Photophysical properties of light harvesting molecules: three different approaches (of increasing complexity and accuracy) to foresee the harvesting behaviour are reviewed with a highly didactic flow. Design principles are highlighted. A supplementary document explaining the details is available among my uploads. This set of slides collects a self-made research I did for a photochemistry course. I don't own part of the shown material and references for many public images are collected at the end of the presentation.

1. Photophysics of dendrimers
Design principles for light harvesting systems
Giorgio Colombi – giorgio.colombi@studenti.unipd.it
Master student in Material Science
University of Padova [IT]
A.A. 2016/2017 - Aarhus University
Photochemistry (Prof. Peter Ogilby)

2. Catalysis
Supramolecular
The general picture
PHOTOPYSICS
Equilibrium
statistical
model
Propensity
matrix
model
Numerical
multiscale
approach
PROCESSING
Linear
E.T.
Nonlinear
effects
APPLICATIONS
and Devices
STRUCTURE
and
CONSTITUENTS
DENDRIMERS
1/37
Light harvesting
SINTESYS

3. DENDRIMERS
Today’s speach 2/37
Catalysis
Supramolecular
PHOTOPYSICS
Equilibrium
statistical
model
Propensity
matrix
model
Numerical
multiscale
approach
Linear
E.T.
Nonlinear
effects
APPLICATIONS
and Devices
Light harvesting
PROCESSING
STRUCTURE
and
CONSTITUENTS
SINTESYS

4. Introduction 3/37
• A wide world of categories and cases
• A photochemical prospective
• Natural antenna systems

5. Introduction_categories and cases 4/37
Dendrimer Dendron Dendritic Nanoparticle
Dendronic & Dendritic surface Dendronized polymer Dendriplex

6. Introduction_categories and cases 5/37
• Based on Metal Complexes • Based on Porphirines
• Based on Fullerenes
• Based on conjugated units
• Based on Azobenzene and Azomethine
• Based on PPV and PTs

7. Introduction_A photochemical prospective 6/37
• Solar cells
• Reverse photochemical cell
(Artificial Photosyntesis)

8. Introduction_A photochemical prospective 7/37

9. Introduction_The lesson from nature 9/37
• Involvement of supramolecular structure with a precise organization in the dimension of:
• SPACE: relative location of the components
• TIME: rates of competing processes
• ENERGY: excited states energies & redox potentials
Rin ∼ 1.8 nm

10. 10/37
• Key photophysics
• Organization in space, time and energy
• Equilibrium Statistical model
• Dynamics of the energy transfer
• Recap of FGR
• Adsorption
• Foster energy transfer
• Propensity matrix model
• Numerical approach
Dendrimers for light harvesting

11. Photophysics_Organization in space 11/37
• C = coordination number of the core
• z = coordination number at each node
• (z-1) = branching
• g = number of generations
• n= n-th generation ∈ [1,g]
• Ω 𝑛 = 𝐶 𝑍 − 1 𝑛−1
= nodes in the nth generation
0
1
2
3
4g=

12. Photophysics_Organization in time & energy 12/37
Tb
3+
TIME
• Ultrafast (ps) energy transfer between generations
• Mean free passage time (MFPT, 𝜏 ) as a quality index
ENERGY
• E*(n-1) ≤ E* (n)
• …and a lot more to come! 

13. Eq. Statistical model 13/37
• Statistical model
• Ideal highly symmetric structure
• Thermodynamic equilibrium
• High number of generations
• No mechanism
Energy vs geometry (entropy)
𝜏 = 𝜏 (𝐸𝑛𝑒𝑟𝑔𝑦 𝑙𝑒𝑣𝑒𝑙𝑠, 𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑦, 𝑇, 𝑟𝑎𝑡𝑒𝑠)

14. Eq. Statistical model_Geometry vs Energy 14/37
kup
kup
kdown
E*
• Geometry (entropy) and Energy compete in defining the direction of energy transfer
• Energetic funnel
E*
n0 1 2 3 .... g
∆
𝜀
𝑈
𝑈
kout kin
• Geometrically induced bias
𝑘 𝑜𝑢𝑡
𝑘𝑖𝑛
=
𝑧 − 1 𝑘 𝑢𝑝
𝐾 𝑑𝑜𝑤𝑛

15. Eq. Statistical model_Statistical ensambles 15/37
Fixed
variables
Partition Function Probability distribution Bridge equation
Equilibrium
condition
Isothermal Isobaric 𝑁, 𝑇, 𝑃 𝑍 =
𝑖
𝑒−𝛽(𝐸 𝑖−𝑝𝑉 𝑖)
𝑃 =
1
𝑍
𝑒−𝛽(𝐸 𝑖−𝑝𝑉 𝑖) 𝐺 = −𝐾 𝐵T ln Z Min G
Microcanonical 𝑁, 𝑉, 𝐸 𝑍 =
𝑖
1 𝑃 =
1
𝑍
𝑆 = 𝐾 𝐵 ln 𝑍 MAX S
Canonical 𝑁, 𝑇, 𝑉 𝑍 =
𝑖
𝑒−𝛽𝐸 𝑖
𝑃 =
1
𝑍
𝑒−𝛽𝐸 𝑖 𝐹 = −𝐾 𝐵 𝑇 ln 𝑍 Min F
Gran Canonical 𝜇, 𝑇, 𝑉 𝑍 =
𝑖
𝑒−𝛽(𝐸 𝑖−𝜇𝑁 𝑖)
𝑃 =
1
𝑍
𝑒−𝛽(𝐸 𝑖−𝜇𝑁 𝑖) 𝑊 = −𝐾 𝐵 𝑇 ln 𝑍 Min W
• Fixed number of atoms in the dendrimer
• Adiabatic approximation (B.O.)
• Fixed temperature
• Eq. position of the excitation: 𝑍
𝐹 = 𝐸 − 𝑇𝑆
𝑃
min 𝐹

16. Eq. Statistical model_Geometry vs Energy 16/37
• The Partition Function
E*
n0 1 2 3 .... g
∆
𝜀
𝑈
𝑈
Ω 𝑛 = Ω 𝐸 𝑛 = 𝐶 𝑧 − 1 𝑛−1
= nodes in the nth generation
= 𝑒−𝛽ε +
𝑛=1
𝑔
𝐶 𝑧 − 1 𝑛−1 𝑒−𝛽(𝜀+Δ+ 𝑛−1 𝑈)
𝑍 =
𝑖
𝑒−𝛽𝐸 𝑖
𝑍 = 𝑒−𝛽ε + 𝐶𝑒−𝛽(𝜀+Δ)
𝑛=1
𝑔
[𝑒ln(𝑧−1)−𝛽𝑈 ] 𝑛−1
=
𝑛=0
𝑔
𝑒−𝛽𝐸 𝑛 Ω(𝐸 𝑛)
= 𝑒−𝛽𝐸0 Ω 𝐸0 +
𝑛=1
𝑔
𝑒−𝛽𝐸 𝑛 Ω(𝐸 𝑛)
Ω0 = 1 = central node
𝐸 𝑛= 𝜀 + Δ + 𝑛 − 1 𝑈

17. High Temperature:
Geometric Bias
dominates!
Eq. Statistical model_Geometry vs Energy 17/37
𝐾 𝐵 𝑇 <
𝑈
ln(𝑧 − 1)
Energy funnel
dominates!

18. Eq. Statistical model_ MFPT 18/37
• 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑅𝑒𝑙𝑎𝑡𝑒𝑠 𝐾𝑜𝑢𝑡 𝑎𝑛𝑑 𝑘𝑖𝑛 𝑘 𝑔→1 = 𝜏 −1
kup
kup
kdown
E*
kout kin

19. Eq. Statistical model_ MFPT 19/37
𝑘 𝑛→𝑛+1= 𝑘 𝑛+1→𝑛 𝑧 − 1 𝑒−𝛽𝑈
1
𝑍
𝐶 𝑧 − 1 𝑛−1
𝑒−𝛽(𝜀+Δ+ 𝑛−1 𝑈)
𝑘 𝑛→𝑛+1 =
1
𝑍
𝐶 𝑧 − 1 𝑛
𝑒−𝛽(𝜀+Δ+𝑛𝑈)
𝑘 𝑛+1→𝑛
𝑘 𝑜𝑢𝑡= 𝑘𝑖𝑛 𝑧 − 1 𝑒−𝛽𝑈
𝑃𝑛 𝑘 𝑛→𝑛+1 = 𝑃𝑛+1 𝑘 𝑛+1→𝑛 𝑃𝑛 = Ω 𝐸 𝑛 𝑃𝑖 =
1
𝑍
Ω 𝐸 𝑛 𝑒−𝛽𝐸 𝑛
• 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑅𝑒𝑙𝑎𝑡𝑒𝑠 𝐾𝑜𝑢𝑡 𝑎𝑛𝑑 𝑘𝑖𝑛
Geometric bias Energy funnel

20. Bias to the perifery No bias = Ramdom walk Bias to the trap
𝐾 𝐵 𝑇 >
𝑈
ln(𝑧 − 1)
𝐾 𝐵 𝑇 =
𝑈
ln(𝑧 − 1)
𝐾 𝐵 𝑇 <
𝑈
ln(𝑧 − 1)
𝑘𝑖𝑛 < 𝑘 𝑜𝑢𝑡 𝑘𝑖𝑛 = 𝑘 𝑜𝑢𝑡 𝑘𝑖𝑛 > 𝑘 𝑜𝑢𝑡
𝜏 ∝ 𝑒 𝑔
𝜏 ∝ 𝑔2 𝜏 ∝ 𝑔
Eq. Statistical model_ MFPT 20/37
• 𝑘 𝑜𝑢𝑡, 𝑘𝑖𝑛 𝑘 𝑔→1 = 𝜏 −1
𝑈
𝐾 𝐵 𝑇
MFPT
g

21. Dynamics of energy transfer 21/37
• Statistical model → Guidelines:
• Harvesting efficiency (up to 80%)
• Key parameters
• Working regimes
• Dynamics of E.T. → Follow the motion of the excitation
• No need to assume Equilibrium
• Propensity matrix model
• Involved photopysical mechanisms ABS
RET
• Absorption: σ 𝐷𝑒𝑛~ 𝑁 𝜎 𝑎𝑏𝑠
• Foster resonant energy transfer (RET)FGR

22. FGR_ABS and RET 22/37
• Time dependent 1th order perturbative theory
• Transition rate between two states under a perturbation U(t)
𝑘𝑖→𝑓 =
2 𝜋
ℏ
𝑓 𝑈 𝑡 = 0 𝑖 2δ(𝐸𝑓 − 𝐸𝑖)
ABS
RET

23. FGR_ABS: 𝐸𝑖 + ℎ𝑣 → 𝐸𝑓 23/37
𝑘𝑖→𝑓 =
2 𝜋
ℏ
𝑓 𝑈 𝑡 = 0 𝑖 2
δ(𝐸𝑓 − 𝐸𝑖)
𝑘𝑖→𝑓 ∝ 𝜓 𝑓 𝜇 𝜓𝑖
2
𝑣 𝑓|𝑣𝑖
2
𝑆𝑓|𝑆𝑖
2
+ 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + ⋯
𝑈(0) = Σ𝑖 𝑞𝑖 𝑉 𝑟𝑖 ≈ Σ𝑖 𝑞𝑖[𝑉 0 + 𝑟𝑖 𝛻𝑉 + ⋯ ]
≈ E Σ𝑖 𝑞𝑖 𝑟𝑖
≈ E( 𝜇 + 𝜇 𝑁)
𝑘𝑖→𝑓 ∝ 𝑓 𝜇 𝑖 2
+ 𝑣𝑖𝑏𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 + ⋯

24. FGR_RET: 𝐴 + 𝐷∗
→ 𝐴∗
+ 𝐷 24/37
𝑘𝑖→𝑓 =
2 𝜋
ℏ
𝐴∗
𝐷 𝑈 𝑡 = 0 𝐷∗
𝐴 2
δ(𝐸𝑓 − 𝐸𝑖)
𝑘𝑖→𝑓 ∝
𝑘2
𝜀𝑅6
𝐴∗
𝐷 𝜇 𝐴 𝜇 𝐷 𝐷∗
𝐴 2
𝑈 0 = 𝜇 𝐴 ∙ 𝐸 𝐷 =
𝜇 𝐴 𝜇 𝐷
4𝜋𝜀 𝑅3 cos 𝜃1 cos 𝜃2 − sin 𝜃1 sin(𝜃2) cos 𝜑
Orientational factor k
∝
𝑘2
𝜀𝑅6
𝐷 𝜇 𝐷 𝐷∗ 2
𝐴∗
𝜇 𝐴 𝐴 2
∝ ABS rate of A
∝ Fluorescence rate of D
∝
𝜎𝐴
𝑤
∝
𝑘 𝑓,𝐷
𝑤3
For a single frequency 𝑤
Einstain coefficients

25. FGR_RET: 𝐴 + 𝐷∗
→ 𝐴∗
+ 𝐷 25/37
• For a single frequency 𝑤
𝑘𝑖→𝑓 ∝
𝑘2
𝜀𝑅6 𝐷 𝜇 𝐷 𝐷∗ 2
𝐴∗
𝜇 𝐴 𝐴 2
∝
𝑘2
𝜀𝑅6 𝑘 𝑓,𝐷 𝜎𝐴
1
𝑤4
∝
𝑘2
𝜀𝑅6
𝜙 𝑓,𝐷
𝜏 𝐷
𝜎𝐴
1
𝑤4
𝑘 𝑓,𝐷 =
𝜙 𝑓,𝐷
𝜏 𝐷
• RET over all the spectrum
𝑘𝑖→𝑓 ∝
𝑘2
𝜀𝑅6 𝜏 𝐷
0
∞
𝜙 𝑓,𝐷 𝑤 𝜎𝐴(𝑤)
𝑑𝑤
𝑤4 Spectral Overlap ( 𝐽𝑖→𝑓) ← Energy Funnel + Stoke shift
• Ensure the harvesting behavior

26. Harvesting_Spectral overlap ← Energy Funnel + Stoke Shift 26/37
= = =
• Without Energy Funnel: 𝑈 = 0
E*
n0 1 2 3 .... g
∆
𝜀
Abs
Fluo
• With Energy Funnel: 𝑈 ≠ 0
E*
n0 1 2 3 .... g
𝜀
Abs
Fluo
E*∆
𝑈
𝑈

27. Harvesting_Spectral overlap ← Energy Funnel + Stoke Shift 27/37
• Relative directional efficiency (𝜖)
kup
kup
kdown
kout
kin
𝑘 𝑢𝑝 ≈ 𝑘 𝑛→𝑛+1
1
∝
𝑘2
𝜀𝑅6 𝜏 𝐷,𝑛
𝐽 𝑛→𝑛+1
𝑘 𝑑𝑜𝑤𝑛 ≈ 𝑘 𝑛+1→𝑛
1
∝
𝑘2
𝜀𝑅6 𝜏 𝐷,𝑛+1
𝐽 𝑛+1→𝑛
÷
𝜖 =
𝑘 𝑑𝑜𝑤𝑛
𝑘 𝑢𝑝
=
𝜏 𝐷,𝑛
𝜏 𝐷,𝑛+1
𝐽 𝑛+1→𝑛
𝐽 𝑛→𝑛+1
Without Energy Funnel 𝜖 = 1 Random Walk
With Energy Funnel 𝜖 > 1 Biased search

28. Dynamics of energy transfer 28/37
Dynamics of E.T. → Follow the motion of the excitation
• Foster resonant energy transfer
• Propensity matrix model
• Introduction
• 1g Dendrimer
• 2g Dendrimer

29. Propensity Matrix Model (PMM) 29/37
• Propensity matrix model
• Ideal highly symmetric structure
• RET only among adjacent chromophores
• Few physically meaningful parameters
• Straightforward to complicate at will

30. 𝑆𝑡: State vector at time 𝑡
Propensity Matrix Model (PMM) 30/37
• Time discretized in intervals Δ𝑡
• The energy migration is followed over all the dendrimer for each Δt
𝑆1
𝑆2
⋮
𝑆 𝑛
𝑆0 𝑡+Δ𝑡
=
𝐶1→1 𝐶2→1 … 𝐶0→1
𝐶1→2 𝐶2→2 … 𝐶0→2
⋮
𝐶1→𝑛
𝐶1→0
⋮
𝐶2→𝑛
𝐶2→0
𝐶𝑖→𝑗
⋯
⋯
⋮
𝐶 𝑛→𝑛
𝐶0→0
𝑆1
𝑆2
⋮
𝑆 𝑛
𝑆0 𝑡
𝑆𝑡+Δ𝑡
𝐶: Propensity matrix
Propensity 𝐶𝑖→𝑗: Probability of E.T. 𝑖 → 𝑗 in time Δ𝑡
• Operating 𝐶 n-times → Exitation flow over 𝑡, 𝑡 + 𝑛Δ𝑡
𝑆𝑡+𝑛Δ𝑡 = 𝐶 𝑛 𝑆𝑡

31. 𝑓
𝑎𝑎𝜖−1
PMM_1g Dendrimer 31/37
𝑆1
𝑆2
𝑆3
𝑆0 𝑡+Δ𝑡
=
1 − 2𝑓 − 𝑎
𝑓
𝑓
𝑎
𝑓
1 − 2𝑓 − 𝑎
𝑓
𝑎
𝑓
𝑓
1 − 2𝑓 − 𝑎
𝑎
𝑎𝜖−1
𝑎𝜖−1
𝑎𝜖−1
1 − 3𝑎𝜖−1 − 𝛾
𝑆1
𝑆2
𝑆3
𝑆0 𝑡
𝛾
𝑎 0.2
𝜖 0
𝛾 0
𝑎 0.2
𝜖 50
𝛾 0
𝑎 0.2
𝜖 50
𝛾 0.2

32. PMM_2g Dendrimer 32/37
𝑓
𝑎
𝑎𝜖−1
𝛾
𝑆1
𝑆2
𝑆3
𝑆4
𝑆5
𝑆6
𝑆7
𝑆8
𝑆9
𝑆0 𝑡+Δ𝑡
=
1 − 2𝑓 − 𝑎
𝑓
0
0
0
𝑓
𝑎
0
0
0
⋯
𝑓
0
0
0
𝑓
1 − 2𝑓 − 𝑎
0
0
𝑎
0
𝑎𝜖−1
𝑎𝜖−1
0
0
0
0
1 − 2𝑓 − 𝑎 − 2 𝑎𝜖−1
𝑓
𝑓
𝑎
⋯
0
0
0
0
𝑎𝜖−1
𝑎𝜖−1
𝑓
𝑓
1 − 2𝑓 − 𝑎 − 2 𝑎𝜖−1
𝑎
0
0
0
0
0
0
𝑎𝜖−1
𝑎𝜖−1
𝑎𝜖−1
1 − 3𝑎𝜖−1
− 𝛾
𝑆1
𝑆2
𝑆3
𝑆4
𝑆5
𝑆6
𝑆7
𝑆8
𝑆9
𝑆0 𝑡
G2 G1 Trap

33. PMM_2g Dendrimer 33/37
𝑆1
𝑆2
𝑆3
𝑆4
𝑆5
𝑆6
𝑆7
𝑆8
𝑆9
𝑆0 𝑡+Δ𝑡
=
1 − 2𝑓2 − 𝑎21
𝑓2
0
0
0
𝑓2
𝑎21
0
0
0
⋯
𝑓2
0
0
0
𝑓2
1 − 2𝑓2 − 𝑎21
0
0
𝑎21
0
𝑎21 𝜖21
−1
𝑎21 𝜖21
−1
0
0
0
0
1 − 2𝑓1 − 𝑎10 − 2𝑎21 𝜖21
−1
𝑓1
𝑓1
𝑎10
⋯
0
0
0
0
𝑎21 𝜖21
−1
𝑎21 𝜖21
−1
𝑓1
𝑓1
1 − 2𝑓1 − 𝑎10 − 2𝑎21 𝜖21
−1
𝑎10
0
0
0
0
0
0
𝑎10 𝜖10
−1
𝑎10 𝜖10
−1
𝑎10 𝜖10
−1
1 − 3𝑎10 𝜖10
−1
− 𝛾
𝑆1
𝑆2
𝑆3
𝑆4
𝑆5
𝑆6
𝑆7
𝑆8
𝑆9
𝑆0 𝑡
G2 G1 Trap

34. Propensity Matrix Model (PMM) 34/37
• FAILS for:
• Large systems
• Long linkers
• Soft linkers
• Low steric hindrance
• Highly symmetric 𝐶
• 𝑘 𝑅𝐸𝑇,𝑛 ; 𝜖 𝑛 , 𝛾
• Propensity matrix model
• Ideal highly symmetric structure
• RET only among adjacent chromophores
• 3D Folding and geometrical distortion

35. Dynamics of energy transfer 35/37
PHOTOPYSICS
Equilibrium
statistical
model
Propensity
matrix
model
Numerical
multiscale
approach
Linear
E.T.
• Fundamental understanding
• Explore key parameters
• Design principles
Address the specific dendrimer
• Realism
• Time, money, effort, skills…

36. Numerical approach 36/37
Single
Cromophore
Dendrimer
DFT
• Ψ𝑞 → 𝜇 𝑞
• E 𝑞
MC
MD
Semiempirical CG
• Structure
• Ψ𝑞 𝑡0 𝑖
Coarse grained
graph
• 𝐻 𝑑𝑑(t)
Q.M. Propagator
algoritm
Ψi 𝑡 = Ψi 𝑡0 𝑒
−
𝑖
ℏ
𝐻 𝑑𝑑(t) 𝑡
𝐸𝑥𝑐𝑖𝑡𝑎𝑡𝑖𝑜𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 (𝑛) ∝ Σ𝑖∈𝑛 Ψi 𝑡 2
Ψ𝑔𝑠 Ψ𝑒𝑥
≈ Ψi 𝑡0
1 −
1
2
𝑖 𝐻 𝑑𝑑Δ𝑡
1 +
1
2 𝑖𝐻 𝑑𝑑Δ𝑡

37. Non Linear effects 37/37
• Dendrimer structure → Exceptoinal 𝜎 𝑇𝑃𝐴
• Direct TPA
∝ 𝑁
• Cooperative pooling
∝ 𝑁2
• Accretive pooling
𝑧 > 4
𝑁
RET

38. Bibliography
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• Balzani et al., Harvesting sunlight by artificial supramolecular antennae, Solar Energy Materials and Solar Cells 38 (J995)
159-173
• Balzani et al., Designing light harvesting antennas by luminescent dendrimers, NewJ. Chem., 2011, 35
• Balzani et al., Forward (singlet–singlet) and backward (triplet–triplet) energy transfer in a dendrimer with peripheral
naphthalene units anda benzophenone core, Photochem. Photobiol. Sci. , 2004, 898-905
• Scholes et al., Lessons from nature about solar light harvesting, Nature chemistry 2011, 3
• Scholes, Andrews, Resonance energy transfer: Beyond the limits, Laser Photonics Rev. 5, 114–123 (2011)
• Klafter et al., Geometric versus Energetic Competition in Light Harvesting by Dendrimers, J. Phys. Chem. B 1998, 102,
1662-1664
• Klafter et al., Dendrimers as Controlled Artificial Energy Antennae, J. Am. Chem. Soc. 1997, 119, 6197-6198
• Astuc et al., Dendrimers designed for functions, Chem. Rev. 2010, 110, 1857–1959
• Andrews, Energy flow in dendrimers: An adjacency matrix representation, Chemical Physics Letters 433 (2006) 239–243
• Andrews, Light harvesting in dendrimer materials: Designer photophysics and electrodynamics, Journal of Materials
Research 2012 , 27(4), pp. 627–638
• Andrews et al., Development of the energy flow in light-harvesting dendrimers, The Journal of Chemical Physics
127,2007
• Press, The Art of Scientific Computing, second edition
• Mansfield et Klushin. Monte Carlo Studies of Dendrimer Macromolecules, Macromolecules 1993, 26, 4262-4268