## 1. Introduction

The absorption of light is an important phenomenon which has many applications in all the natural sciences. One can say that all the chemical elements, molecules, complex substances, and even galaxies, have their own “fingerprint” in the light absorption spectrum, as a consequence of the allowed transitions between all electronic and vibronic levels.

The UV-Visible (UV-Vis) light (200-800 nm) has an energy comparable to that typical of the transitions between the electrons in the outer shells or in molecular orbitals. Each atom has a fixed number of atomic levels, and therefore those spectra are composed of narrow lines, corresponding to the transitions between these levels. When molecules and macromolecules are considered, the absorption spectrum is no longer characterised by thin lines but by wide absorption bands. This is due to the fact that the electronic levels are split in many vibrational and rotational sub-levels, which increase in number with the increasing complexity of the molecules. IR spectroscopy is often used to investigate these lower energy modes, but for very complex biological molecules not even this technique can resolve each line precisely because the energy split between the various levels is too small. One possible way to obtain higher resolution spectra is to lower the sample temperature, in order to suppress many of the vibrational and rotational modes. For biological molecules, though, lowering the temperature can be a problem if one wants to study, for example, the activity of enzimes, which only work at physiological temperatures. One of the advantages of absorption spectroscopy (IR and UV-Vis) is to be a non-disruptive technique, also for “delicate” molecules like polymers and biomolecules.

In the process of light absorption by molecules, once a photon with the right energy is absorbed, the molecule goes into an excited state at higher energy [Born and Wolf 1999, Dunning & Hulet 1996]. Eventually, it spontaneously returns to the ground state, but it can relax following several mechanisms. When excited, the molecule reaches, in general, one of the sub-levels of a higher electronic state. The first process is then, generally, a relaxation to the lower energy state of that electronic level (schematised in figure 1). This process is usually very fast (in the femtosecond scale) and not radiative. From this level, there are several pathways to dissipate the energy: a radiative transition from the lower level of the excited state to the ground state (fluorescence), accompanied by the emission of a photon at lower energy than the absorbed one; a flip of the electronic spin, which leads to a transition between singlet and triplet state (intersystem crossing), often associated with another

radiative process (phosphorescence); a non radiative decay where the energy is released by heat dissipation. In some molecules the relaxation pathway following the excitation is more complex, and it can involve interaction with other molecules. In such cases the energy can be transferred to other molecules via radiative or non radiative processes: azobenzene, for example, is a photosensitive molecule which, after excitation, undergoes a conformational change; a more common molecule, like chlorophyll in plant cell chloroplasts, transfers the excitation to the neighbouring molecules until the energy reaches the photosynthetic complex where the photosynthesis takes place.

The common characteristic shared by fluorescent molecules, molecules with a triplet state and photosensitive molecules like azobenzene, is that the lifetime of the excited state is long compared to the time it takes for the excitation to occur. This brings us to the subject of this chapter, which deals with a phenomenon, closely associated with the lifetime of the excited state, which we called “dynamic photobleaching”. In general usage, the term “photobleaching” has been taken to refer to permanent damaging of a chemical, generally due to prolongued exposure to light. Here, we will not consider this, but rather a reversible phenomenon whereby the number of molecules in the ground state is depleted as a consequence of the long lifetime of the excited state.

This effect has important consequences for UV-Visible spectroscopy measurements. In practical use, UVVis light absorption experiments are simple and straightforward: a collimated beam of light is sent onto a sample, the transmitted light is collected by a spectrometer and the ratio between the incident and the transmitted light is measured. Its simplicity means that this technique is widely used in many areas of science. The information one can get from these measurements concerns the allowed electronic transitions. On the other hand, once the electronic structure of a substance is known, computer simulations are able to reproduce absorption spectra.

A very common use of UV-vis spectroscopy is to measure the concentration of substances, and this requires the celebrated Lambert-Beer (LB) law. This semi-empirical law states that the light propagating in a thick absorbing sample is attenuated at a constant rate, that is, every layer absorbs the same proportion of light [Jaffe & Orchin 1962]. This can be expressed simply as the remaining light intensity at a depth *x* into the sample is: *I*(*x*) = *I*_{0} exp(−*x/D*) where *I*_{0} is the incident intensity and *D* is a characteristic length which is called the “penetration depth” of a given material. If an absorbing dye is dispersed in a solution (or in an isotropic solid matrix) this penetration depth is inversely proportional to the dye concentration. In this way it is possible to determine a dye concentration *c* by experimentally measuring the absorbance, defined as the logarithm of intensity ratio

where *x* is the thickness of the sample (the light path length), *D* is the penetration depth, *c* the concentration of the chromophore, and *δ* the universal length scale characteristic of a specific molecule/solvent. One should note that in chemistry and biology one often uses base-10 logarithm in defining the Absorbance, rather than the more intuitive natural logarithm. If *c* is in molar units, the constant of proportionality *ε* is the “molar absorption coefficient” and it is inversely proportional to the characteristic length *δ* defined above.

The derivation of this empirical law is straightforward. It assumes that the fraction of light absorbed by a thin layer of sample (thickness *dx*) is proportional to the number of molecules it contains (see figure 2), expressed as the volume fraction *n* times the volume of the thin layer (*Area*
*dx*)

where *I* is the intensity of the incident light. Introducing the cross section *σ*, which is a measure of the probability of a photon being absorbed by a chromophore, the differential equation becomes

Solving the equation from 0 to *x* (total thickness of the sample), with a light *I*_{0} incident on the front of the sample, one has

and we obtain equation 1 (rearranging the units opportunely).

Thanks to the Lambert-Beer law, UV-visible absorption spectroscopy is a useful and practical tool in many areas of science [Serdyuk et al. 2007]. The technique is widely used in organic chemistry and biology, as macromolecules often have a characteristic absorption in the UV and, more rarely, in the visible region of the EM spectrum. For example, all proteins have a characteristic absorption band around 190nm, due to the molecular orbital formed by the peptide bond, and another band around 280nm due to the aromatic side chains of aminoacids. Usually, this band is used to determine the concentration of proteins in a compound. Nucleic acids also absorb in the UV region and have a strong absorption band at 260 nm. The ratio between the absorption peak at 260 and 280 nm can give information about the relative quantity of DNA and protein in a biological complex, like ribosome. In atmospheric sciences, absorption spectroscopy is used to identify the composition of the air [Heard 2006 ]. Because the concentration of the species is very low, the light path must be very big to yield a detectable signal. Because *L* is so large and the concentration can change over the long range, a generalised Lambert-Beer law is preferred:

where *σi* is the absorption cross section of each species *i*. Visible absorption can even be applied as a diagnostic tool. In medicine, for example, it is used to measure microvascular hemoglobin oxygen saturation (StO2) in small, thin tissue volumes (like small capillaries in the mouth) to identify ischemia and hypoxemia [Benaron et al. 2005].

All these applications rely on the validity of the LB law. However, this empirical law has limitations, and deviations are observed due to aggregation phenomena or electrostatic interactions between particles. The simpler form of the LB law also fails to describe the two-photon absorption and the excited state absorption process, and it must be substituted by a generalised Lambert-Beer law [Nathan et al. 1985]. These phenomena are usually present only at very high incident light intensity. Also, highly scattering media, very relevant for the medical and geological applications, produce large deviations from LB law.

This chapter addresses the topic of deviations from the LB law occurring in photosensitive media due to self-induced transparency, or photobleaching [McCall & Hahn 1967, Armstrong 1965]. This effect has been reported in a number of different biological systems such as rhodopsin [Merbs & Nathans 1992], green fluorescent protein [Henderson et al. 2007] and light harvesting complexes [Bopp et al. 1997] stimulated with strong laser radiation.

In figure 1, we showed how the excitation/disexcitation of a molecule is essentially a 3-state (or more!) process. Some of the energy loss, however, occurs very quickly and only involves vibrational levels. Considering the different time scales, one can simplify this into a 2-state model: an excitation process which promotes the molecule into a long-lived metastable state and its relaxation to the ground state. The origin of the long life of the metastable state depends on the particular system under study. In the case of spin flip of the excited electron, the physical reason underlying the stability of the triplet state is to be found in the selection rules, which practically forbid the transition between two different spin states (excited triplet state- ground singlet state). This process has raised a vivid interest in the scientific community in the last few decades, because triplet state is often a big problem in organic semiconductor devices [Wohlgenannt & Vardeny 2003]. Alternatively, the molecule, excited by light, gets “trapped” in a metastable state, separated from the ground state by an energy barrier. This is the case for azobenzene, a small molecule which exists in two different forms (isomers *trans* and *cis*). The transition between the two isomers requires breaking a double bond. UV light with a certain energy induces this double-bond breakage and lets the molecule rotate around its axis; with a certain probability, the bond will reform when the molecule is in a metastable *cis* isomer. The relaxation to the ground (lower energy) state can only happen if there is enough energy to break the double bond again. This can occur if the molecule is excited with a light at a different wavelength, or if the thermal fluctuations provide the molecule with enough energy to overcome the energy barrier and return to the ground state. The thermal relaxation is very slow and the characteristic lifetime depends on the nature of the chromophore and of the surrounding environment. This is a classical Kramers problem of overcoming an energy barrier (the breakage of the double bond) between the metastable and the ground state. In the case of this simple molecule, the Lambert-Beer law is no longer accurate because of a phenomenon which we call here “dynamic photobleaching” or saturable absorption. It means that the photons which shine on a sample are absorbed by the chromophores in the first layers. If these molecules don’t return to their ground state immediately, when new photons fall on the sample they can’t be absorbed anymore in the initial layers and therefore propagate through the sample with a sub-exponential law. So, the effective photo-bleaching of the first layers allows a further propagation of light into the sample and this leads to nonlinear phenomena which are interesting both from the theoretical [Andorn 1971, Berglund 2004, Statman & Janossi 2003, Corbett & Warner 2007] and from the experimental point of view [Meitzner & Fischer 2002, Barrett et al. 2007, Van Oosten et al. 2005, Van Oosten et al. 2007].

The aim of this chapter is to explore the effect that this phenomenon has on the typical absorption measurements which are commonly performed on these kinds of molecules. We will propose a new theory which can mathematically describe this effect and then we will give experimental evidence of its validity both on azobenzene, a molecule with a very long-lived excited state and whose kinetics of transition can be followed, and on more common fluorescent molecules, like chlorophyll, focussing on the absorption of light at equilibrium.

## 2. Materials and methods

### 2.1. Azobenzene

The molecule 4’-hexyloxy-4-((acryloyloxyhexyloxy)azobenzene (abbreviated as AC_{6}AzoC_{6}) was synthesized in our lab by Dr. A.R. Tajbakhsh. Its molecular structure is shown in figure 3 and its synthesis is described in [Serra & Terentjev 2008a]. All azobenzene-based molecules exist in two isomers, *trans* and *cis*: the transition between the trans isomer, more stable, to the cis isomer, metastable, is stimulated by UV light, while the opposite reaction can be spontaneous. The isomers of the described molecule are shown in fig. 3

The two isomers of this molecule absorb light at different wavelengths: the trans isomer has a peak around 365 nm, while the cis isomer absorbs at 440 nm. It is thus possible to monitor the kinetics of *trans-cis* transition.

Monitoring a conversion process in real-time presents difficulties for a traditional spectrophotometer, because measurements over the whole spectrum of wavelengths take a long time, and moreover it is often difficult to access the sample in order to provide the UV illumination for isomerisation. For this reason, we chose a spectrometer equipped with a CCD camera, which is able to collect signal across the whole visible spectrum simultaneously. This technique works by illuminating the sample with white light; a system of gratings then splits the transmitted light into its various spectral components, whose intensity is measured by an array of photodiodes. This type of spectroscope does not require a fixed or enclosed sample holder, therefore placing another source of illumination close to the sample is easy.

For the measurements of light absorbtion a Thermo-Oriel MS260i (focal length 260mm) spectrometer was used. The apparatus consists of a quartz probe lamp with an adjustable slit, a quartz cuvette with 1cm optical path, an optical liquid lightguide to conduct the light from the cuvette to the spectrometer, a 50 *μ*m slit at the entrance of the spectrometer, and the Andor linear-array CCD camera connected to a computer. The simultaneous measurement of all spectral frequencies allows for a response as fast as 0.021 s and the possibility of reducing noise by averaging over many measurements. Before every absorption spectrum, a background and a reference spectrum were collected: the background is the spectrum collected without the illumination from the probe lamp, and the reference was the spectrum collected with the probe lamp illuminating the cuvette filled with solvent (without the chromophore dye). The absorbance was then calculated from the counts of the detector as:

For all the experiments, it was important to verify that the linear relation between absorbance and concentration held for the value of absorbances considered. It was shown that the absorption-concentration relation was linear below *A* ≈ 1.2 in the base-10 defined absorbance. At the intensity used for this experiment, for a concentration *c* expressed in moles, the penetration depth at 365nm was *δ* = *c* * 480nm.

At higher concentrations, the linearity fails because of various phenomena, including aggregation of the molecules (especially with molecules like proteins or polymers), the scattering of light from big particles and stray light in the spectroscope.

We provide monochromatic illumination to stimulate the isomerisation of azobenzene using a Schott KL 1500 LCD lamp, placed at 90 degrees with respect to the incident probe light and the optical fiber that collects the light from the sample. In this way the cuvette is irradiated with UV light while the absorption spectrum is recorded along the perpendicular beam path, so the absorption can be followed in real time without interference of the illumination light. The intensity of the monochromatic light was in the order of a few tens of *μ*W cm^{−2}.

All isomerisation reactions were followed for 90 minutes, which was a sufficient time for reaching the respective photostationary states. After this, the lamp was switched off and the absorbance was measured during thermal isomerisation in the dark. An example of spectrum measured with this technique is shown in fig. 4.

### 2.2. Other chromophores

Chlorophyll was extracted from *Commelina Communis* leaves[1] -
. The leaves were first boiled in distilled water, in order to kill the enzymes which digest chlorophyll once the leaf is cut from the plant. The leaves were then dried and ground up with a pestle, with a few drops of acetone, and then left in a 50% hexane/water mixture (the hexane forms a layer on top of the water), to separate the chlorophyll from the water-soluble compounds (vitamins, etc). The extracted solution was filtered to avoid impurities, like dust particles or even intact chloroplasts which could be responsible for light scattering effects. The whole extraction process was carried out in the dark. This method of extraction does not allow the separation of chlorophyll from the carotenoids which could be present in the leaves. However, the collected spectrum shows that the stronger absorption bands are those of chlorophyll, and this means that the other compounds are present only in low concentrations. Moreover, for the purpose of our experiment, an highly purified chlorophyll is not needed, because the analysis focusses on the absorption band around 660nm, far from the absorption band of carotenoids (blue region). What is important to remark is that there are different kinds of chlorophyll, whose relative content varies from species to species. The two main chlorophyll components, called “chlorophyll a” and “chlorophyll b”, differ by a carboxylic group, attached only to the porphyrine ring in chlorophyll b. The two molecules have slightly different absorption spectra, but this is not relevant for the experiments, provided that the plant species from which the chlorophyll is extracted is always the same (and has therefore the same percentage of chlorophyll a and b).

From the discussion above, it is clear that chlorophyll is a very special molecule, and has many peculiarities. In order to demonstrate that the theory is more general, a commercial dye with a strong absorption in the visible region is also investigated. Nile Blue A, a dye commonly used for staining DNA, but with spectral properties similar to chlorophyll, was selected. The chemical structure of chlorophyll and Nile Blue is shown in figure 5.

The spectroscopy experiments were conducted with an Ocean Optics USB 4000 spectrometer, equipped with optical fibers. A 25W halogen lamp with spectral range 400-880 nm, whose intensity could be tuned, was used as illumination source. The light was focussed with collimating lenses onto a 1 cm cuvette containing 3 ml of solution.

For comparison with more “conventional” spectroscopes (meaning, with a fixed sample holder and a fixed intensity of incident light), a Cary UV-Vis Spectrophotometer was used to measure spectra and absorbances of the two substances at various concentrations. The intensity of the incident light from the spectrometer was also measured with a power meter.

In order to measure the intensity of the incident light, key quantity in our experiments, the light from the source was shone onto the detector directly, in the absence of any sample or cuvette in between. Knowing the characteristic response of the spectrometer detector, it is possible to measure the intensity of light. Three different values, the number of counts at a single wavelength or the integral of the intensity over the range of wavelength which correspond to the absorption peak, and the integral over all the wavelengths could be used to quantify the incident light intensity. In all cases the outcome of the experimental results was the same. Using as a value of intensity the intensity at the single peak-wavelength made it possible to compare it to the conventional spectroscope (which produce monochromatic light). It was verified that the detector had a linear response in counts versus intensity over the range of intensities we used.

For the absorption spectra, measured as *A* = log_{10}(*I*_{0}*/I*) reference spectra for the pure solvent were taken before each measurement at every light intensity. The absorption and fluorescence spectra of these materials are shown in figure 6. We chose to refer all the absorption values to the wavelengths of the peaks in the yellow-red region.

For each solution, the linearity of the absorption/concentration curve was verified, in order to avoid falling into a trivial nonlinear regime. The experiments were conducted in random order of light intensity, and the reproducibility was verified. The absorption of each chlorophyll solution at 660 nm was stable over a range of hours at constant incident light intensity, indicating the absence of chemical irreversible bleaching. Fluorescence from the dye was also ruled out as a possible source of disturbance, because at the light intensity we used it is not detectable with our equipment.

## 3. Theory

Here we present a description of the dynamical photobleaching effect in the case of azobenzene isomerisation, which was previously discussed. We will then generalise the discussion to all the molecules with a long-lived excited state, and show how this affects the measurements of light absorption.

The non-Lambertian propagation of light through a medium has important consequences for the analysis of photo-isomerization kinetics: when the photo-bleaching becomes important, the measured absorbance no longer follows a simple (traditionally used) exponential law. In photosensitive molecules like azobenzene, irradiation with light of a certain wavelength induces a conformational change (isomerization) from an equilibrium *trans* state where the benzene rings are far apart to a bent *cis* state where they are closer. The isomerization process follows first-order kinetics. Calling the fraction of molecules in the two states *trans* and *cis nt* and *nc* we have

where *I* is the intensity of light, *k* is the *trans-cis* isomerisation rate, *kb* the stimulated back transition rate (*cis* to *trans*), and *γ* the thermal relaxation rate. In the experiments on azobenzene described below we use an illuminating light monochromated at the *trans-cis* transition wavelength. In this case the stimulated *cis-trans* isomerization is negligible (that is, *Ikb* → 0) and, remembering that *nc* = 1 − *nt* the kinetic equation reduces to

In this equation the intensity *I* = *I*(*x*) is the light intensity at a certain depth into the sample. It is convenient to define a non-dimensional parameter *α* = *I*_{0}*k/γ*, which represent the balance of photo- and thermal isomerization at a given incident intensity *I*_{0}. In this notation, the amount of molecules in the *trans* conformation in the photostationary state, when the balance between *nt* and *nc* is stable, and therefore *dnt/dt* = 0 the equation reduces to simply

therefore

To express mathematically the reversible photobleaching phenomenon, it can be assumed that the change in light intensity across a thin layer of sample (thickness d*x*) is proportional to the number of molecules which are excited, i.e. the number of chromophores which absorbed light in a small volume of sample of thickness *dx*; neglecting the stimulated *cis-trans* isomerization (which is appropriate in our study), the model can be much simplified to give, per unit time:

Then, combining all the parameters, such as the photon cross section and the transition rates, we recover the penetration depth *D* = *δ*/*c* as the relevant parameter of the relation, and the final expression is:

with *D* the penetration depth, inversely proportional to concentration. In order to study this problem at the photostationary state, one needs to combine the equations (4) and (3).

Solving the differential equation, the stationary-state light intensity at a depth *x* is given by the relationship [Corbett & Warner 2007]

Looking at the equation above some important insights can be gained. The most important is that in the limit *α* = 0 the equation reduces to the Lambert-Beer law, i.e. an exponential decay in the transmittance through the medium. Therefore all the nonlinearity is included in *α*. The opposite limit, when *α* is very big, leads to a linear relation between transmittance and sample thickness. Figure 7 is a representation of equation 6 and it shows the intensity variation *I*(*x*) for several values of the parameter *α*: from the plot of transmittance as a function of *x*, it is clear that, if the incident intensity, and therefore *α*, is low enough, the Lambert-Beer law is valid and the decrease is exponential, but if the incident intensity is high the bleaching of the first layers becomes progressively more relevant such that they become partially transparent to the radiation. The decay thus tends to become linear in the bulk of the sample, *I*(*x*) ≈ *I*_{0}(1 − *x*/*αD*).

In order to model the dynamics of photoisomerisation, which is evidently inhomogeneous across the sample, it is not enough to model photobleaching with equation 6, but instead the equations (2) and (4) should be coupled. Calculating a time derivative of equation 4 leads to

In the right hand side expression, equation 2 can be substituted, giving

This equation can be solved with the following method [Corbett et al. 2008]. Introducing the variable *y* = ln(*I*/*I*_{0}), which is a very sensible variable, being also the inverse of the absorbance, the left-hand side is greatly simplified and one obtains

Also, from equation 4

Substituting *nt* in equation 9 and rearranging, one finds

In the next step, one has to keep in mind that

Rearranging equation 11 and exchanging the order of derivatives on the left-hand side, the equation reduces to

It is now possible to integrate this expression. Integrating between 0 and *x*, the integral and the derivative on the left-hand side cancel out and one finds:

The factor *γα* comes from the solution of the definite integral for *x*=0 (the lower integration limit). In fact, if *x*=0, *I* = *I*_{0} and therefore *y*=0 by definition.

The last step is a time integration. At time zero, the absorbance *A* = −*y* must be equal to the Lambert-Beer law value *x*/*D*. Including these considerations, the integral expression for the intensity *I*(*x, t*) becomes:

The upper limit of this integral is the measurable absorbance *A* = ln[*I*_{0}/*I*(*x, t*)] from a sample of thickness *x*. Figure 8 shows a simulation predicting the time-evolution of intensity transmitted along the path *x/D*. Note that at *t* = 0 all curves converge to the Lambert-Beer *I/I*_{0} = exp(−*x/D*), while at long times a significant portion of chromophore is bleached and the transmitted intensity increases.

We should note that the problem of non-linear photo-absorption dynamics is not only restricted to azobenzene isomerisation. Even ordinary dye molecules that do not undergo conformational changes stimulated by photon absorption, still follow the same dynamic principles, but with electronic transitions in place of *trans-cis* isomerization. Therefore, the results of this paper should be looked upon as widely applicable to other systems. In particular, the two key conclusions, that the crossover intensity into the non-linear photo-absorption regime is independent of dye concentration and that the rate of the transition is independent on solvent viscosity, are probably completely general.

The model we propose only assumes a two-configuration system, and it does not imply anything about the nature of the two states. Therefore, it is important to verify that this model has a wider and broader validity, and, in detail, that it helps to understand the behaviour of a large class of chromophores, like fluorescent molecules. Azobenzene molecule exists in two physical states, *trans* or *cis*; for chlorophyll, one could make an analogy and, considering the electronic transition, call the two states “ground” and “excited”, we still find the same formula at the photostationary state

where *x* is the path length of light through the sample. It is important to see here that the absorbance has a nonlinear dependence on the incident light intensity. The limits where the LB law is recovered are either very low incident intensity (practically, it can never be achieved) or a very fast recovery to the ground state compared to the excitation.

Equation 15 can has important implications for the interpretation of absorption data. Solving the equation for the absorbance *A* = log(*I*_{0}*/I*) leads to the expression

from which it is clear that the value of the absorption does not only explicitly depend on the concentration and optical path length, but also on the intensity of the incident light *I*_{0}.

## 4. Non-linear kinetics of photobleaching.

Azobenzene, as previously discussed, is a small molecule with two double-bonded nitrogen atoms linked to two benzyl rings. It is photosensitive, because exposure to light induces an isomerisation *trans-cis* (indicating two possible spatial arrangements of the benzyl rings) around the double bond between the nitrogens, and this results in a molecule shape change. The process is fully reversible either by stimulated backward photoconversion (with a light at a different wavelength), or by spontaneous relaxation to the equilibrium *trans* configuration.

The isomerization of azobenzene and its derivatives has been extensively studied for the last fifty years [Sudesh Kumar & Neckers 1989, Renner & Moroder 2006, Rau 1990], because this molecule has many interesting features and its applications range from electronics to biomedicine. It has been used as a model molecule for all the biological processes that involve similar reactions, like the isomerization of retinal in rhodopsin, or as a probe for measuring the free volume in polymers [Victor & Torkelson 1987]. More recently, its characteristic response to the polarization state of light made it a suitable molecule for surface patterning [Nathanson et al. 1992, El Halabieh et al. 2004]. Finally, azobenzene-containing elastomers can give rise to inhomogeneous photo-mechanical effects and their applications as photoactuators and artificial muscles are under study [Hogan et al. 2002, Finkelmann et al. 2001, Yu et al. 2004].

However, in spite of the large literature on the subject, many fundamental mechanisms and effects have not been clarified yet. It is assumed that the isomerization reaction is very sensitive to both electrical and mechanical characteristics of the environment which surrounds the molecules, but identifying and separating these effects is a difficult and often ambiguous task.

Of the two possible isomers of azobenzene, the *trans* form is the lowest energy form, since the benzyl ring electron clouds are far apart (see fig. 3), but under UV light an isomerisation occurs; once in the *cis* state, the molecules can return back to their equilibrium trans state both by stimulated isomerisation with visible light or by thermal relaxation [Rau 1990]. The rate constants of these two processes are usually different, thermal isomerisation being slower. The microscopic mechanism that leads to the isomerisation is still not clear, but there are suggestions for both rotational and inversional [Asano & Okada 1984] mechanisms may be competing.

The isomerisation of azobenzene can be monitored by UV-Vis (ultraviolet and visible light) spectroscopy, because the two *trans* and *cis* compounds have different absorption spectra in this range: the *trans* isomer absorbs around 365 nm, while the *cis* isomer at around 440 nm [Rau 1990]. Irradiation with light at the wavelength of the *trans* peak progressively depletes the molecules in this conformation. This reaction can be “followed” by measuring the intensity of the absorbance of the spectrum peak at 365 nm, which decreases as the *trans-cis* photo-isomerisation reaction proceeds. An example of spectral evolution as isomerisation proceeds can be found in figure 4.

The models that were proposed for reaction kinetic are basically first order models, with the important exception of azobenzene in polymer matrices. The fraction of isomers in the *cis* state, *nc*, varies as [Zimmerman et al. 1958, Mechau et al. 2005]:

where *I* is the irradiation intensity, *kb* and *k* are the *cis-trans* and *trans-cis* constant of photoisomerisation respectively, *γ* is the thermal *cis-trans* isomerisation and *nt* represents the fraction of molecules in the *trans* state, and it is equal to (1 − *nc*). In the derivation of the formula the sensible assumption was made that the *trans-cis* thermal isomerisation constant is negligible.

A basic characteristic of the photoisomerisation problem is the rate of spontaneous thermal *cis-trans* isomerisation *γ*. For a given azobenzene derivative, at fixed (room) temperature and sufficiently low dye concentrations to avoid self-interaction, this rate is approximately the same for all our solutions. We measured this rate after monitoring the relaxation of the spectrum after the UV illumination is switched off (see [Serra & Terentjev 2008a] for detail) and obtained *γ* ≈ 1.25 10^{−4}s^{−1} (or the corresponding relaxation time of ~ 8000s).

In order to test the predictions of the theory, dynamic absorption measurements were performed for different dye concentrations and different light intensities. Considering equation (14) this is equivalent to changing *x*/*D* (where *D* is inversely proportional to the dye concentration) and *α*, which is proportional to the incident intensity *I*_{0}. With our experimental setup it was possible to follow all the isomerisation kinetics and thus the time dependence of *I*/*I*_{0} [Serra & Terentjev 2008b].

We prepared three different dye solutions with (non-dimensional) weight fractions *c* = 2.5 10^{−3}, 0.01 and 0.025, resulting in values of penetration depth ranging from *D* = 36 mm, to *D* = 3.6 mm. We recall here the physical meaning of the penetration length, which is the distance through the sample over which the light falls across a sample to 1/10 of its original intensity. The cuvette containing the sample is 1 cm long; therefore a sample with *D* equal to a few millimeters is almost completely opaque.

The measurements were performed using the Thermo-Oriel spectroscope described in the materials and methods section. Illumination was provided by a Nichia chip-type UV-LED, emitting at 365nm (bandwith about 10nm wide) whose output power was accurately regulated by a power supply. The LED light was attenuated by passing through a black tube of controlled length, placed in front of a quartz cuvette with 1 cm optical path. Several values of intensity were used in reported measurements, ranging between *I*_{0} = 4 and 60 *μ*W/cm^{2}. It is important to point out that these intensity values are very low and that most experiments on azobenzene isomerization are performed with intensities which are orders of magnitude higher, making the photobleaching much more of an issue. The low values of intensity allowed us to have a kinetics slow enough to detect the features which the theory predicts at short times. Every point of the spectrum was collected as an average of 100 measurements. All isomerization reactions were followed for several hours, until a photostationary state was reached. The measurements were repeated at different illumination intensities (regulated with the power supply) and at different dye concentrations.

In all cases it was important to verify that the dye concentration remained in a range where, at time *t* = 0, the linear proportionality between absorbance and concentration (Lambert-Beer law) held. This is important because the concentration of molecules in the *trans* state at every instant was determined from the absorbance at 365 nm. Absorbance was measured at several concentrations. The deviation from linearity started at *A* ≈ 1.2, which corresponds to the dye weight fraction of *c* = 0.03 (3 wt%) in our 1 cm cuvette. After this point, aggregation effects start playing a role and the basic Lambert-Beer law is no longer valid, undermining the theoretical relationship given by the equation (14). We always kept the concentrations below this value, so that the linearity at *t* = 0 was maintained, with *A* = *x*/*D* where the penetration depth is inversely proportional to chromophore concentration, expressed as weight fraction, *D* ≈ *δ*/*c* with *δ* = 91 *μ*m.

For our detailed dynamic experiments, a very important issue was the viscosity of the solution. In fact, at high illumination intensity we have encountered an unexpected problem. Figure 9 shows that the transmission of light through a low-viscosity dye solution (in pure toluene) displays a characteristic oscillatory behaviour. Detailed analysis of this phenomenon is under further investigation. Whether the oscillations are linked to the local convection due to the heating of the sample spot [Nitzan & Ross 1973] or to the diffusion of the less dense *cis* molecules – or whether they are intrinsic to the non-linear photochemical process [Borderie et al. 1992] – is not clear at this stage.

In order to avoid this difficulty, the dye solutions were prepared in a mixture of toluene and polystyrene of high molecular weight. Adding polystyrene increases the viscosity of the solution by over 2 orders of magnitude, and in this way prevents fluid motion in the cuvette on the time scales of our measurements. Polystyrenetoluene solutions were prepared at a fixed weight ratio. Adding polystyrene to toluene increases the Rayleigh scattering of the solution, but we felt that we could safely do that because on one hand the absorption dynamics is not affected (we used the same concentration for all the measurements and for the reference spectrum), and on the other we measured the transmittance of the toluene-polystyrene solution, which is almost equal to the pure toluene solution at 365nm.

In figure 10 the representative experimental results are shown for the solution with the highest chromophore concentration (*D* = 4.6 mm, leading to *x*/*D* = 2*.*2) and three values of incident intensity. One finds that all curves converge to the same initial value corresponding to the *I*/*I*_{0} = exp[−*x*/*D*], which for this concentration means quite a low transmission (*I*/*I*_{0} ≈ 0.11). If the isomerisation didn’t take place, the sample absorption would be constant in time according to the classical Lambert-Beer. A traditional description of the kinetics of isomerisation would predict an exponential decay of the absorption over time, but from figure 10 we see a strong deviation. We fit the data with the theoretical model given by equation (14) where we input the values of *γ*(thermal relaxation) and *x/D*, leaving only *α* = *I*_{0}*k/γ* free. Two data sets at higher intensity show the transmitted *I*(*x, t*) reach saturated values. In this case we are confident of the fit because we have to match both the slope and the amplitude of the curve. We obtain *α* ≈ 60 for *I*_{0} = 60 *μ*W/cm^{2}, *α* ≈ 20 for *I*_{0} = 20 *μ*W/cm^{2}, and *α* ≈ 4 for *I*_{0} = 4 *μ*W/cm^{2} (the matching of values is pure coincidence). We found that one particular output of experimental recording, the absorbance plateau value (photo-bleached) at long times, was extremely sensitive to the reading of reference intensity *I*_{0}. The latter measurement could be affected by various stray factors and in a few cases we had to rescale the raw absorbance readings with a proper reference value. This issue did not have any effect on absorbance at *t* = 0, or the kinetics.

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*
*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

which gives in the stationary state the correct solution of equation (6) approximated at small *x/D*:

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}*μ*W^{−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 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.

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.

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 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:

then

Knowing *α*, one can simply determine the unknown concentration *c*_{1} as

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 ± 20000*M*^{−1}*cm*^{−1}. The literature reports ε = 77000*M*^{−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:

If a set of concentrations and relative absorptions are known, one can plot the quantity *c*(1 − 10^{−A})^{−1} as a function of *A*ln(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

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±120*M*^{−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.

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 *kb* the rate of the stimulated back transition.

At the stationary state, the left hand side of the equation is zero and *n* becomes

This is the value which should be inserted in the expression for the photobleaching 4, giving

This can be simplified by dividing by *k*, introducing the parameter *φ* = *kb*/*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:

It is convenient here to reintroduce our usual non-dimensional parameter *α* = *I*_{0}*k*/*γ*

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

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* = 114000*M*^{−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

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. Our experiments were deliberately carried out in a highly viscous solvent to eliminate the additional complexities, presented in figure 9, caused by the possible local convection flows of different isomers. Certainly, a much more in-depth study will be required to take such effects into account.

The second result is related to the extension of the model to all fluorescent molecules, or indeed any molecule which has a long-lived excited state. We showed that, even for those molecules where the conversion between the two states is too fast to be followed by the spectroscope, the nonlinearity has an important influence. The absorption values at the stationary state were sensitive to the experimental conditions, and particularly the intensity of the incident light.

It must be said that in the literature the light intensity of the spectrometer light source is very rarely mentioned, therefore it is possible that many values of absorption coefficient reported are wrong or meaningless. The reason why, to our knowledge, no one took this phenomenon into consideration before is that many commercial spectroscopes always work with the same light intensity, so the results are self-consistent. Also, weighing proteins or other materials is often a difficult task, therefore a discrepancy between the expected value and the literature values could be easily explained away. In fact, we showed that there is a much deeper reason.

Moreover, this raises also problems of interpretation of data obtained comparing, for example, the intensity of two different absorption peaks, because, even at the same incident light intensity, the nonlinearity also depends on the factor (k/γ) which is different at each wavelength. All these problems could be overcome using the method we suggested, which is simple and straightforward and it allows reproducibility of results.