Difference between revisions of "Förster Theodor"

Revision as of 05:14, 19 July 2017

Erich Sackmann

Professor Emeritus, Physics Department E22, Technische Universität München, D85747Garching, Germany, EU.

1 Introduction

Our development as a scientist is guided by scientific giants who show us how complex scientific questions can be solved by smart experiments and thinking. One of these scientific lighthouses in photophysics and photochemistry is Theodor Förster; not only due to his great scientific discoveries but also owing to his devotion to science and his benevolent attitude towards students and young scientists. Looking back to my time as a student I find this most remarkable since in the post war years his personality and outstanding scientific competence contributed much to the renaissance of science in post-war Germany. Förster’s discoveries and his way of thinking influenced the research of generations of scientists working on photophysical and photochemical properties of molecules, including several of those obtaining Nobel Prizes, (see (Weller 1980) and (Porter 1976)). Most remarkably, Förster belongs to a tiny group of scientists whose fame is growing steadily after their death. The only other scientist of this group in Germany that comes to my mind is Alfred Wegener, the discoverer of the tectonic structure of the earth. One reason for Försters growing recognition is that his discovery of energy transfer and of fluorescent excited complexes (excimers and exciplexes) provided us with molecular rules that allow us to study dynamic processes in complex fluids and cells with nanometer resolution and time scales of 10nsec. It should also not be forgotten that Försters’ interest in applied science paved the way for the rational photochemical synthesis of polymers and pharmaceuticals in industry. In this overview, I first summarize some aspects of Förster’s life. Then I describe his ground breaking discoveries and their impact on photochemistry and photobiology. In the third part I describe the application of FRET and excimer forming probes as molecular rulers allowing us to study the dynamics and structure of complex materials cells and chromatin with nm spatial resolution on 10 nsec time scales.

2 Biographical Sketch

Theodor Förster was born May 15th, 1910 in Frankfurt am Main. In 1929 he finished school in his hometown and studied Theoretical Physics and Mathematics at the Wolfgang Goethe University of Frankfurt a. M. He did his PHD (on the polarisation of electrons by reflection) at the Institute of Prof Erwin Madelung, who was one of the most prominent solid state physicist in Germany in his time. Indeed, he finished his Ph.D. after only four years at the age of 24. The reason for this astonishing dead was that, at that time, students could study mathematics and physics without making specific examinations. They finished their studies with an oral examination after having completed their Ph.D. work (Weller 1980)

After finishing his Ph.D. Förster moved to the University of Leipzig, together with his young mentor Karl-Friedrich Bonhöfer, one of the discoverers of ortho- and para-hydrogen who (at the age of 32) had been appointed full professor and director of the Institute of Physical Chemistry at the 29-year-old University of Leipzig. This first Institute for Physical Chemistry in Germany had been founded by Walter Ostwald, one of the founding fathers of physical chemistry and colloid research. At the same time Peter Debye, Werner Heisenberg, and Friedrich Hund worked as professors at the Physics Faculty. For that reason, years Leipzig became for several an attractive center for physicists from all over the world. Most likely, Förster’s ongoing interest in applied science was stimulated by his contact with Peter Debey.

During the Leipzig years (1934-1942) Förster published about 10 papers on various fields of molecular physics, including the stabilization of organic molecules by the carbon valency and double bonds and on light absorption by aromatic molecules. In 1942, at the age of 32, Förster became Professor of Physical Chemistry at the University of Posen. This university had been founded in 1919 by the king of Poland after the reunification of the province Posen with Poland in 1914. It had been taken over by Germany during the occupation of Poland. Most remarkably, Förster did not publish any papers during his four years in Posen. He was certainly not idle and most likely spent his time to create a family, to establish a curriculum on Physical Chemistry and to think about new scientific directions.

In 1947 Förster returned to Karl-Friedrich Bonhöfer who had become director of the newly founded Max Planck Institute for Physical Chemistry in Göttingen. Here he accomplished his ground breaking theory on energy transfer between organic molecules. Moreover, he wrote his monography “Fluoreszenz organischer Verbindungen” which for many years became the bible of the photochemists and photophysicists, at least in German speaking countries. In this book, Förster demonstrated his outstanding ability to explain complex quantum mechanical concepts to chemists and experimental physicists.

In 1951 Förster accepted the chair for Physical Chemistry at the Technical University Stuttgart where he worked until his premature death in 1974. Förster’s life ended in a tragic way. While returning from swimming in his car he had a heart attack. His car went into the left lane and was hit by a truck. Most likely he was dead before the truck hit his car.

3 The conception of the dipolar model of intermolecular energy transfer

After the publication of Dirac's “The Quantum Theory of Emission and Absorption of Radiation” theory in 1927 the question of energy exchange between molecules was revisited by many physicists. It was generally thought to be determined by collisions between atoms or molecules. Around 1925 Jean Baptist Perrin (the man who proofed Einstein’s theory of Brownian motion) had estimated that two molecules (one of which is excited) can exchange energy when they approach a critical distance of ~15 nm (Perrin1927). However, sensitized fluorescence and fluorescence depolarization experiments with chromophores (such as Fluorescein) strongly suggested that energy exchange between chromophores can occur over distances of 50nm. In his first estimate, Perrin had assumed that the molecules are two oscillators with sharp frequencies. He conjectured that in order to explain this discrepancy one has to consider the Stokes shift of fluorescence spectra as well as the shape of the absorption and emission spectra of the energy exchanging molecules (see (Perrin 1927) and introductory remarks in (Förster 1946)).

Stimulated by Perrins suggestion Förster developed a classical model of energy transfer in 1946 (Förster 1946) and a rigorous quantum mechanical theory in 1948 (Förster 1948). Both theories are based on the assumption that the energy transfer is mediated by dipolar interaction between an excited electron (initially located at the donor D) and a ground state electron at the acceptor A. The classical theory is beautifully described in a review by Hans Kuhn (Kuhn1982), the second European hero of photophysics and photochemistry. Here I briefly focus on the salient features of Förster’s quantum mechanical theory which shows that he was a keen scientific pioneer.

To calculate the transfer rate in the quantum mechanical model he keenly applied the Dirac transition theory (Dirac 1927) which is also often attributed to Fermi and is then called Fermi’s Golden Rule. He wrote down the following expression for the rate of energy transfer between two molecules A and B

[math]k_{ET}=\frac{2\pi}{h}\int \int d\overrightarrow{r}_{k}d\overrightarrow{r_{l}}\varphi _{A}^{*}(\overrightarrow{r}_{k})\varphi _{B}(\overrightarrow{r}_{k})H(\overrightarrow {r}_{k},\overrightarrow{r}_{l})\overrightarrow{r_{l}}\varphi _{A}(\overrightarrow{r}_{k})\varphi _{B}^{*}(\overrightarrow{r}_{k})\tag{1}[/math]

Where [math]\varphi _{A}[/math] and [math]\varphi _{A}^{*}[/math] are the wave functions of the electrons (k and l) in the ground and excited state. For electron distances large compared to the size of the molecules, Förster assumed that the Hamiltonian H is determined by the dipolar interaction between the electron in the excited state of the donor and the electron located at the acceptor in the ground state

[math]H(\vec{r}_{k},\vec{r}_{l})\frac{e^{2}}{\epsilon \left | r_{k}-r_{l} \right |}\approx \frac{e\vec{p}\vec{r}}{\varepsilon r^{3}}\tag{2}[/math]

where [math]\vec{p}=e\vec{r}[/math]is the electric dipole moment and [math]\varepsilon[/math]the dielectric constant of the solvent.

The second outstanding achievement of Förster is the establishment of a correlation between the energy transfer rate [math]k_{ET}[/math] and the overlap integral between the emission spectrum of the donor and the absorption spectrum of the acceptor and he derived the famous equation for the transfer rate

[math]k_{ET}=\frac{1}{\tau _{D}}\left ( \frac{R_{0}}{R_{kl}} \right )^{6}\tag{3}[/math]

In this equation [math]R_{0}[/math] is the critical distance over which energy transfer is efficient, which is now called the Förster Radius. [math]\tau_{D}[/math] is the life time of the donor in the excited state. In the earlier classical paper from 1946, Förster had already shown that [math]R_{0}[/math] is a function of three independent quantities: an orientation factor [math]\kappa [/math], the fluorescence quantum yield [math]\eta _{F}[/math] and the overlap between the absorption and the emission spectrum [math]\varepsilon _{A}\left ( \nu \right )[/math] and [math]\Phi_{D}(\nu)[/math], respectively(see Förster 1959)

[math]R_{0}^{6}=\left ( \frac{3\, c^{4}\, \kappa ^{2}\ln 10}{128\, \pi ^{5}\, N_{A}\, n^{4}} \right )\Phi_{F} \int_{0}^{\infty }\frac{\epsilon _{A}(\nu )\Phi _{D}\nu }{\nu ^{4}}\, d\nu \tag{4}[/math]

The situation would have been hopeless complicated if Förster would not have found a simple method to measure the radius [math]R_{0}[/math] by determining the fraction [math]w_{DA}[/math] of energy transferred from the energy donor to the acceptor.

[math]w_{DA}=\frac{1}{1+(r/R_{0})^{6}}\tag{5}[/math]

A compelling quantitative verification of the Förster theory was provided by Hans Kuhn (the other giant of photochemistry and photophysics in after-war Europe) in beautiful experiments. He measured the energy transfer between two chromophores anchored in lipid monolayers via hydrophilic tails. By embedding the monolayers in multi-lamellar Langmuir-Blodgett-Kuhn films, the distance of the chromophore containing monolayers could be varied by the intercalation of various numbers of dye-free monolayers between the dye exposing monolayers. Since one deals with a two-dimensional system the transfer rate depends on the fourth power of the Förster radius (Kuhn 1982). Thus Kuhn not only provided a beautiful proof of the Förster theory but also clearly demonstrated that the energy transfer rate depends on the dimensionality of the system studies.

4 Why was the Förster theory so revolutionary: the idea of supramolecular states.

Förster tacitly assumed that the excited states of complexes (AB)* can be described as super-molecules with fixed nuclear positions and with the two valence electrons moving in a supramolecular potential whereby the initial and final states (D*A and DA*), can be represented by the wave functions [math]\Psi _{initial}=\varphi _{k}(D^{*}A)[/math] and [math]\Psi _{final}=\varphi _{k}(DA^{*})[/math]. The squares of these function are measures for the probability that the valence electron at the acceptor and the donor is excited (see (Hopfield 1974)).

With his keen concept of super-molecular states of molecular complexes, Förster paved the way for numerous theories of photophysical processes of molecular complexes including exciton formation in organic crystals, the red shift of the absorption band of chromophore complexes. This concept formed also the basis for the theories of photochemically induced electron transfer reaction between an excited electron donor (D*) and an acceptor in the ground state (A) by electron tunneling (Hopfield 1974; Jortner 1985) and intersystem crossing (Jortner 1980), (Markus and Stutin1985).

Försters keen dipolar theory had to go through a critical phase and was criticized by famous colleagues, such as Davydov (known for his treatment of excitation of crystals (Knox 2012)). However, during the ongoing discussion, the FRET theory stood its test. Försters theory was later extended for more tightly packed molecules by Dexter. By introducing electron exchange terms more complex cases such as energy transfer between triplet states could be explained. The history of this development is well described in common paper. In fact, Förster had proposed to treat situations of the closely packed chromophore by introducing quadrupole terms and discussed the transition between weak and strong coupling. An interesting discussion of the extension of the Förster theory to tightly packed chromophores can be found in a paper jointly written by Dexter, Förster and Knox (Knox 2012).

5 Försters impact on photobiology-Energy and quantum efficiency of photosynthesis

Theodor Förster was always interested in photobiology as I remember from many visits by famous photochemists who came to Stuttgart to discuss with him the possible role of energy transfer reactions in photosynthesis. In fact, before he wrote his famous quantum mechanical theory in 1948 he had published a paper “on the theory of photo synthesis (Förster 1947). Based on estimations of the chlorophyll density he conjectured that the excitation energy captured by a chlorophyll molecule in the photosynthetic membrane can move over some 104 chlorophyll molecules by dipolar energy transfer and exciton diffusion. Most importantly he pointed out that carotenoids can be involved in the energy transfer process.

Today we know that that energy transfer processes play a key role for the light harvesting by bacteria and plants. Figure 2 (in the BOX) shows the present view of the arrangement of chlorophyll and carotenoids in the light harvesting complexes of purple bacteria based on electron microscopy, spectroscopic studies and sophisticated theories by the group of Klaus Schulten (for references see (Ritz, Damjanivic and Schulten 2002). Two Light harvesting complexes (LHC1 and LHC2) surround the photosynthetic reaction centres. They guide the light, absorbed by any chlorophyll molecule in a most effectively to the chlorophyll dimer, of the RZ which is often called the special pair).

Försters prominent pupil Klaus Schulten explained the role of carotenoids as protector of the photosynthetic reaction center and inhibitor of plant cell death. His theoretical studies provided strong evidence that a unique feature of excited carotenoids is their transition to long living triplet states by ultrafast internal conversion (in 0.2 psec). These states lie lower than the excited singlet states of oxygen [math]^{1}O_{2}^{*}[/math] and can thus quench these photo-damaging states. The low lying triplet states of Chlorophylls ([math]^{3}B-Chl[/math]) can also be populated by energy transfer from chlorophyll triplet states according to

[math]BChl^{*}+^{1}Car\rightarrow ^{1}BChl+^{3}Car^{*}[/math]

This intersystem crossing is enabled by strong interaction by exchange mechanisms and requires the tight packing of chromophores as already anticipated by Förster in 1947.

Here, another remark is appropriate. Today we assume that our knowledge of the molecular structure of the photosynthetic reaction centre is basically a result of X-ray and Electron Microscopy studies. However, our knowledge on the organisation and the function of the chromophores is due to spectroscopic studies and the wealth of knowledge about photophysical processes collected over a hundred years. One major triumph of spectroscopists was the discovery of the special chlorophyll pair, long before the structure determination by Deisenhofer and coworkers.

Förster-Schulten model of energy transfer in photosynthetic reaction centers.

Figure 2 shows the present view of the arrangement of the chlorophyll and carotenoids in the light harvesting complexes which surround the photosynthetic reaction centres and guide the light, absorbed by any chlorophyll molecule in most effective way to the h-receptor, of the reaction centres. More recent theoretical studies of the energy transport in photosynthetic systems by the Schulten group has shown that the energy transfer also involves the carotenoids embedded in the Light Harvesting Complexes (LHC). They play a key role for the protection of photosynthetic membranes from excess light intensities. In fact the subtle control of the energy transfer from the location of absorption to the reaction centres is one of the most stunning examples how nature exploits physical concepts to control life processes.

Today we assume that our knowledge of the molecular structure of the photosynthetic reaction centre is due to X-ray and Electron Microscopy studies. However, our knowledge on the organisation and the function of the chromophores, is due to spectroscopic studies based on the wealth of knowledge about photophysical processes collected over a hundred years. One major triumph of spectroscopists was the discovery of the special chlorophyll pair long before the structure determination by Deisenhofer In fact the array of chromophores in the reaction centre is based on spectroscopic studies.

6 The Förster Cycle

Chemistry in the excited state and how Förster measured excited state lifetimes long before the development of flash photolysis.

A surprising feature of the theoretical physicists Förster was his ongoing interest in the industrial application of photochemical techniques. He liked discussions with industrial researchers who frequently asked for his advice about all questions of photochemical synthesis of organic molecules or about strategies to avoid the bleaching of fabrics. This may have stimulated him to study the physical basis of photochemical reactions after he had moved to the University of Stuttgart in 1951. At that time photochemical reactions were utilized in industry to activate organic molecules by free radical mediated chlorination and nitrosation (replacement of hydrogen by groups with R-NO functionality) which serves the generation of activators of polymerization. An example is a caprolactam a precursor for the production of polyamides, such as perlon.

Förster realized that excited molecules can be considered as electronic isoforms of the ground state molecules with strongly modified chemical properties, such as the acidity and basicity of polar molecules. He concentrated on the photochemical modification of the chemistry of [math]\beta[/math]-Naphtol (see Figure 3a) which was applied for the production of dyes (such as Sudan). In a ground breaking experiment, he showed that when [math]\beta[/math]-Naphtol (ROH in Fig. 3a) is excited at very low pH~2-3 one observes the fluorescence of the anion (RO-). In striking contrast, in the ground state, the molecule dissociates only in very basic solutions (of pKa~9.6). This simple observation prompted him to develop the powerful concept of the Förster cycle allowing us to determine thermodynamic parameters and pK values of excited molecules (Förster 1950). Most interestingly, it allowed Förster to measure the life time of excited states (which was generally considered as measurably small) several years before flash photolysis was invented by Porter and Eigen.

In this ground breaking study, Förster showed that for a situation in which the fluorescence and absorption spectra are centro-symmetrically located about a wavelength [math]\lambda_{0}[/math] called the symmetry rule) the life time can be measured by application of Einstein’s law of induce excitation and spontaneous emission. He derived the following equation for the determination of the lifetime [math]\tau_{e}[/math]

[math]\frac{1}{\tau _{e}}=2.88\times 10^{-9}\, n^{2}\int_{0}^{\infty }\frac{(2\tilde{\nu }_{0}-\tilde{\nu })}{\tilde{\nu }}\, \epsilon (\tilde{\nu })\, d\tilde{\nu }\tag{6}[/math]

In this equation, n is the refractive index and is the wave number of the symmetry axis between the long wavelength absorption and the short wavelength emission band. For Naphtol in acidic aqueous solutions he obtained [math]\tau_{e}[/math]=5.8[math]\times[/math]10[math]^{-8}[/math] sec which is astonishingly close to the result measured by flash photolysis [math]\tau_{e}[/math]=4[math]\times[/math]10[math]^{-8}[/math]sec. This equation was derived in a more rigorous way a decade later and the validity of this method was established experimentally for several aromatic molecules (Stickler and Berg 1962).

The concept of the Förster cycle was extended by his pupil Ernst Lippert to study the color changes of organic molecule due to a change in solvent polarity which enabled him for the first time to measure dipole moments of molecules in the excited states (Lippert 1955).

The major value of this discovery of Förster lies in the fact that it showed for the first time how to measure thermodynamic properties of photochemical reactions, such as heats of transition or dissociation constant of complex formation in the excited state, He developed a thermodynamic concept, now called Förster-cycle, for the evaluation of excited state equilibria, such as acid-base equilibria. This concept paved the way for the determination of the equilibrium constants [math]K_{a}[/math] in the excited states by considering the cycle processes shown in Figure 3a. The basic idea for measuring molar enthalpies was to consider only the difference in molar enthalpy changes in the ground and excited states, neglecting the changes of the entropy associated with the reaction. Consider the free energy change of deprotonation in the ground (0) and excited state (1) by assuming that [math]\Delta S_{1}=\Delta S_{0}[/math]. The dissociation constants in the ground and excited state are then determined by the difference in the molar heats of transition [math]\Delta H_{i}[/math]

[math]\delta \Delta H=\Delta H_{1}-\Delta H_{0}=-RT\ln \left \{ \frac{K_{a}^{*}}{K_{a}} \right \}.\tag{7}[/math]

By considering the energy diagram of Fig 3a one obtains

[math]\Delta H+N_{A}h\nu _{1}-\Delta H^{*}-N_{A}h\nu _{2}=0\tag{8}[/math]

and the dissociation constant is given by

[math]\ln K_{a}^{*}=\ln K_{a}+\frac{h(\nu _{1}-\nu _{0})}{k_{B}T}\tag{9}[/math]

The thermodynamic basis of the Förster cycle was later refined by Grabowski and Rubaszewska ( Grabowski and Rubaszewska 1976).

The interest in the Förster cycle has gained new impetus with the application of photophysical techniques in cell biology after it became possible to genetically express green fluorescent proteins (GFP) from jelly fish, reviewed in (Tsien 1998). The chromophore is embedded in a barrel like protein shell where it is protected from hydrolysis by enzymes. If the molecule is excited at 395 nm, the phenol (which has a pKa~0.1 in the ground state) dissociates, in close analogy to the behavior of [math]\beta[/math]-naphtol. In contrast to the short lifetime of isolated phenols, the dissociated state of GFP is metastable and lives for extended periods of time. This fortunate situation allows us to average over many excitations of single molecules enabling the measurement of distances between molecules with nm precision.

There are still many open questions concerning the structure and function of the green fluorescent proteins and its isoforms. The chromophore can form a network with the amino acid scaffold by hydrogen bridges as indicated by arrows in Figure 3b. The ionic state is likely stabilized by formation of a link between the ionized phenolate and the histidine group of the protein (see Figure 3b). At present, several other models as that shown in Figure 3 are discussed in the literature. To the authors knowledge a systematic quantitative analysis of the function of GFP in terms of the thermodynamics of the Förster cycle is still missing. Such analyses would certainly necessary to test the validity of the various models.

7 From Excimers to Exciplexes –New applications in life science

Some years after he had moved to Stuttgart, Förster opened another new field of photophysics with the discovery of excited complexes between a ground state and an excited pyrene molecule (called excimers). Together with his pupil X. Kasper he observed the appearance of a new broad band (centred at 450nm) at the long wavelength end of the fluorescence spectrum (with the 0-0 transition located at 395 nm). The fluorescence band showed no vibrational fine structure and increased with increasing concentration (Förster and Kaspar 1955). Förster called this effect “Fluoreszenzumschlag” (english concentration transformation). With his unmatched insight into the behaviour of excited molecules he realized that this is due to the formation of sandwich-like complexes between a molecule in the ground and in the excited state, formed according to

[math]P^{*}+P\Rightarrow {PP}^{*}[/math]

With the discovery of excimers Förster had found a case of strong interaction between excited and ground state molecules which he most likely had anticipated many years before. (see Förster 1957 and Förster 1969). Again Förster generalized the discovery of excimers formed by homodimers to excited hetero-complexes (now called exiplexes) and this concept paved the way for the new field of charge transfer reactions in the excited state. In his typical generosity he left the field to younger colleagues, in particular to his most important pupil Albrecht Weller who led the field of photochemistry of charge transfer complexes to the highest possible standard. In the paper written in 1969 Förster discussed many aspects of excimer formation including measurements of the enthalpies and entropies of excited complex formation and excimer formation in triplet states (Förster 1965, Förster 1969). In the impressive study of 1963 he cited 143 publications and this review most certainly summarizes the state of the art around 1970.

Both, the Förster energy transfer (FRET) and excimer forming molecules had been constantly used by material scientists and solid state physicists to study chemical reactions and dynamic properties of organic and biomimetic materials. The interest in these discoveries exploded in recent years after biophysicists and cell biologists realized that they provide us with molecular rulers that can dynamically measure molecular distances in the 5-50 nm range by labelling biomolecules either with energy donors and acceptors (FRET-techniques) or with excimer forming probes. The dyes can also serve as molecular beacon to localize nanomole molecules in solutions complex fluids and cells .The main advantage of fluorescent probes in biological physics applications, compared to classical super-resolution microscopy techniques, is the possibility to measure dynamic processes down to nanometer scales and with 10 nsec time resolution which, in the authors view, this is a major task of biological physicists.

The application of FRET as molecular ruler in living matter research is described in many reviews (references see (Knox 2012)) and I restrict myself to some critical remarks which had always been pointed out by Förster. First, due to the geometric factor [math]\kappa[/math] in Eq. (3), the energy transfer depends on the orientation of the chromophores and under certain conditions ([math]\theta =45^{\circ},\, \xi =90^{\circ}[/math]) the geometric factor and thus the energy transfer probability can become very small. Moreover if the chromophores are confined to planar or linear targets the [math]R^{-6} - dependence of Eqs. (4) and (5) does not hold as shown in the above mentioned Kuhn experiments (Kuhn 1982). For quantitative studies of molecular interaction by FRET or the excimer technique it is thus important to carefully consider the topology of the system studied. In the following, I will address the problem of dimensionality by considering the use of excimer probes for studying dynamic properties in bio-membranes and new applications in genetic engineering. The excimer probe technique is not as popular as FRET. However, its usefulness and power have not been fully exploited yet. In other words, there is much room left for the application of excimer probe technique to study dynamic properties and dissipative processes in biomaterial research. After switching to biological physics the author was encouraged by Förster to apply excimer probe technique for measuring micro-viscosities and diffusivities \ltmath\gtD_{lat}[/math] in artificial and biological membranes, doped with a few mole% of pyrene or pyrene-labelled lipids. From the point of view of physics of condensed matter lipid membranes are of great interest as prototypes of two-dimensional fluids (Galla et al 1979). Diffusivities can be measured by stationary analysis of the excimer quantum yield and with non-stationary techniques, enabling the observation, of the formation and decay of excimer fluorescence. Hans Joachim Galla (a scientific grandson of Förster) first solved the problem by a stationary technique (neglecting the decay of excimers). It is based on the measurement of the jump frequency [math]\nu_{j}[/math] of pyrene labelled lipids between two neighboring sites (separated by a distance [math]\lambda[/math]) in the lipid matrix. By application of a random walk model in 2D lattices, he could determine the two-dimensional jump frequency from which the diffusion coefficient [math]D_{lat}[/math] in the membrane plane obtained according to

[math]D_{lat}=\frac{1}{4}\nu _{j}\lambda ^{2}[/math]

In order to measure diffusivities [math]D_{lat}[/math] and microviscosities of membranes with high precision nonstationary measurements of the time dependent reaction rate of excimer formation are required. The theoretical basis for such measurement in organic solvents was laid by Martinho and Winnik (Martinho and Winnik 1987). Birks and coworkers solved the experimental problems and measured the excimer formation rate r(t) (defined in Figure 4b) in organic solution (Birks, Dyson and Munro 1963). The diffusion coefficients were obtained by application of the Smoluchowski equation of chemical reactions in solutions.