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Calculation of the viscosity dependence of the excited-state lifetime in pinacyanol solution
Volume
CHEMICAL
106, number 5
CALCULATION
OF THE VISCOSITY
OF THE EXCITED-STATE
Department,
Received
1 February
DEPENDENCE
and B.
Riedrich-Sdriller-Uni~~e~~~
WILHELM1
Jean, DDR-6900
Jena. German
Denzocmtic
Republic
1984
The viscosity dependence of the Sr liretime of pinacyanol in solutions Planck equation considering two relaxation channeb - internal conversion state barrier. A mmparison with experimental data is given.
I_ Introduction Picosecond-resolved spectroscopy offers the possibility to obtain information about molecular motion in excited states of dissolved molecules. The dependence of excited-state lifetimes of organic dyes on solvent viscosity has been a subject of numerous experimental investigations (see e.g. refs. [l-3]). In particular, picosecond-laser studies of pinacyanol were carried out by several authors [2-s]_ Sundstrom and Cillbro [2,3] measured S, lifetimes of pinacyanol solutions in a relatively small viscosity range (solvent viscosity range: )I-alcohols OS-10 cP; glycerol-methanol 2-20 cP; glycerol-water 2-10 cP). It was found that the St lifetime is proportional to q7, where -y depends on the type of solvent (rr-alcohols: -y = 0.9 1; glycerol-methanol: -y = 0_45;glycerol-water: -r = 0.57). Mialocq et al. [4] investigated pinacyanol solutions (solvents: glycerol-water, ethylene glycolwater) in a viscosity range between 2 and 1500 cP. The linear relationship between r-l and q-1 obtained [4] in the 77region from 2 to 60 CP is equivalent to a value 7 = 1. They obtained y = 0.27 at higher viscosities, that is for q = 60- 1500 CP [4] _ The application of the Fokker-Planck equation to the interpretation of internal rotation was first discussed by McCaskiB and Gilbert [6,7]. The model [6] explains successfully the viscosity dependence of the conformational relaxation time of 1 ,I-binaphthyl [7]. in the present paper, the viscosity dependence of 428
4 May 1984
LETFERS
LIFETIME IN PINACYANOL SOLUTION
M. KASCHKE, J. KLEINSCHMIDT Physics
PHYSICS
is calculated by solving the appropriate Fokkerand conformational
relax&ion
over an escited-
the S, lifetime of pinacyanol solution is calculated in a rather broad range of 9 (2-200 cP) by solving the appropriate Langevin or Fokker-Planck equation. Two relaxation processes, which depopulate the S, level - internal conversion and conformational change - are assumed. The results are compared with experimental data given by Mialocq et al. [4]_ By fitting the calculated dependence of the lifetime on r~ to measured data, characteristic parameters of the appropriate model may be determined.
2. Model
of internal
motion
Within the approximation of classical statistics, the internal rotation of large molecular parts around one axis can be described by the Langevin equation [6]:
(1) ~3is the angle around the rotation axes and I the reduced moment of inertia of the moiecule with respect to the internal motion along y; (Yrepresents a drag coefficient, which is taken to be proportional to the macroscopic viscosity 77,which is justified for stick as well as slip boundary conditions [8,9]. Uiisthe energy of the state i, which depends on the configurational coordinate cp. Since the molecule may in general be hopping between different electronic states. Ui is a random function of time. F(t) is a random fluctuating 0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume
106, number
CHEMICAL
5
torque. If F(r) is h-correlated, ensemble-averaging of eq. (1) gives the following Fokker-Planck equation for the probability Pj(q,~, r) of finding the molecule at time t in the ith electronic state at the angle rp and with angular velocity $ :
(
a
a.
kTa
a2
Iduia
1 (s4c=o11’2
9
IE
_a
The ki,(+7, (j) are rate coefficients for internal conversion or intersystem crossing between electronic states. An approximate solution of eq. (2) in a single electronic state i can be obtained by taking the ensemble average ofa large number of trajectories that are solutions of eq. (I), which are gained by means of computer simulation techniques [IO-l?]. Transitions to other electronic states are handled as follows: During each time step [E, t + &I. in addition to generating a random number x that gives the random fluctuating torque F(r), another homogeneously distributed random number)’ E [O,l ] is generated, where a transition to the electronic state j is supposed to occur during that time step if kijAt >,y_ If transitions between more than two states are possible, several random numbersyk E [O,i ] must be generated. Here, in the case of pinacyanol, the calculation is restricted to only one transition, namely S, + So (klo, rate coefficient of internal conversion). The second relaxation channel will be assumed to be a conformational change over a barrier to a distorted form of pinacyanol in the S1 state (fig. 1). The S, potential is constructed as piecewise harmonic with a barrier of barrier height Eg at coordinate pB. WI = [I-W
1 Alay 1954
PHYSICS LE-ITERS
cd; = [I-1u;‘(p)),,,,]“2
are the angular frequencies of the harmonic potential segments in the initial well and over the barrier. respectively. It must be noticed that ti; is not an additional parameter but is a function of the parameters wl, yB and Eg. The simulation is carried out on the assumption that no reverse process takes place over the barrier in the St state. The internal conversion rate (St + So) is calculated in the Franck-Condon approximation [ 13-15]_ The transition rate kli oj is proportional to the squared overlap integral:
Fig. 1. Scllematic potenti
=o
energy duam.
,
orherwise
G?
where u1 i and VQ are vibrational wavefunctions of the considered soft torsional mode in the S, and So states respectively, Eoi, Eli are Ihe total vibronic energies. A 1 o is factorized into the electronic parr of 111~ transition moment and into matrix elements of promoting modes. The non-radiative balance needs at least several molecular vibrations (with angular frequencies greater of the considered
than the angular frequencies wo_ w1 torsional mode) to secure conserva-
tion of energy El + iiiw, x E,, +ifiwg, where E\i is the energy
contribution
of these
molecular
vibrarions
modes). The overlap integral (u, Juoj) is calculated within the semiclassical approsimation 1161. The angle p* ar which the (ul i + uoi) FranckCondon transition occurs (if Eli z Eoi) is the one (or ones) for which the angular momentum p is conserved p(g*. i)=p(p*. i)_ The Franck-Condon factor is then given by the following espression [ 161: (promoting
I[ de1(iI/ dil [deoG')/dil l~UlilUOj)l"
=
nfi IP I IA Qp*)l
=
l+(P*)I-
l ,
(4)
where AI”= IdO,/dp - dUg/d&*;,, (i),eov)are the vibrational energies of torsional vibrations in the S1 429
Volume 106, number 5
CHEMICAL
PHYSICS
and So states respectively, i, j are the quantum numbers. Transitions take place at cp* for all states i, j with Eil “Eio.
3. Results
The object of the present paper consists in the explanation of experimental data of Mialocq et al. [4] _ As explained above, we consider two relaxation channels, where the conformational change over the barrier is predominant in the low viscosity region (9 < 60 cP) and where the internal conversion exceeds the conformational relaxation at high r] values (q > 60 cP). By use of known molecular data, the moment of inertia I and the quotient a/p for pinacyanol were estimated to be I = 2 X lO-44 kg m2, (Y/Q= 2.5 ps-1 cP-1. To get a first estimate of the excited-state parameters WI, qB, EB, we employ bimers’ approximate relation for the Fokker-Planck equation [6] in the low-viscosity region (I) < 60 cP), where the neglect of IC processes seems to be justified. By using these values, we start the simulation procedure which takes into account both the relaxation channels. The inter-
nal conversion plays the predominant role at high viscosities (q > 60 cP). To fit the experimental data, we
LETTERS
4 May I+84
varied a c-independent factor denoted by C in the transition rate (3) and the angle y*, where the FranckCondon transition takes place. The measured points [4] and the calculated viscosity dependence of the lifetime arc shown in fig. 2. Calculations were carried out at various temperatures (T= 200-400 K). The temperature dependence of the internal conversion rate A-t, and the rate of conformational relaxation k, for 1) kept constant can be fitted by Arrhenius plots with intrinsic activation energies E,(K) = 150 cm-l and EA(conf. relax.) = 250 cm-l. It is obvious from fig. 2 that the statistics treatment used here yields results which agree better with measuring points than the results (dashed curve) obtained by the use of rate equations. The dashed curve corresponds to the expression 7 = (ktc + kR)-’ where X-t, and kn are obtained by fitting the experimenta! results in the high- and the low-viscosity range, respectively (ktc = o.o17(?7/cP)-o.‘7 ps-1, kn = 0_2(17/cP)- 1 ps- I)_ References S_P. Velsko and G.R. Fleming, J. Chem. Phys. 76 (1982) 3553. V. Sundstrom and T. GiBbro, J. Phys. Chem. 86 (1982) 1788. V. Sundstrom and T. Gillbro. Chem. Phys. 61 (1981) 257. J.C. hlialocq, P. Goujon and hl. Arvis, J. Chim. Phys. 76 (1979) 1067. J.C. Mialoq, J. Jaraudias and P. Goujon, Chem. Phys. Letters 47 (1977) 123. J.S. McCaskihaod R.G. Gilbert. Chem. Phys. 44 (1979)
389.
3.5
15 25
50
100
250
2500
p IcP Q]pS-’
Fig. 2. Viscosity dependence of the S1 lifetime. 0. measur@ points; 0. calculated values (wl = 4 pP’ , EB = 250 an-‘,
ps-r,
~FJr: 2S”, C = 0.002 v* = 100); 0, calculated &es (wt = 4 ps-‘, Eg = 200 cm-‘, qB = 25”, C = 0.003 psf’ p* = 8-1, T = 293 K.
430
,
(71 C.V. Shank, E.P. Ippen. 0. Teschkeand K.B. Eisenthal, J. Chem. Phys. 69 (1977) 5547. [S] G.R. Aims, D.R. Baner, J.I. Baumann and R. Pecora, J. Chem. Phys. 58 (1973) 5570. 191 C. Huand R. Zwanzig, J. Chem. Phys. 60 (1974) 4353. [lo] M.R. Pearand J.H. Weiner, J. Cbem. Phys. 69 (1978) 785. [ll] J.H. Weiner and R.E. Forman, Phys. Rev. BlO (1971) 315. [ 121 M. Kaschke, J. Kleinschmidt and B. Wiihehni, to be published. [ 131 B. DiBartolo. ed.. Radiationless processes (Plenum Press, New York. 1980). [ 14 ] C-W. Struck and W.H. l%nger, J _Luminescence 10 (1975) 1. 1151 E.U. Condon, Phys. Rev. 32 (1928) 858. [i6] W.H. hfilkr, Advan. Chem. Phys. 30 (1975) 77.
CHEMICAL
106, number 5
CALCULATION
OF THE VISCOSITY
OF THE EXCITED-STATE
Department,
Received
1 February
DEPENDENCE
and B.
Riedrich-Sdriller-Uni~~e~~~
WILHELM1
Jean, DDR-6900
Jena. German
Denzocmtic
Republic
1984
The viscosity dependence of the Sr liretime of pinacyanol in solutions Planck equation considering two relaxation channeb - internal conversion state barrier. A mmparison with experimental data is given.
I_ Introduction Picosecond-resolved spectroscopy offers the possibility to obtain information about molecular motion in excited states of dissolved molecules. The dependence of excited-state lifetimes of organic dyes on solvent viscosity has been a subject of numerous experimental investigations (see e.g. refs. [l-3]). In particular, picosecond-laser studies of pinacyanol were carried out by several authors [2-s]_ Sundstrom and Cillbro [2,3] measured S, lifetimes of pinacyanol solutions in a relatively small viscosity range (solvent viscosity range: )I-alcohols OS-10 cP; glycerol-methanol 2-20 cP; glycerol-water 2-10 cP). It was found that the St lifetime is proportional to q7, where -y depends on the type of solvent (rr-alcohols: -y = 0.9 1; glycerol-methanol: -y = 0_45;glycerol-water: -r = 0.57). Mialocq et al. [4] investigated pinacyanol solutions (solvents: glycerol-water, ethylene glycolwater) in a viscosity range between 2 and 1500 cP. The linear relationship between r-l and q-1 obtained [4] in the 77region from 2 to 60 CP is equivalent to a value 7 = 1. They obtained y = 0.27 at higher viscosities, that is for q = 60- 1500 CP [4] _ The application of the Fokker-Planck equation to the interpretation of internal rotation was first discussed by McCaskiB and Gilbert [6,7]. The model [6] explains successfully the viscosity dependence of the conformational relaxation time of 1 ,I-binaphthyl [7]. in the present paper, the viscosity dependence of 428
4 May 1984
LETFERS
LIFETIME IN PINACYANOL SOLUTION
M. KASCHKE, J. KLEINSCHMIDT Physics
PHYSICS
is calculated by solving the appropriate Fokkerand conformational
relax&ion
over an escited-
the S, lifetime of pinacyanol solution is calculated in a rather broad range of 9 (2-200 cP) by solving the appropriate Langevin or Fokker-Planck equation. Two relaxation processes, which depopulate the S, level - internal conversion and conformational change - are assumed. The results are compared with experimental data given by Mialocq et al. [4]_ By fitting the calculated dependence of the lifetime on r~ to measured data, characteristic parameters of the appropriate model may be determined.
2. Model
of internal
motion
Within the approximation of classical statistics, the internal rotation of large molecular parts around one axis can be described by the Langevin equation [6]:
(1) ~3is the angle around the rotation axes and I the reduced moment of inertia of the moiecule with respect to the internal motion along y; (Yrepresents a drag coefficient, which is taken to be proportional to the macroscopic viscosity 77,which is justified for stick as well as slip boundary conditions [8,9]. Uiisthe energy of the state i, which depends on the configurational coordinate cp. Since the molecule may in general be hopping between different electronic states. Ui is a random function of time. F(t) is a random fluctuating 0 009-2614/84/S 03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume
106, number
CHEMICAL
5
torque. If F(r) is h-correlated, ensemble-averaging of eq. (1) gives the following Fokker-Planck equation for the probability Pj(q,~, r) of finding the molecule at time t in the ith electronic state at the angle rp and with angular velocity $ :
(
a
a.
kTa
a2
Iduia
1 (s4c=o11’2
9
IE
_a
The ki,(+7, (j) are rate coefficients for internal conversion or intersystem crossing between electronic states. An approximate solution of eq. (2) in a single electronic state i can be obtained by taking the ensemble average ofa large number of trajectories that are solutions of eq. (I), which are gained by means of computer simulation techniques [IO-l?]. Transitions to other electronic states are handled as follows: During each time step [E, t + &I. in addition to generating a random number x that gives the random fluctuating torque F(r), another homogeneously distributed random number)’ E [O,l ] is generated, where a transition to the electronic state j is supposed to occur during that time step if kijAt >,y_ If transitions between more than two states are possible, several random numbersyk E [O,i ] must be generated. Here, in the case of pinacyanol, the calculation is restricted to only one transition, namely S, + So (klo, rate coefficient of internal conversion). The second relaxation channel will be assumed to be a conformational change over a barrier to a distorted form of pinacyanol in the S1 state (fig. 1). The S, potential is constructed as piecewise harmonic with a barrier of barrier height Eg at coordinate pB. WI = [I-W
1 Alay 1954
PHYSICS LE-ITERS
cd; = [I-1u;‘(p)),,,,]“2
are the angular frequencies of the harmonic potential segments in the initial well and over the barrier. respectively. It must be noticed that ti; is not an additional parameter but is a function of the parameters wl, yB and Eg. The simulation is carried out on the assumption that no reverse process takes place over the barrier in the St state. The internal conversion rate (St + So) is calculated in the Franck-Condon approximation [ 13-15]_ The transition rate kli oj is proportional to the squared overlap integral:
Fig. 1. Scllematic potenti
=o
energy duam.
,
orherwise
G?
where u1 i and VQ are vibrational wavefunctions of the considered soft torsional mode in the S, and So states respectively, Eoi, Eli are Ihe total vibronic energies. A 1 o is factorized into the electronic parr of 111~ transition moment and into matrix elements of promoting modes. The non-radiative balance needs at least several molecular vibrations (with angular frequencies greater of the considered
than the angular frequencies wo_ w1 torsional mode) to secure conserva-
tion of energy El + iiiw, x E,, +ifiwg, where E\i is the energy
contribution
of these
molecular
vibrarions
modes). The overlap integral (u, Juoj) is calculated within the semiclassical approsimation 1161. The angle p* ar which the (ul i + uoi) FranckCondon transition occurs (if Eli z Eoi) is the one (or ones) for which the angular momentum p is conserved p(g*. i)=p(p*. i)_ The Franck-Condon factor is then given by the following espression [ 161: (promoting
I[ de1(iI/ dil [deoG')/dil l~UlilUOj)l"
=
nfi IP I IA Qp*)l
=
l+(P*)I-
l ,
(4)
where AI”= IdO,/dp - dUg/d&*;,, (i),eov)are the vibrational energies of torsional vibrations in the S1 429
Volume 106, number 5
CHEMICAL
PHYSICS
and So states respectively, i, j are the quantum numbers. Transitions take place at cp* for all states i, j with Eil “Eio.
3. Results
The object of the present paper consists in the explanation of experimental data of Mialocq et al. [4] _ As explained above, we consider two relaxation channels, where the conformational change over the barrier is predominant in the low viscosity region (9 < 60 cP) and where the internal conversion exceeds the conformational relaxation at high r] values (q > 60 cP). By use of known molecular data, the moment of inertia I and the quotient a/p for pinacyanol were estimated to be I = 2 X lO-44 kg m2, (Y/Q= 2.5 ps-1 cP-1. To get a first estimate of the excited-state parameters WI, qB, EB, we employ bimers’ approximate relation for the Fokker-Planck equation [6] in the low-viscosity region (I) < 60 cP), where the neglect of IC processes seems to be justified. By using these values, we start the simulation procedure which takes into account both the relaxation channels. The inter-
nal conversion plays the predominant role at high viscosities (q > 60 cP). To fit the experimental data, we
LETTERS
4 May I+84
varied a c-independent factor denoted by C in the transition rate (3) and the angle y*, where the FranckCondon transition takes place. The measured points [4] and the calculated viscosity dependence of the lifetime arc shown in fig. 2. Calculations were carried out at various temperatures (T= 200-400 K). The temperature dependence of the internal conversion rate A-t, and the rate of conformational relaxation k, for 1) kept constant can be fitted by Arrhenius plots with intrinsic activation energies E,(K) = 150 cm-l and EA(conf. relax.) = 250 cm-l. It is obvious from fig. 2 that the statistics treatment used here yields results which agree better with measuring points than the results (dashed curve) obtained by the use of rate equations. The dashed curve corresponds to the expression 7 = (ktc + kR)-’ where X-t, and kn are obtained by fitting the experimenta! results in the high- and the low-viscosity range, respectively (ktc = o.o17(?7/cP)-o.‘7 ps-1, kn = 0_2(17/cP)- 1 ps- I)_ References S_P. Velsko and G.R. Fleming, J. Chem. Phys. 76 (1982) 3553. V. Sundstrom and T. GiBbro, J. Phys. Chem. 86 (1982) 1788. V. Sundstrom and T. Gillbro. Chem. Phys. 61 (1981) 257. J.C. hlialocq, P. Goujon and hl. Arvis, J. Chim. Phys. 76 (1979) 1067. J.C. Mialoq, J. Jaraudias and P. Goujon, Chem. Phys. Letters 47 (1977) 123. J.S. McCaskihaod R.G. Gilbert. Chem. Phys. 44 (1979)
389.
3.5
15 25
50
100
250
2500
p IcP Q]pS-’
Fig. 2. Viscosity dependence of the S1 lifetime. 0. measur@ points; 0. calculated values (wl = 4 pP’ , EB = 250 an-‘,
ps-r,
~FJr: 2S”, C = 0.002 v* = 100); 0, calculated &es (wt = 4 ps-‘, Eg = 200 cm-‘, qB = 25”, C = 0.003 psf’ p* = 8-1, T = 293 K.
430
,
(71 C.V. Shank, E.P. Ippen. 0. Teschkeand K.B. Eisenthal, J. Chem. Phys. 69 (1977) 5547. [S] G.R. Aims, D.R. Baner, J.I. Baumann and R. Pecora, J. Chem. Phys. 58 (1973) 5570. 191 C. Huand R. Zwanzig, J. Chem. Phys. 60 (1974) 4353. [lo] M.R. Pearand J.H. Weiner, J. Cbem. Phys. 69 (1978) 785. [ll] J.H. Weiner and R.E. Forman, Phys. Rev. BlO (1971) 315. [ 121 M. Kaschke, J. Kleinschmidt and B. Wiihehni, to be published. [ 131 B. DiBartolo. ed.. Radiationless processes (Plenum Press, New York. 1980). [ 14 ] C-W. Struck and W.H. l%nger, J _Luminescence 10 (1975) 1. 1151 E.U. Condon, Phys. Rev. 32 (1928) 858. [i6] W.H. hfilkr, Advan. Chem. Phys. 30 (1975) 77.