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Triplet decay functions of aromatic molecules in the presence of heavy-atom perturbers after singlet excitation
Volume 90. number 2
DECAY FUNCTIONS
TRIPLET
23 July 1962
CHEMICAL PHYSICS LIXTERS
OF AROMATIC
IN THE PRESENCE OF HEAVY-ATOM
MOLECULES
PERTURBERS
AFTER
SINGLET
EXCITATION
Jan NAJBAR ImIlhlfe
Rcccwed
of Cl~emlmy, Japellmiarr
Ufliversrfy, 3 Karasra, 30-060 Krakbw, Poland
18 blarch 1982
Phosphoresccncc decay functions and decay funcllons of the trlplet stale of orgamc molcculcs in the presence of mndomly dlslrlbuled pcrlurbcrs ax dewed. The intlucncc orpcrlurbcrson intersystem crossing IStaken ducctly into account. The formulnc are cwct and valid for my time and any pcrturbcr concentration m rigid solutions. Model calculations of trIplet decay funchons arc prcscnlcd.
1 Introduction
The phosphorescence decay functions are usually measured using indirect excitation of the molecules to the triplet state VKIa singlet excited state [l-3]. In the presence of perturbers in a rigid solutron the observed decay functions represent mean values averaged over all possible perturber configurations [2,3]. Heavy-atom perturbers influence the rate constants of S,%+ T intersystem crossing, kTs, phosphorescence, kpT, and TI%+ S,, intersystem crossmg, k,,. The decay functions of the trlplet state are usually derived under the assumption that the mtersystem crossing S1%+= T quantum yield IS independent of the presence of the perturber [3,4]. Ths assumption may be a reasonable assumption for molecules with high intersystem crossing quantum yield, close to one. The Increased intersystem crossmg due to perturbers produces an Increased fraction of the molecules in the tnp-
let state with perturber molecules m close vlcmity, the prepared system not being random. In this paper we derive the decay functions of the phosphorescence and population of the triplet state for the general cas? where the influence of perturbers on the S,%-+ T intersystem crossing IS directly taken mto account. Model calculations will show the importance of Increased SIG T mtcrsystem crossing on the decay characteristics of the trrplet state of aromatic molecules in the presence of heavy-atom pcrturbcrs.
2. Decay kinetics
Consider the aromatIc molecule and perturbmg molecule to be separated by a distan!e R m a rigid solution The perturber Influences the rate constants for mtersystem crossing processes and phtisphorescence. The singletstate quenching St* S0 by perturbers may easdy be included in the considerations Let .S(t) represent the probability that at tune f the aromatlc molecule in the presence of a perturber in a particular configuration is in the excited smglet state, and let T(t) be the correspondmg probability for the triplet state. Let 0srepresent the dbsorption coefficient of the aromatic molecule at the frequency of excitation and let I(f) represent the intensity of excitation irradiation as a function of time. We assume that the optlcal density of the system IS small enough for the excitation beam not to be greatly attenuated, and that system diffusion or migration is unimportant. The kinetic equations for S(t) and T(f) are given by
l
154
0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
Volume 90, number 2 dS(r)/dr
23 July 1982
CHEMICALPHYSICS LEITERS
= -[kFS
+ k,,
dT(r)/df = -[km
+ k&(t)
+ k,,]
+ E@),
T(t) + k,&)
0)
,
(2)
where kFs = k&, the radiative rate constant of the S, + SO transitron, kCS =k&, the rate constant of internal conversion; contributions from drfferent perturbers to the rate constants k,,, kcp~and kCT are assumed to be additive so that kTS = k& + ZrAk~g(Rr), kpy = k”a + C,AkpT(Rr), kGT =k& + Z,AkC@,); k, =km +kCT, kS =k& +k& +kTS. The general solution of eq. (1) IS
s(t) =
J E;I(i)
G,(t - r’) dt’ ,
(3)
_oD where GS(r -t’)
=u(r-f’)
exp[-(r
u(t-t)=O
-r’)ks],
forr
For the population of eq. (2) is T(r) = /
fortat’.
(4)
ofthe trrplet state, T(t), the excitation source function is grven by k&(f)
kTS S(s) GT(t -s)
The general solution
ds ,
(5)
--s)kT]
(6)
_B where G,(t -s)
= u(f -s)
From eqs Q)-(6)
exp[-(r
we obtam
J
T(f) = j +kTSI(t’)u(s-m __
c’) cxp[-(s
From eq. (7) we obtam the response function
-r')k~]u(t--s) exp[-(r-s)kT]
for the populatron
dr’ds.
(7)
of the triplet state of we assume I(l) =1,8(t).
Integrating over t’ we have
T(t) = E: i kTS exp [-tkT0
s(ks - kT)] ds = E! i I’(r, s) ds , 0
(8)
where I’(r, s) = k,, exp [-kT(f -s) - kSf] . Forrigid soluttons the decay functionsofthe population of the perturber molecules may be expressed by
q(t)
= (T(t)) =
e;]
a?@, s)) d5.
(9) of the triplet state averaged over drfferent configuratrons
(10)
0
Superscripts indrcate pulse(E)
and indirect (I) excitation. Here we apply the averaging procedure introduced by Blumen [5,6] for the derivation of the donor decay functions in the presence of energy acceptors. We assume that rn our system there areN sites accessible for n perturber molecules. Let p be the probability 155
23 July
CHEhlICAL PHYSICS LETTERS
Volume 90, number 2
1982
that a site is occupred by a perturber, then assuming that occupancies of the sites are random and uncorrelated we can use the brnominl drstrrbution for the probabilrty P,,,, of a configuration with II perturbing molecules in the system P” ,,), =p”(l
-p)N-‘1,
m = 1, . , ;
(11)
0
The subscript nz indicates different distnbutions of n perturbers the configuratron of perturbers, the main value is given by
overN sates. For a function K, R1 dependent
on
N
c Kn, pn.
(K) = II
=o
n1
(12)
111 -
For the derivation of averaged decay functions titles: N 0 5f
N ngO P”(l -p)N-‘l
of the triplet population
c Akra(Rr) G(n,m)
n
JE(rl,
171)
and phosphorescence,
we need two iden-
exp[-AkX(Rj)r] I
Akrs(Ri>exp[-Ak~(R,)tl ,EFmj exP[--Ak~(Rj)fI] j*i
and
=
N Ak&iW&,)
P,$
x g
exP[-AkX(Rr)r]
*km(Ri)w[-*kx(R,)rl
,=I I -p
N Ak~s(Rr)
1 -p fp exp[-AkX(R$]
+p exp[-Akx(Ri)t]
I-
_ 5
AkTs(Ri)Akpr(Ri)
I=1
Cl -P +P
exp[-Ak#-)tI
p + R exp [-Akx(Ri)f] exp[-AkX(Ri)2t]
evWk_&Pl12
II
N X,Gr (1 --P +P
exP[-Ak&)~lI.
(14)
The proof of relations (13) and (14) follows by expanding the product and multrplication of the terms. These formulae guarantee automatrcahy that each lattice site can at most be occupied by only one perturber molecule. Usmg identrty (13) it can be shown that the averaged function (9) IS W(r, s)) = exp[-k$-s)-k$] where
156
[k& +pKT&,
F, s)]@cb. t, s),
(19
KTscP.
23 July 1982
CHEMICALPHYSICSLETTERS
Vc&me 90, number 2
N
8) cN AkTs(RI)E(Rir 1 -p +pE(R, r, s)’ C
r, s)= ,=I
E(R,, t, S) = e~p{-Ak~(Ri)f
HP, t, s)=i2
- [AkTS(Ri) - Ak#Zi)]r)
[I-P
+PE(R;, 6
4 1
_
The phosphorescence intensrty in the presence of the heavy-atom perturbers is given as the mean value of the product k,,$-(f):
$!I@) =UcpTT(f)) = eijJW,(f,
s)J d.r,
(I71
0
exp[-kTf - (ks -kT)s,
where Wt&, s)) = Uc,k,, (I?&,
Usmgidentities (13) and (14) it can be shown that
r; $1+ k&&J.
s), = {k; s k0pT +p &&&%
6 s) + K’rs,pT@, C JII
-&I +P2[&S@,&@+&% f, $ -KTS,&, 2, 2s)ll O@,4 4 expl-$r - @OS
,
(18)
where
K&P,
N A.kn(Ri)E(R,, t, 8)
N Ak,,(R,)Akn(R,)QR,,
r. r) = c i=r 1 -p +pE(R,, t, S)’
C $1= 1;
k&&,
1 -P +PQR,.
r, $1 ’
C s)
N AkTs(Ri)Ak~(RI)E(RI,26 2s)
X_rS,pTIR 25 2) = ,g
I1 -P +PW,, t, s)12
If we nedect the influence of the perturbers on the intersystem crossing rate constants, ANTE= tions over s in eqs. (10) and (17), using eqs. (IS) and (lg), respcctrvely, give
0, the mtegra-
N i!,%> = [~~k~s~(k~ -k$l
expl-kicj
iFt 11 -P +P exP[-AkT(R~)~]}
(19)
for the decay functron of the population of the triplet state, and e”ko
$1(f) = E k&k;
exp(-kt
r)
fl(l-p +p 1#=I w4--AkT(R,)~l
N AkpyR, exp[-Ak@,-)f]
+ p 22
1=~1 -_p+P
N
exp[-AkT(RI)c]}
(20)
for the phosphorescence decay function The product function in eqs (19) and (20) may be written as N IncP(p, f)=is ln{l--P+P exP[-AkT(R$]].
(~1)
The formu‘fac for the decay functions (lo), (15), (17)-(20) are exact and valid for all concentratl~ns of the pcrturber The sums in eqs. (Is)-(21) defining drfferent functions In #@, f, s) and KXy(p, C,s) run over all sites accessible for the perturber molecules The geometrica arrangements of the sttes may to a large extent be arbitrary. The sites may be arranged in line, in a plane or in threedimensional space. The number of sates may be finrte or infmrte. Assuming a continuous distribution of sites and spherical symmetry, the sums over i may be given as e g., In #$p, f) = 4zrp f In{1 - p +p exp [-A~=~~}~]}~* dR , Jh
(221 157
Volume 90,
CHEhllCAL PHYSICS LETTIXS
number 2
23 July 1982
where p is the densrty of the sites, R, IS the drstance of closest approach between an aromatic molecule and perturber In the three-drmensional case, under the assumption of a contmuous distnbution of sites in the low concentration limrt p Q 1, we obtam In $01. t) *In
d(C, f) = -W(t),
(23)
where
H(f)=(4~J’J,-,/lOOO)~
(1-exp[-Ak-r(R)~]}R2dR, Ro
and
Km@,
f, O)+KpT(‘)
= (41iNo/1000)
p
Ak&?)
exp[-AkT(R)r]R2
dR,
(24)
Ro where C
1s the
molar concentratron of the perturber andNO IS Avogadro’s number. using eqs. (23) and (24) rn the low-concentration hrnit, we obtain
From eqs (19) and (20),
i?(t) = [rg k&/(ki
- k;)]
i,c’(t)= [EgkQ(k;-
exp [-kt t - M(t)]
kt)] [k& + CK&t)]
,
exp [-ktr
(25) - CH(r)]
(26)
The decay functron of thepopulatron ofthe trrplet state (25)is sirmlar to that derived e.g., by Inokuti and Hirayama (71 for the triplet-trrplet energy transfer Detaded drscussion of the formulae (19) and (25) was given by Blumen [5,6]. The phosphorescence decay function (26) was derrved by Bazhin [4] using standard averagmg procedures Thus our formulae lead to the correct expressions for these cases-where such expressions are known.
3. Numerical example The decay function of the phosphorescence of aromatic molecules u-rthe presence of the perturber under indirect excitation of the molecules to the triplet state is givenby eqs (17),(18). The decay functionswere calculated from the photophysical parameters of table 1. The orders of magnitude of the parameters charactedzing the influence of the perturber on the rate constants km and kGT are similar to these found for aromatic molecules perturbed by l- ions in hydroxylic glasses [8]. The decay functions were calculated for two values of the triplet hfetime,r; = 1 0 and 10.0 s, to model the situation for aromatic hydrocarbons in theirperproto andperdeutero forms Numerrcal cvaluatrons of the sums over R [functions In @(p, r, s), K-&p. r, s), KPT.(P, !, s), KTsv,(p, t, s) and R,,,,,(p, 2t. its)] were performed over a limited volume of radius 40 A wrth cut-ofF parameter R, = 5 A, assuming a continuous distrrbution of sites in three-dimensional space and assummg that the distance-dependent contrrbutions Ak,&R) may be represented by AJ+(R)
= kiy
exp(-ox@),
where kiy and oXy are constants. The number of parameters was reduced by assuming that om = CQ- = orS The phosphorescence decay functions were calculated for drfferent values of k&: 0 0, lo* k& and lo9 k& A computer program in FORTRAN was used The step for integrations ovetR was 0.1 A. The step for s was ~$40 and integrations were lrmrred to 5 7:. The computing time necessary for the calculation of 50 pornts of the decay function was ~10~ s (CYBER 72). The calculated decay functions of the phosphorescence are shown in fig. 1 for 0
CHEMICALPHYSICS
Volume 90, numbcr 2
23 July 1982
LEnERS
Table 1 Photophyscal constants of unperturbed system
Pnrnmctcrs charactcr~mg the external hcnvq-atom cffcct
7”s = 10 ns
aTS=+T=UGT=143A
h-&=0
k’pT = 5000 5-l
k&=5x +=
10’s_1
,,‘:“” ,$I01
ki;T = 1460 s-l kkS = 0.0, IO8 kkl
1.001100s
kFT = 0 03 s-’
or lo9 kkT
p = 0 OOSl
k_ 0
0‘
u
12
l/m
‘r
I‘g 1. The phosphorcsccncc decay functions. 0 - decays of thcunpcrturbcdsystcms, 1,2and 3 -dccnysofthcpcrlurbcd systcn1s ior k’ - 0 0, IO8 kbT and IO9 k&, respecrwcly (1, 2 and 3 fo::‘- T - 1 s; I’, 2’ and 3’ for 7: = 10 0 s).
crossing gives an important contribution to the phosphorescence decay. This contrlbufion is especrally important for systems with low intersystem crossing quantum ylclds. The contrlbutlon is also dependent on the decay constants
of the tnplet
state.
Acknowledgement This work was supported
by the Pohsh Academy of Sciences under the 03.10 program.
References [I] [2] [3] [4] [S] [6] [7] [S]
J B Buks, Photophysics of aomatlc molecules (Mlcy-Intcrscwncc, NW York, 1970) J Nqbar, J Lumm. 11 (1975/76) 207. J. NaJbar, J-B. Buks and T.D S. tlamdton, Chem. Phys. 23 (1977) 281 NM. Bazi-un, Chem Phys. Letters 69 (1980) 580. A. Blumcn. J. Chem. Phys 72 (1980) 2632.74 (1981) 6926. A. Blumen, Nuovo Clmento 63 (1981) 50. hl lnokub and F Huaynma, J. Chcm. Phyn 43 (1965) 1978 J. NaJbxr and J Rodakiewicz-Now& Chcm. Phys. Letters 58 (1978) 545
159
DECAY FUNCTIONS
TRIPLET
23 July 1962
CHEMICAL PHYSICS LIXTERS
OF AROMATIC
IN THE PRESENCE OF HEAVY-ATOM
MOLECULES
PERTURBERS
AFTER
SINGLET
EXCITATION
Jan NAJBAR ImIlhlfe
Rcccwed
of Cl~emlmy, Japellmiarr
Ufliversrfy, 3 Karasra, 30-060 Krakbw, Poland
18 blarch 1982
Phosphoresccncc decay functions and decay funcllons of the trlplet stale of orgamc molcculcs in the presence of mndomly dlslrlbuled pcrlurbcrs ax dewed. The intlucncc orpcrlurbcrson intersystem crossing IStaken ducctly into account. The formulnc are cwct and valid for my time and any pcrturbcr concentration m rigid solutions. Model calculations of trIplet decay funchons arc prcscnlcd.
1 Introduction
The phosphorescence decay functions are usually measured using indirect excitation of the molecules to the triplet state VKIa singlet excited state [l-3]. In the presence of perturbers in a rigid solutron the observed decay functions represent mean values averaged over all possible perturber configurations [2,3]. Heavy-atom perturbers influence the rate constants of S,%+ T intersystem crossing, kTs, phosphorescence, kpT, and TI%+ S,, intersystem crossmg, k,,. The decay functions of the trlplet state are usually derived under the assumption that the mtersystem crossing S1%+= T quantum yield IS independent of the presence of the perturber [3,4]. Ths assumption may be a reasonable assumption for molecules with high intersystem crossing quantum yield, close to one. The Increased intersystem crossmg due to perturbers produces an Increased fraction of the molecules in the tnp-
let state with perturber molecules m close vlcmity, the prepared system not being random. In this paper we derive the decay functions of the phosphorescence and population of the triplet state for the general cas? where the influence of perturbers on the S,%-+ T intersystem crossing IS directly taken mto account. Model calculations will show the importance of Increased SIG T mtcrsystem crossing on the decay characteristics of the trrplet state of aromatic molecules in the presence of heavy-atom pcrturbcrs.
2. Decay kinetics
Consider the aromatIc molecule and perturbmg molecule to be separated by a distan!e R m a rigid solution The perturber Influences the rate constants for mtersystem crossing processes and phtisphorescence. The singletstate quenching St* S0 by perturbers may easdy be included in the considerations Let .S(t) represent the probability that at tune f the aromatlc molecule in the presence of a perturber in a particular configuration is in the excited smglet state, and let T(t) be the correspondmg probability for the triplet state. Let 0srepresent the dbsorption coefficient of the aromatic molecule at the frequency of excitation and let I(f) represent the intensity of excitation irradiation as a function of time. We assume that the optlcal density of the system IS small enough for the excitation beam not to be greatly attenuated, and that system diffusion or migration is unimportant. The kinetic equations for S(t) and T(f) are given by
l
154
0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
Volume 90, number 2 dS(r)/dr
23 July 1982
CHEMICALPHYSICS LEITERS
= -[kFS
+ k,,
dT(r)/df = -[km
+ k&(t)
+ k,,]
+ E@),
T(t) + k,&)
0)
,
(2)
where kFs = k&, the radiative rate constant of the S, + SO transitron, kCS =k&, the rate constant of internal conversion; contributions from drfferent perturbers to the rate constants k,,, kcp~and kCT are assumed to be additive so that kTS = k& + ZrAk~g(Rr), kpy = k”a + C,AkpT(Rr), kGT =k& + Z,AkC@,); k, =km +kCT, kS =k& +k& +kTS. The general solution of eq. (1) IS
s(t) =
J E;I(i)
G,(t - r’) dt’ ,
(3)
_oD where GS(r -t’)
=u(r-f’)
exp[-(r
u(t-t)=O
-r’)ks],
forr
For the population of eq. (2) is T(r) = /
fortat’.
(4)
ofthe trrplet state, T(t), the excitation source function is grven by k&(f)
kTS S(s) GT(t -s)
The general solution
ds ,
(5)
--s)kT]
(6)
_B where G,(t -s)
= u(f -s)
From eqs Q)-(6)
exp[-(r
we obtam
J
T(f) = j +kTSI(t’)u(s-m __
c’) cxp[-(s
From eq. (7) we obtam the response function
-r')k~]u(t--s) exp[-(r-s)kT]
for the populatron
dr’ds.
(7)
of the triplet state of we assume I(l) =1,8(t).
Integrating over t’ we have
T(t) = E: i kTS exp [-tkT0
s(ks - kT)] ds = E! i I’(r, s) ds , 0
(8)
where I’(r, s) = k,, exp [-kT(f -s) - kSf] . Forrigid soluttons the decay functionsofthe population of the perturber molecules may be expressed by
q(t)
= (T(t)) =
e;]
a?@, s)) d5.
(9) of the triplet state averaged over drfferent configuratrons
(10)
0
Superscripts indrcate pulse(E)
and indirect (I) excitation. Here we apply the averaging procedure introduced by Blumen [5,6] for the derivation of the donor decay functions in the presence of energy acceptors. We assume that rn our system there areN sites accessible for n perturber molecules. Let p be the probability 155
23 July
CHEhlICAL PHYSICS LETTERS
Volume 90, number 2
1982
that a site is occupred by a perturber, then assuming that occupancies of the sites are random and uncorrelated we can use the brnominl drstrrbution for the probabilrty P,,,, of a configuration with II perturbing molecules in the system P” ,,), =p”(l
-p)N-‘1,
m = 1, . , ;
(11)
0
The subscript nz indicates different distnbutions of n perturbers the configuratron of perturbers, the main value is given by
overN sates. For a function K, R1 dependent
on
N
c Kn, pn.
(K) = II
=o
n1
(12)
111 -
For the derivation of averaged decay functions titles: N 0 5f
N ngO P”(l -p)N-‘l
of the triplet population
c Akra(Rr) G(n,m)
n
JE(rl,
171)
and phosphorescence,
we need two iden-
exp[-AkX(Rj)r] I
Akrs(Ri>exp[-Ak~(R,)tl ,EFmj exP[--Ak~(Rj)fI] j*i
and
=
N Ak&iW&,)
P,$
x g
exP[-AkX(Rr)r]
*km(Ri)w[-*kx(R,)rl
,=I I -p
N Ak~s(Rr)
1 -p fp exp[-AkX(R$]
+p exp[-Akx(Ri)t]
I-
_ 5
AkTs(Ri)Akpr(Ri)
I=1
Cl -P +P
exp[-Ak#-)tI
p + R exp [-Akx(Ri)f] exp[-AkX(Ri)2t]
evWk_&Pl12
II
N X,Gr (1 --P +P
exP[-Ak&)~lI.
(14)
The proof of relations (13) and (14) follows by expanding the product and multrplication of the terms. These formulae guarantee automatrcahy that each lattice site can at most be occupied by only one perturber molecule. Usmg identrty (13) it can be shown that the averaged function (9) IS W(r, s)) = exp[-k$-s)-k$] where
156
[k& +pKT&,
F, s)]@cb. t, s),
(19
KTscP.
23 July 1982
CHEMICALPHYSICSLETTERS
Vc&me 90, number 2
N
8) cN AkTs(RI)E(Rir 1 -p +pE(R, r, s)’ C
r, s)= ,=I
E(R,, t, S) = e~p{-Ak~(Ri)f
HP, t, s)=i2
- [AkTS(Ri) - Ak#Zi)]r)
[I-P
+PE(R;, 6
4 1
_
The phosphorescence intensrty in the presence of the heavy-atom perturbers is given as the mean value of the product k,,$-(f):
$!I@) =UcpTT(f)) = eijJW,(f,
s)J d.r,
(I71
0
exp[-kTf - (ks -kT)s,
where Wt&, s)) = Uc,k,, (I?&,
Usmgidentities (13) and (14) it can be shown that
r; $1+ k&&J.
s), = {k; s k0pT +p &&&%
6 s) + K’rs,pT@, C JII
-&I +P2[&S@,&@+&% f, $ -KTS,&, 2, 2s)ll O@,4 4 expl-$r - @OS
,
(18)
where
K&P,
N A.kn(Ri)E(R,, t, 8)
N Ak,,(R,)Akn(R,)QR,,
r. r) = c i=r 1 -p +pE(R,, t, S)’
C $1= 1;
k&&,
1 -P +PQR,.
r, $1 ’
C s)
N AkTs(Ri)Ak~(RI)E(RI,26 2s)
X_rS,pTIR 25 2) = ,g
I1 -P +PW,, t, s)12
If we nedect the influence of the perturbers on the intersystem crossing rate constants, ANTE= tions over s in eqs. (10) and (17), using eqs. (IS) and (lg), respcctrvely, give
0, the mtegra-
N i!,%> = [~~k~s~(k~ -k$l
expl-kicj
iFt 11 -P +P exP[-AkT(R~)~]}
(19)
for the decay functron of the population of the triplet state, and e”ko
$1(f) = E k&k;
exp(-kt
r)
fl(l-p +p 1#=I w4--AkT(R,)~l
N AkpyR, exp[-Ak@,-)f]
+ p 22
1=~1 -_p+P
N
exp[-AkT(RI)c]}
(20)
for the phosphorescence decay function The product function in eqs (19) and (20) may be written as N IncP(p, f)=is ln{l--P+P exP[-AkT(R$]].
(~1)
The formu‘fac for the decay functions (lo), (15), (17)-(20) are exact and valid for all concentratl~ns of the pcrturber The sums in eqs. (Is)-(21) defining drfferent functions In #@, f, s) and KXy(p, C,s) run over all sites accessible for the perturber molecules The geometrica arrangements of the sttes may to a large extent be arbitrary. The sites may be arranged in line, in a plane or in threedimensional space. The number of sates may be finrte or infmrte. Assuming a continuous distribution of sites and spherical symmetry, the sums over i may be given as e g., In #$p, f) = 4zrp f In{1 - p +p exp [-A~=~~}~]}~* dR , Jh
(221 157
Volume 90,
CHEhllCAL PHYSICS LETTIXS
number 2
23 July 1982
where p is the densrty of the sites, R, IS the drstance of closest approach between an aromatic molecule and perturber In the three-drmensional case, under the assumption of a contmuous distnbution of sites in the low concentration limrt p Q 1, we obtam In $01. t) *In
d(C, f) = -W(t),
(23)
where
H(f)=(4~J’J,-,/lOOO)~
(1-exp[-Ak-r(R)~]}R2dR, Ro
and
Km@,
f, O)+KpT(‘)
= (41iNo/1000)
p
Ak&?)
exp[-AkT(R)r]R2
dR,
(24)
Ro where C
1s the
molar concentratron of the perturber andNO IS Avogadro’s number. using eqs. (23) and (24) rn the low-concentration hrnit, we obtain
From eqs (19) and (20),
i?(t) = [rg k&/(ki
- k;)]
i,c’(t)= [EgkQ(k;-
exp [-kt t - M(t)]
kt)] [k& + CK&t)]
,
exp [-ktr
(25) - CH(r)]
(26)
The decay functron of thepopulatron ofthe trrplet state (25)is sirmlar to that derived e.g., by Inokuti and Hirayama (71 for the triplet-trrplet energy transfer Detaded drscussion of the formulae (19) and (25) was given by Blumen [5,6]. The phosphorescence decay function (26) was derrved by Bazhin [4] using standard averagmg procedures Thus our formulae lead to the correct expressions for these cases-where such expressions are known.
3. Numerical example The decay function of the phosphorescence of aromatic molecules u-rthe presence of the perturber under indirect excitation of the molecules to the triplet state is givenby eqs (17),(18). The decay functionswere calculated from the photophysical parameters of table 1. The orders of magnitude of the parameters charactedzing the influence of the perturber on the rate constants km and kGT are similar to these found for aromatic molecules perturbed by l- ions in hydroxylic glasses [8]. The decay functions were calculated for two values of the triplet hfetime,r; = 1 0 and 10.0 s, to model the situation for aromatic hydrocarbons in theirperproto andperdeutero forms Numerrcal cvaluatrons of the sums over R [functions In @(p, r, s), K-&p. r, s), KPT.(P, !, s), KTsv,(p, t, s) and R,,,,,(p, 2t. its)] were performed over a limited volume of radius 40 A wrth cut-ofF parameter R, = 5 A, assuming a continuous distrrbution of sites in three-dimensional space and assummg that the distance-dependent contrrbutions Ak,&R) may be represented by AJ+(R)
= kiy
exp(-ox@),
where kiy and oXy are constants. The number of parameters was reduced by assuming that om = CQ- = orS The phosphorescence decay functions were calculated for drfferent values of k&: 0 0, lo* k& and lo9 k& A computer program in FORTRAN was used The step for integrations ovetR was 0.1 A. The step for s was ~$40 and integrations were lrmrred to 5 7:. The computing time necessary for the calculation of 50 pornts of the decay function was ~10~ s (CYBER 72). The calculated decay functions of the phosphorescence are shown in fig. 1 for 0
CHEMICALPHYSICS
Volume 90, numbcr 2
23 July 1982
LEnERS
Table 1 Photophyscal constants of unperturbed system
Pnrnmctcrs charactcr~mg the external hcnvq-atom cffcct
7”s = 10 ns
aTS=+T=UGT=143A
h-&=0
k’pT = 5000 5-l
k&=5x +=
10’s_1
,,‘:“” ,$I01
ki;T = 1460 s-l kkS = 0.0, IO8 kkl
1.001100s
kFT = 0 03 s-’
or lo9 kkT
p = 0 OOSl
k_ 0
0‘
u
12
l/m
‘r
I‘g 1. The phosphorcsccncc decay functions. 0 - decays of thcunpcrturbcdsystcms, 1,2and 3 -dccnysofthcpcrlurbcd systcn1s ior k’ - 0 0, IO8 kbT and IO9 k&, respecrwcly (1, 2 and 3 fo::‘- T - 1 s; I’, 2’ and 3’ for 7: = 10 0 s).
crossing gives an important contribution to the phosphorescence decay. This contrlbufion is especrally important for systems with low intersystem crossing quantum ylclds. The contrlbutlon is also dependent on the decay constants
of the tnplet
state.
Acknowledgement This work was supported
by the Pohsh Academy of Sciences under the 03.10 program.
References [I] [2] [3] [4] [S] [6] [7] [S]
J B Buks, Photophysics of aomatlc molecules (Mlcy-Intcrscwncc, NW York, 1970) J Nqbar, J Lumm. 11 (1975/76) 207. J. NaJbar, J-B. Buks and T.D S. tlamdton, Chem. Phys. 23 (1977) 281 NM. Bazi-un, Chem Phys. Letters 69 (1980) 580. A. Blumcn. J. Chem. Phys 72 (1980) 2632.74 (1981) 6926. A. Blumen, Nuovo Clmento 63 (1981) 50. hl lnokub and F Huaynma, J. Chcm. Phyn 43 (1965) 1978 J. NaJbxr and J Rodakiewicz-Now& Chcm. Phys. Letters 58 (1978) 545
159