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
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
Photovoltaïc Cell Separation o holes and electrons
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Fluorescence Phosphorescence K 06 09 s- K 0-06 s-
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
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
Photoinduced Electron Transer Nonradiative Energ Transer (ollowed b emission)
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
Scheme o the ctive Orbitals in Energ Transer General Scheme: ctive Orbitals:
igression: Indiscernabilit o Electrons and Pauli Principle "Energetic space" "Phsical direct" space
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
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. 50-00 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
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. 50-00 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 =. 5 0 7 n 0 F ν α ν 4 E dν ν 4
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 =. 5 0 7 n 0 F ν α ν 4 E dν ν 4
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.
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
H ' Förster Mechanisms d t d t dt dt β C = = H ' ¹ 0 *(singlet) + 3 + 3*(triplet) llowed H ' β = = C H ' dr dr dt β dt w β w d w α w α w d w ¹ 0
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
H ' Förster Equation dt dt dt dt β C = = H ' ¹ 0 spin-allowed transitions: * (singlet)+ * (singlet) + 3 + * (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.
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.
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.
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
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
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
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!
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)
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
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
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
Useul Relations ev = 8,066 cm- E ev =4. 5 λ nm
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