Luminescence Kinetics of Linear Polymer Molecules with Chromophores Randomly Distributed along the Chain

Transcription

1 Optics an Spectroscopy, Vol. 9, No. 5,, pp Translate from Optika i Spektroskopiya, Vol. 9, No. 5,, pp Original Russian Text Copyright by Bounov, Berberan-Santos, artinho. OLECULAR SPECTROSCOPY Luminescence Kinetics of Linear Polymer olecules with Chromophores Ranomly Distribute along the Chain E. N. Bounov*,. N. Berberan-Santos**, an J.. G. artinho** *Russian State Hyrometeorological University, St. Petersburg, 959 Russia bounov@mail.amiral.ru **Center of Physical Chemistry of olecules, Technical University, Lisbon, 49- Portugal Receive arch 3, Abstract Luminescence kinetics of chromophores ranomly attache to an isolate flexible polymer chain is stuie. In the static case, when the polymer chain oes not change its conformation uring the onor excitestate lifetime, analytical equations are erive for the luminescence kinetics of both eal an noneal polymer chains. The conitions are formulate when the inhomogeneous broaening shoul be taken into account. For the ynamic case, the iffusion an hopping mechanisms of the polymer motion are stuie. AIK Nauka/Interperioica. INTRODUCTION The energy transfer an migration of electronic excitations in polymers is wely stuie an commonly use to analyze their structure [, ]. The reason is that the time epenence of the luminescence intensity of onor an acceptor chromophores can be connecte with the interchromophore istance by means of Forster s energy transfer rate [3]. Such experiments are usually performe at very low polymer concentrations, which makes it possible to neglect the energy transfer between ifferent polymer molecules. There are several ways of attaching chromophores to a polymer chain [,, 4 9]. In this paper, we stuy the following moel. Each link of the polymer chain may contain, with a certain probability, a onor or acceptor chromophore so that all the chromophores are istribute along the chain in a ranom way [, 4, ]. Such polymer molecules can be synthesize by esterification of a long chain of ialcohols by acs containing onors or acceptors an showing the same reactivity with respect to the alcohol. The goal of this paper is to stuy the irect incoherent energy transfer from an excite onor to another onor or acceptor uner the above conitions. We also stuy the effect of inhomogeneous broaening of the chromophore spectrum an the motion of the polymer chain (for the iffusion an hopping mechanisms of conformation changes) on luminescence kinetics. We will focus our attention on a single polymer molecule isolate from other molecules (the polymer concentration is assume to be sufficiently low). This assumption is mae in orer to simplify the problem an neglect the interaction between neighboring polymer chains, which may give rise to the energy transfer between chromophores localize on ifferent chains. We will also assume that the solvent is strong an the temperature of the polymer is above the θ-point of the transition, so that we can use known istribution functions for the istance between the ens of the polymer chain. Finally, we will conser the case of the isotropic ipole ipole interaction between the chromophores, when the energy transfer rate w(r) can be written in the form wr ( ) -- R, τ r () where τ is the lifetime of the onor chromophore excite state, R is the Forster raius, an r is the istance between the chromophores. STATIC ODE As was mentione above, there are several moels of chromophore attachment to a polymer molecule. In this paper, we assume that there is a certain probability that a chromophore can attach to any link of the polymer chain. To neglect the energy migration over the onors, we assume that the concentration of the chromophore-onors is low. Then, we have the following ifferential equation for the ecay of the excite onor state p( ue to the energy transfer to the acceptors: ---- pt t () wr ( i )pt (), i () where r i is the istance between the onor an the ith acceptor an is the number of acceptors attache to the polymer chain. Equations () o not contain any 3-4X//95-94$. AIK Nauka/Interperioica

2 LUINESCENCE KINETICS OF LINEAR POLYER OLECULES 95 loss of excitation relate to a finite lifetime of the excite onor state, since this channel of the ecay has nothing to o with the energy transfer process. Thus, the total kinetics of the luminescence ecay can be represente as a prouct of two factors: exp( t/τ) an the ecay kinetics relate to the energy transfer to the acceptor. Note that r i in Eq. () is inepenent of time because, in this part of the paper, we neglect the motion of the polymer chain uring the lifetime of the excite onor state. The istances r i evently epen on the chain conformation. Assuming that the onor is initially excite with the probability equal to unity, we obtain from Eq. () pt () exp t w( r i ) i i exp[ tw( r i )]. (3) The onor luminescence kinetics I(, relate to the energy transfer to acceptors, is obtaine by averaging (3) over all conformations with the istribution function (K, r) [( (K, r) is the probability ensity use to fin the istance between the onor an the acceptor in the polymer, consisting of K links, equal to r, an N is the number of links in the polymer chain between the onor an the acceptor]. In the -imensional space we have It () K (4) where V π / /Γ( + /) an Γ(x) is the gamma function. Note that the istance between two points of the polymer chain (which are not too close to each other), separate by N links, an the istance between the ens of the polymer chain consisting of N links satisfy one an the same istribution function []. Therefore, by enoting (r) (N, r), we can simplify Eq. (4) an use the following equality: (5) This equality imposes restrictions upon the results obtaine below: they are val at a low chromophore concentration when the average number of links of the polymer chain between the onor an the nearest acceptor is sufficiently large. For the eal (Gaussian) polymer chain, (r) has the form [5, ] --- V K r ( K, r)e tw() r K N r --- V K r ( K, r) ( e tw() r ) r N ( K, r) ( r). ( r) A exp( B r ),, () where For a noneal chain (a chain with the volume interaction), we have [5, ] where A --- /, B π R g R g ( r) A r ϑ exp( B r δ ), ( + ϑ)/δ δb A -, V Γ[ ( + ϑ)/δ] Γ[ ( + ϑ + )/δ] δ/ B ---. Γ[ ( + ϑ)/δ] R g (7) Here R g is the raius of gyration of a polymer ball ( a R N for an eal chain an a N ν g R g for a noneal chain, where a is the statistical length of a link) an ϑ an δ are the imension-epenent critical inices. These critical inices can be written as ϑ (γ )/ν an δ ( ν), where γ an ν are two universal critical inices epening on. The self-consistent fiel metho (Flory theory) in a space with the imension yiels [] ν 3/ ( + ), 4, (8) γ ( ), γ 4/3 ( ), γ 7/ ( 3). Therefore, for the three-imensional space (which is the case uner conseration), we have from Eqs. (7) an (8) ν 3/5, ϑ 5/8, δ 5/, A.8958/ R g 59/3, B.7/ R g 5/4. (9) The functions 4πx (x) (where x r/ ) for the eal an noneal polymer chains are shown in Fig.. Note that the results of the Flory theory (9) are very close (within two percen to those obtaine by computer simulation (ν.59 []), with the accuracy being sufficient for any real polymer chain. When stuying the excitation transfer in a system of finite size (the polymer chain has K links), one has to make two spatial averages. The first is the average over all possible positions of acceptors with respect to the position of the initially excite onor (localize, e.g., on the ith link of the polymer chain). This will give us kinetics of the luminescence ecay I i ( for the given initial conitions. The secon is the average over all possible positions of the initially excite onor (in the system of finite imensions, these positions are not R g OPTICS AND SPECTROSCOPY Vol. 9 No. 5

3 9 BODUNOV et al. 4πx (x) x r ( R g ) Fig.. The istribution function for the (sol line) eal an (ashe line) noneal chains. I( t/τ Fig.. Luminescence kinetics for the (sol line) eal an (ashe line) noneal polymer chains. R /a, c.. equivalen. This problem was solve numerically for an eal polymer chain in [, 7] using the GAF metho [] an in [4, 8 ] using the Huber approximation [, 3]. The problem of excitation transfer in a polymer chain becomes much easier for an infinite chain (K, uner the conition that /K cons. In this case, there is no necessity to calculate the secon average, since all positions of the initially excite onor are equivalent. Thus, with allowance for formula (5), we have It () exp c N r4πr ( r) ( e tw( r) ), () where c is the mean number of acceptor chromophores falling into one link of the polymer chain (c ). For the case of ipole ipole interaction, both integrals in Eq. () can be calculate exactly. As a result, for the luminescence kinetics I ( of the eal polymer chain, we have I () t Γ 3 -- R c t τ - /3 exp () an for the luminescence kinetics I n ( of the noneal chain, we have I n () t 3.87c R () a 5/3 t - τ 3/54 exp. The luminescence given by Eq. () ecays slower than Eq. () (see Fig., R /a, c.), since, for the noneal polymer chain, (r) when r [see a 8.5c R t τ - /3 exp, a Eq. (7)], while for the eal chain (r) const when r [see Eq. ()]. In polymers where only the onor onor energy transport occurs, a funamental quantity, interesting in both a theoretical an a practical sense, is G s ( the ensemble-average probability that the initially excite chromophore still remains excite at the instant t. This probability contains contributions from excitations that never abanone the initially excite chromophore an from excitations that returne to that chromophore after one or several energy transfer events. As before, G s ( oes not contain any loss of excitation relate to a finite lifetime of the excite chromophore state. The probability G s ( is relate to the time epenence of luminescence epolarization: r( r G s (, where r( is the anisotropy of luminescence an r is its initial anisotropy [4,, 4]. Conser two chromophores (the zeroth an firs attache to ifferent links of the polymer chain. The ynamics of their excite states p ( an p ( is escribe by the equations ---- p t ( r, wr ( )[ p ( r, p ( r, ], ---- p t ( r, wr ( )[ p ( r, p ( r, ], (3) where r is the istance between the chromophores. Solutions of these equations, uner the assumption that at the initial instant only the zeroth chromophore is excite with the unit probability, has the form p ( rt, ) exp[ tw( r )]. (4) In the Huber approximation [, 3], G s ( is obtaine by averaging the prouct of functions (4) for all pairs OPTICS AND SPECTROSCOPY Vol. 9 No. 5

4 LUINESCENCE KINETICS OF LINEAR POLYER OLECULES 97 of chromophores (with each pair containing one excite chromophore): G s () t i exp[ tw( r i )] -- [ e tw( r) ] (5) Here, is the number of pairs, equal to the number of chromophores (onors) attache to the polymer chain. The broken brackets enote averaging over all conformations. Assuming, as when eriving Eq. (), that the polymer chain is infinitely long, we obtain G s ( c exp -- N r4πr g () N () r ( e tw( r) ), where c is the mean number of onor chromophores falling into one link of the polymer chain. In the case of ipole ipole interaction, the integrals entering Eq. (), can be calculate exactly. Note that Eq. () iffers from () by the substitutions c c/ an R R. Therefore, from Eq. (), using the above substitutions, for the eal polymer chain we have s G (7) Expression (7) was obtaine in [8, ] within the framework of the Huber approximation an in [] using the GAF metho in a two-particle approximation. A three-particle approximation of the GAF metho leas to the function G s ( exponentially growing in time. This is a common problem of the GAF metho in a low-imensional system [5]. For the noneal polymer chain, using formulas (7) an (), we have s G n (8) Formula (8), again, can be obtaine from formula () by the substitutions c c/ an R R, in the same way as formula (7) was obtaine from (). INHOOGENEOUS BROADENING In recent years special attention has been pa to polymer molecules containing onors an acceptors. () t /3 3Γ 3 -- R c t τ - /3 exp a 5.8c R t τ - /3 exp. a () t.347c R a 5/3 t - τ 3/54 exp. which exhibit inhomogeneous broaening of their spectral lines []. In such polymers, one can observe the epenence of the luminescence kinetics on the excitation wavelength, on the exciting pulse wth, an on the temperature. In this paper, we try to answer the following question: what is the smallest inhomogeneous wth σ where we can notice this effect? In the case of inhomogeneous broaening of the spectra, the luminescence kinetics of onors I inh ( (uner nonselective excitation of the chromophores attache to a long polymer chain) can be written as I inh () t E g D ( E ) exp c E g A ( E ) N r4πr ( r) e tw ( E E, r) ( ), (9) where w(e E, r) is the rate of energy transfer from the onor (with the transition energy E ) to the acceptor (with the transition energy E ), an g D (E ) an g A (E ) are the normalize istributions of onors an acceptors over the transition energies. For simplicity, we will make the following assumptions. First, let these istributions be Gaussians with equal wths an let their peaks be locate at the transition energies E D an E A : g D ( E) g A ( E) ---- exp πσ ---- exp πσ E D ( E D E) --, σ ( E A E) --, σ () E A E. Secon, let the homogeneously broaene luminescence spectra of onors an the absorption spectra of acceptors also be of Gaussian shape, with their wths being the same an equal to δ/. Then, the energy transfer rate controlle by the ipole ipole interaction, proportional to the spectral overlap integral [3], can be written as we ( E, r) -- RE ( E ) τ r R ( E ( E E ) R E ) exp --- () where R is the Forster raius of the onor acceptor energy transfer, with the onor an acceptor energies being E E A an E E D. Note that the thir an fourth integrals in Eq. (9) were alreay calculate [see, e.g., formula () for the δ, E - +, δ OPTICS AND SPECTROSCOPY Vol. 9 No. 5

5 98 I inh (. BODUNOV et al. I inh ( cr /a (t/τ) / cr /a (t/τ) /3 Fig. 3. Luminescence kinetics of the eal polymer chain for inhomogeneously broaene spectra. The maximum of the onor emission spectrum coinces with that of the acceptor absorption spectrum (E D E A ). The numbers near the curves are the inhomogeneous/homogeneous linewth ratios σ/δ. Fig. 4. Luminescence kinetics of the eal polymer chain for inhomogeneously broaene spectra. The maxima of the onor emission spectrum an acceptor absorption spectrum are locate at ifferent transition energies (E D E A + σ). The numbers near the curves are the inhomogeneous/homogeneous linewth ratios σ/δ. eal polymer chain]. Using this result an substituting () unto (9), we obtain I inh () t E g D ( E ) 8.5c R t τ - /3 exp () for the eal polymer chain. The luminescence kinetics () was calculate numerically. The results of the calculations are shown in Fig. 3 for the case E D E A (or E ). As is seen from Fig. 3, the increase of the inhomogeneous broaening of the spectra (i.e., the ratio σ/δ) slows own luminescence ecay, because inhomogeneous broaening reuces the energy transfer rate for all onor acceptor pairs [this is event from Eq. ()]. The luminescence kinetics for the case E σ is shown in Fig. 4. The luminescence ecay becomes faster at shorter times an slower at longer times. This results from the fact that when E σ, the energy transfer rate in the onor acceptor pairs, for which E E < E, increases, while in the pairs with E E > E, it ecreases as compare to the case where inhomogeneous broaening is absent. It follows from Figs. 3 an 4 that the inhomogeneous broaening of the spectra shoul be taken into account when σ/δ >.. CONFORATION OTION OF A POLYER CHAIN The effect of the motion of a polymer molecule in ilute solutions on the rate of intramolecular reaction between reactive groups attache to the ens of a flexi- a ( E E g A ( E ) E ) E exp δ 3δ ble polymer chain was stuie in [ 33]. In [ 8], it was assume that the reaction occurs when the reactive groups approach close enough to one another (contact reaction), an that the resulting polymer motion was escribe using the generalize iffusion equation. The asymptotic behavior of the reaction at short an long times was calculate. The energy transfer (luminescence quenching) is a similar process. The only istinction (from the theoretical point of view) is in the epenence of the reaction rate on the istance between the chromophores. This is why the stuies of the effect of polymer motion upon luminescence kinetics [9, 3] also employe the iffusion equation. For polymer molecules with a ranom arrangement of chromophores along the polymer chain, the following equation for luminescence kinetics is val: I if ( exp c N r4πr ( () r *( rt, )), (3) r where * (r, is the istribution function for istances between an excite onor an a nonexcite acceptor an N is the number of links in the polymer chain between the onor an the acceptor. Initially, * (r, t ) (r). In the static situation, * (r, (r)exp[ tw(r)] [see Eq. ()]. It is known [, 9, 3] that free links perform Brownian motion in a solvent with a certain iffusion coefficient D, an that the istribution function may be treate as an energy term by introucing the interaction energy U(r): V( r) Ur ( )/kt ln ( r), (4) OPTICS AND SPECTROSCOPY Vol. 9 No. 5

6 LUINESCENCE KINETICS OF LINEAR POLYER OLECULES 99 where k is the Boltzmann constant an T is the absolute temperature. Thus, we can write the equation [9 33] ----g t N *( rt, ) D r ---- g r r r N *( rt, ) + D r g r r N * V - r wr ( ) *( rt, ) with the bounary an initial conitions ---- g r N * ( rt, ) V( r) r r r *( r, ) ( r), where r is the istance of closest approach. (5) The first two terms on the right-han se of Eq. (5) escribe changes in the istribution function * (r, ue to the Brownian motion of the links in the spherical fiel U(r), while the thir term escribes changes in the same function ue to the energy transfer from the excite onor to the acceptor. The initial conition signifies that the chromophores cannot approach each other at a istance closer than r. Equation (5) was solve numerically in [9, 3, 33] for the case of short polymer chains an a special form of the istribution function (, an in [3, 3] for the case of the eal chain. A comparison with experimental ata has shown that the iffusion coefficient can be as high as 5 cm s [3 33]. In all the papers cite above, another moel of attachment of a chromophore to a polymer molecule was stuie; namely, the onor an the acceptor were attache to the ens of the polymer chain. For the polymer with a ranom istribution of chromophores along the chain one has, first, to numerically solve Eq. (5) separately for each value of N, an then perform averaging over N [see Eq. (3)]. This is a very teious proceure. For this reason, we obtaine the analytic solution to Eq. (5) in the limit of slow iffusion. To o that, we use a metho teste in [34], where the solution of the iffusion equation was sought in the form of an iteration series in terms of the small parameter D, an only the two first terms were retaine (the zeroth approximation an the first-orer correction proportional to the iffusion coefficien. In this case, for the ipole ipole mechanism of the energy transfer, we have, where the first-orer correction from the equation (r, can be foun (7) The action of the operator H in Eq. (7) upon the function f(r, (r)exp( R /r t τ) is efine as (8) Taking into account that (r) exp[ V(r)] [see Eq. (4)], expression (8) can be rewritten in the form (9) When computing the integrals entering into Eq. (3), we also use the following formulas. It is known [, ] that for the eal polymer chain (3) For the case of the noneal chain, our calculation yiele where C t ( ) ( rt, ) ( ) R ( t t' ) R - Hg r N ( r) t' exp exp --- t'. τ r τ Hf ( r) D r ---- f( r, r r r + D r f( r, ---- V( r). r r r Hf ( r) D r g r r N ( r) ---- R t exp --. r r τ ( r) N ( r) N 3. πa r C ---- a -- /ν r a, Γ ---- ν( + ϑ) Γ - + ϑ + νδ δ -- νv Γ -- + ϑ - Γ -- + ϑ δ δ ν( + ϑ δ) --- ν (3) Omitting intermeiate manipulations, fulfille for the three-imensional space ( 3, C.5747), we present the final result. For the luminescence kinetics ( of the eal polymer chain, we obtain I if. R *( rt, ) ( r) t ( ) exp -- + g r N ( rt, ), τ () I if () t 8.5c R t τ - /3 exp c Dt, a a (3) OPTICS AND SPECTROSCOPY Vol. 9 No. 5

7 7 BODUNOV et al. an for the noneal chain I n if () t 3.87c R a 5/3 t - τ 3/54 exp.535c Dτ a - a R /3 t - τ 7/8. (33) Equations (3) an (33) are correct when the secon term in square brackets is smaller than the first one (within the time span of the orer of the excite-state lifetime τ). This means that the inequality Dτ < R shoul be satisfie. The first terms in Eqs. (3) an (33) escribe static quenching, an the secon terms are the first-orer corrections relate to the iffusion motion of the polymer chain links. As one can see, the polymer motion enhances the luminescence quenching. Equation (5) can also be solve in the approximation of extremely fast iffusion (the case of total mixing [35]). In this case, the iffusion is so strong that a ecrease of the function * (r, at small r is immeiately counterbalance by iffusion from larger istances. For this reason, the istribution function * (r, retains its shape [coincent with that of (r)], while its magnitue exponentially ecays in time [34, 35]. Thus, taking into account the normalization conition 4π g (r)r N r an omitting intermeiate calculations, similar to those presente in [34], we have *( rt, ) ( r) exp( k if ( N), k if ( N) 4π τ R g r N ( r)r r, (34) To obtain the rate constants from these equations for n the eal ( k if (N)) an noneal ( k if (N)) polymer chains, we replace the istribution function (r) with a rectangular function with the wth r I if () t c N e k if( N)t exp ( ). R g an assume that r. Therefore, the equations obtaine below for k if (N) will be correct to within a numerical factor of the orer of unity. Thus, for the eal chain [see Eq. (), R a ], we obtain k if ( N) R g R R , τ r 3 R g 3/ τ r 3 a 3 N 3/ I if () t 8π -c R ---- t 3 3Γ( 5/3) r a τ - /3 exp 5.358c R ---- t r a τ - /3 exp, (35) an for the noneal chain [see Eq. (7), a N ν, ν 3/5], we have I n if () t k n if ( N) (3) As is seen, the luminescence kinetics is not exponential even in the limit of extremely fast iffusion [see Eqs. (35) an (3)]. The reason for this is that the links containing chromophores perform Brownian motion inse a confine space of ifferent volume. The size of the latter is etermine by N, i.e., the number of links between the onor an acceptor. Note that this (iffusion) approximation is val when the istance between the chromophores attache to the polymer chain changes (upon motion of the links) by small steps compare with R, i.e., the Forster energy transfer raius. It is only in this case that the iffusion equation can be use to escribe the luminescence kinetics. In principle, it is possible to suggest another moel of polymer motion, alternative to that of iffusion. One can assume that conformation transitions may occur (with corresponing changes of the istances between the onor an the acceptor), an that these transitions between stable conformations are suen an ranom both in time an in space [3,, 3, 37]. The istance between the chromophores changes, in this case, in such a way that the initial conformation of the system appears to be completely forgotten after a certain time. This moel of polymer motion may be calle the hopping moel. The istance between the chromophores, after each jump, appears to be conserably change as compare to the Forster raius. 4 R 4 R g -- - τ 3 5/8 r R g ( 3 + 5/8)/ R , τ 3 5/8 r a 3 + 5/8 N 59/3 π exp Γ( 89/59) sin( 3π/59) - R c t τ - 3/59 r 3 5/8 a 3 + 5/8 R 3.5c t + τ - 3/59 exp. 3 5/8 r a 3 5/8 OPTICS AND SPECTROSCOPY Vol. 9 No. 5

8 LUINESCENCE KINETICS OF LINEAR POLYER OLECULES 7 Note that Eq. (3) is val for any type of polymer motion (rather than only for iffusion). In the case of the hopping mechanism, this equation for the luminescence kinetics I hop ( can be rewritten as I hop () t exp c N( I hop ( N,. (37) Here, we use the normalization conition 4π g (r)r r an introuce the function N I hop (38) This function has the meaning of luminescence kinetics of a polymer chain comprise of N links an possessing one onor an one acceptor on opposite ens of the chain [9 33, 37]. By introucing the mean time of the conformation change (τ conf ) an assuming that the process of conformation changes is Poissonian, we obtaine the following equation for the luminescence kinetics I hop (N, [37]: I hop t (39) Here, I(N, is the luminescence kinetics in the absence of conformation transitions. For the eal polymer chain, this kinetics is escribe by the equation [37] where T --- t ( a N). The function I(N, T) R g 3 τ - R g is approximate, to within one percent, by the function [37] while for the noneal chain ( is approximate by Eq. [37] ( N, 4π *( rt, )r t. ( N, I( N, exp( t/τ conf ) I( N, t ) exp( t /τ conf )I hop ( N, t t ) t. τ conf I( N, R π -- y / y 7T ---- exp y, / 8y 3 I( N, ( +.45T /4 ) exp( 5/4 T /4 ), I( N, (4) a N ν, ν 3/5), it which, in turn, to within three percent, is reprouce by the function [37] R g - Γ( + 4/45) y 4/45 Γ( 9/9) y - 3 T exp ---- y, Γ( 59/45) y /5 I( N, ( +.75T 73/3 ) exp(.t 5/7 ). (4) (Note that Eq. (39) was obtaine in [, 3] for another moel of attachment of chromophores to the polymer chain an for another type of conformation transition but that erivation appears to be val in the case uner conseration as well). Equation (39) is to be solve separately for each value of N. After that, one shoul calculate the luminescence kinetics (37). This is a very time consuming task. Analytical results can be obtaine only in the limit of very small τ conf. It was shown in our paper [37] that when τ conf reuces, kinetics (39) becomes exponential: I hop (N, ~ exp( k hop (N), where k hop (N) is the rate constant obtaine from the equation k hop (4) For this reason, in the limit of small τ conf, Eq. (37) can be rewritten in the form (43) Using Eq. (4), we obtaine [37] the rate constant (N) for the eal polymer chain k hop an the rate constant ( N) --- I( N, τ conf exp( t/τ conf )t / I( N, exp( t/τconf ) t. (N) for the noneal chain (44) k n hop ( N) -- R (45) τ conf N 59/3 a 59/8 τ --- conf 59/8. τ Substituting these results in Eq. (43) an integrating, we obtain the luminescence kinetics I hop ( for the eal polymer chain I conf an the luminescence kinetics chain I n hop I hop () t c N e k hop exp ( ( N)t ). k hop ( N) (4) ( for the noneal () t 3.5c R --- τ conf --- 5/8 t exp --- 3/59. (47) τ Note again that Eqs. (4) an (47) are val in the limit of small τ conf. --- R τ conf N 3/ a 3 τ --- conf / τ n k hop () t 5.358c R τ conf --- /3 t exp --- /3 τ a 5/3 a 5/3 n I hop τ conf τ conf OPTICS AND SPECTROSCOPY Vol. 9 No. 5

9 7 BODUNOV et al. The time τ conf is the moel parameter: it equals the inverse rate constant of conformation transitions an can be obtaine by the comparison of theoretical an experimental ata. By comparing the luminescence kinetics (4) an (47) obtaine in the hopping limit of the polymer motion, with kinetics (35) an (3) obtaine in the iffusion limit, we see that they similarly epen on time but ifferently on R. This fact can be use to make clear (by replacing chromophores attache to the polymer chain) the kins of motion of a polymer in a solvent. CONCLUSION In this paper, we stuie the kinetics of luminescence of chromophores ranomly istribute over an isolate flexible polymer chain (a chromophore can be attache with a certain probability to each link of the polymer chain). The energy transfer between the chromophores was assume to be of the ipole ipole type. In the static case, when the polymer chain oes not change its conformation uring the excite-state lifetime, we obtaine, for the eal an noneal chains, formulas () an () for the luminescence kinetics an formulas (7) an (8) for the emission anisotropy. It was shown that the inhomogeneous broaening of the luminescence spectra of onors an absorption spectra of acceptors shoul be taken into account when an inhomogeneous broaening σ is large enough (σ >.δ, where δ is the homogeneous broaening). In the ynamic case, when a flexible polymer chain changes its conformation uring the excite state lifetime, we consere the iffusion an hopping limits of the polymer chain motion. In the iffusion limit, we erive formulas (3) an (33) for the luminescence kinetics of the eal an noneal chains, respectively. This type of motion obviously is not of great importance when the iffusion coefficient is sufficiently small (Dτ R ). We have shown that even in the limit of very fast iffusion (D ) the luminescence kinetics remains nonexponential [see Eqs. (35) an (3)]. This results from the fact that acceptors attache to ifferent links of the polymer chain perform Brownian motion in a confine space of ifferent size. We also stuie the hopping limit of changes in the polymer chain conformation (an alternative to the iffusion limi. This type of polymer motion is important when the time of existence of a stable conformation is sufficiently short. We obtaine analytical expressions for the luminescence kinetics in the limit of extremely short τ conf. The kinetics remains nonexponential. The reason for this is the same as in the case of iffusion. Note once again that the luminescence kinetics iffers from that given in this paper by the factor exp( t/τ), which takes into account a finite lifetime of the excite chromophore states. ACKNOWLEDGENTS E. N. Bounov is grateful to INVOTAN (ICCTI, Portugal) for financial support. REFERENCES. S. E. Webber, Chem. Rev. 9, 49 (99).. B. ollay an H. F. Kauffmann, in Disorere Effects on Relaxation Processes, E. by R. Richert an A. Blumen (Springer-Verlag, Berlin, 994), pp Th. Forster, Ann. Phys. (Leipzig), 55 (948). 4. A. H. arcus, D.. Hussey, N. A. Diachun, an. D. Fayer, J. Chem. Phys. 3, 889 (995). 5. A. K. Roy an A. Blumen, J. Chem. Phys. 9, 4353 (989).. G. H. Frerickson, H. C. Anersen, an C. W. Frank, J. Chem. Phys. 79, 357 (983). 7.. D. Einger an. D. Fayer, acromolecules, 839 (983). 8. K. A. Petersen an. D. Fayer, J. Chem. Phys. 85, 47 (98). 9. K. A. Petersen, A. D. Stein, an. D. Fayer, acromolecules 3, (99).. G. H. Frerickson, H. C. Anersen, an C. W. Frank, J. Polym. Sci., Polym. Phys. E. 3, 59 (985).. G. H. Frerickson, H. C. Anersen, an C. W. Frank, acromolecules, 45 (983).. R.. Pearlstein, J. Chem. Phys. 5, 43 (97). 3. P. D. Fitzgibbon an C. W. Frank, acromolecules 5, 733 (98). 4. G. H. Frerickson an C. W. Frank, acromolecules, 57 (983). 5. B. Ya. Balagurov an V. G. Vaks, Zh. Éksp. Teor. Fiz. 5, 939 (973) [Sov. Phys. JETP 38, 98 (974)].. T. Palszegi, I.. Sokolov, an H. F. Kauffmann, acromolecules 3, 5 (998). 7. I.. Sokolov, J. ai, an A. Blumen, J. Lumin. 7 77, 377 (998). 8. B. ovaghar, G. W. Sauer, an D. Wurtz, J. Stat. Phys. 7, 473 (98). 9. A. I. Onipko, L. I. alysheva, an I. V. Zozulenko, Chem. Phys., 99 (988).. A. Yu. Grosberg an A. R. Khokhlov, Statistical Physics of acromolecules (American Inst. of Physics Press, New York, 994).. C. R. Gochanour, H. D. Anersen, an. D. Fayer, J. Chem. Phys. 7, 454 (979).. D. L. Huber, Phys. Rev. B, 37 (979). 3. D. L. Huber, Phys. Rev. B, 5333 (979). 4.. N. Berberan-Santos an B. Valeur, J. Chem. Phys. 95, 848 (99). 5. I. Rips an J. Jortner, Chem. Phys. 8, 3 (988).. G. Wilemski an. Fixman, J. Chem. Phys., 8 (974). 7.. Doi, Chem. Phys. 9, 455 (975). 8. B. Frman an B. O Schaughnessy, acromolecules, 4888 (993). OPTICS AND SPECTROSCOPY Vol. 9 No. 5

10 LUINESCENCE KINETICS OF LINEAR POLYER OLECULES E. Haas, E. Katchalski-Katzir, an I. Z. Steinberg, Biopolymers 7, (978). 3. G. Liu an J. E. Guillet, acromolecules 3, 99 (99). 3. G. Liu an J. E. Guillet, acromolecules 3, 973 (99). 3. J. R. Lakowicz, J. Kusba, W. Wiczk, et al., Chem. Phys. Lett. 73, 39 (99). 33. J. R. Lakowicz, J. Kusba, I. Kryczynski, et al., J. Phys. Chem. 95, 954 (99). 34. E. N. Bounov,. N. Berberan-Santos, an J.. G. artinho, Chem. Phys. Lett. 97, 49 (998). 35. V. L. Ermolaev, E. N. Bounov, E. B. Sveshnikova, an T. A. Shakhverov, Nonraiative Transfer of Electron Excitation Energy (Nauka, Leningra, 977). 3. I.. Sokolov, J. ai, an A. Blumen, J. Lumin. 7 77, 377 (998) N. Berberan-Santos, E. N. Bounov, an J.. G. artinho, Opt. Spektrosk. 89, 953 () [Opt. Spectrosc. 89, 87 ()]. Translate by V. Zapasskiœ OPTICS AND SPECTROSCOPY Vol. 9 No. 5