Quantum Theory of Polymers II.b Energy transfer in polymers: Förster and Dexter mechanisms

Transcription

1 Quantum Theor o Polmers II.b Energ transer in polmers: Förster and exter mechanisms Jean Marie ndré EC Socrates Erasmus programme FUNP, Namur Universit o Warsaw

2 Light emitting devices: charge injection charge transport to a recombination region 3 ormation o excited states rom recombination o radical-ion (polaron) states 4 light emission. Energ harvesting / light transducing devices such as photovoltaic cells: optical absorption photoinduced charge separation 3 charge transport 4 charge collection at the electrodes

3 Photovoltaïc Cell Separation o holes and electrons

4 OLE

5 Fluorescence Phosphorescence K s- K 0-06 s-

6

7 Franck-Condon Shit The rate o non-radiative phonon emission a* a0*c0* is much larger than the radiative transition rate. Thereore, radiative transitions are most likel to occur rom the lowest vibrational level o an electronic state KSH s rule Franck-Condon energ = red shit = Stokes shit 0.5 ev

8 Non-radiative Electronic Energ Transer Förster & exter Mechanisms uring energ transer between molecules, an excited donor molecule * transers its energ to an acceptor which in turn is promoted into an excited state: * + + * In some doped organic thin ilms, the exciton energ can be transerred rom the host to the guest molecule, quenching luminescence o the host while increasing that o the guest. I and are the same = energ migration Electrostatic interaction Exchange interaction

9 Photoinduced Electron Transer Nonradiative Energ Transer (ollowed b emission)

10 From Fermi's Golden Rule to exter & Förster Non-Radiative Energ Transer Equations * + + * k m n = p p m V n ρ E n = β ρ E h h H = electrostatic interaction o all electrons and nuclei β = H ' i = i =å = =å H ' dt dt β =β C = β = H ' dt dt E

11 Scheme o the ctive Orbitals in Energ Transer General Scheme: ctive Orbitals:

12 igression: Indiscernabilit o Electrons and Pauli Principle "Energetic space" "Phsical direct" space

13 ψ(,) = ψ(,) ψ(,) = -ψ(,) ψ(,) = +ψ(,) antismmetr Pauli principle or ermions smmetr Pauli principle or bosons i: then a correct antismmetric wave unction would be: Y, = = Y, Y, µ Y =, µ

14 Förster Equation Electronic energ transer process = one-step transer o electronic excitation rom an excited ONOR molecule (*) to an CCEPTOR molecule () in separate molecules (INTERMOLECULR ENERGY TRNSFER) or in dierent parts o the same molecule (INTRMOLECULR ENERGY TRNSFER) + hν * * + + * Förster = Non-radiative resonance excitation (dipole-dipole) energ transer occurs when an excited molecule (*) can transer its excitation energ to an acceptor () molecule over distances much greater than collisional diameters (e.g Å). The Coulomb term represents the classical interaction o the charge distributions: Q() = e φ*() φ() and Q() = e φ() φ*() that can be expanded into multipole terms : dipole-dipole, dipole-quadrupole, t not too small distances R between the donor and the acceptor, the dominant dipole term is: β Coulomb dipole dipole» MM 3 R

15 Förster Equation Electronic energ transer process = one-step transer o electronic excitation rom an excited ONOR molecule (*) to an CCEPTOR molecule () in separate molecules (INTERMOLECULR ENERGY TRNSFER) or in dierent parts o the same molecule (INTRMOLECULR ENERGY TRNSFER) + hν * * + + * Förster = Non-radiative resonance excitation (dipole-dipole) energ transer occurs when an excited molecule (*) can transer its excitation energ to an acceptor () molecule over distances much greater than collisional diameters (e.g Å). Thus, ET can be approximated to occur via dipole (donor)-dipole (acceptor) interaction (Coulombic interaction). The oscillating dipole o the excited donor (*) causes electrostatic orces which can be exerted on the electronic sstem o the acceptor: k m n k ET = t R0 R = p p m V n ρ E n = β ρ E h h 6 6 R 0 = n 0 F ν α ν 4 E dν ν 4

16 Förster Equation τ = average donor exciton lietime or recombination in the absence o energ transer R0 = Förster critical distance given b the overlap integral over all energies hν φe = quantum eicienc o donor emission n = reractive index o the host F = normalized emission spectrum o the donor α = molar extinction coeicient (absorption spectrum) o the acceptor equation valid i and are well separated (0 Å) and exhibit broadened relativel unstructured spectra spectral overlap is insigniicant no important medium or solvent interactions solvent excited states ling much higher than those o and «overlap» integral: k ET = t R0 R 6 6 R 0 = n 0 F ν α ν 4 E dν ν 4

17 igression: 4-coordinates-spinorbitals H ' dt dt dt dt β C = = H ' ¹ 0 = x,, z s w = x,, z { α β Orthogonalit o spin unctions: α w α w dw= β w β w dw= α w β w dw= β w α w dw=0 Thus ater integration, ψ*()ψ() will give a result dierent o zero onl i the spins o ψ* and ψ are the same. and ater integration, ψ*()ψ() will give a result dierent o zero onl i the spins o ψ* and ψ are the same.

18 H ' Förster Mechanisms d t d t dt dt β C = = H ' ¹ 0 *(singlet) + + *(singlet) llowed H ' β = = C H ' dr Spins o and * and o and * are the same dr dt β dt w β w d w β w β w d w ¹ 0

19 H ' Förster Mechanisms d t d t dt dt β C = = H ' ¹ 0 *(singlet) *(triplet) llowed H ' β = = C H ' dr dr dt β dt w β w d w α w α w d w ¹ 0

20 H ' Förster Mechanisms d t d t dt dt β C = = H ' ¹ 0 *(triplet) + + 3*(triplet) Not allowed 3 H ' β = = C H ' dr dr dt α dt w β w d w β w α w d w =0

21 H ' Förster Equation dt dt dt dt β C = = H ' ¹ 0 spin-allowed transitions: * (singlet)+ * (singlet) * (singlet) + 3* (triplet) Forbidden transitions (triplet-triplet, singlet-triplet) 3 * (triplet) + * (singlet) + Spin conservation o both and = llowed transitions on and + 3* (triplet) + 3* (triplet) The triplet-singlet transition 3 * (triplet) + + * (singlet) is orbidden, but sometimes observed since 3* has a long lietime and although slow ket can be larger than the 3* transition rate.

22 exter Equation Electronic energ transer process = one-step transer o electronic excitation rom an excited ONOR molecule (*) to an CCEPTOR molecule () in separate molecules (INTERMOLECULR ENERGY TRNSFER) or in dierent parts o the same molecule (INTRMOLECULR ENERGY TRNSFER) + hν * * + + * exter = Non-radiative electron exchange energ transer (EEET) occurs when an excited donor molecule (*) and an acceptor molecule () are close enough (0-5 Å) to be in molecular contact. I their electron clouds suicientl overlap each other, an exciton could diusivel hop rom one molecule to the next with no change o spin. lso called overlap or collision mechanism. The exchange term represents the interaction o the exchange charge distributions: Q() = e φ*() φ*() and Q() = e φ() φ(). The vanish i the spin-orbitals φ*() φ*() or φ() φ() contain dierent spin unctions and exponentiall decrease with distance.

23 exter Equation Electronic energ transer process = one-step transer o electronic excitation rom an excited ONOR molecule (*) to an CCEPTOR molecule () in separate molecules (INTERMOLECULR ENERGY TRNSFER) or in dierent parts o the same molecule (INTRMOLECULR ENERGY TRNSFER) + hν * * + + * exter = Non-radiative electron exchange energ transer (EEET) occurs when an excited donor molecule (*) and an acceptor molecule () are close enough (0-5 Å) to be in molecular contact. I their electron clouds suicientl overlap each other, an exciton could diusivel hop rom one molecule to the next with no change o spin.. lso called overlap or collision mechanism. For transers between states with allowed transitions, exter transer is tpicall overwhelmed b long-range Förster dipole-dipole processes. But since triplet-to-triplet energ transer is orbidden b spin conservation in Förster mechanism, exter is the onl mechanism permitting triplet energ transer.

24 pz k ET = h 0 F ν F ν dν F = normalized emission spectrum o the donor F = normalized absorption spectrum o the acceptor Z, determined b the molecular overlap and related to the intermolecular spacing R Z µ exp L = eective average Bohr radius R = distance between the centers o and J= 0 F ν F ν dν spectroscopic overlap integral, measure o the donor-emission and acceptor absorption R L

25 H ' exter Mechanisms H' dt dt dt dt β E = = ¹ 0 *(singlet) + + *(singlet) llowed β = = E H ' H ' dr dr dt β dt w β w d w β w β w d w ¹ 0

26 H ' exter Mechanisms H' dt dt dt dt β E = = ¹ 0 *(triplet) + + 3*(triplet) llowed 3 β = = E H ' H ' dr dr dt α dt w α w d w β w β w d w ¹ 0

27 nother model: exter transport = simultaneous transer o an electron and a hole between * and. Mobilit o the the triplet exciton = product o electron (Ke) and hole (Kh) transer rates: K ET = K e K h Thermall activated!

28 exter Equation H ' H' dt dt dt dt β E = = ¹ 0 spin-allowed EEET: * (singlet) + 3 * (triplet) + Spin conservation o total spin o * sstem + * (singlet) (also allowed under dominating long-range Förster mechanism ) + 3* (triplet) triplet-triplet energ transer (orbidden b resonance-excitation energ transer)

29 Summar: TPT: k = p β ρe h ρe = densit o states, related to spectral overlap J β Förster (95): exter (953): k ET Coulomb» 6 R ν k ET Exchange» e J R / L J

30 Förster: Coulomb term = interaction o charge distributions : and ma be expanded in multipole expansion, limited to dipole-dipole β µ C MM 3 R k Cb µ 6 R ν J

31 exter: Exchange term = interaction o exchange charge distributions : and vanishes i Ψ* and Ψ* or Ψ and Ψ correspond to dierent spin unctions depend on the spatial overlap o orbitals o and decreases exponentiall with increasing internuclear distance β E µ exp R L k Exch µ exp R L J

32 Useul Relations ev = 8,066 cm- E ev =4. 5 λ nm

33 Reerences V. Balzani,. Juris, M. Venturi, S. Campagna, S. Serroni, Luminescent and redoxctive Polnuclear Transition Metal Complexes, Chem. Rev., 996, 96, P.F. Barbara, T.J. Meer, M.. Ratner, Contemporar issues in Electron Transer Research, J. Phs. Chem., 996, 00, R. Farchioni, G. Grosso (Eds.), Organic Electronic Materials, Conjugated Polmers and Low Molecular Weight Organic Solids, Springer, Berlin (00) M. Klessinger, J. Michl, Excited States and Photochemistr o Organic Molecules, VCH Publishers, New York (995) R.. Marcus, Nobel lecture (99), Rev. Modern Phs., 993, 65, 599 J.F. Rabek, Mechanisms o Photophsical Processes and Photochemical Recations in Polmers, Theor and pplications, J. Wile, New York (987) G.C. Schatz, M.. ratner, Quantum mechanics in Chemistr, Prentice Hall, Englewood Clis (993)