present investigations which give more detailed information about the dynamics of TBT in PIB. These are measurements of the fluorescence autocorrelation function and peak frequency trend measurements together with the calculation of the two-point-frequency autocorrelation function (TPFCF). Using these data, it was possible to get insight into the rates of spectral diffusion in the time regime between milliseconds and seconds. In previous experiments in polymers [ l&13,14], these methods have already been used. To our knowledge, this is the first time that both kinds of investigations have been applied to the same system. First, we want to introduce and compare these two different types of measurements. To measure the fluorescence autocorrelation function, the laser was held at a fixed frequency, the resonance frequency of a given molecule, and the autocorrelation function was derived by a digital correlator which records all photon pairs in a time region between us and hundreds of seconds. The normalized fluorescence intensity autocorrelation function is given by [20]:

(1) where If is the fluorescence intensity. The temporal distribution of the fluorescence photons reflects the spectral diffusion dynamics of a single molecule. A TBT molecule absorbing the laser light at a fixed frequency emits fluorescence light. After a tunnelling process of a TLS in the matrix the resonance frequency of the molecule is shifted (spectral diffusion) and the fluorescence ceases. If the tunnelling process occurs in the backward direction the molecule is shifted back into resonance and absorbs again. The fluorescence light of such a molecule, shifting in and out of resonance, is emitted in bunches which are

64 (I 995) I-1 I

5

separated by dark intervals. The decay of the autocorrelation function itself is characteristic for the dynamics seen by the molecule. In previous investigations [12,13] the decay of the fluorescence autocorrelation function due to spectral diffusion was measured and single exponential decays, biexponential decays and logarithmic decays were reported. In our measurements we also observed autocorrelation functions showing single exponential steps as well as linear decays on a logarithmic time scale. In the case that a molecules dynamics is dominated by a single TLS the absorption frequency of the molecule changes between two distinct spectral positions. Such a molecule is called a two-state jumping molecule. As shown by Fleury et al. [13], in this case the correlation function is given by: gc2’(t) = 1 + Cexp( - R,t), R,, = RI2 + R21.

(2)

RI2 is the transition rate from state 1 to the state 2 and R2 1 is the transition rate from state 2 to state 1; C is the contrast. Fitting a single exponential to the correlation function, one can determine C and RO. In principle photon bunching caused by intersystern crossing into the metastable triplet state could also lead to a decay of the correlation function and interfere with spectral diffusion measurements. Recent experiments of terrylene in p-terphenyl [19] have shown that the intersystem crossing yield is extremely low for terrylene. For TBT we would expect similar internal photophysical dynamics and therefore at the power levels of the experiments reported here, the contrast in the correlation function due to intersystem crossing basically goes to zero. The second powerful method for investigating dynamical processes in polymers is to record the peak frequency trend of a spectrally diffusing molecule. The procedure to record this kind of data was described earlier in this paper. A peak frequency trend displays the spectral position of the resonance peak of a single molecule at a given time. As was proposed by Reilly et al., from such peak frequency trends the two-point-frequency autocorrelation function (TPFCF) can be calculated [S]. It is

J. Tiuel et al. / Journal of Luminescence64 (I 995) I-I I

6

given by:

0

(v(O), v(t)) =

2w

400

ml

800

1003 -

T

3.0

;tn;jv(r)v(r+

-

t)dr.

(3)

2.5

0

-

2.0

-

Because of our experimental conditions we can reduce Eq. (3) with Vi= v(ti), i = 1,2,3, . . . ,N; (Vi) E 0; t = m x 2.7 s to:

& y$r

(v(O),v(t)) = (vo, vm> =

ViVi+m.

-

N

C4)

As was shown by Reilly et al. [S] the TPFCF decays exponentially. By fitting a single exponential to the decay of the TPFCF one can determine the time scale of the spectral dynamics of a given molecule. While the fluorescence intensity auto-correlation function analyzes the fluctuations of the fluorescence intensity caused by spectral diffusion at a fixed frequency, the TPFCF analyzes the temporal evolution of the spectral peak position. It is clear that both correlation functions can be quite different for an arbitrary molecule with the exception of a two-state jumping molecule. In this special case the rates determined from both autocorrelation functions should be the same, since now the fluorescence intensity fluctuations caused by jumps into and out of resonance are triggered by exactly the same process as spectral jumps analyzed by the TPFCF. Besides the principle difference between the two types of autocorrelation functions, there is also the issue of the two different time scales of the corresponding experiments. The time resolution of peak frequency trends in this work was limited to 2.7 s. An increase of the scan rate would require shorter counting intervals and therefore would lead to an unacceptable decrease of the signal-to-noise ratio (SNR). On the other side it is difficult to measure the fluorescence intensity autocorrelation function in a time regime of seconds because this results in unreasonable long measurement times. Summarizing, it was difficult in the particular system TBT in PIB to measure both autocorrelation functions for one and the same single molecule. 3.3.1. Two-point-frequency-autocorrelation function We now first want to present peak frequency trend measurements and the corresponding

8 8 :z z a

3.0 2.5

-

2.0

-

Time[s] Fig. 3. Peak frequency trend of a single TBT molecule in PIB at different exciting laser intensities: (a) At I = 7.5 mW/cm* the visits 4 spectral positions (see arrows); molecule (b) I = 25 mW/cm*; 4 spectral positions; (c) I = 250 mW/cm2; 5 spectral positions.

TPFCF for a single TBT molecule at three different laser intensities. It is obvious from Fig. 3 that for this molecule the time resolution of 2.7 s was sufficient to analyze its spectral diffusion dynamics with the TPFCF. At an intensity of I = 7.5 mW/cm2 (Fig. 3(a)) the TBT molecule occupied 4 spectral the laser power to Increasing positions. I = 25 mW/cm2 (Fig. 3(b)), does not appear to change the dynamics appreciably. After raising the intensity to I = 250 mW/cm’ (Fig. 3(c)) the dynamics became faster and a fifth spectral position appeared. In Fig. 4 the calculated two-point-frequency autocorrelation functions are displayed together with exponential fits to the data. At I = 7.5 mW/cm’ and I = 25 mW/cm2 the rates determined were similar (R. = 0.07 s-l and R. = 0.08 s- I, respectively), which is qualitatively expected when looking at the trend data in Figs. 3(a) and (b). At I = 250 mW/cm2 the rate increases to R. = 0.33 s- ‘, which is clearly higher than in the previous measurements. These data show that by

J. Tittel et al. /Journal of Luminescence 64 (199s) l-l I

1 0

50

150 loo Time [s]

200

250

Fig. 4. Calculated two-point-frequency autocorrelation functions using the data in Fig. 3. A single exponential decay has been fitted to the data, showing an increase of rates R for increasing the exciting laser power: (a) I = 7.5 mW/cm*, R. = 0.07 s - ‘; (b) I = 25 mW/cm’, R,=O.O8s-‘; (c) I = 250mW/cm*, R. = 0.33 ss’.

increasing the exciting intensity the dynamics becomes faster and manifold. Similar trend data were observed for many different single TBT molecules and in most cases the jump rates increased with increasing laser intensity (see also [21]). 3.3.2. Fluorescence autocorrelation measurements In Fig. 5 the fluorescence autocorrelation functions of a single TBT molecule in PIB at four different laser intensities are shown. A part of these data has already been published [ 151, but here we will present a more detailed quantitative analysis of these data. At the lowest intensity (I = 2.5 mW/cm* (0)) the autocorrelation function shows a clear step between 0.1 and 1 s. A single exponential decay with a rate parameter R, = 1.5 s-l could be fitted to the measured data. Then the exciting intensity was raised to 7.5 mW/cm* (A). Now the autocorrelation function changed its shape: the step began to smear out towards shorter times. At the highest

10

loo

1000 10000 Time [ms]

Fig. 5. Normalized fluorescence intensity autocorrelation function g’*‘(t) for a single TBT molecule in PIB. The sequence of measurements at different intensities was: (I) I = 2.5 mW/cm* (0); (2) I = 7.5 mW/cm* (A); (3) I = 25 mW/cm’ (0); (4) I = 0.75 mW/cm* (+); (T = 1.4 K). The lines drawn through the measurement at I = 2.5 mW/cm’ and at (0) I = 0.75 mW/cm* (a) are the results of single exponential fits using Eq. (2) The lines drawn through the 2nd and 3rd measurement are fits obtained by averaging over a distribution of rates cc l/R plus a single exponential (Eq. (6)). The traces are shifted along the ordinate for clarity.

intensity (25 mW/cm*, Cl), the step almost vanished and the shape of the autocorrelation function comes close to a logarithmic like decay seen as a straight line on the logarithmic time scale. Finally, the intensity was lowered to 0.8 mW/cm* (+) and the step in the autocorrelation function reappeared. Here the rate was determined to be R,=0.7s_‘. It has been assumed [13] that whenever a single exponential step occurs in the autocorrelation function, the molecule is a spectral two-state jumper, which is dominated by a single TLS nearby in its amorphous environment. Since an exponential step in the fluorescence autocorrelation function was observed, we recorded a peak frequency trend to check if the molecule is a two-state jumper. In the present case, the peak frequency trend recorded at

8

J. Tittel et al. /Journal

-0.2’

8 0

0

a 200

200

8 400

400

2 600

600

of Luminescence

’ 3 ’ 800 1000 1200 1400

800

loo0

1200

1400

Time [s] Fig. 6. (a) Spectral peak frequency trend of the molecule which was also the source of the data in Fig. 5. The laser was repeatedly scanned over a frequency range of 1.7 GHz in 2.7 s. The frequency jitter was smaller than + 17 MHz. The intensity of the laser was I = 2.5 mW/cm2. The investigated single molecule visits numerous spectral positions. (b) Two-point-frequency autocorrelation function calculated according to Eq. (4) and using the data in (a). The lack of correlation indicates that the molecule is spectrally diffusing on a timescale faster than the time resolution (2.7 s) of the measurement.

the spectral peak frequency trend. There may be the possibility that the molecule jumps with rate R, = 1.5s’ between the frequency position 0 GHz and a second position outside the frequency range shown in Fig. 6(a). This, however, is unlikely because in this case we would have missed the frequency position of the molecule in many of the successive laser scans. Before we give a tentative explanation of the contradictory scenario - single exponential decay of the fluorescence autocorrelation function (single TLS) and jumping between a large number of frequency positions as seen in the peak frequency trend (many TLSs) - we will analyze the data taken at higher laser intensities. This will also help to understand the data at the lowest intensity. With increasing laser intensity the step in the fluorescence autocorrelation function (Fig. 5) began to be superimposed by a linear decay on the logarithmic time scale. This leads to the assumption that for higher exciting laser power a large number of TLSs with different rates were activated. Fluctuations of a single TLS lead to an exponential decay of the autocorrelation function. Now, when we stay within the picture of TLS transitions occurring spontaneously and independently from each other, we can write down the contributions of many fluctuating TLSs to the autocorrelation function as an average over a distribution of rates: 4 I”’

the intensity of I = 2.5 mW/cm* (Fig. 6(a)), at which a single exponential decay could be fitted to the fluorescence autocorrelation function, showed that the molecule is visiting a large number of frequency positions. The TPFCF calculated from this peak frequency trend is shown in Fig. 6(b) where it is seen that the data points are completely uncorrelated indicating a dynamical process which is faster than the time resolution of the peak frequency trend, i.e. faster than 2.7 s. This observation is consistent with the result of the fluorescence autocorrelation measurement shown in Fig. 5 (0) which was done at the frequency position 0 GHz in Fig. 6(a) and which gives an average jump time (r = l/R,) of z 700 ms. The jump rate in this case was just too fast to be analyzed quantitatively by

64 (1995) I-II

= 1 + C* exp( - Rt)P(R)dR, s

(5)

P

where C* is a normalizing constant, P(R) is the distribution of rates and p and q are the limits over which the integration was performed. In investigations of spectral diffusion in glasses with spectral hole-burning [8] and photon-echo experiments [22-241 two different types of distribution functions were established to interpret experimental data in the time regime between ms and s. These are a l/R rate distribution and a log-normal distribution of rates. As the l/R distribution gave a good agreement for a larger number of glasses, we prefer to use it here. In Fig. 7 we show a simulation of Eq. (5) using a increasingly broader distribution P(R) cc l/R.

J. Tittel et al. /Journal of Luminescence 64 (1995) I-11

9

or narrow rate distribution is additionally superimposed on a broad distribution of rates. The latter possibility, however, is very unlikely, because it is difficult to imagine to have some narrow distribution of rates superimposed on some broad distribution of rates whereby both distributions have to originate from the same reservoir of TLSs. Therefore, we fitted a decay with an average over a distribution of rates plus a single exponential to the data. The averaged autocorrelation function now becomes: Time [logarithmic, a.u.] Fig. 7. Simulation of the fluorescence intensity autocorrelation function with an increasingly broader distribution of rates according to Eq. (5). The distribution of rates was P(R) CCl/R. The ratio q/p was set to 1.1; 10; 100; loo0

gC2)(t)= 1 +

{exp( - Rt)P(R)dR}

The integration was solved numerically. For a l/R distribution p corresponds to the lower limit of rates (l/t,,_) and q to the higher limit (l/tmin). The lower integration limit p and the upper integration limit q determine the shape and the decay in the following way. The ratio of q/p determines the shape of the decay. A small ratio of q/p results in an exponential like decay because in this case the l/R distribution becomes naturally very narrow. For high values of the ratio q/p a logarithmic like decay is observed (Fig. 7). The absolute values of p and q are controlling the decay with respect to time. Qualitatively, the simulation in Fig. 7 shows a similar behaviour as the intensity dependent measurements in Fig. 5. It can be seen that a narrow distribution of rates makes the intensity autocorrelation function decaying like a single exponential similar to the decays in the measurements at I = 2.5 mW/cm’ (0) and I = 0.75 mW/cm’ (+). A broad distribution results in a logarithmic like decay which looks qualitatively very similar to the measurement at I = 25 mW/cm’ (Cl). Similar shapes were obtained by a simulation with a log-normal distribution of rates. The simulation is not shown here. We fitted Eq. (5) to the data shown in Fig. 5 using a l/R rate distribution. The results of these fits are not shown here, because it was not possible to get agreement between the fits and measurements on the entire data set. The disagreement indicated that formally a process with a single rate

here, R. is the rate of the single exponential decay and A and B are normalization parameters whose values depend on the contribution of each term to the contrast [13] of the autocorrelation function. We fitted Eq. (6) to the measured data with both types of distributions. Because of a rather crude numerical fitting procedure, the values of the fit parameters are connected with large errors which we cannot specify. The fits of Eq. (6) to the measured data using a distribution of rates a l/R are shown in Fig. 5 (4, Cl). These fits resulted in a much better agreement than those with a log-normal distribution. For the measurement at I = 7.5 mW/cm’ (+), R. was determined to be 2.2 s-i. The integration over the distribution was done in the limits of rates from 80 to 4OOOs-‘. For the measurement at I = 25 mW/cm2 (O), R. was found to be 6 s- ‘. The integration limits here were R = 60 to 5000 s-i. The difference in the values in the lower integration limit may be due to the fitting error. We also observed an increase of the rate of the single exponential decay from R,, = 1.5 s-l at I = 2.5 mW/ cm2 to R. = 6 s-l at I = 25 mW/cm2 (0). The values of all fitting parameters are listed in Table 1. In a recent publication [15] the contradictory scenario - single exponential decay of the fluorescence autocorrelation function and jumping between a large number of frequency positions - was tentatively explained by the assumption that the fluctuation times of the TLSs involved lie within

P

+

(Bexp(- ROE)),

(6)

10

J. Tittel et al. / Journal of Luminescence

Table 1 Numerical values of the parameters for fitting Eq. (6) to the measurements in Fig. 5. If the value of A is zero, Eq. (6) becomes a single exponential decay Laser intensity

B

0.75 mW/cm2 2.5 mW/cm’ 7.5 mW/cm’ 25 mW/cm’

0.07 0.7s-’ 0.096 1.5 s-l 0.0725 2.2 s- ’ 0.0825 6 s- ’

RO

A

P

4

0 0 0.0185 8Os~‘4OUOs-’ 0.021 6Os-’ 5OOOs-’

64 (1995) I-i I

intensity dependence of the autocorrelation function as seen in Fig. 5. This indicates that there may be a stable distribution of TLSs, the activation of which depends on the light intensity. The mechanism of the intensity dependence may be again thermal activation of TLSs by local heating near the chromophore through non-radiative decay of the optical excitation.

4. Conclusions the limits of a narrow distribution of rates. The more detailed analysis of the data from Fig. 5 as described above, however, has now led to the conclusion that at low intensities the molecules dynamics is mainly dominated by a single TLS. Obviously, the coupling to a single TLS or, to be more specific, a single rate can dominate the fluorescence autocorrelation function even when the spectral peak frequency trend shows that the molecule is influenced by many TLSs (Fig. 6). Increasing the exciting intensity, a broad distribution of TLSs is additionally activated. Our rather crude fitting procedures indicated that a rate distribution P(R) a l/R can describe the experimental data. Summarizing, the spectral diffusion dynamics of a single molecule can be interpreted as being influenced by a nearest neighbour TLS as well as by a large number of more distant TLSs. The relative contributions of both may differ from molecule to molecule and additionally may depend on the probing intensity. Therefore, the decay of the autocorrelation function can be described as a sum of a single exponential, the contribution of the nearest neighbour TLS, and an average over a distribution of exponential decays with different rates, the contribution of the distant TLSs. The experiments showed that by increasing the laser intensity not only a distribution of TLSs is activated but also the rate of the single exponential contribution increases. The latter effect was already observed in other investigations and its intensity dependence could be explained by a simple kinetic model assuming a photo induced mechanism as e.g. tunnelling in the excited state or activation by heat released in non-radiative transitions [13]. What seems to be remarkable is the reversibility of the

In this paper we presented an investigation of the dynamics of single TBT chromophores in an amorphous matrix of poly(isobutylene). We measured the line widths for many single TBT molecules in PIB and found a broad distribution giving evidence of the wide variety of locally varying dynamics seen by the probe chromophores. From similar measurements with TBT in semicrystalline PE we derived a distribution of line widths which was essentially the same. We concluded that the bulky tert-butyl substituents of TBT prevent any crystalline ordering of the PE polymer chains around the TBT molecules rendering the local environments in the two polymers, which are pure hydrocarbons, very similar. To get more detailed information on spectral diffusion, we also measured the peak frequency trajectories of spectrally diffusing single molecules from which the two-point-frequency autocorrelation function could be calculated. Additionally, the fluorescence intensity autocorrelation function was recorded which allowed the investigation of spectral diffusion in the time regime between ms and s. Using these methods, we studied in detail a specific single molecule, the dynamics of which at low intensity was mainly dominated by a single TLS and at higher intensities additionally by a distribution of TLSs. We could show that the logarithmic like decay of the fluorescence autocorrelation function at higher intensities could be reproduced by assuming the well-known l/R distribution of rates for the fluctuating TLSs. Additional work about the temperature dependence of spectral diffusion processes in TBT/PIB is now in progress.

J. Tiitel et al. /Journal

of Luminescence

Acknowledgements We would like to thank able hints on the sample also like to thank Prof. a sample of PIB. Financial Forschungsgemeinschaft ledged.

Dr. K. Fischer for valupreparation. We would Dr. M. Antonietti for support by the Deutsche is gratefully acknow-

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