Ultrafast excited-state dynamics in biological and in organised environments

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Thesis

Reference

Ultrafast excited-state dynamics in biological and in organised environments

FÜRSTENBERG, Alexandre

Abstract

La dynamique d'états excités de sondes fluorescentes placées dans des environnements biologiques et organisés a été étudiée par spectroscopie optique stationnaire et par spectroscopie laser femtoseconde. En particulier, l'influence de tels environnements sur des processus ultrarapides tels que la solvatation, la relaxation vibrationnelle, l'inhibition de fluorescence et la dépolarisation de fluorescence a été suivie. Les systèmes étudiés sont : ( 1) les protéines avidine et streptavidine au ligand naturel desquels, la biotine, différentes sondes fluorescentes ont été attachées ; (2) des intercalateurs d'acides nucléiques fluorescents en présence de différentes séquences d'ADN et d'ARN ; (3) des systèmes photosynthétiques artificiels composés de 32 chromophores identiques qui s'autoassemblent dans des membranes lipidiques. Les nombreux exemples expérimentaux permettent de démontrer la puissance de la spectroscopie de fluorescence femtoseconde en tant qu'outil pour obtenir des informations sur l'environnement local de sondes fluorescentes et ainsi des informations structurales sur des [...]

FÜRSTENBERG, Alexandre. Ultrafast excited-state dynamics in biological and in organised environments. Thèse de doctorat : Univ. Genève, 2007, no. Sc. 3924

URN : urn:nbn:ch:unige-5122

DOI : 10.13097/archive-ouverte/unige:512

Available at:

http://archive-ouverte.unige.ch/unige:512

Disclaimer: layout of this document may differ from the published version.

1 / 1

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Département de chimie physique Professeur Eric Vauthey

Ultrafast Excited-State Dynamics in Biological and in

Organised Environments

THÈSE

présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention chimie

par

Alexandre FÜRSTENBERG de

Ballmoos (BE)

Thèse N°3924

GENÈVE

Atelier de reproduction de la section de physique 2008

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de l’Université de Genève (Suisse) sous la direction de M. le Prof. Eric Vauthey. Je tiens à lui exprimer mon infinie reconnaissance pour sa confiance et sa disponibilité de tous les instants, pour la liberté qu’il m’a accordée et pour l’enthousiasme avec lequel il a partagé son savoir et sa passion. Je ne saurai imaginer de meilleures conditions pour préparer une thèse de doctorat que celles qu’il m’a offertes.

Ma gratitude va également à Mme le Dr Anita C. Jones de l’Université d’Edinburgh (Royaume-Uni), à M. le Dr Thomas Gustavsson du Com- missariat à l’Energie Atomique de Saclay (France) et à M. le Prof. Stefan Matile de l’Université de Genève (Suisse) qui ont accepté de s’engager comme experts lors de la soutenance et à ce titre de lire et juger le conte- nu de cet ouvrage.

La réalisation de ce travail de thèse n’aurait pas été possible sans les compétences admirables et les précieux conseils dans le domaine de la synthèse, de la modélisation ou de la biochimie de nombreux collabora- teurs et amis : Dr Julieta Gradinaru et Prof. Thomas R. Ward de l’Université de Neuchâtel (Suisse), Prof. Todor G. Deligeorgiev, Dr Aleksey A. Vasilev et Dr Nikolai I. Gadjev de l’Université de Sofia (Bul- garie), Dr Thibault Dartigalongue et Dr François Hache de l’Ecole Poly- technique de Palaiseau (France), Dr Sheshanath Bhosale, Dr Adam L.

Sisson, Dr Naomi Sakai, Prof Stefan Matile, Cyril Nicolas, Dr Benoît Laleu, Prof. Jérôme Lacour, Giovanni La Macchia, Dr Guillaume Bollot, Dr Jiri Mareda, Dr François Pellissier et Dr Vincent Ossipow de l’Université de Genève (Suisse).

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Je remercie très chaleureusement mes collègues et amis de l’Université de Genève qui ont hautement contribué à faire de chaque jour de travail aussi un jour de rencontres et d’échanges amicaux : Gonza- lo Angulo, Natalie Banerji, Patrick Barman, Pierre Brodard, Nathalie Dupont, Guillaume Duvanel, Aude Escande, Didier Frauchiger, Gaëlle Gassin-Martin, Jakob Grilj, Laure Guénée, Sophie Jacquemet-Gobet, Cécile Jaggi-Chevalley, Marc Julliard, Oksana Kel, Bernhard Lang, Max Lawson Daku, Dominique Lovy, Nathalie Mehanna, Mia Milos, Omar Mohammed, Ana Morandeira, Cyril Nicolas, Benjamin Nottelet, Olivier Nicolet, Tiphaine Penhouët, Didier Perret, Angela Punzi, Thomas Riis- Johanessen, François Rouge, Patrick Ryan, Alexandra Spyratou, Simon Verdan, Bruno Vitorge.

Un merci particulier à Stefan Matile, Naomi Sakai et les membres de leurs groupe pour m’avoir ouvert les portes de leurs « Journal Clubs » et pour la collaboration si enrichissante tant scientifiquement qu’humai- nement que nous avons eue. Un merci tout aussi particulier à Andreas Hauser l’amitié et le soutien qu’il m’a témoignés tout au long de mon passage à Genève. ainsi que pour les nombreuses portes qu’il m’a ouver- tes en toute confiance.

Durant toute la durée de ce travail, j’ai pu compter sur le soutien sans faille de nombreux amis proches dont beaucoup partagent ma passion pour la musique et l’art choral : Caroline, Laure, Xavier, Blaise, Sophie, Emmanuel, Véronique, Nathalie, Stève, Nicolas, Marie, Marie, Stéphane, Lucile, Pascal, Benoît, Salomé, Emmanuelle, Bertrand, Silène, Jérôme, Claudine, Emmanuelle, Jean-Michel, Bernhard, Uta, Claire. Qu’ils soient ici chaleureusement remerciés ! J’ajoute un merci spécial à Claudine et Laure qui, certainement plus que d’autres, ont supporté le poids de cette thèse avec moi, ainsi qu’à Caroline pour son indéfectible et si précieuse amitié.

J’adresse enfin d’immenses remerciements à ma famille, à mon frère et à mes parents en particulier. Ceux-ci m’ont non seulement transmis leur savoir, investi sans hésitation dans mon avenir et témoigné une confiance et un soutien à tout moment, mais ont surtout le grand mérite de m’avoir transmis les gènes de deux grands-pères… chimistes !

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Amesgrands-parents Amonfrère

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l’essentielestinvisiblepourlesyeux. Antoine de Saint-Exupéry

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Preface

Over millennia, Man has had in mind to explore phenomena at the fron- tier of time and space, in a perpetual thirst of extending the very limited senses with which he is endowed. In the realm of size, microbalances, microscopes and telescopes have permitted him to go beyond what he can see or feel. In the temporal dimension, his observations are naturally confined to times between the response time of his eye, about 50 milli- seconds, and his lifetime, about 2 billions of seconds [1]. At the lower limit of time for chemistry and biology, however, lies the femtosecond time scale (1 fs 10" ^!¹⁵ s), which finally became experimentally available about 25 years ago. It constitutes the fundamental time scale for following the dynamics of the chemical bond: the speed of atomic motion being about 1 km/s, the time required to monitor atomic-scale dynamics over a distance of 1 Å is in the range of 100 fs. Thus, only on the femtosecond time scale can the very act at the heart of chemistry, that of bond breaking and bond making of molecular structures, be observed directly. This is the key task conferred to !"#$%&'"#()$*+ [2, 3].

From initial investigations on atoms and simple molecules in the gas phase, femtochemistry has evolved to study complex processes and systems in the condensed, mesoscopic, or solid phases, in the bulk, at sur- faces, or at interfaces, and has grown to a mature field. New methodolo- gies for probing and following in real time molecular dynamical phenomena, such as multidimensional spectroscopies or ultrafast X-ray and electron diffraction, have appeared within the last decade and have already yielded fruitful results. Another promising area of research is that of

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complex biological systems at the new limits of spatial and temporal reso- lution. Contributions to !"#$%,(%-%.+ include studies of the elementary steps of vision, photosynthesis, protein dynamics, and charge transport in DNA [4, 5]. In proteins like cytochromes, haemoglobins, or rhodopsins, endowed with an intrinsic light absorbing unit — the &'*%#%/'%*" —, a photoinduced femtosecond event such as bond breaking, twisting, or electron transfer occurs and can be monitored spectroscopically, provid- ing valuable information on the mechanism of the biological processes they are involved in.

A second important segment of femtobiology is formed by studies in which the centre of interest is the influence of a biological macromolecule on spectroscopic properties of a chromophore rather than a particular biological problem. The way and the rate at which chromophores excited by light relax back to their initial state depend strongly on their local molecular environment, especially if the chromophores, then also termed

!-0%*%/'%*"), can deactivate by emitting light (or fluorescence). Whereas it is common to have chromophores in a disordered solvent environment, the highly organised nature of biological macromolecules such as proteins or nucleic acids makes biomolecular environments very special. Despite the growing number of reports [4, 5], the field is too young and the available information too scarce for a full understanding of the influence of biological environments on the dynamics of excited chromophores to be achieved. A comprehensive picture would in turn enable the use of chromophores to infer new knowledge on systems as well as the devel- opment of new routes to access very local information spectroscopically.

The thesis work presented here falls within this scope and aims at con- tributing to a better understanding of how organised macromolecular environments affect the relaxation dynamics of excited fluorophores. It is built up as follows. Chapter 1 provides the reader with the theoretical tools and basic concepts required for a good understanding of the other chapters of this book. Chapter 2 gives an overview of the state of the art of femtobiology. Experimental results obtained in the course of this thesis work and describing how the photophysics of fluorescent probes is af- fected by protein systems, nucleic acid systems, and artificial photosys- tems are discussed in Chapters 3, 4, and 5, respectively. Details of the experimental techniques and methods are finally given in Chapter 6.

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Chapter 1 Basic Concepts in Photophysics and Photochemistry

1.1. Light-Matter Interaction: Spectroscopy

1/"&$*%)&%/+ is the study of the interaction of electromagnetic radiation with matter. It makes use of light of any wavelength, from radio waves to X-rays, to report on molecules and molecular processes. For example, ultraviolet (UV) and visible spectroscopy probes transitions between electronic states; infrared spectroscopy monitors vibrational states; circular dichroism (CD) spectroscopy measures the differential absorption of left- and right-circularly polarised light; nuclear magnetic resonance (NMR) spectroscopy is based on the absorption of radiofrequency electromagnetic radiation by nuclei placed in a static magnetic field. Spectroscopy can be used to characterise quantitatively, structurally and dynamically chemical and biological systems and is thus a very powerful tool. All the experiments described in Chapters 3–5 of the present work are based on spectroscopic techniques.

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1.1.1. The Light

At the heart of spectroscopy lies the interaction between light and matter.

In its classical description, light is an electromagnetic wave obeying Max- well’s laws. It consists of an electric field, ^{2 *3$}^{! !}

# $

, and of a magnetic field,

# $

4 *3$! !

, oscillating perpendicularly to each other at angular frequency % and propagating along the direction of the wave vector 5!

:

# $ # $

0

, ,

, .

sin sin

2 * $ 2 5 * $

4 * $ 4 5 * $

%

" & ! &

! ! ! ! !

! ! ! ! ! (1.1)

2!0

is the wave amplitude, (5 *! !& ! &% $)

its phase at time $, and *!

the position vector. The amplitude of the wave vector is given by

5 5 2'

" " (

! , (1.2)

where ( is the wavelength. The angular frequency, the wavelength, the frequency ), the period 6, and the speed of light & are related to each other by

2 2 ^& 2 6

% ') ' '

" " ( " . (1.3)

In the case of linearly polarised light, the oscillation plane of each of both fields is constant and equation (1.1) becomes (for light propagating along the 7 direction)

# $ # $

# $

₀⁰

# $

, ,

, .

sin sin

2 7 $ 2 5 7 $

4 7 $ 4 5 7 $

%

" & ! &

" & ! & (1.4)

The phase velocity 8_*, that is, the velocity at which a point of the wave of constant phase propagates is independent of the frequency in vacuum and is equal to the speed of light:

#

_{0 0}

$

^{1 / 2}

8_*" "& + , ^! , (1.5)

where +₀ is the vacuum permeability and ,₀ the vacuum permittivity.

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In its quantum mechanical description, light is emitted or absorbed in discrete quanta known as /'%$%9). The photon energy 2 is given by

2"') "'&), (1.6)

where ' is Planck’s constant and ) is the wavenumber defined as ) 1

"(. (1.7)

When light propagates through a dielectric medium (propagation through metals will not be considered), it induces an electric dipole moment in the molecules of the material. On a macroscopic scale, this gen- erates a polarisation ^{: $}^!

# $

in the medium which oscillates at the light frequency and which is directly proportional to the electric field as long as the field intensity is low (for high field intensities, see section 6.1.2):

# $

0

# $

: $! " , 2 $

"

- . (1.8)

The proportionality factor -^", a second order tensor, is termed the electric susceptibility and is a complex quantity.¹ The phase velocity of light in a dielectric medium is related to its electric susceptibility:

# $

^{1 / 2}

0 0 1

8"_* " "& .1+ , 0-" /2^! . (1.9) The complex refractive index 9" is defined as

# $

^{1 / 2}

( )

( ) 1

vacuum material 9 8

8

*

" " 0 " !

"

" - . (1.10)

This complex quantity is usually separated into its real and imaginary parts, the refractive index 9 and the attenuation index ;:

9"" 09 (;. (1.11)

1 Complex quantities are denoted with a tilde.

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The refractive index is also expressed as the ratio of the speed of light in vacuum and of the light phase velocity inside the dielectric material

# $

Re 8_* " 8"_* :

&

9"8_*. (1.12)

The effect of the refractive index of a material on light is mainly to alter its phase velocity. The attenuation index, on the other hand, describes the effect of the material on the amplitude of the electromagnetic wave. It is directly connected to the absorption of light by the material. The refractive and attenuation indices are related to each other by the Kramers- Kronig relationship [6].

1.1.2. The Absorption of Light

Light is absorbed by a molecule if energy is transferred from the electromagnetic field to the molecule. A beam of light passing through a solution of absorbing molecules thus decreases progressively in intensity. This decrease in the intensity < over the course of a small volume element is proportional to the intensity of the light entering the considered volume, to the concentration & of the absorbers, and to the length of the path d7 through the element:

#

,

$

'

# $ # $

,

d^< ( ⁷ " !, ( ^< ( ^{7 & 7}d . (1.13)

The proportionality constant ,' depends on the wavelength of the light and on the structure of the absorber, its orientation, and its environment.

Integrating equation (1.13) over a pathlength = yields the Bouguer- Lambert-Beer law:

# $

0

# $

exp

#

'

$

0

# $

10 ^&= 0

# $

10 ^>

< ( "< ( !, &= "< ( ^!^, "< ( ^! . (1.14)

<0 is the light intensity before passing through the sample, >",&= is the absorbance or optical density of the sample, and , is called the molar extinction coefficient ( ', "ln10&,). The dependence of , and > on the light frequency can be displayed by plotting these quantities as a function

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of the frequency, the wavelength, or the wavenumber in what is termed an

?,)%*/$(%9 )/"&$*0#. The absorbance is proportional to the concentration of the absorber, to the pathlength of the sample, and is related to the nature of the absorber. It is directly connected to the attenuation index ;:

ln10 4

; >

= (

" ' . (1.15)

Light can only be absorbed by a molecule if the light oscillation frequency corresponds to a resonance of the oscillating dipole induced in the molecule. In a quantum mechanical picture of the absorption process, a molecule in a stationary state a of energy 2_a can only change to a state

b of energy 2_b by absorption of light of frequency ) if the relation

b a

2 !2 "') (1.16)

holds true. In other words, for a photon to be absorbed by a molecule, its energy must correspond to the difference in energy between two stationary states of the molecule. Absorption requires light to interact with the transition dipole moment of the molecule, +!_ab

, a quantity which couples stationary states a and b together, described by wavefunctions ³_a and 3b respectively, while the molecule passes through a succession of super- position states. This quantity is defined as

ab b ˆ a b ˆa

+! " 3 + 3 " +

. (1.17)

+ˆ is an operator that usually corresponds either to the electric dipole moment, to the magnetic dipole moment, or to the electric quadrupole moment. For an allowed transition in the visible region, the relative mag- nitude of the electric dipole, magnetic dipole, and electric quadrupole moment operators are roughly in the ratio 10 :10 :1 [7]. In optical spec-⁷ ² troscopy, the discussion is thus most commonly confined to electric dipole transitions, unless one deals with CD spectroscopy or considers static magnetic field effects. The magnetic field term in equation (1.1) is therefore usually neglected.

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The macroscopic observable related to the transition dipole moment through a quantity ! termed the oscillator strength is the extinction coefficient:

# $

² ²

0 0

2 d 2 ab

3 ln10 8

3

" "

>

# & #

! @ " '"

, , ) ) ' ) +

"

4

" ^! ^, ^(1.18)

where #_" is the mass of the electron, @_> Avogadro’s constant, " the elementary charge, )₀ the central frequency (barycentre) of the absorption band, and where the integration is performed over the full absorption band. For a single electron, the oscillator strength varies between 0 and 1 and describes the probability of occurrence of a transition upon interaction between light of a given wavelength range and a given molecule.

1.2. Excited-State Population Dynamics

1.2.1. The Jablonski Diagram

Once a molecule has absorbed a light quantum, it is said to be in an excited state. Absorption of a photon inducing an electronic transition by a molecule can be described as the jump of an electron from one molecular orbital to another. When considering a closed-shell organic molecule, the first transition most commonly takes place between the highest occu-

Figure 1.1. Examples of electronic configurations corresponding to the lowest electronic states of a typical organic molecule. Electrons are symbolised by arrows with spin up or down. It should be noted that several different configurations can contribute to a given state. The vertical scale is an energy scale.

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pied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). In this case, the excited states may exist in two different multiplicities as shown in Figure 1.1, either singlet (S) or triplet (T), de- pending on whether the two unpaired electrons have the same spin or not. The states are numbered according to their energy sequence (from 1 to 9), the ground state being given the index 0 and the lowest singlet and triplet states being denoted S and 1 T . ₁

The electronic state energies of molecules are far greater than 56, implying that the so-called partition function in the Maxwell-Bolzmann distribution of classical thermodynamics, describing the relative population of a state of a given energy when the molecule has many states available, is practically unity. In other words, even if a molecule has one electronic ground state and many excited states, only the ground state is actually available at equilibrium. Although higher energy levels could, in principle, be reached by thermal activation solely, this is impossible in practice because the required temperatures would cause the molecule to decompose far before these are reached. This implies that electronic excited states

Figure 1.2. Typical Jablonski diagram representing the possible pathways for population and relaxation of excited states. Full arrows correspond to radiative transitions, dashed arrows to non-radiative ones.

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can only be populated by the intervention of external energy (light, chemical reaction).

Consequently, electronically excited states have a finite, usually very short lifetime. Several different photophysical or photochemical processes are generally responsible for the excited-state energy dissipation.

Jablonski diagrams, such as the one in Figure 1.2, are well-suited to depict the excited-state deactivation pathways of a molecule. Distinction between two main relaxation categories is usually made. If the excited-state deactivation involves emission of light, it is said to be *?=(?$(8" and is repre- sented by full arrows in the Jablonski diagram, whereas energy dissipation through heat is termed 9%9-*?=(?$(8" and symbolised by dashed (or sometimes curled) arrows.

A-0%*")&"9&" (see section 1.2.2) is associated with the radiative transition between two states of the same spin multiplicity, /'%)/'%*")&"9&"

with that between states of different spin multiplicities. With purely organic molecules, fluorescence, in principle, takes place from a singlet state while phosophorescence from a triplet state. <9$"*9?- &%98"*)(%9 (see section 1.2.3) refers to a non-radiative process by which the molecule makes a transition from one electronic state to a vibrationally excited state of a lower electronic state of identical spin multiplicity; this process is usually followed by 8(,*?$(%9?- "9"*.+ *"-?7?$(%9 to the lowest vibrational level of the new state. Molecules can also change spin and convert from an excited singlet state to a triplet state through (9$"*)+)$"#

&*%))(9. (see section 1.2.4) followed by vibrational energy relaxation.² Inter- or intramolecular processes which may transiently or perpetually alter the chemical nature of the molecule such as electron transfer, proton transfer, energy transfer, or photochemical reactions can also contribute to deactivating the excited state. Intramolecular excited-state deactivation pathways are discussed in sections 1.2.2–1.2.5, whereas intermolecular pathways are described in section 1.4.

2 Very often, the denominations “internal conversion” and “intersystem crossing” are used to designate the process itself together with the vibrational energy relaxation step. Strictly speaking, however, internal conversion and intersystem crossing correspond to “horizon- tal” processes, that is, processes in which the energy of the system does not change.

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1.2.2. Fluorescence

Following light absorption, a standard organic fluorophore is usually excited to some higher vibrational level of either S₁ or S₂. In the condensed phase, molecules generally rapidly relax to the lowest vibrational level of S by vibrational energy relaxation and, if higher excited states are popu-1

lated, internal conversion, typically within 10^!¹² s. As fluorescence lifetimes are often on the order of 10^!⁹–10^!⁸ s, internal conversion and vibrational energy relaxation are generally complete prior to emission. Hence, fluorescence is commonly considered to originate in a thermally equili- brated excited state, that is, from the lowest vibrational level of S and ₁ does not depend on the excitation wavelength. This statement, which underlies Kasha’s rule [8], is no more valid for molecules with short excited-state lifetimes; azulene or zinc tetraphenylporphine (ZnTPP) are two well-known exceptions to this rule as they display S fluorescence ₂ emission [9-11].

Return to the electronic ground state by fluorescence normally occurs to an excited vibrational ground-state level, which then quickly reaches thermal equilibrium by vibrational energy relaxation. As a consequence, the emission spectrum is in principle a mirror image of the absorption spectrum of the S –₀ S transition. This similarity occurs because the spac-₁ ing of the vibrational energy levels of the S₁ state is similar to that of the ground state, since electronic excitation often does not significantly alter the nuclear geometry.

Examination of the Jablonski diagram reveals an important character- istic of fluorescence spectra. As energy dissipation processes such as vibrational energy relaxation and internal conversion occur prior to emission, and as emission also takes place to higher energy levels of the electronic ground state, fluorescence typically occurs at lower energies, that is longer wavelengths, than absorption. This phenomenon was first docu- mented by Sir G. G. Stokes in 1852 [12] and is termed as 1$%5"))'(!$. a) The Radiative Rate Constant

Einstein demonstrated in 1917 that the rate constant for emission can be related to the dipole strength for absorption. Be a collection of two-level systems with lower level a and upper level b , their associated energies 2a and 2b, and their populations @a and @b. It is assumed that these

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systems are all at temperature 6 and are surrounded by a bath of ambient radiation density ^{5 )}

# $

. One can show that the rate of upward transition from a to b due to absorption is expressed as

# $

b

ab b

d d

@ 4 @

$ " &5 ) & , (1.19)

where 4, the rate constant associated with this transition, is called the Einstein coefficient for absorption. 4 is directly proportional to the dipole strength (see [13] for more details):

2 2

2 2 ba

2 3 4 !

9

' +

" !

# . (1.20)

Light also stimulates the downward transition from b to a and the coefficient for this must be identical to the Einstein coefficient for the upward transition. If the system had no other way to decay, the transition rate due to stimulated emission would be

# $

b

ab b

d d

@ 4 @

$ " ! &5 ) & . (1.21)

Obviously, an excited system can also emit light spontaneously. The transition rate due to spontaneous fluorescence occurs with a rate constant >, the Einstein coefficient for fluorescence:

b b

d d

@ > @

$ " ! & . (1.22)

The Einstein coefficient for fluorescence is a direct measure of the excited-state lifetime 6 of the system in absence of radiation density:

> 1

"6. (1.23)

The total rate of downward transition is given by summing equations (1.21) and (1.22):

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# $

b ab b

d d

@ 4 > @

$ " !.1 &5 ) 0 /2& . (1.24)

At equilibrium, the transition rates of the system are equal and one can relate the populations by

# $ # $

^ab

b

a ab

@ 4

@ 4 >

5 ) 5 )

&

"

& 0 . (1.25)

But at thermal equilibrium, one also knows from thermodynamics that

b b a ab

a

exp exp

@ 2 2 '

@ 56 56

)

7 ! 8 7 8

" 9! :" 9! :

; < ; <. (1.26)

Combining both expressions of the population ratio reveals a relationship between > and 4. By using Planck’s expression for the energy density of black-body radiation [13], one finally finds

3 3 3 2 3

ab ab 2

3 3 ba

1 8 32

3

'9 9!

> 4

& &

' ) ' ) +

" "6 & " & !

# . (1.27)

This equation states that the rate constant for spontaneous emission is directly proportional to the dipole strength for absorption, implying that strong absorbers are inherently also strong emitters. Furthermore, as > is proportional to the cube of the fluorescence frequency, the emission strength will increase as absorption and emission move to higher frequen- cies, that is, to shorter wavelengths.

The Einstein relationship between absorption and fluorescence is strictly valid only for a system that absorbs and emits at a single frequency, which is clearly not the case of molecules in the condensed phase since they display broad absorption and emission spectra. In this case, the rate of fluorescence can nonetheless be related to the integrated absorption strength by corresponding expressions developed successively by Lewis and Kasha (1945), Förster (1951), Strickler and Berg (1962), Birks and Dyson (1963), and Ross (1975) [13].

If a set of molecules with ground and electronic excited states a and b in equilibrium with black-body radiation and at thermal equilibrium is

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considered, the rates of absorption and emission at each frequency must balance. Hence, the rate constant for fluorescence at frequency ), 5fl

# $

) , can be related to the rate constant for excitation at the same frequency,

# $

5ex ) , and to the population ratio of the two states:

# $ # $

^a

# $

^a ⁰⁰

fl ex ex

b b

@ B exp '

5 5 5

@ B 56

) " ) & " ) & & ⁷9 ) ⁸:

; <. (1.28)

Ba and B_b are the vibrational partition functions of electronic states a and b ,³ and ')₀₀ the energy difference between the lowest vibrational levels of the two electronic states (the so-called 0–0 transition energy). By using for 5ex

# $

) an expression for the rate at which a material with a speci- fied concentration and molar coefficient ^{, )}

# $

absorbs energy from a black-body radiation field, and by scaling the measured fluorescence amplitude at each frequency relative to the fluorescence A and to the extinction coefficient , at the 0–0 transition frequency (see [13] and [14] for a complete derivation), one obtains a way to calculate the overall rate constant for fluorescence, 5_rad, if the emission spectrum, the ratio of the vibrational partition functions and the extinction coefficient at a reference wavelength are known:

# $

rad fl d

5 "

4

5 ) )^. ^(1.29)

# $ # $

2

00 00 00

rad a

A b 00

8000 ln10

9 d

5 B A

@ B & A

) ) , )

' ) )

)

7 8

" & &9 : &

; <

4

^. ^(1.30)

The rate constant 5_rad is a molecular analogue of the Einstein > coefficient for fluorescence. If the mirror-image law holds, one can also relate the radiative rate constant to the absorption spectrum instead of the emission spectrum. This expression is usually found in terms of the wavenumber ):

# $

2 a 3 1

rad f

A b

8000 ln10

9 & B d

5 @ B

' ) , ) )

)

! !

" & &

4

^. ^(1.31)

3 The ratio can differ from unity if the energy of one or more vibrational modes or if the spin multiplicity differs in the two electronic states [13]. The partition function was omit- ted in the Einstein relationship because a system without any sublevels was considered.

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The factor )f^!³ ^!¹ is the reciprocal average of )^!³ over the fluorescence spectrum:

# $

3 1

f 3

d d A

A

) )

) ) ) )

! !

"

4

!

4

^. ^(1.32)

Except for the ratio of the vibrational partition functions which Ross added later [13], equation (1.31) is the expression proposed by Strickler and Berg [14].

b) The Fluorescence Lifetime and Quantum Yield

The fluorescence lifetime and quantum yield are among the most important characteristics of a fluorophore. The natural fluorescence lifetime, 6rad, also called radiative lifetime, is the reciprocal of the radiative rate constant defined before:

1 rad 5rad

6 " ^! . (1.33)

As a general rule, a lifetime is defined as the reciprocal of a rate constant:

( 5(1

6 " ^!. (1.34)

The radiative rate constant actually represents the probability per unit time that a molecule relaxes from state b to state a by emitting a photon:

ba rad

d d

: 5

$ " . (1.35)

The temporal variation of the spontaneous fluorescence emission intensity, <fl

# $

$ , follows a first-order kinetic law. If a single population of fluorophores contributes to the emission, the decay kinetics is given by

# $ # $ # $

fl fl 0 exp rad

< $ "< !5 &$ . (1.36)

The natural fluorescence lifetime is thus an “average” value of the time spent by the molecule in the excited state in the absence of any other

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deactivation channel than fluorescence; it corresponds to the time neces- sary for the fluorescence intensity to decay from <fl

# $

0 to ^<^fl

# $

^{0 e}^& ^!¹^{, that}

is, by about 63 %.

As the Einstein > coefficient, the radiative rate constant can also be related to the oscillator strength and thus to the transition dipole moment responsible for absorption:

2 2 2 2

rad 03

0

2 2

" 3

" 9 9

5 !

# &

' ) ,

7 0 8

" &99; ::< & . (1.37)

As ! is directly proportional to )₀, the barycentre of the absorption band, it follows again that 5_rad depends on the cube of the frequency.

If the excited state decayed only through fluorescence, its population would decrease exponentially with time with a time constant, 6_S₁, equal to 6rad. Most of the time however, as explained before (see section 1.2.1), additional decay channels which cause the excited state to decay non- radiatively are operative. These competing processes include for example internal conversion with rate constant 5_IC, intersystem crossing (5_ISC), electron transfer (5_ET), or energy transfer (5_EET). The excited-state deactivation rate through non-radiative channels, 5_nr, is defined as the sum of the rate constants for the individual processes:

nr IC ISC ET EET ...

5 "5 05 05 05 0 . (1.38)

The total rate constant, 5_S₁, for the excited-state deactivation through all ( deactivation channels is then simply given by

S1 rad nr (

(

5 "5 05 "

=

5^. ^(1.39)

The actual excited-state lifetime of most molecules, often called the fluorescence lifetime, is thus shorter than the radiative lifetime (also called intrinsic or natural fluorescence lifetime). For allowed transitions in small organic molecules, it is typically of about 10 ns.

The fluorescence quantum yield, >_fl, is defined as the number of emitted photons relative to the number of absorbed photons. Since both radiative and non-radiative processes contribute to the excited-state deac-

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tivation, the fraction of fluorophores that decay through emission, and hence the fluorescence quantum yield, is given by

1 1

rad rad S fl

rad nr S rad

5 5

5 5 5

6

> " " "6

0 . (1.40)

The fluorescence quantum yield can be close to unity if the rate of the non-radiative processes is much lower than that of the radiative decay. It should also be noted that, in terms of energy, the yield is always less than unity for fluorescence because of the losses induced by the Stokes shift.

In practice, the non-radiative and radiative rate constants are most of the time not known, so that the fluorescence quantum yield cannot be determined by a simple lifetime measurement. However, since the fluorescence intensity is directly proportional to the number of emitted photons, the integral of the fluorescence intensity over the whole fluorescence spectrum wavelength range yields a quantity which is proportional to the total number of emitted photons. To determine the fluorescence quantum yield, one can therefore compare the integrated fluorescence intensity of the sample with that of a standard of known fluorescence quantum yield, >^S_fl, which absorbs the same number of photons as the sample in the experiment, that is, which displays the same absorbance at the excitation wavelength:

# $

fl Sfl

d

1 d A A

) )

> "

4

) ) &>

4

^, ^(1.41)

where ^A

# $

⁾ is the measured fluorescence spectrum of the sample and

# $

A1 ) the measured fluorescence spectrum of the standard. Most accu- rate results are obtained if the standard and the sample absorb and emit over the same wavelength range and if the comparison is done in the same solvent. A correction term of 9²/9_S², where 9 is the refractive index of the sample solution and 9_S² the refractive index of the standard solution, should be included in equation (1.41) if the solvents in which the sample and the standard solutions are prepared differ [15, 16].

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c) The Fluorescence Polarisation Anisotropy

When considering a transition between two electronic states of a molecule, the probability that the transition will take place is proportional to the square of the cosine of the angle between the transition dipole moment associated with that transition and the electric field vector of the light [16]. Hence, illumination of a sample of randomly oriented fluorophores with polarised light (be it polarised along the C axis) results in a photoselection: right after excitation, the population of excited fluorophores will not be random but partially oriented along the electric field vector (the C axis), with most of the fluorophores aligned close to the C axis and very few of them having their transition dipole moment oriented in the 7+

plane. An orientational anisotropy, *, is thus generated in the sample, which can be quantified — in a similar way enantiomeric excesses are for example quantified in organic chemistry — by considering the excess number of molecules having their transition dipole parallel to the excitation light polarisation with respect to the molecules having their transition dipole perpendicular to the light electric field vector, @_$!@_?, compared with the total fluorophore population:

2

@ @

* @ @

?

" ! 0

$

. (1.42)

Fluorescence emission from a molecule is related to its transition dipole for emission and will be polarised along this transition dipole. Very often, the transition induced upon absorption of light is the same as that arising upon emission, so that the transition dipole moments for absorption and emission are the same. Consequently, the light emitted by a sample of immobilised fluorophores illuminated with polarised light will also be polarised. Because the emitted light intensity is directly proportional to the number of emitters, it is possible to directly access the orientational anisotropy by measuring the fluorescence polarisation anisotropy, that is, by comparing the light intensity emitted by the sample with a polarisation parallel to the excitation light electric field, <_$, and that with perpendicular polarisation, <_?:

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2

< <

* < <

?

" ! 0

$

. (1.43)

This measurement is easily achieved by using a set of two polarisers placed right before and after the sample (see section 6.2.2).

The anisotropy value depends on the relative orientation of the transition dipoles for absorption and emission. For a single-photon transition,

3cos2 1

*_" 5@!

, (1.44)

where @ is the angle between the two transition dipole moments. The anisotropy value can vary between 0.4 for parallel dipoles and –0.2 for perpendicular dipoles. It is equal to zero for an angle of 54.74°, the so- called magic angle. Anisotropy values exceeding 0.4 can be reached by multiphoton excitation [16] or with degenerate transitions [17].

In solution, the fluorophores are in principle not immobilised but can diffuse and reorient. This will cause the anisotropy to change with time and ultimately to go to zero when the orientation of the excited fluorophore population is again random. Energy transfer also leads to the fast depolarisation of the excited-state population and fluorescence anisotropy measurements can be used to detect it. The time-dependent anisotropy is measured from the polarised fluorescence intensity decays [16, 18]:

# $ # $ # $

# $

²

# $

< $ < $

* $ < $ < $

?

" ! 0

$

. (1.45)

The decays of the parallel and perpendicular components of the emission are related to the fluorescence intensity at magic angle conditions, ^{< $}

# $

^,

by [16]

# $

¹

# $

1 2

# $

< $^$ "3< $ .1 0 * $ /2, (1.46)

# $

¹

# $

1

# $

<_? $ "3< $ .1 !* $ /2. (1.47)

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The time-dependent anisotropy decay is then analysed to determine which model is most consistent with the data. Very often, ^{* $}

# $

is described as a multiexponential decay. In many cases however, as for an isolated spherical molecule, the anisotropy evolves in time according to a single exponential law:

# $

0exp

#

/ r

$

* $ "* !$ 6 , (1.48)

where *₀ is the initial anisotropy value, from which the angle between the considered transition dipoles can be estimated using equation (1.44), and 6r is the rotational correlation time which describes how the spherical solute undergoes rotational diffusive motion. The rotational correlation time is related to the viscosity of the solvent, A, to the temperature, 6, and to the volume of the molecule, D_m, by the Stokes-Debye-Einstein relationship [19, 20]:

r

r m

1 56

5 D

6 ^" ^" A. (1.49)

The molecular volume can be estimated using atomic increments based on the van der Waals radii [21].

Additional information on the system and on the initial anisotropy value can generally be gained by steady-state anisotropy measurements (see section 6.2.2). The steady-state anisotropy, *_ss, can be calculated from an average of the anisotropy decay over the fluorescence intensity decay.

A single-exponential intensity decay yields

# $ # $

# $

1

1 1

0 S

ss 0 0

S S

0

1 /

d

d ^* ^*

< $ * $ $ * 5

* *

5 5

< $ $ 6 6

B

" B " " &

0 0

4

^. ^(1.50)

Equation (1.50) shows that there are two limiting values for the steady- state anisotropy value. If the depolarisation takes place on a time scale much faster than the excited-state lifetime, that is, if 5*%5S₁, no steady- state anisotropy will be detected (*₎₎C0). If, on the other hand, the reori- entation of the molecules is much slower than the excited-state decay, that is, if 5*&5S₁, then *))C*0.

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A measurement of the steady-state anisotropy as a function of the wavelength yields an anisotropy spectrum. The anisotropy is usually independent of the emission wavelength,⁴ so that only excitation anisotropy spectra are reported. The variation of *0 — and therefore of *ss — with the excitation wavelength can be understood by considering that, as excitation is performed further and further away from a 0–0 transition, the more important will become the admixture from the next lying electronic state, with a different transition dipole, so that the transition dipoles involved in the transition get more and more displaced. In other words, an anisotropy spectrum is a way to evaluate the “purity” of an electronic transition at a given wavelength. It can be useful to identify the different transitions observed in electronic spectra.

In some cases, the anisotropy decay cannot be described by a single exponential decay. Several reasons can account for this, a distribution of correlation times or the hindered rotation of a probe bound to a macromolecule itself undergoing a rotational motion being the most frequently encountered. The latter case is commonly described with the “wobbling- in-a-cone” model [22, 23] which assumes the probe to experience a motion restricted to a cone and its transition dipole to diffuse freely inside this cone fixed on a macromolecule itself rotating on a slower time scale (Figure 1.3). The anisotropy is then approximated by

# $ 0

#

1

$

e ^$^/ ^r e ^$^/ ^d

* $ "* .1 !>_B ^! ⁶ 0>_B/2 ^! ⁶ , (1.51)

where 6r is the rotational correlation time associated with the probe molecule and 6d that associated with the slowly tumbling macromolecule. >B is an order parameter describing the degree of motional restriction and is related to the semicone angle @max by

# $

²

0

max max

1 1 cos cos

2

> *

* @ @

B B

. /

" "D 0 E

1 2 . (1.52)

4 The independence on emission wavelength is expected since emission almost always occurs from only one state, the lowest singlet state, so that only one dipole is involved in the emission process.

Ultrafast excited-state dynamics in biological and in organised environments

Thesis

Reference

Ultrafast excited-state dynamics in biological and in organised environments

Ultrafast Excited-State Dynamics in Biological and in

Organised Environments

Preface

Table of Contents

Chapter 1

Basic Concepts in Photophysics and Photochemistry

1.1. Light-Matter Interaction: Spectroscopy

1.1.1. The Light

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1.2. Excited-State Population Dynamics

1.2.1. The Jablonski Diagram

1.2.2. Fluorescence

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