J. Phys. Chem. 1993,97, 3668-3670
3668
Rotational Depolarization of Fluorescence in Low-Viscosity Solutions R. Alicki,' M. Alicka, and A. Kubicki Luminescence Research Group, Institute of Experimental Physics, University of Gdahk, Wita Stwosza 57, PL-80-952 Gdahk, Poland Received: December 4, 1992; In Final Form: January 26, 1993
The experimental investigation of rotational depolarization carried out for prolate molecules with short lifetimes (0.1-1 .O ns) and in low-viscosity solutions (0.1-1 .OcP) explores the region where lifetimes become comparable with time scales of the slowly varying tails of the random torque autocorrelation functions. Therefore, the simDle model based on the "white noise" random torque must be revised. The new phenomenological model illustrated by experimental data. of r'otational depolarization is proposed and .~~ ,
1. Introduction In recent years a large number of experimental data were collected on the dependence of the fluorescence anisotropy r for prolate luminescent molecules in solutions on the viscosity 7 and the mean lifetime T . I - ~ Although the deviations from the simple Perrin formula were always observed in the low-viscosity region the experimental results (for strongly varying q and almost constant T ) fit very well into another quite simple formula : ( T
1>' = kT"'+ verr t,
Here, ro is a limit value of r in rigid solutions, T i s temperature, and Veff and t , are two phenomenological parameters of the dimension of volume and time, respectively. Similar relation is valid for the ?-dependence of a second-rank orientational correlation time 72 in liquids measured by different techniques9-'
Remembering that
(3) where r ( t ) is a time-dependent fluorescence anisotropy and that T* = LJomr(t)d t r0
(4)
we obtain eq 2 from eq 3 with T~
= lim t , 7-m
The physical meaning of Veff is well understood as a volume of the molecule multiplied by a dimensionless factor (typically of order 1) describing molecular shape and the boundary conditions.I0 However, the origin of the intercepts t , and TO is still a matter of d i s c u ~ s i o n . ~Some - ~ ~ of the authors interpret TO (or t , ) as a "free rotator relaxation time"8*9 T~ = W T (5) while the othersl2.l3(cf. ref 15 also) consider the inertial effect using the Langevin equation with the "white noise" random torque which leads to the formula
t , = I/6kTz (6) with I as the moment of inertia of the molecule. Obviously, eq 6 yields T~ = 0. ~
* Address correspondence to this author at the Institute of Theoretical Physics and Astrophysics. University of Gdahsk.
0022-365419312097-3668%04.00/0
One should stress here the difference between the possible theoretical explanations and experimental justifications of the formulas (1) and (2). The latter involves time integrals of the autocorrelation functions generally related to the transport coefficients (like q ) while the former is formulated in terms of the Laplace transforms of the autocorrelation functions calculated a t the point 117. Therefore, changing the lifetime T and using eq 2 we may extract informations about the actual timedependence of the autocorrelation functions. By quenching fluorescence we explore the region of small T , and hence, the accurate determination of the value of TO by measurements of the fluorescence depolarization in stationary regime is impossible. The theoretical models based on a generalized Langevin equation predict T~ = 0. This is related to the assumption of the statistical independence of a random torque on an actual state of the molecule. This is strictly true in the limit of infinitely heavy molecule only and hence for a real molecule cross correlations between random torque and kinetic terms may lead to TO # 0 as has been proposed in ref 16. In the present paper we do not discuss the theory for TO but rather consider the case of t , >> 1701
= 0. The agreement between the formulas ( 5 ) or ( 6 ) and experimental results is rather poor. For some molecules eq 5 gives reasonable values of T O (or t,) while for the others the measured values are much smaller or even negative.I0 Moreover, there exists no dynamic theory based on the general principles of nonequilibrium statistical mechanics which reproduces the formula ( 5 ) . The inertial effect described by eq 6 should play a role at least for molecules with short lifetimes, but generally the measured values of t , for prolate molecules are larger by 3-4 orders of magnitude than those obtained using reasonable fits for the moment of inertia I . Another argument against such simple interpretations of ti has been provided by the measurements of the fluorescencequenched by nitrobenzene.' In those experiments T was varied and 7 was kept constant and low. The results fit well to the following parabolas (7) where y = T / ( r o / r - l ) , x = l / r , and T~ H 72 are positive parameters of the dimension of time. The parameter is again a certain effective volume close to Vefr from eq 1. Obviously assuming the validity of eq 5 we obtain y independent of x while eq 6 yields a linear relation between y and x. 0 1993 American Chemical Society
Rotational Depolarization of Fluorescence
The Journal of Physical Chemistry, Vol. 97, No. 15, 1993 3669
2. Theoretical Considerations
0.6
We start with a basic result of ref 12 which can be written as (3)
0.5 n
N
where &(A) = J;e-"G(t) dt and G(t) = ( w ( t ) w ( O ) ) is the autocorrelation function of an angular velocity of the fluorescent molecule. The idea of the present paper is to calculate G(t) using the generalized Langevin equation for w ( t ) I 7 with a general random torque M(r)
-IT($
d 1 = - S)O(S) ds dt Here, I is an actual moment of inertia, ( M ( t ) )= 0, and according to the fluctuation dissipation theorem -u(t)
+ -p(t)
At) ( M ( t ) M ( O ) )= ZkTy(t) (9) Therefore, after simple calculations we obtain from (8) and (9) &(A) = kT[(l/k7')BX) Z
+ A]-'
0.4
[I)
2
'0
0.3
i U
h 0.2 0.1
0.0
0.5
1.5
2.0
2.5
x [10-12kg m2] Figure 1. The dependence of y = rz(ro/r- 1) on x = TT for PPO (v), POPOP (+), and a-NOPON (m) in n-paraffins CnH2"+2at 293 K with n = 5-1 1 for PPO and a-NOPON and with n = 5-16 for POPOP.
Then indeed
and finally
JOmflt)dt
Remark. If M ( t ) is assumed to be a white noise as in ref 12 thennX) = 6kTVefmand we obtain eq 1 with t, given by eq 6. Generally, one expects that the autocorrelation functions like (M(t)M(O))in fluids possess a rapidly decaying positive part and slowly varying tails which may be positive or negative.18 Using Taylor expansion for the function AX) and the standard relation
=BO)
BO)= Jom(M(t)M(0))dt = 6kTVeffv
(12)
we obtain eq 1 with t,
1.0
= (6kT)-'[I(6kTZ T - Jomtf(r) dt) + -(Jm(f2/2)flt) 12 0 dt) 7
+
...I
(13)
In order to explain the experimentaldata described by the relations (1) and (7) in the case of rather large prolate molecules the following conditions must be satisfied: (a) J;tf(t) dt C 0 and is 7-independent and (b) J;tzf(t) dr C 0 and tends to zero for large 7.
It means that the random torque autocorrelation functions possess negative slowly varying tails. Obviously, the conditions a and b cannot fix uniquely the parametrization of the relevant autocorrelation function f ( t ) in terms of hydrodynamical parameters but they may suggest simple models. Consider the function At) of the following form
At) = (1
+ ~)6kTV,ff$(t) - c$,,(t), t > 0
(14)
with the function I#J,, satisfying the scaling condition
= a2$(7t)
(15) Morover, we assume that $ > 0, and the normalization is chosen such that $,(t)
Jom$(x)dx = 6kTVeff
(16)
= -cKx$(x) d x < 0,
and eq 12 holds. The first term proportional to Dirac-b describes the rapidly decaying (comparing to the lifetime T ) part off(t) while #,,(t) is its slowly varying tail. The particular scaling property of the tail (15) could be explained as follows. The random torque acting on the molecule in fluid consists of the part due to almost independent collisions with the neighboring molecules and the second part due to the interaction with different hydrodynamical modes. The later may be responsible for the existence of tails with relaxation times proportional to Q-' which are characteristic for the viscous damping of the transverse hydrodynamical modes most effectively coupled to the rotating molecule. The magnitude of the coupling should be proportional to q what gives the factor q2 in (15). The existence of a slowly varying component of a random torque was discussed in ref 16 also.
3. Examples of the Experimental Results We present now a few examples of experimental results. Figure 1 shows the dependence of T2/(r/ro- 1) on TT for PPO, POPOP, and a-NOPON in different parafines. The data for PPO and a-NOPON were taken from ref 8 while for POPOP they were obtained by the authors using the photon counting apparatus described in ref 19. Figure 2 shows the dependence of T/(r/rO - 1) on 1/ T for the same molecules with the fluorescence quenched by nitrobenzene. In this case the data were taken from ref 7,and the results for POPOPwereconfirmed by theauthorsusingslightly different technique. For the same molecules Table I summarizes the results of calculations of the phenomenological parameters. The effective volumes Verf and V:ff were obtained from the different solvents method and quenched fluorescence, respectively. The values of a volume V calculated from the geometry of the molecules are presented for comparison also. The parameters T I and 7 2 were obtained from quenched fluorescence data with the fit given by eq 7. The dimensionless parameter u denotes the ratio
where t , is obtained from the different solvents method while V:,
Alicki et al.
3670 The Journal of Physical Chemistry, Vol. 97, No. 15, 1993
predictions based either on the Perrin formula or on the models of the rotational Brownian motion with a white noise random torque. These discrepancies can be explained assuming the existence of slowly varying tails of the random torque autocorrelation function. The new model with a physically realistic form of random torque was proposed and tested by experimental data for a number of substances. The further investigations based on the microscopic kinetic theory of fluids are needed to support this purely phenomenological Ansatz.
i 0.25
1
0 3 U
t
h
1 0.10L" 0.5 1.0
"
1.5
"
"
2.0
2.5
9 -1
x [lo s
"
'
3.0
1 3.5
702.
in cyclohexane from the measurements of fluorescence quenched by nitrobenzene.
TABLE I
PPO POPOP a-NOPON
vx IO"' v,,,x IO"' (ml)
(m')
505 1320 1542
175 624 823
v,,,x IO"' (m1)
172
580 712
References and Notes ( I ) Kawski, A.; Kamifiski, J.; Kukielski, J. Z . Naturforsch. 1979, 340,
3
Figure 2. T h e dependence of y = T / ( r o / r - 1) on x = 1 / for ~ PPO (V), POPOP (+), and a - N O P O N (H). T h e values of T a n d r were obtained
molecule
Acknowledgment. The authors are deeply grateful to Dr. W. Wiczk and Mrs.L. Gruzdiewa for the assistance by the lifetime measurements, Mr. J. Lichacz and Dr. Z. Kojro for making accessible their experimental data, and to Prof. A. Kawski for discussions. Theworkissupported by theGrantof thecommittee for Scientific Research BW/5200-5-0060-2.
(ii) (ii) 0.17 0.38 0.56
0.32 0.44 0.40
u
1.0 0.44 0.36
and rl from the quenched fluorescence method. Theoretically u = 1, The smaller values of u for POPOP and a-NOPON are probably due to the essential deviations from straight lines of the curves on Figure 2 that make the second-order approximation (1 2) less accurate.
4. Conclusions Fluorescence anisotropy data for prolate fluorescent molecules with lifetimes in the range of 0.1-1.0 ns and in the solutions of kg/ms) disagree with the the viscosities 0.1-1.0 CP(1 CP=
(2) Kawski, A.; Kojro, Z.; Alicka, M. Z . Narurforsch. 1980,35a, 1197. (3) Griebel, R. Ber. Bunsenges. Phys. Chem. 1980, 84, 919. (4) Salamon, Z.; Skibifiski. A.; Celnik, K. Z . Naturforsch. 1982, 37a, 1027. (5) Kawski, A.; Kojro, Z.; Kubicki, A. Z . Narurforsch. 1985,40a, 313. (6) Kawski, A.; Kojro, Z.; Bojarski, P.; Lichacz, J. Z . Narurjorsch. 1990, 45, 1357. (7) Kojro, Z . Ph.D. Thesis, University of Gdadsk, 1985. (8) Kawski, A.; Bojarski, P.; Kubicki, A. Z . Narurforsch. 1992, 470, 971. (9) Alms, G.; Bauer, D.; Brauman, J.; Pecora, R. J . Chem. Phys. 1973, 59, 5310.
(IO) Kivelson, D. In Rorarional Dynamics ofSmalland Macromolecules; DorfmBller, Th., Pecora, R., Eds.; Springer Verlag: Berlin, 1987. ( I 1) Dorfmiiller, Th. In Rorarional Dynamics of Small and Macromolecules; Dorfmiiller, Th., Pecora, R., Eds.; Springer Verlag: Berlin, 1987. (12) Alicki, R.; Alicka, M.; Kawski, A. Z . Narurforsch. 1981,36a, 1158. (13) Alicki, R.; Alicka, M. Z . Naturforsch. 1983, 380, 835. (14) Gaisenok, W.A.;Solnerevich,J. J.;Sarshevskii, A. M. 0pr.Specrrosc. (Engl. Trans.) 1980, 49, 714. (15) Evans, G.;Kivelson, D. J . Chem. Phys. 1986, 84, 385. ( I 6) Freed, J. E. In Rotational Dynamics of Small and Macromolecules; Dorfmiiller, Th., Pecora, R., Eds.; Springer Verlag: Berlin, 1987. ( I 7) Kubo, R.; T d a , M.; Hashitsume, N. Starisrical Physics 11;Springer Verlag: Berlin, 1985. (18) RBsibois, P.; De Leener, M. Classical Kinetic Theory of Fluids; Wiley: New York, 1977. (19) Kubicki, A. Exp. Tech. Phys. 1989, 37, 329.
