Diffusion and long-range energy transfer

Volume 34, number 3

CHIEF.!ICX

PHYSICS

LETTERS

1 August 1975

DIFFUSION AND LONG-RANGE ENERGY TRANSFER U. CoSELE blstimtJZrPhysik am Mar-Planck-Insritur fiir Merallforschung, Sturrgart, and Institut f2r theoretische und angervandre Physik der Universitlit Stuttgart,

Stuttgart,

Gmnany

and M. HAUSER, U.K.4. KLEIN and R. FREY Insn’tur w ~~ysikafischc aernie drr Llniversit;it Stuttgart, Received

Stuttgart,

Germny

18 April 1975

Bxed OH a pair probability method for the statistics of resonance energy transfer the effect of additional diffusion is studied. Simple approximate formulae for luminescence quenchin g are derived and compxed with results prtioudy $-en in the literature.

1. Introduction

2. Mathematical

The theory of the statistics of resonance energy transfer by dipole-dipole interaction between excited donors and unexcited accepiors in ri@d solutions has

Let us assume that after S-excitation in the volume V there areNfD excited donor molecules (or excitons

been developed by Fkster [l] and Galanin [2] based on different methods. kter on additional diffusion of the acceptor or donor-molecu!es was investigated

theoretically refs. [3, $1.) and not

and experimental!y.

(For references see

Methods

were

and results

partly

different

in accordance.

Recently

some of the present authors have developed

new method for the treatment of rhe statistics of long-range energy transfer and have also derived the 2

basic equations for additional diffusion [5]. In this it is OUT intention to give simple approximate anaIytical solutions based on this statistical method of pair probability densities already used with succass for diffusion controlled bimolecular reactions [6-91 or formally similar physical situations [lo]. Comparison with already existing formulae shows that our simp!e approximations are partly more accurate than previous more complex results. Our result is also applicable to exciton dipoledipole enera transfer in molecular cryst& if the mction of the exciton may be described in terms of diffusion [II, 121.

paper

formulation

of fbe problem

in molecular crystals) which m2y trvlsfer their excim tation energy with a certain probability by dipoledipole interaction ceptor molecules

to some of- the IV: (unexcited) ac(IV! > ~4:). The transfer probability

per second depends on the distance r of the interactpair and may be witten as [ 1,131 ing donor-acceptor TV(r) = CYp .

(1)

a! characterizes the efficiency of the resonance energy transfer of the special,donor-acceptor combination. Often (1 is replaced by &r, where 7 is the mean lifetime of the excited donor molecules for the acceptor concentration C, = 0. We define pair probability densities Pii [S, 61 for the excited donor molecules and the acceptor molc-

cules, where Pij(rD,rA, t) dVDdVA/V’ is the probability of findiq at time r the donor DI (f = l..Ng) in a volume

element dVD at rD and the aCC@tor Ajo= 1 ;.. N$ in dvA at rA. If surface effects are negected Pij depends on the reIative position of D;

and Aj.,r = rD+A, only. Up to a cetiain collision distance rAD the following partial differential Of&j

equation

determines

the behaviour

Lsl 519

VoIume 34, number 3

CHEMICAL

PHYSICS

1 August 1975

LETTERS

and with appropriate boundary conditions. We chose those of Smo!uchowski [ 141 where D = D, + D, is the sum of the diffusivities of the acceptor and donor molecules (or in organic crysrals the diffusivity of the exciton). fu(t> pii is the approximative probability change of pij due to energy transfer from the donor D,- to any acceptor Ak with k t’i or from any donor 01, (m # i) to the acceptor $_ In deriving the &(i(r) term one had to replace a triplet probabiiity by a product cf pair Frobabilities 161. We restrict ourselves to the case of randomly distributed donors and acceptors without initial correlabecome independent of i, j at tion. Then all pjj and fii all times. Suppressing the subscripts i, j we arrive with the substitution [5,6,8] : p(r, t) = U(r, t) exp

[I -

f

,f(r’) dr’ - +

0

at

the modified

av;lat=

DAU-

differential

1

(3)

equation

W(r) U.

(4)

U(T=r*+)=O.

(99)

Eq. (4) cannot be solved analytically in general. Therefore we give treatments for long times and for short times a5 well as interpolation formulae.

3. Approximation

for long times

For long times we may assume steady

cording to (I), eq. (4) may then be written for spherical symmetry as a modified Bessel differential equation of order l/4: zZdZ,+z da2

dv --(&+22)u=o. dz

(10)

Takin into account (8) and (9) and with z (11 2 ‘&(WP w e g et the solution of (l”o>,s

X,,(z) 23’114 4C

U(Z)=-2

Using the relation [5]

-

Jx,

f(t')d{

(3

(Ci, C$ : initial concentration of acceptors and excited donors res ectively after Gexcitation, J C’, * Ci 3 CD ) and assuming spherical symmetry we obtain for the concentration Cs of excited donors after 6excitation dCc/dt = -Ci;fT - C~C,~~(f),

(6)

with

state condi-

tions for U(r, r): NI/i3r= 0 [8,15]. With the substitution U = z_~r-~/~,z = (1/29)(o~/D)~/~ and W(r) ac-

where K,,Q and I,,,

Kl/4

(20)

b/4

(20)

are mod&d

~$I(z)

1

I (11)

Bessel Functions of

order l/4. In order to obtain simple solutions for the long-time limit of Q(t) - designated as 9’, - we distinguish between z. < 1 and zO > 1. For z. -=%1 we get with (11) and (7) up to terms of higher order @: = 4irDrAD .

(12)

(12) is also the long-time result for pure diffusion controlled reactions without any lcng-range energy transfer [6,14] i For z. > 1) where resonance energy transfer plays

an Lssential role, we arrive at @,R= 4zD+, The first term of (7) describes the hnninescence quenching of the donors due to direct collisions and the second term ‘Aat due to long range energy transfer. For the determination of @e(t) we must solve (4) with the~initial condition for randomly distributed donors

and acceptors

U(r. r=O)= ..

1

forr7r1,*

(8)

(13)

with + = 0.676(0r/D)r/~

[l - 4.3 exp(-2z0)]

.

(141

The formal interaction radius + should be used only for comparisori with ‘AD or in connection with (13), especially for D + 0 where rF + m but @E -+ 0. The second term in the bracket of (14) may be neglected for z. S 1. FomaUy (12) and (13) are analogous but

CHEMICAL PHYSICS LETTERS

Volume 34, numbEz 3

the radius rF depends on the diffusivity D and decreases with increasing D and fmally becomes smaller than the constant ‘AD. The transition region from (13) to (12) is defined by ‘F x ‘AD or ~0 C= 1 or in terms of a diffusion length by (2Dr)‘l’

= 0.7 ‘0 @O/r&*

.

(15)

For essentially higher values of D the quenching effect does no longer depend an 0: and may be described by (12). Using a completely different scattering lengthmethod Yokota and Tanimoto [16] arrived at the same result as (13) and (14) but without noticing the necessary condition z. > 1 because they did not take into account the collision term 4m2,D(aujarj,,,,, correctly throughout their calculations.

which is also knotin to be the correct diffusion [6,8,14].

behaviollr and interpolation

fonndae

For short times and z0 < 1 we have a;

(t) = 4&

the usually fusion and [6,8,14]. known to

(D/n@

(16)

immeasurable transition term for pure difSmoluchowski boundary conditions For z. > 1 the result is easily obtained and be [I, 2,131

@)0’(t) = $ n(iTa/r)l/z Eq. (17) the form

,

.

may be written formally analogous

(17)

and Smoluchowski

boundary

or instead analogous

result for pure conditions

For z. 9 1 we obtain

QR (t) ;= 4irDP [y + r”/(iTDt)q

,

of 7’= 0.93 approximately form to (21)

(22) 7 =Z 1 finally

the

where only IAD is replaced by the ra.dius p which depends on D and o! according to (19). A rough approximation over the whole range of

z. or D is given by (24)

rem=:~D

4. Short.time

1 August 197.5

fr*.

(25)

The quantity rea should be used only in connection with (24). Eq. (24) contains all limiting cases such as D+O,~+O,Dverylargeororverylarge,t+Oor .t + w and is certainly a helpful approximation over the whole range of diffusion and resonance energy transfer parameters in quenching experiments. We restrict ourselves now to (21)-(23) for discussion and comparison with previous results.

5. Discussion and comparison

to (16) in

@; (t) = 4nr’2 (D/?Tt)‘/2 ,

(18)

where r* = 0.724(@)‘~

(1%

denotes a radius which except for a factor 7 = 0.93 is the same as the formal interaction radius ‘F introduced for the long time behaviour @E according to (14). Instead of a more complicated interpolation method which we deal with in section 5, we propose a simple linear superposition of +‘o(t) and &_ B(t) = @‘, f @()($ .

Eq. (20) leads in the case of dominating diffusion (z. < 1) to

(20)

Using the short-time behaviour of the integrated form of Q(t) up to higher order terms in I Yokota and Tanimoto [ 161 derived with a Pad&approximzt method an interpolation expression for CE and hexcitation over the whole range of time (and implicitly 20 s=- 1)

(26)

B= [(I + 10.87~ f 1550x*)/(1

t 8.743~)]~1~

,(27)

with x = h-1/3 t213. Al most identical numbers have been given by Swenberg and Stacy [17] for exciton

diffusion. The disadvantage of this interpolation is that (26) and (27) lead fdr @ to 0.91 X 4ti3/4rr’/4 f Due to B numerical stead of 0.91.

error Yokoti

and Tnnimoto

t

give 0.51 in-

521

Yolume

instead of 0.676 X 47iD3/r.,1/4 according to (13) and (14). This is an-error of about 35% in aR(t) for the long-time measured

behaviour difference

which may possibly between experiment

cause the and theory

[181We may improve (27) by taking into account the known value for @E in the Pad&approximant method. We obtain instead of (27) a modified value

B=

[(l

f 5.47x

+4.00x7-)/(1

+ 3.34x)13j4

,

(28)

er-

compared with the best interpolation expression (28) is less than 5% for (22) and less than 8% for (23). We propose to use instead of the more complicated but not more accurate relations (26), (27), the simple equations (22) and (23). With (22) we have for example instead of (26) therefore

Cg = Cs” exp[-t/r--41rLlrFCAt--

$r CA(nat)‘j2]

(29) We compare our results

(22) and (23) and hence (29) with the result of Voltz et al. which is essentially given by 1 + ‘o/(:iT~r)1/2]

.

(30)

It is easily seen that for r* < r0/2 (rO < O.S@T)~/~) the Voltz formula leads to too high a value for Q,(t), for 0.5(Dr)1/2 < r. to too low a value _ For D + 0 the

iroltz result is not applicable contrary to what is true for (22) and (23). A second interpolation formula aven by Voltz et al. has similar disadvantages, but will not be discussed here. A comparison with the results of Kurskii and Se!ivanenko [20] shows that they are also not in accordance with our results (21)-(23) and (29).

We summarize our results: (i) In the diffusivity region r* < ‘ADI which means ong-range energy transfer (or)‘/” % 70 (P&&l may be n&glected completely and pure diffusion governs the luminescence quenching effects. (ii) For the whole region of the ratio ‘*/‘AD a rough approximation is given by eq. (24) with an effect& mteractlon radius reK z ‘AD + 7+. (iii)

For r* > ‘AD (or (OT)~/~ < ‘0 (rO/rADj2)

we

simply have to nplace the constant collision radius .raD by the diffusion depe-ndent radius Faccording 522

to (19). The expressions (22), (23), (29) will be sufficiently accurate for all practical purposes. A slightly more accurate formula we have with (28) in connection with (26). If in addition an attractive long-range potential p(r) =A/? between donors and acceptors is effective

we must use instead of the collision radius TAD an effective collision radius (rA,-,jeff = (A/FsZ-)1/” [2 I,22 j (k: Boltzmann’s constant, T: absolute temperature), which

now has to be compared

with

r*.

The formulae derived in this paper may also be extended to acceptor fluorescence and to any shape of

which should lead to a better agreement between theory and experiment than (27). Numerical calculations show that the maximum ror in aR (t) of the simple formulae (22) and (23)

q)(t) = 47;D(ro/Z)[

1 August1975

CHEMICAL PHYSICS LETTERS

3

34, numlxr

excitation

by simple linear convolution.

We thank Professor Dr. A. Seeger, D.J. Miller, B.A. and Dr. M.L. Jenkins for critically reading the manuscript. References [l] Th. FGrster, Z. Naturforsch. 4n (1949) 321. [2] M.D. Galanin, Soviet Phys. JETP 1 (1955) 317. [3] J-B. Birks, Fhotophysics of aromatic molecules (Wiley, New York, 1970). [4] I.B. Berlman, Energ transfer paameiers of aromatic compounds (Academic Press, New York, 1973). 151 U. Gijsele, hi. Hauser and U.K.A. Klein, 2. physik. Chem. NF (1975), to be publishxl. 161 T.R. Waite, Phys. Rev. 107 (1957) 47 1, r71 H.M. Simpson and A. Sosin, Radiation Eff. 3 (1970) 1. IPI LJ. Gi%ele and A. Seeger, 10 be published. PI T.R. Waite, J. Chem. Phys. 28 (1958) 103. IlO1 W. Frank and A. Seeger, Appl. whys. 3 (1974) 61. [111 H.C. Wolf, in: Advances in atomic and molecular physics, Vol. 3, eds. D.R. Bates and I. Estermann (Academic Press, New York, 1967) p. 119. [I21 R.C. Powell, whys. Rev. B2 (1970) 1159. [I31 Th. Fiirster, Ann. Fhysik 2 (i948) 55. Z. Physik. Chem. 92 (1917) 192. [I41 M. van Smoluchowski, f1.51 H. Reiss and V.K. La Mer, J. Chem. Phys. 18 (1950) 1. [161 hl. Yoltotz and 0. Tanimoto, J. Phys. SOL Japan 22 (1967) 779. [ 17 ] C.E. Swenberg and W-T. Stacy, Phys. Stat. Sol. 36 (1969) 717. [ 181 J-B. Bkks and M.S.S.C. Leite, J. &ys. 83 (1970) 513. [ 191 R. Voltz, C. Lzustriat and A. Cache, 5. CXm. Phys. 63 (19661 1253. .Yu:A.- Kurskii 2nd A.S. Selivanenko, Opi. SpctrY. USSR 8 (2960) 340. A. Kraut, F. Ih;rorschak and H. Wollenberger, fiys. Stat. Sol. 44b (l971) 805. K. Schrzder, Diffusion Reactions of Point Defects, Repr. Jiil- 1063-FF (1774) p. 118.