CHM 5175: Part 2.7 Emission Quantum Yield Source Detector hn F(em) = # of photons out # of photons in Sample Ken Hanson MWF 9:00 – 9:50 am Office Hours MWF 10:00-11:00 1 High Efficiency Emitters Metal Ion Sensing High Efficiency Emitters High Efficiency Emitters Biological Labeling High Efficiency Emitters OLED Samsung (10/9/13) High Efficiency Emitters Expression Studies 4-(p-hydroxybenzylidene)- imidazolidin-5-one Green Fluorescent Protein High Efficiency Emitters High Efficiency Emitters http://www.glofish.com/ Emission Quantum Yield Source Emission Quantum Yield (F) Detector hn F= # of photons emitted # of photons absorbed Sample Ground State (S0) hn Singlet Excited State (S1) hn Excited State Decay Radiative Decay Excitation F= # of photons emitted # of photons absorbed Non-emissive Decay Non-radiative Decay Excited State Decay Reaction Kinetics S1 Energy kA S0 kA knr kr S0 kA = excitation rate kr = radiative rate knr = non-radiative rate S1 kr + knr S0 If it is assumed that all processes are first order with respect to number densities of S0 and S1 (nS0 and nS1 in molecules per cm3) Then the rate Equation: dnS1 dt k A nS0 (k r k nr )nS1 Excited State Decay S1 Rate equation: dnS1 Energy kA knr kr S0 dt k A nS0 (k r k nr )nS1 Sample is illuminated with photons of constant intensity, a steady-state concentration of S1 is rapidly achieved. dnS1/dt = 0 nS1 kA = excitation rate kr = radiative rate knr = non-radiative rate nS0 k A k F knr Substitution for photon flux and the relationship between kA and kr then (math happens): kr F knr kr F = Emission Quantum Yield Quantum Yield kr F knr kr = # of photons emitted # of photons absorbed kr knr Non-radiative Rates kr FF = kr + knr kr FF = kr + kchem + kdec + kET + ket + kpt + ktict + kic + kisc … Rate constants: kr = radiative kchem = photochemistry kdec = decomposition kET = energy transfer ket = electron transfer ktict = proton transfer ktict = twisted-intramolecular charge transfer kic = internal conversion kisc = intersystem crossing Emission Quantum Yield Quantum Yield: kr F k r kisc k nr Emission Quantum Yield Emission Quantum Yield F= # of photons emitted # of photons absorbed = 0 to 1 Fluorescence Quantum Yield kr FF = kr + kchem + kdec + kET + ket + kpt + ktict + kic + kisc … 1) Internal conversion (kic) -non radiative loss via collisions with solvent or via internal vibrations. 2) Quenching -interaction with solute molecules capable of quenching excited state (kchem, kdec, kET, ket ) 3) Intersystem Crossing Rate 4) Temperature - Increasing the temperature will increase of dynamic quenching 5) Solvent - viscosity, polarity, and hydrogen bonding characteristics of the solvent -Increased viscosity reduces the rate of bimolecular collisions 6) pH - protonated or unprotonated form of the acid or base may be fluorescent 7) Energy Gap Law Intersystem Crossing S1 T2 T1 E S0 Excitation Fluorescence Intersystem Crossing Phosphorescence Atom Size ISC Increase in strength of spin-orbit interaction FFluorescence FPhosphorescence Temperature Dependence Measuring Quantum Yield Source Detector hn Sample kr F knr kr = # of photons emitted # of photons absorbed We don’t get to directly measure F, kr or knr! We do measure transmittance and emission intensity. Measuring Quantum Yield Relative Quantum Yield “Absolute” Quantum Yield Relative Quantum Yield kr F knr kr = # of photons emitted # of photons absorbed If is proportional to the amount of the radiation from the excitation source that is absorbed and Ff . If = Ff I0 (1-10-A) If = emission intensity Ff = quantum yield I0 = incident light intensity A = absorbance I f ∝ Ff If ∝ A Relative Quantum Yield I f = Ff I 0 (1-10-A) Ff ∝ I f If ∝ A Compare sample (S) fluorescence to reference (R). Reported (1-10-AR) IS FS x = x -A IR FR (1-10 S) Measured F I A n = quantum yield = emission intensity = absorbance = refractive index nS2 nR2 Known Relative Quantum Yield (1-10-AR) IS FS x = x -A IR FR (1-10 S) Absorption Emission Reference Emission AR AS nS2 nR2 I Sample Emission Same instrument settings: excitation wavelength, slit widths, photomultiplier voltage… Relative Quantum Yield (1-10-AR) IS FS x = x -A IR FR (1-10 S) nS2 nR2 ? Excitation Snell’s Law: Sinqi ni = no Sinqo Detector Sinqi2 nS2 = 2 nR Sinqo2 Emission References Relative Quantum Yield Emission Detector Source Absorption Detector IF P0 P Sample Reflectance Scatter (1-10-AR) IS FS x = x -A IR FR (1-10 S) nS2 nR2 Relative Quantum Yield [Ru(bpy)3] 2Cl in H2O F = 0.042-0.063 [Ru(bpy)3] 2PF6 in MeCN F = 0.062-0.095 Relative Quantum Yield (1-10-AR) IS FS x = x -A IR FR (1-10 S) Minimizing Error: • Sample/Reference with similar: - Emission range - Quantum yield • Same Solvent • Known Standard • Same Instrument Settings - Excitation wavelength - Slit widths - PMT voltage nS2 nR2 Absolute Quantum Yield F= # of photons emitted # of photons absorbed Source Detector Integrating Sphere Absolute Quantum Yield Hamamatsu: C9920-02 (99% reﬂectance for wavelengths from 350 to 1650 nm and over 96% reflectance for wavelengths from 250 to 350 nm) Fig. 2. Schematic diagram of integrating sphere (IS) instrument for measuring absolute fluorescence quantum yields. MC1, MC2: monochromators, OF: optical fiber, SC: sample cell, B: buffle, BT-CCD: back-thinned CCD, PC: personal computer. Absolute Quantum Yield Absolute Quantum Yield Instrumentation Hamamatsu: C9920-02G Absolute quantum yield measurement system Absolute Quantum Yield Horiba QY Accessory Data Acquisition 1) Set excitation l 2) Insert reference - holder + solvent 3) Irradiate Reference 4) Detect output across - excitation and emission 5) Insert Sample Holder + solvent + sample 6) Repeat 3 and 4 Absolute Quantum Yield Absolute Quantum Yield Self Absorption/Filter Effect Anthracene Fluorescence intensity Self Absorption/Filter Effect Inner filter effect If = Ff I0 (1-10-A) Concentration (M) (1-10-AR) IS FS x = x -A S IR FR (1-10 ) nS2 nR2 Single Crystal Measurements Quantum Yield and Lifetime Substitution for photon flux and the relationship between kA and kr then (math happens): kr F knr kr Intrinsic or natural lifetime (tn): lifetime of the fluorophore in the absence of non-radiative processes 1 kr tn = Radiative Rate and Extinction Coefficient: Extinction Coefficient 1 n n n dn kr = 8 t no 3.42 10 Radiative Rate 2 max 2 Relationship between absorption intensity and fluorescence lifetime Strickler and Berg “Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules” J. Chem. Phys. 1962, 37, 814. Strickler-Berg relation The relation of the radiative lifetime of the molecule and the absorption coefficient over the spectrum [ref. 5] 2 n2 1 n max n dn kr = 8 t no 3.42 10 n: refractive index of medium n: position of the absorption maxima in wavenumbers [cm-1] : absorption coefficient Relationship between Einstein A and B coefficients Suppose a large number of molecules, immersed in a nonabsorbing medium with refractive index n, to be within a cavity in some material at temperature T, The radiation density within the medium is given by Planck’s blackbody radiation law, 8 hn 3n3 (n ) c3 hn exp( ) 1 kT 1 -(1) Blackbody Radiation By the definition of the Einstein transition probability coefficients, the rate of molecules going from lower state 1 to upper state 2 by absorption of radiation, N1a B1a2b (n1a2b ) -(2) N1a : number of molecules in state 1a v 1a 2b : frequency of the transition Einstein transition The rate at which molecules undergo this downward transitionprobability is given bycoefficients N2b [ A2b1a B2b1a (n 2b1a )] -(3) spontaneous emission probability induced emission probability At equilibrium the two rates must be equal, so by equating (2) and (3), Relationship between Einstein A and B coefficients A2b1a N [ 1a 1] (n 2b1a ) B2b1a N 2b -(4) According to the Boltzmann distribution law, the numbers of molecules in the two states at equilibrium are related by N 2b hn exp[ 2b1a ] N1a kT -(5) Ratio of molecules in the ground and excited state Substitution of Eqs. (1) and (5) into (4) results in Einstein’s relation, A2b1a 8 hn 2b1a n3c3 B2b1a 3 -(6) Relationship B coefficient to Absorption coefficient The radiation density in the light beam after it has passed a distance x cm through the sample, the molar extinction coefficient (n ) can be defined by (n , x) 10 (n )Cx e 2.303 (n )Cx (n , 0) Radiation Density Photons per distance C: concentration in moles per liter If a short distance dx is considered, the change in radiation density may be d (n ) 2.303 (n ) (n , 0)Cdx -(7) For simplicity, all the molecules will be assumed to be in the ground vibronic state, 10 Cdx 1000N10 N A 1 -(8) The number of molecules excited per second with energy hv is given by N (n ) c d (n ) n hn -(9) Excitations/second Relationship B coefficient to Absorption coefficient Combining Eqs. (7), (8), and (9), it is found N (n ) [2303c (n ) / hn nN A ] (n , 0) N10 -(10) Probability of a single excitation transition the probability that a molecule in state 10 will absorb of energy hv and go to some excited state To obtain the probability of going to the state 2b, it must be realized that this can occur with a finite range of frequencies, and Eq. (10) must be integrated over this range. Then N102b 2303c [ (n )d lnn ] (n 102b ) N10 hnN A -(11) If the molecules are randomly oriented, the average probability of absorption for N1a B1a2b (n1a2b ) a large number of molecules, Eq (2) give a similar relation to (11) -(2) 2303c B102b -(12) d lnn hnN A for all The probability coefficient for all transitions to the electronic state Probability 2 excitation transitions B102b B102b b 2303c d lnn hnN A -(13) Lifetime relationship for molecules The wavefunctions of vibronic states are functions of both the electronic coordinates x and the nuclear coordinates Q, -(14) 1a ( x, Q) 1 ( x, Q)F1a (Q) If M(x) is the electric dipole operator for the electrons, the probability for induced Absorption or emission between two states is proportional to the square of the Matrix element of M(x) between two states B1a 2b B2b1a K | 1a ( x, Q) M ( x) 2b ( x, Q)dxdQ |2 * -(15) Using (14), the integral in this expression can be dQ Excitation 1a ( x, Q)M ( x) 2b ( x, Q)dxdQ 1a (Q)M 12 (Q) 2b (Q)Relating * * where M 12 (Q) 1 ( x, Q) M ( x) 2 ( x, Q)dx * Probability to Relaxation Probability of atoms Electronic transition moment integral for the transition Assuming the nuclei to be fixed in a position Q Lifetime relationship for molecules It can be expand in a power series in the normal coordinates of the molecule M12 (Q) M12 (0) (M12 / Qr )0 Qr -(16) r For strongly allowed transitions in a molecule, the zeroth-order term is dominant Then (15) reduces to B1a2b B2b1a K M12 (0) 2 F * 1a F2b dQ 2 -(17) Expanding the model to molecules Taking the appropriate sums, we find B102 B102b K M 12 (0) b F 2b comprise F * 1a F 2b dQ 2 b B102 K M 12 (0) Since the 2 2 a complete orthonormal set in Q space -(18) Lifetime relationship for molecules The rate constant for emission from the lowest vibrational level of electronic state 2 to all vibrational levels of state 1, A201 , can be written by using Eqs. (6) and (17) A201 A201a (8 hn / c ) K M12 (0) v201a 3 2 3 a F 3 * 1a a It is desirable to be able to evaluate the term v 201a a 3 F * 1a F 2b dQ 2 -(19) 2 F 2b dQ experimentally . If the fluorescence band is narrow, v3 can be considered a constant and removed from the summation, the remaining sum being equal to unity Relating relaxation 2 * probabilities to lifetime By dividing by a F1a F20 dQ 1 for a single transition v201a a 3 * F 1 a F20 dQ F * 1a a F 20 dQ 2 2 Lifetime relationship for molecules The sums over all vibronic bands can be replaced by integrals over the fluorescence spectrum, so the expression reduces to I (n )dn n 3 I (n )dn n f 3 Av 1 Expanding to all transitions Now, by combining Eqs. (13), (18), and (19), we obtain Finally a relationship -(20) between lifetime and extinction coefficient 8 2303 n2 3 1 A201 n d lnn f Av 2 c NA It is convenient to write this equation in terms of the more common units 1 t0 1 A201 8 2303 cn2 N A n f Natural Lifetime 2.880 109 n2 n f 3 3 Av Av 1 1 Extinction Coefficient g1 d lnn g2 g1 d lnn g2 g1 and g2 : degeneracies of the 1, 2 states -(21) Quantum Yield and Lifetime Math happens: kr F knr kr Intrinsic or natural lifetime (tn): lifetime in the absence of nonradiative processes (F = 1) = # of photons absorbed Experimental lifetime (t): lifetime with radiative and nonradiative processes 1 kr tn = F = # of photons emitted t tn 1 t = kr + knr Algebra happens: kr = F/t knr = (1 − F)/t F and t from experiment, calculate kr and knr Radiative vs Non-radiative kr F knr kr kr = Φ/τ X = Br X= I knr = (1 − Φ)/τ kr 2.0 x 108 s-1 2.1 x 108 s-1 knr 1.1 x 108 s-1 1.4 x 109 s-1 • Same extinction coefficient • Same radiative rate • knr larger with I Quantum Yield and Lifetime Unquenched Emission kr F= kr + kic t = 1/(kr + kic) With an acceptor molecule kr F= kr + kic + kET t = 1/(kr + kic + kET) kET knr t F Quantum Yield and Lifetime Unquenched Emission kr F= kr + kic t = 1/(kr + kic) With a quenching molecule kr F= kr + kic + kq[C] t = 1/(kr + kic + kq[C]) [C] knr t F Side Note: Energy Gap Law kr F= kr + knr Large Gap Small Gap Higher energy absorption/emission E1 Lower energy absorption/emission knr E E E1 knr E0 E0 Energy Gap Law Energy Gap (E) knr Poor overlap Strong overlap S1 S0 S1 S0 S1 S0 Energy Gap Law knr (1011 s-1) Energy Gap (E) knr kr F knr kr 833 667 555 476 416 cm-1 nm F Energy Gap knr = 1013e-aE (sec-1) If you red shift emission efficiency goes down! Implications of Energy Gap Law Organic Solar Cells Charge Separation Excitation hn A A C C* kcs knr A C* kcs h ∝ kcs + knr h Solar cell efficiency A- C+ Implications of Energy Gap Law kcs kcs h ∝ kcs + knr knr Irradiance (W M-2 nm-1) knr h h Solar cell efficiency C* A Energy Gap (E) AM1.5 (Global tilt) 1.5 To increase efficiency: 1.0 Decrease E Reduce knr 0.5 Increase kcs 0.0 500 750 1000 1250 1500 1750 2000 Wavelength (nm) Implications of Energy Gap Law Organic Light Emitting Diodes F = 50% N N lmax = 650 nm t = 90 ms Pt N N 500Å Ag 1000Å Mg:Ag 100Å Alq3 400Å PtOEP : Alq3 350Å NPD 60Å CuPc ITO Red OLED >20% efficiency Implications of Energy Gap Law Organic Light Emitting Diodes F = 40% lmax = 765 nm t = 50 ms Visible Image 550 nm EL Intensity (a.u.) LiF/Al AlQ3 AlQ3:Pt(TBP), 6% NPD 300 400 500 600 700 800 900 Wavelength (nm) nIR OLED <7 % efficiency ITO IR monocular which replaces infrared emission (800 nm) with green Side Note: Photochemical Yield Source hn Sample F= # of reacted molecules # of photons absorbed Actinometry Reference hn F = # of events # of absorbed photons hn 100% yield # of absorbed photons = # of events From NMR/UV-Vis/MS of the photoproduct… Variables Photons/second/area Wavelength Absorption Extinction Coefficient Sample hn hn ?% yield From NMR/UV-Vis/MS… # of events = F = # of absorbed photons 3 5 = 60% yield From Actinometer Quantum Yield End Any Questions?