Nonlinear Dynamics Part 2 doc

Nonlinear Dynamics 18 Fig. 11. The same experiment as in figure 10 but with an intermediate dye concentration (x/D = 1.1), and the same values of irradiation intensity. Here the solid lines are not fits, but theoretical plots of equation (14) for α = 60, 20 and 4 for the decreasing I 0 , respectively. [Serra & Terentjev 2008b] At a lower concentration of chromophore, corresponding to D ≈ 9.2 mm and x/D = 1.1 (the transmitted intensity is about 1/10 of the incident intensity), figure 11 shows the similar features of the non-linearity, which are especially evident at very short times. Again all curves start at the same I/I 0 ≈ 0.33. At higher irradiation intensities we achieve the saturation and the steady-state value I(x) corresponding to the solution of equation (6). The change of curvature, notable in figures 8 and 10, is not so clear here even at the highest I 0 . However, in the comparative analysis of data we now take a different approach. Assuming all the parameters for the curves are now known (γ and x/D from independent measurements, and α from the fitting in figure 10), we simply plot the theoretical equation (14) on top of the experimental data. It is clear that the theory is in excellent agreement with the data. Finally, we study the case of low dye concentration (D ≈ 91 mm, x/D = 0.14) in figure 12: this is also the case which is more relevant for biological spectroscopy studies, where the concentration of chromophore is usually small. In this exemplified case, the initial transmittance is very high: almost 85% of the incident light goes through the sample. Here, the complicated integral equation (14) simplifies dramatically, since at small x/D  1 the difference between A = ln 10 ln(I 0 /I) and x/D (which is the range of integration in (14), is also small. The integration can then be carried out analytically, giving (19) which gives in the stationary state the correct solution of equation (6) approximated at small x/D: Nonlinear Absorption of Light in Materials with Long-lived Excited States 19 Fig. 12. At low dye concentration (x/D = 0.14) the sample is relatively transparent. The data are for the same three values of irradiation intensity as in the earlier plots (but note that the I/I 0 axis starts from 0.8). The solid lines are theoretical fits for α = 60, 20 and 5. The inset shows the plot of exponential relaxation rate τ −1 against I 0 , with the linear fit. [Serra & Terentjev 2008b] The fits of the data for I(t) are again in good agreement with the full theory. More importantly, we also see that that the rate of the process described by the approximation (19) is given by the simple exponential, τ −1 = γ(1 + α) = γ + kI 0 . This is in fact the rate originally seen in the kinetic equation (2). Therefore, if we instead fit the family of experimental curves in figure 12 (and several other data sets we measured) by the simple exponential growth of the absorbance, we can have an independent measure of the relaxation rates obtained by this fit. The inset in figure 12 plots these rates for all the I 0 values we have studied. A clear linear relation between the relaxation rate and I 0 allows us to independently determine the molecular constant: The measurement of k, with high accuracy, gives the ratio k/γ ≈1 cm 2 s −1 μW −1 , which explains the fitted values of the non-dimensional parameter α = I 0 k/γ. The consequences of this nonlinear behaviour have in the last year raised an interested in some research groups who studied the azobenzene-based actuators. The original work by Corbett and Warner, in fact, focused only on the steady-state behaviour, could lead to accurate prediction about the effect of dynamic photobleaching on the bending angle of elastomers [Corbett & Warner 2007]. In fact, the dynamic photobleaching is the reason why heavily doped cantilevers, where the penetration depth is very small, can still bend if irradiated with sufficiently intense beams. Because the contraction of cantilevers is due to the force generated by the differential contraction of the top and bottom layers, if the light was propagating exponentially in the medium the bending would be impossible, because the thin layer where the light propagates is too small to generate enough force. A non- exponential propagation of light due to photobleaching, instead, can explain this effect. Subsequent work by Van Oosten, Corbett et al. [Van Oosten et al. 2008, Corbett & Warner Nonlinear Dynamics 20 2008] and White et al. [White et al. 2009] shown experimental evidence of this effect on the bending of cantilevers. Lee et al. also shown the nonexponential kinetics on a different azobenzene-based molecule [Lee et al. 2009]. 5. Absorption of fluorescent molecules. Because absorption spectroscopy is so widely used in biology, we want to show the effect of dynamic photobleaching on a biological molecule, and we chose chlorophyll, an important substance in biology (and in everyday life). Chlorophyll has a very recognisable absorption spectrum, which shows two clear peaks, one in the blue and the other in the red region (which procures its green colour) of the electromagnetic spectrum. It is also fluorescent in the far red and the characteristic lifetime of its excited state is about 4 ns [Hipkins 1986, Jaffe & Orchin 1962]. If it is irradiated by UV light or very strong visible light it undergoes a photo-chemical bleaching which degrades the molecule irreversibly and leads it to precipitate from solution, as many studies reported [Mirchin et al. 2003, Mirchin & Peled 2005, Carpentier et al. 1987]. We wish to observe a dynamical reversible bleaching due to the absorption of light, rather than this chemical degradation process. In the previous section, the theoretical model was verified in the case of azobenzene, a molecule with a very long lived excited state. Because the kinetics of transition could be followed by a spectrometer, it was also possible to model it with the kinetics law (equation 14). The model, as we said, does not make any hypothesis on the nature of the transition, and can therefore be extended to all “two-state” (or more realistically, to the simplified 3- state) systems. Fluorescent molecules have an excited state with a characteristic lifetime of a few nanoseconds, which is still much slower than the typical time of excitation. These characteristic times, though, are too short to be followed with conventional spectroscopes, and the transition kinetics cannot be followed as in the previous case. The model, however, also makes predictions also about the transmittance at the photostationary state, which differs from the LB law transmittance. To clarify, in figure 10 the Beer limit would be the transmittance at time zero, and the stationary state the transmittance at long times. It was important, for our experiments, to rule out all possible mechanisms leading to failure of LB law. As it was previously discussed, LB law has many limitations. It fails at high concentration of dyes, when they start to interact with each other and form aggregates; it fails if the stray light is high and the apparent absorption seems to reach a saturation level; it can fail at high intensity of the incident light if nonlinear effects like multiple photon absorption, or saturable absorption occur [Abitam et al. 2008, Correa et al. 2002]; it fails for highly scattering samples because the light is sent out at a non-zero angle. In order to rule out all these possible effects, we place ourselves in the most favourable experimental conditions: low concentration of dye and low illumination intensity. According to the model, the behaviour at the stationary state is described by equation 16. The important thing to observe is that the absorbance (or, equivalently, the transmittance) also depends on the intensity of the incident light I 0 . In order to experimentally verify this dependence, five different solutions of chlorophyll at known concentrations were measured at various light intensities. In this section all the absorbances will be reported in base-10 logarithmic form. Figure 13 shows the outcome of measurements of chlorophyll absorption of the same solution using different incident intensities. The result was striking: the change in the measured absorbance was very substantially affected by this parameter. Nonlinear Absorption of Light in Materials with Long-lived Excited States 21 Fig. 13. Absorption of chlorophyll in ethanol at the same concentration (in fact, exactly the same solutions) measured only changing the incident illumination intensity, I 1 = 6.5, I 2 = 13.1, I 3 = 27.5 μWm −2 s −1 . Some interesting consequences of this effect are shown in figure 14 and 15. The values correspond to the steady-state absorption at the peak wavelength. Indeed it is possible to see a strong dependence on the incident light intensity which is enhanced at high solute concentrations. A change in intensity of about 80% of the maximum value leads to a change in absorbance of about 50%. Figure 14 shows the dependence of the absorbance on the intensity at various concentrations. Equation 16 cannot be explicitly solved for A, but only for I 0 which gives the fits in the plot. Figure 15 shows the same data in the classical absorbance- concentration plot, for different intensities. It is important to remark that the experimental points can be satisfactorily fitted with a straight line in all cases (as the LB law says) but the line slopes are very different. Therefore the absorption coefficient may have different values if it is measured with a different light source. The exchangeability of results between different laboratories is thus in question. We obtained analogous results with Nile Blue, a simpler chromophore. We decided to test this dye, described in the Material and Methods section, because it has an absorption spectrum similar to chlorophyll in the red region, but it is a simpler and well studied molecule. This also proves that the results are general, and that aggregation phenomena which may occur in chlorophyll solutions (giving rise to scattering phenomena from still intact chloroplasts) are ruled out as a possible cause for the observed behaviour. All of the experiments were repeated several times and the behaviour was reproducible. Moreover, the intensity of light was increased and decreased alternatively to exclude the hypothesis of a chemical permanent photobleaching as a reason for absorption decrease. Due to the phenomenon of reversible (dynamic) photobleaching, a simple absorption experiment like the one described in the introduction is in practice impossible. The values of the absorption coefficients are meaningless if they don’t carry the information about the intensity of the incident light. Nonlinear Dynamics 22 Fig. 14. Absorption of chlorophyll as a function of the intensity of incident light. One can see an increase of absorbance at low intensities. The values are reported for five different concentrations. In the figure, the black dotted line corresponds to the intensity of the incident light of a commercial “traditional” spectrophotometer (Cary-UV-Vis). This comparison is done in order to show that the range of intensity of our set-up is the same as a more conventional one. m Fig. 15. Absorption as a function of concentration for the different values of incident light. All the lines have a good “Lambert-Beer” linear form but different proportionality coefficients. The LB limit was extrapolated from the ideal limit of zero intensity. In light of this, can we use the theoretical model to find a new method to determine concentrations using absorption spectroscopy, removing this dependence on the incident light intensity? Looking at equation 15, knowing the ratio of the concentrations of two solutions makes it possible to measure the combined ratio of parameters α = (k/γ)I 0 . If one solution has an unknown concentration c 1 and another solution is obtained by a dilution of Nonlinear Absorption of Light in Materials with Long-lived Excited States 23 the first one, so that c 1 /c 2 = r, measuring the absorbance of the two solutions 1 and 2, at the same incident light intensity and the same path length x, one obtains: (20) then (21) Knowing α, one can simply determine the unknown concentration c 1 as (22) The relation yields the ratio c/δ, and therefore the knowledge of the absorption coefficient δ −1 is required. On the other hand, the same method allows determination of δ once the concentration is known. We took a series of concentrations of Nile Blue solutions and the corresponding absorbance measured at different intensities of incident light. Taking pairs of measurements of absorbance at known concentration, at the same value of light intensity, we extracted the value of the parameter r from each pair, and from that we calculated α and ε according to equation 21 and 22. Averaging over all of them, we obtained the value ε = 120000 ± 20000M −1 cm −1 . The literature reports ε = 77000M −1 cm −1 , but we attribute this to the fact that, at intensities greater than zero, the absorption values are always systematically smaller (see figures 14 and 15). Therefore the deviation from the literature value is still consistent with our findings. The limitation of this method is that it very sensitively depends on the value of the concentration ratio, and therefore the errorbars are quite large. One should not, in fact, rely on the value of α measured only with one pair of measurements. Figure 16 shows the dispersion of the estimate values of α obtained using different pairs of values. The strong dependence of the parameter α on the value of r is evident from formula 21. It is possible to see that the function is divergent when but this happens when r ≈ A 1 /A 2 , which is exactly the region of interest. For this reason, the values of α are very scattered (some of them are even negative, which is physically meaningless), and this formula, although it is correct is principle, is hardly applicable to real experimental data. 6. How to use absorption spectroscopy In order to overcome this disadvantage, a more robust method is suggested to determine the value of the absorption coefficients. The strong dependence of α on r obviously remains, because it comes directly from formula 21. A method based on a linear regression can be used instead to calculate δ from a series of absorbances and known concentrations. Also, a series of measurements at known δ allow the evaluation of a substance concentration, just like in traditional absorption experiments. Rearranging equation 15, one obtains: (23) Nonlinear Dynamics 24 Fig. 16. Values of α obtained for three different values of light intensity, using various pairs of measurements, according to the suggested method. It is important to notice that the values of α are systematically higher at higher intensity, as expected. As one would expect from equation 21, the values of α are very scattered and therefore the calculation of ε is not precise: this is due to the strong dependence of α on r around the point where r ≈ A 1 /A 2 . If a set of concentrations and relative absorptions are known, one can plot the quantity c(1 − 10 − A ) −1 as a function of Aln(10)(1 − 10 − A ) −1 . The result is a line whose slope is δ/x and whose intercept is δ/xα. It is therefore possible to determine all the important parameters. Figure 17 shows this plot obtained for a set of Nile Blue dye. Promisingly, all the lines obtained with this method are well fitted with parallel lines, which indicates that they all converge to the same value of ε. Several lines indicate several values of incident light intensity. From the plot one can find the parameter α for all the intensities, and also δ, which Fig. 17. Application of the suggested method to determine the relevant parameters α and δ. The slope depends on δ, which is the same for all the samples, but the intercept depends on α which changes with the intensity of light. Nonlinear Absorption of Light in Materials with Long-lived Excited States 25 is simply the slope of the lines. Once δ is known, one can then determine, for x = 1, ε = δ −1 , the molar absorption coefficient. Following classical error analysis we obtain ε = 117700±120M −1 cm −1 . The agreement with the previous method is good and this second method has the advantage of greatly reducing the experimental errors. Whilst this method appeared highly successful at a first glance, we discovered that plotting the values of α against the intensity of the incident light, measured from the number of counts on the spectroscope, generated a relation which is not linear. According to the theory, α is simply the product between the incident intensity and some characteristic constant of the material, therefore the nonlinearity shown in figure 18 is not acceptable. Fig. 18. The parameter α extracted from the intercepts of figure 17 as a function of the intensity of the incident light. The theoretical model predicts a linear relationship, which is not what is observed in the figure! Considering the possible causes of this discrepancy, one can see in the model that stimulated emission is completely neglected. Neglecting the light-stimulated back-transition to the ground state was reasonable in the case of azobenzene, where the trans and cis peaks were very far apart, but for fluorescent molecules the same light excites both transitions and this factor should therefore be considered. The calculations become more complicated but the procedure is the same as that described in the introductory section of this chapter. One should now return to the kinetics equation, which we re-write here for simplicity. We call n the fraction of molecules in the ground state and k b the rate of the stimulated back transition. (24) At the stationary state, the left hand side of the equation is zero and n becomes (25) This is the value which should be inserted in the expression for the photobleaching 4, giving (26) Nonlinear Dynamics 26 This can be simplified by dividing by k, introducing the parameter ϕ = k b/k and integrating the equation Note that earlier, neglecting the stimulated back-reaction, we essentially had ϕ = kb/k→0. While the integration on the right-hand side is trivial, the left-hand side splits into a sum The integration gives Final simplification leads to: (27) It is convenient here to reintroduce our usual non-dimensional parameter α = I 0 k/γ (28) This expression is the full and general result. In many cases we expect ϕ to be small, so the expansion at the first order correctly recovers the usual expression 6. Expansion to the second order, instead, gives (29) Using this equation, the fit to the experimental data improved. The expression was readapted to take into account the base-10 logarithm of the absorbance. The parameter space was restricted because we expected δ and α to be in the same range as previously determined. The best fit to the curves was obtained using δ = 8.75 ⇒ ε = 114000M −1 cm −1 and ϕ = 0.3. The value of ϕ, quite substantially greater than zero, is consistent with the need to modify the original equation. Figure 19 shows the concentration c on the y-axis and the absorbance A on the abscissa (different curves for different light intensities): this is because formula 29 can be easily inverted. The values of α, obtained by fitting, increase linearly with the incident intensity, as shown in figure 20. This is good evidence that the stimulated emission cannot be neglected: the theory, thus modified, can well reproduce the experimental data. 7. Conclusions The most important conclusion of our work is that one has to be cautious with the classical concept of light absorption, represented by the Lambert-Beer law. Even without considering Nonlinear Absorption of Light in Materials with Long-lived Excited States 27 Fig. 19. Fitting of the absorbance/concentration curves at different intensities, obtained with the model which considers the stimulated back-transition. Fig. 20. The parameter α extracted from the fit in the figure above as a function of the intensity of the incident light. In this case we observe the linear relationship with the correct intercept in the origin. multi-particle effects at high concentration or multiple photon absorption, even at very low concentrations (corresponding in our case to the low x/D ratio) the illumination intensity above a certain crossover level would always produce a non-linear dynamical effect equivalent to the dynamic photo-bleaching, which increases the effective transmittance of the sample. We emphasise that this is a totally reversible phenomenon, unrelated to the chemical bleaching, which involves irreversible damage to the material. The crossover between linear and strongly non-linear regimes is expressed by the non-dimensional parameter α = I 0 k/γ and is, therefore, an intrinsic material parameter of every chromophore molecule, but not dependent on the dye concentration. Note that the thermal cis-trans isomerization rate γ is strongly temperature dependent, influencing the crossover intensity. Azobenzene is an ideal molecule for this kind of study, because it allows investigation of the transition kinetics using a simple spectrometer. The experimental data we obtained confirm the predictions of the theoretical model, which provided a satisfactory fit to the data. At high illumination intensity one finds a characteristic sigmoidal shape. 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