Modeling of excitonic properties in tubular molecular aggregates Bondarenko, Anna

University of Groningen

Modeling of excitonic properties in tubular molecular aggregates Bondarenko, Anna

DOI:

10.33612/diss.98528598

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Bondarenko, A. (2019). Modeling of excitonic properties in tubular molecular aggregates. University of Groningen. https://doi.org/10.33612/diss.98528598

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Modeling of Excitonic Properties in

Tubular Molecular Aggregates

Zernike Institute PhD thesis series 2019-26

ISSN: 1570-1530

ISBN: 978-94-034-1939-8 (printed version) ISBN: 978-94-034-1938-1 (electronic version)

The work described in this thesis was performed in the research group Theory of Con- densed Matter of the Zernike Institute for Advanced Materials at the University of Groningen, the Netherlands.

Cover image: Artistic representation (triangulated image) of an exciton wavefunction of the tubular aggregate studied in this thesis.

Printed by ProefschriftMaken.

Copyright © 2019 Anna Bondarenko

Modeling of Excitonic Properties in Tubular Molecular Aggregates

PhD thesis

to obtain the degree of PhD at the University of Groningen

on the authority of the Rector Magnificus Prof. C. Wijmenga

and in accordance with the decision by the College of Deans.

This thesis will be defended in public on Friday 11 October 2019 at 11.00 hours

by

Anna Bondarenko born on 28 December 1987

in Kharkiv, Ukraine

Supervisor

Prof. J. Knoester

Co-supervisor

Dr. T.L.C. Jansen

Assessment Committee

Prof. J. Cao Prof. R.M. Hildner

Prof. P.H.M. van Loosdrecht

Contents

1 General introduction 1

1.1 Supramolecular systems . . . . 1

1.2 Collective excited states in molecular aggregates . . . . 2

1.3 Disorder and localization . . . . 6

1.4 Excitation energy transfer . . . . 8

1.5 Multiscale modeling . . . 10

1.6 Tubular cyanine dye aggregates . . . 11

1.7 Aim and outline of this thesis . . . 12

2 Unraveling optical signatures of tubular aggregates altered with halogen exchange 15 2.1 Introduction . . . 16

2.2 Experimental details . . . 17

2.3 Theoretical modeling . . . 19

2.4 The influence of the tube radius on the absorption spectrum . . . 22

2.5 Conclusions . . . 24

2.6 Appendix: Theoretical calculations and modeling . . . 25

2.6.1 Electronic structure calculations . . . 25

2.6.2 Extended Herringbone (EHB) model . . . 25

2.6.3 Model Hamiltonian. . . 27

2.6.4 Linear absorption spectrum . . . 28

2.6.5 Linear dichroism spectrum . . . 29

2.6.6 Parametrization and fitting procedure. . . 29

2.6.7 Couplings in C8S3-Cl and C8S3-Br aggregates . . . 30

3 Nano-confinement of excitons in tubular molecular aggregates 33 3.1 Introduction . . . 34

3.2 Model and Approach. . . 36

v

vi Contents

3.3 Results and Discussion . . . 40

3.3.1 Absorption spectra . . . 40

3.3.2 Degree of localization: participation number . . . 44

3.3.3 Extent of the wave function from its autocorrelation function . . 47

3.3.4 Fractal character of the wave function . . . 50

3.4 Conclusions . . . 50

3.5 Appendix: Additional Information . . . 53

3.5.1 Modeled structures . . . 53

4 Multiscale modeling of complex molecular aggregates 55 4.1 Introduction . . . 56

4.2 Results and discussions . . . 58

4.3 Methods. . . 66

4.4 Appendix: Additional information . . . 68

4.4.1 Obtaining the preassembled structures . . . 68

4.4.2 MD Simulations on the preassembled structures . . . 70

4.4.3 Probing the energetic disorder . . . 72

4.4.4 Absorption spectra calculation. . . 75

5 Comparison of methods to study exciton dynamics 79 5.1 Introduction . . . 80

5.2 Model system. . . 83

5.3 Methods for calculating the EET rate . . . 84

5.3.1 General considerations . . . 84

5.3.2 Treatment of the thermal bath . . . 86

5.3.3 MC-FRET . . . 88

5.3.4 NISE . . . 90

5.3.5 HSR . . . 91

5.3.6 HEOM . . . 92

5.4 Results and discussion . . . 92

5.4.1 High-temperature and fast-modulation limit . . . 93

5.4.2 High-temperature and slow-modulation limit . . . 96

5.4.3 Intermediate regime . . . 100

5.5 Conclusions . . . 105

5.6 Appendix . . . 107

Contents vii

Bibliography 109

Summary 129

Samenvatting 133

List of publications 137

Acknowledgments 139

1

General introduction

Supramolecular aggregates of synthetic dye molecules offer great pos- sibilities to develop nanoscale functional materials for optoelectronic applications. The functionality of such systems is defined by the effi- ciency of absorbing light and excitation energy transport. To study the optical properties of such systems, theoretical modeling of absorption spectra and excitation energy transfer is essential. This chapter intro- duces general concepts of the exciton theory in molecular aggregates.

In essence, it forms the basis for the later chapters of this thesis.

1.1. Supramolecular systems

M anipulating matter at the nanoscale offers great opportunities for solving large- scale societal problems.

The hallmark of nanoscience and nanotechnology is the exploitation of the unique phenomena that occur when matter is organized at the nanoscale, affecting its optical, electrical and magnetic behavior.

One of the strategies to achieve functional nanoscale materials is the possibility of bottom-up engineering, i.e., arranging molecules and atoms in a particular fashion.

This includes self-assembly of molecules into supramolecular structures. Almost unlimited opportunities are offered by using organic molecules as building blocks, as they allow for tuning of the size and properties of materials with high precision.

An important and fascinating feature of assemblies of organic dye molecules, or

1

1

2 1. General introduction

molecular aggregates, is their ability to absorb light and efficiently transport it in the form of excitation energy.

Molecular aggregates can be found in Nature; a notable example is constituted by light-harvesting complexes of plants and bacteria. There, the process of photosynthesis starts with the absorption of solar energy by assemblies of chromophores. Such systems can also be produced artificially. Depending on the intermolecular forces guiding the self-assembly process, molecular aggregates can be obtained in a variety of shapes, such as linear chains,

rings,

films,

or tubes.

Molecules within supramolecular structures are held together via non-covalent forces (van der Waals, electrostatic, hydrogen bonding, etc.), therefore, they keep their chemical identity. However, high order and close packing within molecular aggregates give rise to strong Coulomb interactions which lead to collective excited states. These collective excited states, or Frenkel excitons, are coherently shared by many molecules and are responsible for intriguing optical properties of these aggregates.

This phenomena of molecular aggregates was first discovered more than 80 years ago by Jelley

and Scheibe.

They independently observed that the aggregation of pseudo-isocyanine (PIC) dye molecules is accompanied by a dramatic change in the absorption spectrum: a typical broad band corresponding to the monomer optical response disappears and a characteristic red-shifted narrow band attributed to the collective optical response appears. Today, this red-shifted narrow band is known as J-band after Jelley, and the aggregates exhibiting it are referred to as J-aggregates.

There is another type of aggregates, with blue-shifted absorption band, which are called H-aggregates, where H stands for “hypsochromic” meaning a change of the spectral band position to a higher frequency.

1.2. Collective excited states in molecular aggregates

Frenkel exciton model. The nature of the collective excited states of molecular aggregates can be understood on the basis of the Frenkel exciton model.

Although a molecule usually possesses many excited states, it is often sufficient to consider only one that optically dominates all others. Hence, each molecule can typically be modeled as an effective two-level system—a ground state and a single excited state.

The one-quantum states of the molecular aggregate are then described by the Frenkel exciton Hamiltonian:

H =

X

≤n

|nihn| + X

Jnm

|nihm|, (1.1)

where |ni denotes the state where only molecule

(

) is excited while all

other molecules are in their ground state,

is the transition energy of the molecule

,

and

is the excitonic coupling between molecules

and

. When the molecular

orbitals of the chromophores do not overlap,

is dominated by Coulomb interactions

1.2. Collective excited states in molecular aggregates

1

3

between the transition dipoles of the molecules.

The eigenstates of the Hamiltonian in eq. (1.1) are one-exciton states |ki , i.e., one excitation quantum is shared by the molecules. The state |ki is a linear combination of the molecular excitations |ni :

|ki = X

c_kn

|ni, (1.2)

where the coefficient

, obtained by diagonlization of the Hamiltonian in eq. (1.1), reflects the participation of molecule

in the

th exciton state. The one-exciton manifold is the energetically lowest exciton band consisting of

states. The next higher manifold consists of

two-exciton states in which the molecules share two excitation quanta, etc. For the description of the linear response, like the absorption spectrum, it is sufficient to consider only the one-exciton band, while higher exciton-states are relevant to the nonlinear response.

Ring aggregates. To understand the (linear) optical properties of molecular aggre- gates, it is instructive to consider a simple model of circular one-dimensional geometry, for which many properties can be found analytically.

We consider a homogeneous ring aggregate consisting of

equidistant molecules with equal monomer energies, i.e.,

= ≤ , and equal magnitude of the transition dipole,

. All transition dipoles are arranged in such a way that they have an angle

/2°Ø with the plane of the ring, while their projections on the same plane are tangent to the ring (see Figure 1.1). For the sake of simplicity, we restrict ourselves to nearest-neighbor interactions only, denoted as

.

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z

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k =<latexit sha1_base64="mFqfUNreXOcLsdf4mikRVyiIXYI=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EUIevEYwTwgWcLsZDYZM49lZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSjgz1ve/vZXVtfWNzcJWcXtnd2+/dHDYNCrVhDaI4kq3I2woZ5I2LLOcthNNsYg4bUWj26nfeqLaMCUf7DihocADyWJGsHVSc3TdTUTQK5X9ij8DWiZBTsqQo94rfXX7iqSCSks4NqYT+IkNM6wtI5xOit3U0ASTER7QjqMSC2rCbHbtBJ06pY9ipV1Ji2bq74kMC2PGInKdAtuhWfSm4n9eJ7XxVZgxmaSWSjJfFKccWYWmr6M+05RYPnYEE83crYgMscbEuoCKLoRg8eVl0qxWgvNK9f6iXLvJ4yjAMZzAGQRwCTW4gzo0gMAjPMMrvHnKe/HevY9564qXzxzBH3ifPxMfjsw=</latexit> ±1

k = 0

<latexit sha1_base64="0ut+maMZ7UZsEZK+andb8u0Lemw=">AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9CIUvXisaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz321lZXVvf2CxsFbd3dvf2SweHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbqd+6wmV5rF8NOME/YgOJA85o8ZKD6Nrt1cquxV3BrJMvJyUIUe9V/rq9mOWRigNE1Trjucmxs+oMpwJnBS7qcaEshEdYMdSSSPUfjY7dUJOrdInYaxsSUNm6u+JjEZaj6PAdkbUDPWiNxX/8zqpCa/8jMskNSjZfFGYCmJiMv2b9LlCZsTYEsoUt7cSNqSKMmPTKdoQvMWXl0mzWvHOK9X7i3LtJo+jAMdwAmfgwSXU4A7q0AAGA3iGV3hzhPPivDsf89YVJ585gj9wPn8Aw/ONdA==</latexit>

µ

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µ

<latexit sha1_base64="dQRa/BW56XX2HEIrmTrVrkC3GDA=">AAAB+HicbVDLSsNAFL3xWeujUZduBovgqiRV0GXRjcsK9gFNCJPptB06mYR5CDX0S9y4UMStn+LOv3HaZqGtBy4czrmXe++JM86U9rxvZ219Y3Nru7RT3t3bP6i4h0dtlRpJaIukPJXdGCvKmaAtzTSn3UxSnMScduLx7czvPFKpWCoe9CSjYYKHgg0YwdpKkVsJEhPlQYYl5pzyaeRWvZo3B1olfkGqUKAZuV9BPyUmoUITjpXq+V6mwxxLzQin03JgFM0wGeMh7VkqcEJVmM8Pn6Izq/TRIJW2hEZz9fdEjhOlJklsOxOsR2rZm4n/eT2jB9dhzkRmNBVksWhgONIpmqWA+kxSovnEEkwks7ciMrIZEG2zKtsQ/OWXV0m7XvMvavX7y2rjpoijBCdwCufgwxU04A6a0AICBp7hFd6cJ+fFeXc+Fq1rTjFzDH/gfP4AUESTgw==</latexit>

Wavelength

Absorption

J

n

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exciton band

ground state

Figure 1.1 | Schematic illustration of the Frenkel exciton model for a circular aggregate. The molecules are placed equidistantly along the ring with equal magnitude of the transition dipoles. All transition dipoles make an angleº/2 ° Øwith the plane of the ring, while their projections on the same plane are tangent to the ring. Each molecule is a two-level system, i.e., it has a ground state and one excited state, with equal excitation energy≤. The excited states of the aggregate resulting from the excitation transfer interactionsJ between the molecules are coherently shared between the molecules and are called Frenkel exciton states.

Such exciton states can be classified in bands according to the number of shared excitation quanta.

1

4 1. General introduction

Due to translational symmetry, the exciton eigenstates are Bloch states or “exciton waves” along the ring with wave number

:

|ki = 1 p

X

e^{i k¡n}

|ni, ¡ = 2º/N, (1.3)

with corresponding energies

Ek

= ≤ + 2J cos °

¢

, (1.4)

where

takes integer values,

, assuming

is odd (for

even,

takes the values

).

From eq. (1.4), it follows that the intermolecular interaction

lifts the degeneracy of the energy levels of the monomers

and creates an exciton band of delocalized exciton states. These exciton states are distributed in the exciton band centered around

≤

and the band width is equal to _4|J| for

.

The oscillator strength between the ground state |g i and the exciton state |ki is given by

µ²_k

= |hk| ˆ

= Nµ

µ

cos

+ 1

2 sin

∂

, (1.5)

where

^ˆ P

nµ

ˆ

is the total transition dipole operator given by the sum of the molecular dipole operators and

is the magnitude of the transition dipole moment of a monomer. Eq. (1.5) marks an important result: only three states are optically allowed, namely the

state, which is polarized perpendicular to the plane of the ring, and two degenerate

states, that are polarized within the plane of the ring, perpendicular to each other. These three states collect all the oscillator strength of the individual molecules and all the other states are dark. Due to the giant oscillator strength (proportional to

) collected by these states, they are often referred to as superradiant states. Therefore, the absorption spectrum of the aggregate is dominated by two optical bands.

The

exciton state is the only completely symmetric state, meaning that the oscillations of the transition dipoles of the individual molecules are maximally in phase resulting in a constructive interference of molecular states. For a J-aggregate, as a result of the negative dipole-dipole interactions (

), the

exciton state lies at the bottom of the exciton band at energy

= ≤ ° 2|J| . Hence, the absorption spectrum of the J-aggregate is red-shifted compared to the monomer absorption spectrum.

For an H-aggregate, the opposite is true: positive dipole-dipole interactions (

)

result in the

state lying at the top of the exciton band at energy

= ≤ + 2|J| .

Accordingly, the absorption spectrum of the H-aggregate is blue-shifted compared to

the monomer spectrum. The effect of including long-range dipole-dipole interactions

slightly modifies these numbers, however, it does not significantly change the essential

1.2. Collective excited states in molecular aggregates

1

5

features of the model. The sign of the dipole-dipole interaction

is defined by the angle £ between the transition dipole moment and the vector connecting the two molecules.

The angle £ < 54.7

results in

, i.e., the system is a J-aggregate, while for an H-aggregate, £ > 54.7

resulting in

. The distinction between J- and H-aggregates is schematically depicted in Figure 1.2 using dimers for simplicity. It should be noted, that identification of J and H aggregates based on spectroscopic changes can be misleading, as short-range, non-Coulombic interactions may contribute significantly to the overall shift. A notable example includes aggregates or thin films of poly(3-hexyl thiophene) or P3HT.

|g

|e

monomer dimer

|g

|e

monomer dimer

|g

|e

monomer dimer

Wavelength

Absorbance

J Monomer

Wavelength

Absorption

H

H-aggregate J-aggregate

J > 0 J = 0 J < 0

Parallel

Head-to-tail

= 54.7

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> 54.7

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Figure 1.2 | A schematic exciton energy diagram for a molecular dimer with parallel transition dipoles.

Molecular profiles with the transition dipole moments are depicted by rectangles with the double arrows (l). The dipole-dipole interactionJbetween monomers lifts the degeneracy of the molecular energy level.

The selection rule for light absorption dictates that only in-phase arrangement of the transition dipoles give a dipole allowed state. In the J-aggregate, as a result of a negative dipole-dipole interaction (J < 0), this state is located at the bottom of the exciton band, therefore, the absorption spectrum is red-shifted compared to the monomer spectrum. In the H-aggregate, due to a positive dipole-dipole interaction (J > 0), the in-phase exciton state is located at the top of the exciton band, resulting in the blue-shifted absorption spectrum. The sign of the dipole-dipole interactionJis defined by the angle£between the transition dipole moment and the vector connecting the two molecules.

Tubular aggregates. The analysis of the ring aggregate discussed above can be

extended to a tubular aggregate. The simplest cylindrical aggregate considered here is

composed of

equidistant rings that are stacked on top of each other, with each ring

1

6 1. General introduction

consisting of

equidistant molecules. The orientation of the transition dipoles on each ring is the same as for the single ring considered above. Each molecule is labeled by a two-dimensional vector

,n

) , where

= 1,2,..., N

denotes the ring number and

= 1,2,..., N

denotes the position of the molecule in the ring (denoted

in the previous section).

In the absence of disorder, the cylindrical symmetry of the aggregate governs the Bloch nature of the exciton eigenstates in the ring direction (

coordinate). In the longitudinal direction of the cylinder (

coordinate) with open boundary conditions, the exciton eigenstates are those of a linear chain. The exciton eigenstates of the tubular aggregate are then given by the direct product of the linear chain and the ring wave functions. Restricting to the interactions between neighboring rings, the corresponding expression for the exciton eigenstate has the following form:

|ki = |k

,k

i =

s 2

(N

+ 1)N

X

n

sin(k

)e

|ni, (1.6) where

= º/(N

+ 1) and

= 2º/N

. The wave numbers are defined as

= 1,2,..., N

and

= 0,±1,±2,...,±(N

° 1)/2 (if

is odd).

In the subspace describing the ring direction, the oscillator strength is concentrated in three Bloch states with

= 0 and

= ±1 , as described before. In the subspace describing the linear chain direction, the oscillator strength is mostly concentrated in the non-degenerate state with

= 1 .

Therefore, like in the case of the ring, the cylindrical aggregate has three superradiant states: (k

= 1,k

= 0) , polarized along the axis of the cylinder, and two degenerate states (k

= 1,k

= ±1) , polarized perpendicular to the axis.

The selection rules obtained above are exact only for homogeneous rings and cylindrical aggregates. Clearly, these homogeneous models are an oversimplification of the reality. In the presence of disorder, the exciton states will be localized

on segments of the aggregate, the selection rules will be broken to some extent, and the states will mix.

Disorder can arise from inhomogeneity in the molecular transition energies, the position and orientation of the transition dipoles, or deformation of the tube. In the following section disorder will be discussed in more detail.

1.3. Disorder and localization

In perfectly ordered aggregates, like the homogeneous rings and tubes considered in the previous section, collective excited states are delocalized over the entire aggregate.

In reality, however, aggregates are embedded in a medium, for example, a solvent or

a protein scaffold, where each molecule of the aggregate “feels” a slightly different

environment that fluctuates both spatially and temporally. This results in position-

and time-dependent inhomogeneities in individual molecular transition energies

1.3. Disorder and localization

1

7

and intermolecular couplings, referred to as static and dynamic disorder, respectively.

Both types of disorder may severely affect the delocalized nature of the excitons and, consequently, the observed optical properties. When the coupling to the fast dynamic degrees of freedom, such as vibrations or phonons, is rather small, the effect of static disorder dominates over the dynamic disorder in the environment, and the latter can be omitted. This is usually a good approximation at low temperature or for molecular aggregates where the Stokes shift is small, suggesting a weak coupling to the dynamic environment.

Static disorder. The usual way to treat static disorder is to impose randomness on the individual molecular excitation energies as a consequence of slightly different molecular surroundings. This can be modeled as:

≤n

= ≤

+ ±≤

, (1.7)

where

is an average molecular excitation energy and

is taken randomly from a Gaussian distribution with width of

. This is referred to as static diagonal or energy disorder and the parameter

is the measure of the amount of this disorder. Often, it is assumed that each molecule sees “its own” local environment and, therefore, the energy shifts

are uncorrelated on neighboring molecules. Moreover, physical irregularities of the molecular aggregate, such as changes in positions and orientations of the molecules, cause inhomogeneities in intermolecular interactions. This is referred to as off-diagonal, or interaction disorder. Off-diagonal disorder is in general rarely included, though it was shown to be important for optical properties.

In excitonic systems, disorder breaks the translational symmetry and mixes the homogeneous exciton states. As a result, dark exciton states close to the superradiant state will borrow its oscillator strength. This may lead to a considerable change of the aggregate’s optical properties. For example, a typical characteristic of one-dimensional J-aggregates—asymmetric line shape at low temperatures—arises from the mixing of the superradiant state, lying at the bottom of exciton band, with higher energy states.

A disorder-induced red-shift also originates from mixing of the J-band with higher energy states that push it to lower energy.

Disorder also causes broadening of the absorption band. However, the width of the absorption band of a J-aggregate is typically still much smaller than that of an ensemble of monomers. This commonly observed characteristic of J-aggregates is referred to as exchange narrowing.

This phenomenon originates from the fact that the exciton states delocalized over a delocalization length

effectively average over

uncorrelated Gaussian variables. This reduces the fluctuations in their excitation energy by a factor ^p

as compared to the single molecule. This means that the J-band width is narrowed by a factor ^p

as compared to the monomer absorption spectrum.

The extent of delocalization of the exciton wave function is determined by an

interplay between the intermolecular coupling that delocalizes exciton states and the

1

8 1. General introduction

disorder that tries to localize them.

A common way to quantify the degree of localization is to estimate the number of molecules that coherently share a given exciton state, the so-called participation number.

Another particularly useful measure is the autocorrelation function of the exciton wave function

that estimates not only the size but gives also the directionality of the localization for aggregates with dimension larger than one.

Dynamic disorder. When coupling to the dynamic environmental degree of freedom that causes the disorder is large, and the timescale of the fluctuations of the molecular excitation energies or the intermolecular couplings are fast compared to the exciton processes, the effect of dynamic disorder dominates static frequency disorder.

Dynamic disorder may lead to considerable changes in the absorption spectrum as well. One of such changes, for example, is a result of motional narrowing, i.e., with the decrease of the time-scale of the environment fluctuations, the process of averaging over stochastic processes becomes more effective, thus reducing the width of the absorption band.

It is often a good approximation to account for interactions of the excitons with dynamic degrees of freedom classically. A simple stochastic model is constituted by the overdamped Brownian oscillator.

This model assumes that the classical heat bath randomly pushes the system. This affects the molecular transition energies and this can be expressed by adding a fluctuating term

(t) to the average value:

≤_n

(t) = ≤

+ ±≤

(t), (1.8)

It should be noted that if the coupling to these bath degrees of freedom is large, the latter should be included explicitly as quantum degrees of freedom.

1.4. Excitation energy transfer

Static and dynamic disorder affect the nature of exciton states and as a result excitation energy transfer (EET). Understanding the mechanisms of EET in molecular systems is not only a matter of fundamental scientific interest, but also of relevance for technological applications. EET is one of the key processes regulating the efficiency of photosynthetic light-harvesting systems, and organic electronic devices, such as solar cells.

Coherent vs incoherent dynamics. The nature of the EET in molecular systems is

dictated by a complex interplay of various parameters such as the molecular transition

energy, excitonic coupling

, strength of the disorder

, and coupling to the thermal

fluctuations of the surrounding ° . In the limit of strong excitonic coupling, i.e., when

the excitation transfer interaction between the donor and acceptor molecules, is much

larger than their coupling to the vibrational degrees of freedom, and the strength of

the disorder,

, exciton states are a coherent superposition of the molecular

1.4. Excitation energy transfer

1

9

(localized) states and, therefore, are delocalized over the donor and acceptor. In this regime, the exciton motion is characterized by a wave-like transfer of energy between molecules, referred to as coherent. Coupling to the vibrations of the environment is responsible for the loss of coherences and, consequently, the wave-like transfer is disturbed. In the limit of weak excitonic coupling, i.e., when

, the exciton motion is described as a diffusive hopping from site to site, named as incoherent.

To visualize these two limiting types of exciton motion, it is convenient to consider a site basis representation, in particular to look at the population of the donor and acceptor site as a function of time, as shown in Figure 1.3. Coherent energy transfer is described by the forth and back oscillations in populations of the donor and acceptor.

Incoherent energy transfer is characterized by the exponential decay of the population of the donor and a complementary increase of the acceptor’s population till the equilibrium—in the example, where the energy difference between the donor and acceptor is assumed vanishing, this means till equal population. An intermediate regime of exciton motion is characterized by damped oscillations, as shown in the middle panel.

0 5 10

time (J^-1) 0

0.2 0.4 0.6 0.8 1

site population

0 5 10

time (J^-1) 0

0.2 0.4 0.6 0.8 1

site population

donor acceptor

Coherent Intermediate Incoherent

0 5 10

time (J^-1) 0

0.2 0.4 0.6 0.8 1

site population

donor acceptor

Figure 1.3 | Types of exciton motion between a donor and an acceptor molecule. Coherent motion is characterized by the forth and back oscillations in populations of the donor and acceptor states. Incoherent motion is characterized by the exponential decay of the population of the donor state and exponential increase on the acceptor state till equilibrium is reached. The intermediate regime of exciton motion is characterized by damped oscillations.

Theoretical models used for treating exciton dynamics are commonly based on perturbative approaches. In the regime of strong excitonic coupling, the EET can be described by the Redfield theory.

This is a second-order perturbation theory with respect to the coupling to the bath. In the regime of strong system-bath coupling, the EET is described by the well-known Förster theory.

This is also based on a second-order perturbation theory, but this time with respect to excitonic coupling.

Non-perturbative methods treating both coherent and incoherent energy trans-

fer are available.

However, they are numerically costly, an issue of particular

importance when dealing with multichromophoric systems as molecular aggregates.

1

10 1. General introduction

1.5. Multiscale modeling

Delicate structure-property relationships. So far we used the Frenkel exciton model to explain optical and transport properties of molecular aggregates assuming that we know the molecular arrangement, and as a result the couplings between the molecules and their transition energies. However, typically this molecular arrangement is not known. In the best case, the overall morphology can be resolved by imaging techniques such as cryo-TEM. However, the resolution of this technique is not enough to reach the molecular level.

A long-established way to model optical properties is to combine experimental measurements with phenomenological modeling. Although such approach has proven to be immensely helpful,

the high sensitivity of the optical properties to the details of the molecular packing requires a more accurate and explicit modeling procedure.

In this regard, multiscale approaches that combine different levels of modeling show great potential to help reveal the molecular structure.

Multiscale approaches. A wide range of multiscale approaches exist to couple computational models ranging from coarse-grained to highly accurate quantum mechanical calculations. Classical molecular dynamics (MD) simulations can address questions regarding the plausibility of the overall morphology of the molecular aggregate as well as the underlying molecular packing. Assuming that we know the ‘exact’ force field, MD simulations can predict the intermolecular interaction between the dye molecules and with the solvent at atomistic level of detail. MD trajectories give also access to the level of the structural disorder, i.e., variations in the position and orientation of the molecules. Such structural variations, that give rise to the fluctuations of the intermolecular excitonic couplings—the off-diagonal disorder—can be directly translated into the parameters that determine the Frenkel exciton Hamiltonian. This is particularly interesting, because off-diagonal disorder, being important for the excitonic states and thus the optical properties, is usually not easy to model and, therefore, typically neglected. This arises from the fact that even if the fluctuations of the positions and orientations of the neighboring molecules are uncorrelated, the fluctuations of the arising couplings are correlated.

Furthermore, the fluctuations of the molecular transition energies can be estimated

from microelectrostatic calculations,

that is atomistic calculations which take into

account electrostatic and polarization interactions between the chromophores and

the environment. In such calculations, the interaction between a molecule and the

environment around it can introduce an energy shift to the gas-phase molecular

transition energy. Calculating energy shifts for a large number of molecules results in

a distribution. The mean value and the width of this distribution can then be used to

construct a Frenkel exciton model with diagonal disorder obtained from calculations

rather than from fitting to experiment.

1.6. Tubular cyanine dye aggregates

1

11

Challenges in multiscale modeling. While multiscale approaches promise an enormous progress in understanding complex molecular systems, there are still many challenges which hinder their full adoption. Accurate MD simulations require large- scale calculations of big aggregates made of dye molecule. This is a problematic task since current standard MD force fields are mainly designed to simulate biomolecules, such as proteins, carbohydrates, lipids, and nucleic acids, and are therefore not fitted well to simulate molecules having extended aromatic systems such as dye molecules, fullerenes, or conjugated polymers. The aggregate structure can potentially be obtained from self-assembly simulations. However, the timescale of the self-assembly process is still beyond the reach of current MD tools.

An alternative approach is to start from a guessed structure, for example a phenomenological model fulfilling the boundary condition set by, for instance, the morphology obtained through cryo-TEM imaging.

Ultimately, the success of the multiscale approach is defined by a trade-off between the precision of the computational methods and computational costs. Complicated large-scale molecular systems require complicated modeling. Still, arguably, a simple modeling that gives a clear picture and deep understanding should not be necessarily left behind.

1.6. Tubular cyanine dye aggregates

Aggregation of cyanine dyes is an outstanding example of self-organization of organic molecules. The basic structure of cyanine dyes allows to achieve highly ordered molecular aggregates of various structures and morphologies.

The extended

- conjugated electronic system in the dye’s structure is responsible for the high efficiency of light absorption. In fact, the uniqueness of cyanine dyes as spectral sensitizers in silver halide photography was recognized already in the late 19th century.

The discovery of distinguished optical properties of J-aggregates made it possible to achieve higher levels of photosensitivity.

Since then, cyanine J-aggregates have dominated the field of photography in the past century.

In the last decades, it was discovered that the principles governing cyanine aggre-

gates are very similar to those of the antennae complexes in natural photosynthetic

systems. In this respect, particularly captivating examples are the double-walled

tubular dye aggregates composed of amphiphilic dyes 3,3’-bis(2-sulfopropyl)-5,5’,6,6’-

tetra-chloro-1,1’-dioctylbenzimidacarbocyanine (C8S3).

These tubular aggregates

resemble the light-harvesting complex of green sulfur bacteria

—the most efficient

photosynthetic organisms known. Like natural photosynthetic antennae, these syn-

thetic nanotubes are formed by thousands of closely packed molecules organized in a

tubular geometry. The remarkably uniform supramolecular structure of double-walled

C8S3 aggregates provides an excellent model system for studying how intermolecular

interactions tailor the excitation energy landscape to control and direct energy transfer.

1

12 1. General introduction

1.7. Aim and outline of this thesis

The development of functional nanoscale materials for photonic and optoelectronic applications will be boosted from establishing structure-optical property relationships for biomimetic light-harvesting aggregates. The challenges in establishing these relationships are related to the current lack of information on the microscopic details of the molecular packing within aggregates. Ordering and arrangement of the molecules determine the delicate balance of the excitonic interactions that dictate the optics and exciton dynamics. The details of the molecular packing can be elucidated by a powerful tool—the modeling of the optical spectra combined with comparison to measured spectra.

This thesis is focused on studying the optical properties of large double-walled tubular aggregates of the cyanine dye C8S3, which is an excellent model system for studying optics and energy transfer in biomimetic systems. We establish how structure and molecular packing affect optical signatures. Moreover, we examine the methods applicable to study exciton dynamics in such large multichromophoric systems.

It was shown experimentally that a slight chemical modification in the original C8S3 dye results in the increase of the diameter of the double-walled tubular system in a well-defined fashion. This change of size is accompanied with a change of the optical spectrum. Understanding the origin of the observed changes in the absorption spectra forms the main motivation of Chapter 2. In order to decipher whether the changes in the optical properties arise from the changes in diameter or whether they result from a different molecular packing, we use phenomenological modeling to reproduce the linear absorption and the linear dichroism spectra. An extended herringbone model is used in combination with a Frenkel exciton model to calculate the absorption spectra. Our findings reveal that the observed spectral changes originate purely from the radial growth of the aggregate. This affects the collective optical properties of the supramolecular structure.

The findings of Chapter 2 demonstrate that the physical size of the tubular

aggregates affects the exciton states, in turn resulting in changes of their optical

signatures. It is also known, as explained in Section 1.3, that energetic disorder

destroys the exciton delocalization by breaking the translational symmetry (Section

1.2). Therefore, the following question arises: is it size or is it disorder that matters

most in these systems? Answering this question forms the scope of Chapter 3. A

systematic study of the size effects on the localization properties in disordered tubes is

performed using numerical simulations of the exciton states. Specifically, the radius

and length dependencies of the localized properties are studied for small and large

values of static disorder. The extent of the exciton delocalization is characterized

numerically by calculating the participation ratio and the autocorrelation function of

the exciton wave function. It is demonstrated that the effects of the tube’s radius on

1.7. Aim and outline of this thesis

1

13

the localization properties in the range of parameters relevant to experiment is strong.

Despite the enormous utility of the phenomenological modeling discussed so far, it is limited in its predictive power, which is at the heart of the design of new functional materials, as explained in Section 1.5. The need to improve the available phenomenological modeling formed the basis of Chapter 4. It is demonstrated that accurate insights into the packing of thousands of dye molecules in the complex C8S3 double-walled tubular structure can be obtained by an iterative multiscale approach that combines molecular dynamics and quantum mechanical exciton modeling. It is shown that a thorough understanding of what to use as a starting point and how to validate the structure is required. Both are acquired by modeling of the optical spectra of the obtained structure and comparing it to experimental data. The optical spectra of the thus obtained optimized structure compare very well with experiment.

The microscopic model developed opens the route to accurate predictions of energy transport and other dynamic properties.

Theoretical methods designed to describe energy transfer in extended systems

consisting of closely packed chromophores, such as C8S3 aggregates, should provide

reliable results and be computationally tractable in order to be useful. Such methods

are still not well established, despite a number of available promising methods. A

systematic study of the validity, efficiency, and performance of several popular methods

to study energy transfer in multichromophoric molecular systems is carried out in

Chapter 5. We compare the multichromophoric Förster resonance energy transfer

(MC-FRET) method, the numerical integration of the Schrödinger equation (NISE)

method, and the Haken-Strobl-Reineker (HSR) model, validating them against the

numerically exact Hierarchy of Equations of Motion (HEOM) method. A model system

of a monomeric donor coupled to a multichromophoric acceptor ring of varying size

is considered in two limiting configurations and for a variety of system and bath

parameters, including the regime relevant to biological light-harvesting systems. The

NISE method gives the most reasonable results throughout the parameter regimes

tested, while still being computationally tractable, showing promise in studying exciton

dynamics in large systems, such as C8S3 aggregates.

2

Unraveling optical signatures of tubular aggregates altered with halogen exchange

X

X X

H

C

C

H

X

(CH

)

SO

(CH

)

SO

+

−

−

X = Cl X = Br

Published as B. Kriete,^† A.S. Bondarenko,^† V.R. Jumde,^† L.E. Franken, A.J. Minnaard, T.L.C. Jansen, J. Knoester, and M.S. Pshenichnikov.J. Phys. Chem. Lett. 2017, 8, 13, 2895-2901.

15

2

16 2. Unraveling optical signatures of tubular aggregates

In the field of self-assembly, the quest for gaining control over the supramolecular architecture without affecting the functionality of the individual molecular building blocks is intrinsically challenging. By using a combination of synthetic chemistry, cryogenic transmission electron microscopy, optical absorption measurements, and exciton theory, we demonstrate that halogen exchange in carbocyanine dye molecules allows for fine-tuning the diameter of the self-assembled nanotubes formed by these molecules, while hardly affecting the molecular packing determined by hydrophobic/hydrophilic interac- tions. The findings of this chapter open a unique way to study size ef- fects on the optical properties and exciton dynamics of self-assembled systems under well-controlled conditions.

2.1. Introduction

M olecular self-assembly has proven to be a versatile tool in nanotechnology, as it allows for the autonomous and reproducible assembly of a wide variety of low-dimensional functional systems, extending in size from 10’s of nanometers to microns.

A key challenge in the field of self-assembly is to control the shape and size of the final supramolecular structure with minimal changes of the molecular entities that provide the functionality essential for potential applications.

As the structure of the final assembly is encoded in each individual building block, any modi- fication becomes a highly non-trivial task that requires fine-tuning at the molecular level. It has been shown that tailoring non-covalent molecular interactions such as

-stacking,

hydrogen bonding,

halogen bonding,

or hydrophobic/hydrophilic interactions

provides powerful approaches in directing self-assembly. The coordi- nating nature of hydrophobic/hydrophilic interactions is of special interest, as it may be utilized to tune the supramolecular structure by solely changing the hydrophilic or hydrophobic side groups of the molecules without affecting their functional cores.

Indeed, variations of size and composition of the amphiphilic substituents have been used to change between various structures, such as micelles and bilayers, which is often accompanied by changes in the molecular packing.

In this chapter, we show that even more subtle modifications, namely just replacing a few halogen atoms, may be used to complement hydrophobic/hydrophilic interactions for fine control over the characteristic size of a self-assembled structure, while preserving the molecules’

functional properties and their supramolecular packing.

2.2. Experimental details

2

17

We demonstrate this control of self-assembly for a class of tubular molecular aggregates of amphiphilic carbocyanine molecules that recently have attracted consid- erable interest for their optical functionality.

The close packing of the optically active carbocyanine molecules within the aggregates gives rise to efficient excitation energy transfer and collective optical effects caused by exciton states shared by many molecules.

Changing the amphiphilic side groups results in a wide variety of dif- ferent supramolecular structures,

of which double-walled tubular structures with a diameter in the order of 10 nm have attracted the most attention.

The strong interest in these tubular aggregates stems from their structural resemblance to the light-harvesting antennae of photosynthetic green sulfur bacteria,

which are the most efficient photosynthetic organisms known. Also, the potential of the tubular aggregates as quasi-one-dimensional energy transport wires is of great interest.

Previous attempts to control the diameter of tubular aggregates, including changing solvents or adding surfactants yielded only limited variations of the diameter and often completely changed the supramolecular architecture,

thereby impeding systematic studies of the size effect on the optical functionality and energy transport.

In this chapter, we show how the diameter of the double-walled tubular system may be increased in a well-defined fashion (by 40% for the outer wall and 110% for the inner one) by replacing the four chlorine atoms in the original carbocyanine molecule by bromine atoms. By measurement and simulation of the absorption spectrum, we show that radial inflation of this tubular system is achieved without significantly altering the molecular packing. Besides extending the toolbox of controlling self-assembly, our results pave the road to greater flexibility in controlling of the diameter of tubular aggregates by, e.g., partial substitution of the halogen atoms. This would provide a model system to elucidate the effects of the inherent structural heterogeneity (namely variation of the aggregate radii) encountered in natural chlorosomes.

Moreover, such systematic control also opens up unprecedented opportunities to study size effects on such important photonics properties as exciton dynamics—a crucial aspect of efficient energy transport—and polarization properties, both equally intriguing from theoretical and experimental points of view.

2.2. Experimental details

The dye molecule of interest in this study is the new cyanine dye derivative 3,3’-bis(2- sulfopropyl)-5,5’,6,6’-tetrabromo-1,1’-dioctylbenzimidacarbocyanine, or C8S3-Br, as opposed to its commercially available and much studied counterpart C8S3-Cl (Figure 2.1). The new molecules were produced in a four-step synthesis described in detail in Ref. 103.

Exchanging chlorine with bromine slightly shifts the absorption peak of diluted

molecules towards longer wavelengths, but introduces no other new features (Figure

2

18 2. Unraveling optical signatures of tubular aggregates

X X X

X

X = Cl

X = Br

Figure 2.1 | Schematic representation of the effect of halogen exchange on the radius of the formed aggregates. Chemical structure of C8S3 (on the left) with the halogen substituents abbreviated as X = Cl (C8S3-Cl) and X = Br (C8S3-Br). The radius of the formed tubular aggregates (on the right) is increased by changing the halogen substituent from chlorine to bromine, as revealed by this work.

2.2a,b), which is in line with our electronic structure calculations (see Section 2.6.1).

Addition of Milli-Q water to the methanolic C8S3-Br/Cl stock solutions at room temperature induces a spectral red-shift of about 75 nm ( ª 2400 cm

) and narrowing of absorption bands, both features that are typical for J-type aggregation (Figure 2.2a,b).

The two sharp low-energy bands that both aggregate spectra have in common are broader for C8S3-Br than for C8S3-Cl. In addition, the high-energy flank of the C8S3-Br aggregate spectrum misses the peaks at ª 560 nm and ª 570 nm characteristic for the C8S3-Cl aggregate spectrum. Because the optical properties of molecular aggregates are governed by the interplay of all individual building blocks, the question arises what changes in the aggregate morphology induced by the halogen substitution are responsible for the observed spectral changes.

Experimental evidence for the aggregation of molecules into nanotubes, as sche- matically depicted in Figure 2.1, was found by cryo-TEM. Although thicker bundles of C8S3-Br were occasionally observed (see SI of Ref. 103), there was no apparent mor- phological relation with the isolated tubes. Therefore, the more abundant nanotubes will be the focus of this study.

The cryo-TEM micrograph in Figure 2.2d clearly reveals a double-walled structure of C8S3-Br aggregates, similar to the structure of the C8S3-Cl aggregates (Figure 2.2c).

From the profile scans of the aggregates,

the outer- and inner-wall diameters of C8S3-Br aggregates were obtained as 18.1 ± 0.2 nm and 11.2 ± 0.3 nm, respectively.

This is in striking difference with C8S3-Cl aggregates, where these quantities are

13.1 ± 0.2 nm and 5.4 ± 0.1 nm, respectively. Accordingly, the wall-to-wall thickness

2.3. Theoretical modeling

2

19

500 525 550 575 600 625

Wavelength (nm) 0

0.2 0.4 0.6 0.8 1

Absorption (normalized)

C8S3-Cl

+ H

O

500 525 550 575 600 625

Wavelength (nm) 0

0.2 0.4 0.6 0.8 1

Absorption (normalized)

C8S3-Br

+ H

O

(a)

(b)

50 nm

(d)

50 nm

(c)

Figure 2.2 | Experimental absorption spectra and cryo-TEM micrographs of C8S3 aggregates. Absorption spectra (at room temperature) of molecular aggregates (solid lines) of C8S3-Cl (a) and C8S3-Br (b) together with absorption spectra of both molecules diluted in methanol (dashed lines). Representative cryo-TEM micrographs for C8S3-Br (c) and C8S3-Cl (d) aggregates illustrate the double-walled structure.

of C8S3-Br aggregates amounts to 3.4 ± 0.3 nm, which is thinner than for C8S3-Cl aggregates (3.9 ± 0.1 nm). Here, all error margins refer to the standard error upon averaging.

2.3. Theoretical modeling

It is important to understand whether the changes in the absorption spectra (Figure 2.2a,b) are due mainly to the changes in diameter, or whether they result from different molecular packing in both types of aggregates, which results in differences in excitonic interactions. Since the cryo-TEM micrographs lack sufficient signal and 3D analysis to enhance the signal to noise ratio requires prior information on the symmetry, we retrieve the molecular packing by simulating the absorption spectrum for model structures and determining the structural parameters by fitting the experimental spectrum.

As the basic framework, we use the Extended Herringbone (EHB) model, which

successfully describes the optical transitions of the double-walled tubular aggregates

of C8S3-Cl.

Briefly (see Section 2.6.2 for more details), we consider the dye molecule