Photophysics of dendrimers colombi

Photophysics of Dendrimers
Design principles for light harvesting systems
Giorgio Colombi
Universit`a degli studi di Padova
giorgio.colombi@studenti.unipd.it
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Contents
1 Introduction 3
1.1 Categories and cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The photochemical prospective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Light harvesting 5
2.1 Natural antenna systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Dendrimers as artiﬁcial antenna systems . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Organization in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Organization in time and energy . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Statistical equilibrium model 9
3.1 Energy vs geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Mean free passage time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Propensity matrix model 15
4.1 Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1.1 Foster resonant energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 1G Dendrimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Toward a real system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5 Real systems 22
5.1 Coherent energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Multiscale numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6 Conclusion 25
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1 Introduction
Dendrimers are repetitively branched molecules also known as arborols and cascade molecules. Their
first synthesis has been reported in the 1978 and nowadays they are part of a broader picture of
compounds characterised by a fractal structure. [7]
Figure 1: a)Dendrimer. b) Dendron. c) Dendritic nanoparticle. d) Dendronic and dendritic surface.
e) Dendronized polymer. f) Dendriplex.
Hereafter attention will be given to the highly symmetric dendrimer but similar models and concepts
might be extended also to the other cases. However it can be expected that a decrease in symmetry
demands a significant increase in complexity.
Dendrimers are defined by three components: a central core, an interior dendritic structure (the
branches), and an exterior surface with functional surface groups. While the attached surface groups
affect the solubility and chelation ability, the interior part controls the cavity size, the absorption
capacity, and the capture-release characteristics. Applications highlighted in recent literature [16]
include drug delivery, gene transfection, catalysis, photo activity, molecular weight and size deter-
mination, rheology modification, nanoscale science and technology and light harvesting. This paper
attempts to give a general understanding on the latter point providing a theoretical framework to
relate the fractal structure to the photophysical properties. Doing so, the design principle to obtain
light harvesting molecules are highlighted.
1.1 Categories and cases
The interested reader willing to study the photophysics of this compounds on a more specific level
might find interesting to review some particularly relevant moieties whose constituting units are
based on:
- Porphirines: intense absorption in the VIS range.
- Metal complexes: intense absorption in the VIS range.
- Conjugated units: high electron delocalization and rigid structure.
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- Azobenzene and Azomethine: molecular switches.
- PPV and PTs: organic semiconducting polymers.
- Fullerenes: effecient e−
acceptors. Their anaions are so stable that the electron transfer often
occurs in the Marcus inverted region helped by the presence of six unoccupied levels which
provide a low barriers path from the donor toward the fulleren’s LUMO
In the following the specificities of the different units are not accounted.
1.2 The photochemical prospective
To fully enjoy the struggle of the topic, get a grant and last but not least inspire people, it is always
nice to know the big picture where a particular research takes place. Light harvesting mainly lies in
the field of energy conversion and we might expect it to be a key ingredient of the solar cells coming
in the next future.
Figure 2: Qualitative trends in the electricity market. Reaching the unsubsidised price parity is a
milestone for a new, different approach toward energy production and management. And such new
approach is a milestone itself on the path to reduce inequalities across the world. Credits to the
Institute for Local Self-Reliance (U.S.)
Because of the molecular organic nature of the dendrimers, Organic photovoltaic (OPV) solar cells
(red in the following chart) are the system where their integration is easier. Due to their low efficiency
and relatively short lifetime OPV cells are not direct competitors of the ones based on semiconductors.
They might instead power products of low value but enormous diffusion like the portable devices of
the future. In this context harvesting molecules could:
- Increase the amount of adsorbed light.
- Allow a reduction of the cell’s thickness reducing the probability of charge recombination during
the diffusion toward the collecting electrodes.
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Figure 3: Trends in the solar cells efficiencies. Credits to the National Renewable Energy Laboratories
(U.S.)
A second and even more ambitious application in the field of energy conversion is the so called
Artificial photosynthesis or Inverse photochemical cell [8]. The idea is to produce work thanks to
an external flux of electrons while water gets oxidized to oxygen at a first electrode and CO2 gets
reduced to harmless and useful chemicals like alcohols (fuels). The long shot, however, is to develop
catalysts to make an enantioselective C−C coupling of the CO2 to produce pure enantiomers to be
used as precursors in the drugs’ synthesis.
Figure 4: General schemes for inverse photochemical cells [8]
2 Light harvesting
Generally speaking, an antenna for light harvesting is an organized multicomponent system in which
several chromophoric molecular species absorb the incident light and channel the excitation energy
to a common acceptor component [4].
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Figure 5: Schematic representation of an antenna for light harvesting [9].
2.1 Natural antenna systems
Photosynthetic organism are ubiquitous in nature. They may be quite diﬀerent, but all of them use
the same basic strategy, in which light is initially absorbed by antenna proteins containing many
chromophores, followed by energy transfer (ET) to a specialized reaction center protein, in which
the captured energy is converted into chemical energy by means of electron-transfer reactions [13].
The better known natural antennae are the purple bacteria whose features are reported in the fol-
lowing picture:
Figure 6: Light harvesting in purple bacteria: a) Scheme. b) XRD structure of the complex LH2
The lesson from nature is the involvment of supramolecular structure with a precise organization in
the dimension of:
- Space: the relative location of the components
- Time: the rates of competing processes
- Energy: the excited states energies and the redox potentials
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2.2 Dendrimers as artificial antenna systems
In natural antenna systems the relative positions and orientations of the active molecular compo-
nents are determined by the surrounding protein. Although great progresses have been made on
the assembling of artificial components, it is still difficult (if not impossible) to achieve a full sub-
nanometric control on the structure by means of week intermolecular interactions. Therfore current
attempts to mimic the natural antennas are based on linking molecular components by covalent or
coordination bonds. The bonded components of artificial light harvesting system must satisfy several
requirements. The most important ones are:
- Capability to absorb light in the required wavelength’s range.
- Stability toward photo bleaching.
- Appropriate kinetic factors for ET processes.
- The excited states involved as donors have to be populated with high efficiency, have an high
energy content and be reasonably long-living.
In this picture, dendrimers are ideal candidates, however this set of spectroscopic, excited state,
thermodynamic and kinetic requirements is difficult to satisfy. Extensive investigations carried out
in the past years have shown that porphyrins and metal complexes of the second and third transition
rows with polypyridine-type ligands do satisfy most of the above requirements [4].
Figure 7: Examples of harvesting dendrimers based on porphyrines and metal complexes [8]
A last point not really addressed in fundamental studies is the possibility to process such high weight
molecules. In principle they face the same problem of solubility which is typical of organic semicon-
ducting polymers. Therefore their outer shell has to be functionalized with adequate groups.
Nevertheless all this constraints, Dendrimers are a promising class of molecules to mimic the natural
system and their organization.
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2.2.1 Organization in space
The following picture speaks by itself and shows the remarkable degree of organization achievable in
such molecules.
Figure 8: Denrimer schematic representation [8].
In this paper it is assumed an ideal structure where all the nodes, with the exception of the core, are
equal. In this case only few parameters deﬁne the structure and the following notation is used:
C : Coordination number of the core.
z : Coordination number of each node.
(z − 1) : Branching.
g : Total number of generation.
n : n-th generation ∈ [1, g].
Ωn = C(z − 1)n−1
: nodes in the n-th generation.
2.2.2 Organization in time and energy
From a Jablonski diagram, as the one in the following example, it is straightforward to see the role
of time and energy organization. To channel the exitation toward the center an opposite energy
gradient is needed and some ET processes have to be much faster (ps timescale) than all the other
de-exitation pathways. As a matter of fact, the so called mean free passage time (MFPT, < τ >) is
used as a quality index for the harvesting behaviour of these molecules. The MFPT is the mean time
that has to pass for an excitation to travel from the periphery toward the center of the dendrimer
[11].
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Figure 9: Jablonski diagram for the sketched dendron showing the possible de-excitation paths [12].
3 Statistical equilibrium model
Part of the qualitative arguments given so far, especially what concerns the energetic considerations,
is formally addressed by a statistical approach under the assumption of equilibrium [11] [10]. It is
important to notice the limitation of such model: ET is mainly a kinetic process and in addition
a single dendrimer has not enough degrees of freedom to blindly believe to statistical mechanics.
However a solution of 1023
dendrimers under constant illumination might be expected to reach a
steady state correctly described by this model. Two results are to be expected:
- A solid understanding of the interplay between geometric and energetic aspects.
- An analytical espression for the MFPT as a function of energy levels, geometry, temperature
and equilibrium rates.
The reported table collects some maior results about statistical ensambles:
Figure 10: Statistical ensambles
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The relevant ensamble is easily understood to be the canonical one since N,V,T are fixed variables:
- N: fixed number of nodes inside the dendrimer.
- V: ET occurs on such a short timescale that the nuclei can be regarded as fixed according to
the Born–Oppenheimer approximation.
- T: the molecules are thought to be in a thermal bath. It is a realistic condition needed for the
description of microstates of different internal energy.
In the following sections the results highlighted by the red box are used without any further expla-
nation. The reader is assumed to be already familiar with their meaning.
3.1 Energy vs geometry
Geometry and Energy compete in defining the direction of Energy transfer. In a first approximation
the bond connectivity reflect the possible ET pathways and some rate constant might be defined:
kup: rate to go toward a node belonging to an higher generation
kdown: rate to go toward a node belonging to a lower generation
kout: total rate to go toward an higher generation
kin: total rate to go toward a lower generation
From straightforward probabilistic consideration it is easy to relate the total rates to the multiplicity
of possible pathways and to their local rates.
kin = kdown ; kout = (z − 1)kup
If kup = kdown only the geometry defines the direction of energy migration inside the dentrimer and
kout/kin = (z − 1) > 1. Namely, symmetric dendrimers are characterized by an inherent geometri-
cally induced bias toward the periphery. This is a unique properties which differentiate them from
hyper-branched polymers.
One can now superimpose on the geometric bias an energetic funnel (also said spectral gradient),
descending from the periphery to the origin, which acts as a counterbias to the geometric one. For
the sake of simplicity, the energy difference between the excited states of consecutive generations
is assumed to be constant across the dendrimer. This way only three variables define the energy
gradient:
: Excited state energy of the core
∆: Energy difference between the excited state of the core and the one of the first generation
U: Energy difference between excited states of consecutive generations (Energy funnel)
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Figure 11: Geometric bias and Energetic funnel. The figure shows the used nomenclature.
Statistics allow to formally account for this aspects. As always happens in statistical mechanics,
the knowledge of he partition function allows to derive all the thermodynamical properties of the
system (thanks to the bridge equation) and also the probability distribution associated to a certain
microstate. Under the explained approximations and recalling the energetic and structural variable
so far introduced the partition function is easily derived:
In the canonical ensamble the equilibrium condition is given by the minimum of the Helmoltz free
energy (F). Indeed, evaluating the bridge equation and the probability distribution function, one
finds out that the the probability of finding the excitation is maxim for the position which minimizes
the free energy.
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As an appendix a wolfram code is provided to explore the effects of the parameters all together.
The most important one appears to be the ratio between the spectral gradient U and the thermal
energy β = KBT. Clearly temperature plays alongside with entropy which favour the geometric
bias since the number of microstates (nodes) grows exponentially with the generation number. At
high temperature the free energy is then minimized when the excitation lies at the periphery: the
geometric bias dominates. Instead, when βU > ln(z − 1) the free energy is minimized at the core of
the dendrimer so that the energy funnel dominates and the molecule behave as an harvesting system.
Figure 12: Above: High temperature regime. Middle: Random walk regime. Below: Harvesting
regime. left: Helmoltz free energy as a function of generation number. The green line (T=0K) and
the yellow line (βU = ln(z − 1)) define the boundaries of the harvesting regime and that region is
wider for higher values of U (not shown). Right: Probability of finding the excitation on a certain
generation.
For harvesting dendrimer a spectral gradient is needed and the magnitude of the minimum gradient is
of the order of the thermal energy: U = KBTln(z−1) KBT. At room temperature KBT ∼ 25meV
is very small, typically lower than the energy of vibrational levels. Hence, providing the minimum
spectral gradient is not a complex task and in principle it could be given just by the different local
environment experienced by each generation.
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3.2 Mean free passage time
To derive an analytical expression for the MFPT the problem of the full dendrimer is linearised to a
one-dimensional equivalent.
Figure 13: Dendrimer graph and one-dimensional linearisation
Equilibrium implies a zero energy ﬂux between the generations and, assuming an irreversible ideal
trap (k0→1 = 0), it mathematically corresponds to the system of equations:
Which can be solved thanks to the fact that kin and kout are related:
The exact solution for < τ >= k−1
g→0 for an excitation that starts at the periphery to reach the trap
according to this simple, one-dimensional model is given by [10]:
In the appendix a second wolfram code is provided to graphically evaluate these relations. Obviously
the results are coherent with the previus section and it is possible to recognize once again a region in
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the space of parameters where the dendrimers behave as harvesting systems. As enunciated at the
beginning, the MFPT is a measure of the channelling efficiency and it is interesting to look at its
trend with respect to the total number of generations. The reason is that bigger dendrimers would
better serve the case of antenna system because their absorption cross section is roughly proportional
to the number of chromophors. A scheme with the most important results is provided:
Figure 14: Above: high temperature regime. Middle: Random walk regime. Below: Harvesting
regime.
Figure 15: Qualitative summary of the trends of the previous graphs. The MFPT presents respec-
tively an exponential, quadratic and linear behaviour. It is a clear indication of different dynamics
of energy migration. In the harvesting regime the migration goes straight toward the center and
resemble the law of a constant motion x = vt. The random walk is instead identified by analogy
with the diffusive Brownian motion x =
√
Dt.
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4 Propensity matrix model
The statistical model gives a fundamental understanding of the harvesting efficiency (up to 80%),
of the key parameters and of the working regimes. Nevertheless the relies upon the assumption of
thermodynamical equilibrium while energy transfer is known to be mainly a kinetically controlled
process. The so called Propensity matrix model (PMM) addresses this problem from a stochastic
point of view and describes the out of equilibrium dynamics of the energy absorption, migration and
consumption as an homogeneous Markov chain.
4.1 Markov chain
A Markov chain [21] is a stochastic process which satisfy the Markov property. It states that the
transition probability to a certain state depends exclusively on the immediately precedent state and
not on the whole history of the system. Whatever Markov process is characterized by three elements:
- A timeline, continuous or discrete.
- An ensamble of states.
- A function defining the transition probability among the states. This function behaves accord-
ing to the Markov property.
A Markov process is moreover said homogeneous if the transition probability among the states doesn’t
depend on time itself but only on the specific previous state. The discrete time description is the
easiest to understand and implement. In this case the transition probability is defined according to
the arbitrary timeinterval with respect to whom the time axis is discretized. Homogeneity implies
that the transition probability doesn’t depend on the origin of the time axis but only on the distance
between the two temporal states.
For an homogeneous, time-discrete, Markov chain with a finite ensamble of N possible states a ma-
trix notation is successfully adopted. Such chain might indeed be represented by a transition matrix
C ∈ RN×N
and a vector of initial probability S0 ∈ RN
.
S0 defines the probability that the markov chain starts on each specific state.
The elements of C stand for the transition probabilities among states of the chain. A chain in the
state i has a probability Cij to pass toward the state j in a time interval. The elements Cii on the
principal diagonal stand instead for the probability of the system to remain in the same state.
For such a kind of Markov chain the probability of finding the system at a time tn on each specific
state is given by:
(Sn)T
= (S0)T
Cn
Which means that the vector Sn is reached operating the transition matrix n-times on the initial
vector.
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In the language of dendrimer’s science a slightly different notation is used [6],[5],[2]:
Figure 16: Matrix formulation of the migration of energy across the dendrimer for a single timestep.
The state vectors collects the probability of finding the excitation over each chromophore. Their last
term (S0) refers to the core. Each propensity Cij represents the transition probability defined for the
arbitrary timestep ∆t to transfer the excitation from a first chromophore i to a second chromophore j.
In principle the ET between all the chromophores is accountable. However, one typically disregards
the ET between chromophores kept far apart and only the nearest neighbours have a propensity
Cij = 0. Moreover to keep the number of free parameters as small as possible many elements of the
matrix are equal or related one to each other. This is graphically represented by colours on the right.
In this context, the PMM is a powerful analytical method to track the motion of the excitation
introducing only few phenomenological parameters. In addition, choosing the arbitrary timestep
to conveniently be equal to 1 second, the propensities are described as the physically meaningful
transition rates between the donor and acceptor chromophors. Unlike the statistical model, in the
dynamical one the mechanisms of ET can then be considered.
4.1.1 Foster resonant energy transfer
In a first approximation the chromophors are kept far apart by linkers which in principle are not
even conjugated. Therefore it is easily understood that the excitation migrates mainly thanks to the
Foster resonant energy transfer mechanism (RET).
To be reasonably quantitative and get more insight about the role of the energy funnel a derivation
of the RET transfer rate is given in this section. For a time interval of 1s the RET rate is expected
to describe the propensities Cij kRET,ij.
Described the first time under a classical view by Foster, the RET rate is faster to derive with a
semiclassical approach starting by the Fermi golden rule (FGR). It comes form the first order time
dependent perturbative theory and states that the transition rate between two states interacting by
a time dependant perturbation (U(t)) is given by [17],[18]:
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ki→f =
2π
| < f|U(t = 0)|i > |2
δ(Ef − Ei)
Such law is totally general and has to be specialized with the appropriate perturbation. In this
context I believe it is useful to review also the absorption case (Ei + hv → Ef ) since its results are
to be used to express the RET rate in terms of more friendly parameters.
In this case the perturbation is given by the electrostatic interaction between the radiation’s electric
field and the charges of the molecule. Taking the first order series expansion of U(0) and making use
of the Born–Oppenheimer approximation one finds a well known results which shows the role of the
Frank-Condom factor and the origin of the orbital selection rules and spin conservation.
In the case of RET (A + D∗
→ A∗
+ D), in the absence of net charges the main interaction is the
electrostatic one between the two dipoles. As it is reported below, it depends on the dipols, on the
dielectric function of the medium, on the distance at the power of −3 and on an orientational factor
k. Evaluating the FGR with that, one gets:
This partial result can be now expressed in terms of friendly parameters much easier to associate to
experimental data. To do so, the integral is splitted under the legit approximation that no overlap
is present among the wave functions. The resulting integrals are now easily related to the previous
result for absorption and (equivalently) for fluorescence. Moreover, since the oscillator strength is
known to be proportional to the Einstein coefficient B, for a single frequency they can be further
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expressed as proportional to thermodynamical and phenomenological properties like the absorption
cross section of the acceptor and the fluorescence rate of the donor [20]. It tourns out:
Finally, the RET process can in principle occur on more than one frequency so that the total rate is
the integral of the previous result over all the spectrum:
This result is of fundamental importance to understand dendrimer’s photophysics because the direc-
tionality of energy migration and hence the harvesting behaviour basically depends on the spectral
overlap. It is a measure of the overlap between the absorption spectrum of the acceptor and the
florescence spectrum of the donor.
As already seen in the statistical model, an high rate of ET is not enough to secure an harvesting
regime: the rate toward the core has to be higher than the one toward the periphery. As a matter of
fact the medium and the distance are obviously the same and the orientational factor (k) appears at
the power of 2. This implies that the only parameters which might differ in the ET toward a higher
or lower generation are the spectral overlap and the donor lifetime.
The ratio between kdown and kup takes the name of Relative directional efficiency :
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For a highly symmetric dendrimer where the chromophors are all equals the lifetimes’ ratio goes to
1 and the behaviour is completely defined by the spectral overlaps. The insight which comes out
is complementary to the statistical model and strongly remarks the need of an energy funnel. As
qualitatively explained in the following figure, energy funnel and stoke shift play alongside to ensure
the harvesting behaviour:
Figure 17: The green and red bell-shaped curves stand respectively as an indication of the position
of the absorption and fluorescence spectra with respect to the zero vibrational state of the excited
state. Above: Without energy funnel the spectral overlap is the same in both the directions given
by the black arrows. Below: In the presence of energy funnel the spectral overlap associated to the
inward direction (black arrows) is higher than the one of the outward direction (gray arrows).
In summary, without energy funnel there will be no directionality ( = 1) and the dendrimer doesn’t
behave as an harvesting system. Instead, with its presence higher directional efficiencies ( > 1) are
easily achievable and the flux of energy is said to correspond to a biased search toward the trap.
What said so far is a complementary knowledge which confirms all the results of the statistical models
with the exception of one. Now it is understood that the spectral gradient U has not to be as high
as possible but has to match the condition which maximises the directional efficiency. A too big
energy funnel would result instead in a poor transfer rate so that the other dissipative mechanisms
presented in the Jablonski diagram would prevail.
Quantify the ideal energy gradient is neither straightforward or easy to generalize without knowing
the actual absorption and fluorescence spectra of the specific chromophores in the specific environ-
ment. As a qualitative initial attempt, a good energy funnel value might be of the order of the
stoke shift. However it has to be kept in mind that the highest value of Jn+1→n doesn’t necessarily
coincides with this condition because of the dependence on the intensity of the bands of the two
spectra. Moreover, the most important parameter is the directional efficiency and its maximum will
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be a convolution of an high enough value of Jn+1→n and a low enough value of Jn→n+1.
As a reference, the following scheme graphically reports the typical values of stoke shift for porphyries
and metal complexes. They are the chromophors which best satisfy the requirements explained in
the introduction.
Figure 18: On the table representing the spectroscopic ranges of the electromagnetic spectrum a red
box representing the harvesting regime is over imposed. If the energy funnel has a value covered by
the red box, according to the statistical model, the harvesting regime is secured. The intervals given
by the blue arrows cover the literature-based values of stoke shift [24], [3], [15], [1], [23], [4].
It appears clear that the energy gredient needed to match the stoke shift has to be higher than the
simple thedmodynamical constraint. While the latter can in principle be satisfied only because of
environmental effects ([0, 100]cm−1
), to achieve an high efficiency in terms of kinetic constants the
chromophores themselves have to be modified to tune the S0−S1 energy difference of each generation.
Figure 19: Example of dendrimers made of different units. The arrows indicate the exoergonic energy
transfer steps. Empty and full circles indicate Ru(II) and Os(II), respectively. In the peripheral
positions, circles and squares indicate M(bpY)2 and M(biq)2 components, respectively [4].
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Harvesting system of different constituents are still accountable by the PMM (see section 4.3).
4.2 1G Dendrimer
The solid knowledge about the ET mechanism may be used to compute the propensities of a specific
system and build the propensity matrix. The dendrimer made of only one generation is the easiest
example. Here it is reported the operation corresponding to a single timestep:
Figure 20: Scheme, matrix and nomenclature for the considered mechanisms: ET inside the first
generation (f), ET from chromophores of the 1th generation toward the trap (a), ET from the trap
toward chromophores of the 1th generation (b = a/ ), energy consumption at the trap (γ).
In the appendix a last code to iterate the migration of the energy as a Markov chain is provided.
The following graphs summarize the features for the motion of an excitation which starts on the
chromophore D1 (S0 = (1, 0, 0, 0)):
Figure 21: a) A good transfer rate without directionality is not enough to harvest energy toward the
trap. b) Harvesting regime. c) Harvesting regime in presence of a mechanism of energy consumption.
The curves resemble the A → B → C system constituted by 1th generation, trap and consumed
energy
4.3 Toward a real system
What said for the 1G dendrimer might be straightforwardly generalized to a more complex system
at the price of an higher number of independent propensities.
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Figure 22: 2G dendrimer
Figure 23: 2G dendrimer with differen propensities for different generations
In principle also other mechanisms might be added, for example sources of absorption or ET between
not nearest neighbours. however the PPM is better used to gain a qualitative understanding about
key parameters while fails when it is asked to foresee the behaviour of dendrimer which defies the
initial assumption of ideal high symmetry. Namely, it fails for large dendrimer with long and soft
linkers and/or made by chromophores of low steric hindrance. In such cases the C matrix needs an
un-practical extrimely high number of independent propensities to account for a distorted geometry
and for the interactions among all the chromophors.
5 Real systems
All the insight developed so far relied upon strong assumptions which fall for real system, where the
three dimensional folding and distortion has to be considered.
Figure 24: Examples of six generation dendrimers according to Monte Carlo simulations [7], [19].
To deal with a realistic case and also account for the specificities of the constituting chromophores
and linkers a computational approach is needed.
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Figure 25: possible approaches to describe the energy migration in a linear regime.
5.1 Coherent energy transfer
On the hard path to describe a real system, one has also to face the fact that quantum coherent ET
was found to be relevant and it is now an active research topic [14].
Figure 26: Illustration sketching the diﬀerence between classical and quantum-coherent mechanisms
of excitation energy transfer [14].
Forster theory holds when the electronic coupling is weak, meaning that it is small compared to the
coupling to the environment. As the electronic coupling increases, the so-called strong coupling limit
is reached and ET acquires quantum-coherent character. In the strong coupling limit, the donor and
acceptor electronic states mix strongly to produce new, delocalized states that are little perturbed
by the interaction with the environment. Within these states, known as Frenkel excitons, the energy
is shared quantum mechanically among several chromophores instantaneously. This is the so-called
wave-like transfer.
Without going through all the math, here it is graphically reported a clarifying result of the Frenkel
model for an asymmetric dimer.
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Figure 27: Left: Time-averaged Probability of finding the excitation on the first monomer with
respect to ∆D. ∆D represents how the difference of Van der Waals interactions in the excited and
in the ground state variates during the motion of the excitation. It stands therefore for a difference
in stability between the two extreme cases, which correspond to the completely localized excitation.
Right: Probability in time of finding the excitation on the first monomer. The so called Quantum
Beats are observed. The higher is ∆D the more the excitation is localized and the beats’ period is
reduced. The probability of finding the excitation on the second monomer is given by P2(t) = 1−P1(t)
and hence is obviously out of phase with the first one. Credits to R. Bozio (IT)
5.2 Multiscale numerical approach
In this last section the scheme of a complete computational approach is provided for completeness
[6]. A reader already a bit familiar with computational methods would recognize the need for a
multiscale approach: The aim is to describe ET phenomena which show also a coherent behaviour
in a molecule which can reach the weight of a small-chain polymer.
Figure 28: Scheme of a multiscale computational desctiprion.
24

The idea is to solve by ab-initio methods only the electronic-dependant properties of each single
different chromophore. The actual folded and bended structure of the whole dendrimer is instead
addresed with the molecular dynamics (MD) or Monte Carlo (MC) methods, eventually describing
each chromophore as a whole unit according to a coarse grained picture. The two results are then
combined to build a coarse grained graph. The wavefunctions solved ab-initio are placed in the
positions given by the semiempirical simulation and their interaction is given by an Hamiltonian
which includes all the informations. Knowing the Hamiltonian and the starting wavefunctions their
behaviour in time is given by the time-dependent solution of the Schroedinger equation, which is
conveniently evaluated thanks to a quantum mechanical propagator algorithm. The one reported in
the previous figure is the Cayley propagator, which is a second order expansion of the exponential,
thanks to which the starting solution gets discretized in timesteps. [22]
Finally, an estimation of the excitation density per generation is given taking the square of the
absolute value of the wavefunctions and summing over all the chromophores of each generation.
The graphical result is in qualitative perfect agreement with the propensity matrix model and the
beatings of quantum coherence can be identified at the beginning.
Figure 29: Evolution in time of the excitation density corresponding to an excitation starting on the
third generation [6].
6 Conclusion
In summary, the core of this paper was a review of the key photophysics of dendrimers in the linear
regime. The leading idea is to exploit such molecules to collect and harvest light as nature does. A
highly didactical path was chosen, reporting models of increasing realism and specificity. The points
where they confirm or contradict each other have been stressed across the text. Without entering into
the details of the constituting units (chromophores and linkers) a general but solid understanding of
the properties related to the atypical fractal structure has been provided. It is my personal opinion
that such understanding is a mandatory prerequisite to start a detailed study, either computational
25

or experimental, about whatever harvesting system.
The statistical model gave a first feeling of the role of structural and energetic parameters. A lot of
insight was obtained about the need of an energy gradient to face the inherent geometric bias toward
the periphery. Indeed, according to the ratio U/KBT, three possible regime can be recognized: the
high temperature, the Random walk and harvesting one. Each regime is catheterized by a different
scaling behaviour of the MFPT with respect to the dendrimer size and this is recognized as an evi-
dence of different mechanisms of transport.
The stochastic approach of the PMM goes beyond the assumption of equilibrium and allows to track
the motion of the excitation in time. Doing so, the mechanisms of energy absorption, transfer and
consumption can be implemented. The Foster ET was acknowledged as the main mechanism of ET
and the presence of an energy gradient was understood to be more than just a thermodynamical
prerequisite. It is indeed the most relevant parameter to ensure the directionality of the migration in
terms of a competition between kinetic rates. In this picture, differently from the previous model, the
energy funnel doesn’t have to be as big as possible but has rather to be tuned to maximize the ratio
between the spectral overlaps and hence the ratio between the inward and outward kinetic rates. In
addition, the PMM is the first step to quantify how the coupling between the antenna system and a
reaction center influences the excitation density in time. Results have been shown for the simplest
case of the single generation dendrimer.
Finally, a taste of a complete computational study has been given. Literature results confirmed once
again the insight of the previous models. Furthermore such method was seen to be able to catch the
presence of quantum coherence.
All this elements have been discussed to go as close as reasonably possible to a quantitative description
of light harvesting phenomena. In order to design efficient artificial antennae some guidelines are
now understood:
- Environmental effects are enough to ensure an harvesting behaviour but to achieve the best
efficiency the energy funnel has to be tuned. This can be done by a chemical modification of the
chromophores in the different generations. The ideal value maximizes the relative directional
efficiency and ensures that the spectral overlap for the exoergonic direction is high enough
so that the ET occurs on a timescale which has to be much shorter than that of the other
mechanisms of the Jablonski diagram.
- From a structural poin of view , to have a short ET timescale the two relevant parameters are
the distance between the chromophores (R ∈ [0, ∞])and the orientational factor (k ∈ [0, 1]). A
synthetic chemist struggling in the attempt to control both distance and dihedral angle between
two units might consider the following plots to choose what to prioritize.
26

Figure 30: 3D plot and equivalent 2D color map of the ratio k2
/R6
. The orientational factor spans
all its possible values while the distance has been chosen to span the arbitrary interval [1,2]. The
suggestion is to consider the latter as an arbitrary, relative scale. On the second plot a set of
isolines is over imposed as a guide for the eyes. It appears evident that if the distance between the
chromophores increases of a 50% there will be a loss in the transfer eﬃciency, no matter the increase
of the orientational factor. As a rule of thumb, privileging the distance appears to be a reasonable
choice as long as the synthesis’ work is not constrained by other factor as the steric hindrance. The
case of k → 0 has not to be feared because even in the case of a complete orientational freedom it
would average to a non nil value due to thermal disorder.
- A last parameter easily controllable is the dendrimer’s dimension. In principle, the higher the
number of generations (g) the higher it is its absorption cross section. However two factors
set a boundary. The ﬁrst one is a trivial consideration: due two steric hindrance after nine
generations defects appear to happen in the molecular structure [7]. The second factor to take
in account is the MFPT, which in the harvesting regime scales linearly with the number of
generations. Its value clearly has to remain considerably smaller than the timescale of the
other dissipative processes. There would actually be a third energetic constraints but it is easy
to show that it is much less restrictive than the previous ones. In the hopping picture it must
be considered that at each step toward the center a fraction of the incident energy equal to
the energy funnel (U) is dissipated. For an excitation generated at the periphery the total loss
is then ∆E ≥ gU. A dendrimer of three generation with an ideal energy funnel of the order
of magnitude of its stoke shift (∼ 200cm−1
) would then dissipate ∆E ∼ 600cm−1
. Obviously,
if compared to the initial energy of the excitation (∼ 16000cm−1
in the case of a free base
porphyrine) it is not a big deal.
27

References
[1] Handbook of Porphyrin Science: volume 26. G.Ferreira. World Scintific, 2004.
[2] D. Andrews et al. “Development of the energy flow in light-harvesting dendrimers”. In: The
Journal of Chemical Physics (127 2007).
[3] N. Periasamy et al. “Fluorescence Dynamics of Noncovalently Linked Porphyrin Dimers and
Aggregates”. In: J. Phys. Chem. (99 1995).
[4] Vincenzo Balzani at al. “Harvesting sunlight by artificial supramolecular antennae”. In: Solar
Energy Materials and Solar Cells (38 1995).
[5] D. Andrews. “Energy flow in dendrimers: An adjacency matrix representation”. In: Chemical
Physics Letters (433 2006).
[6] D. Andrews. “Light harvesting in dendrimer materials: Designer photophysics and electrody-
namics”. In: Journal of Materials Research (27 2012).
[7] Didier Astruc, Elodie Boisselier, and Catia Ornelas. “Dendrimers designed for functions”. In:
Chem. Rev. (110 2010).
[8] Vincenzo Balzani, Alberto Credi, and Margherita Venturi. “Photochemical Conversion of Solar
Energy”. In: ChemSusChem (1 2008).
[9] Vincenzo Balzani et al. “Designing light harvesting antennas by luminescent dendrimers”. In:
New J. Chem. (35 2011).
[10] A. Bar-Haim and J. Klafter. “Geometric versus Energetic Competition in Light Harvesting by
Dendrimers”. In: J. Phys. Chem. (102 1998).
[11] A. Bar-Haim, J. Klafter, and R. Kopelman. “Dendrimers as Controlled Artificial Energy An-
tennae”. In: J. Am. Chem. Soc. (119 1997).
[12] G. Bergamini et al. “Forward (singlet–singlet) and backward (triplet–triplet) energy transfer
in a dendrimer with peripheral naphthalene units and a benzophenone core”. In: Photochem.
Photobiol. Sci. (3 2004).
[13] R. E. Blankenship. Molecular mechanisms of photosynthesys. Oxford, 2002.
[14] Elisabetta Collini. “Spectroscopic signatures of quantum-coherent energy transfer”. In: Chem.
Soc. Rev. (2012).
[15] Introduction to Fluorescence Sensing. A. Demchenko. Springer, 2002.
[16] Jean M.J. Frechet and Donald A. Tomalia. Dendrimers and Other Dendritic Polymers. Wiley,
2001.
[17] David J. Griffiths. Introduction to Electrodynamics. 3rd ed. Prentice Hall, 1999.
[18] G. Grosso and G. Pastori. Solid State Physics. Academic Press, 2000.
[19] Mansfield and Klushin. “Monte Carlo Studies of Dendrimer Macromolecules”. In: Macro-
molecules (26 1993).
28

[20] O.Svelto. Principles of lasers. Springer, 2013.
[21] M. Plischke and B. Bergensen. Equilibrium statistical physics. World scientific, 2005.
[22] W.H. Press et al. The Art of Scientific Computing. Cambridge university press, 1992.
[23] D. Walther and G. Vos et al. “A Bibenzimidazole-Containing Ruthenium(II) Complex Acting
as a Cation-Driven Molecular Switch”. In: Inorg. Chem. (39 2000).
[24] S. Zakavi and S. Hoseini. “The absorption and fluorescence emission spectra of meso-tetra(aryl)
porphyrin dications with weak and strong carboxylic acids: a comparative study”. In: RSC Adv.
(5 2015).
29

Energy vs Geometry
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photochem.nb 5

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