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Monitoring energy transfer in hyperbranched macromolecules through fluorescence depolarization
ARTICLE IN PRESS
Journal of Luminescence 111 (2005) 327–334 www.elsevier.com/locate/jlumin
Monitoring energy transfer in hyperbranched macromolecules through fluorescence depolarization A. Blumena,, A. Voltaa, A. Jurjiua, Th. Koslowskib a
Theoretische Polymerphysik, Universita¨t Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany b Institut fu¨r Physikalische Chemie, Universita¨t Freiburg, Albertstr. 23a, D-79104 Freiburg, Germany Available online 4 January 2005
Abstract Fluorescence depolarization under dipolar quasiresonant energy transfer allows to monitor the presence of the excitation on the originally excited chromophore. Moreover, a simple model relates the arrangement of the chromophores to the depolarization decay via the eigenvalues of the corresponding connectivity matrix. Now, for important classes of polymers (such as dendrimers and fractal hyperbranched macromolecules) these eigenvalues can be obtained without the need of diagonalizing the corresponding connectivity matrices, a fact which allows to study very large systems also. We determine the fluorescence depolarization behavior of hyperbranched systems and emphasize the possibilities offered by such measurements in differentiating between the underlying geometries. r 2004 Elsevier B.V. All rights reserved. PACS: 36.20.r; 33.50.j; 05.40.Fb; 61.43.hv Keywords: Dendrimers; Hyperbranched macromolecules; Dynamics; Energy transfer; Random walks; Scaling
1. Introduction The last years have seen, besides a keen interest in dendrimers, a considerable increase in the investigations of general hyperbranched macromolecules [1–10]. In topological terms all these structures are trees, since they do not have loops. Now the Corresponding author. Tel.: +49 761 203 5905; +49 761 203 5906. E-mail address: [email protected] (A. Blumen).
fax:
dendrimers are particular, since they are regular subsets of the infinite Cayley tree [1–10], see Fig. 1. From a central point f branches exit, which in their next generation have at their ends ( f 1) new branches each, where f is the coordination number (number of nearest neighbors). Besides the dendrimers, of much importance are hyperbranched polymers, some of which may be regular fractals [11–15]. A specific example for these is the classical Vicsek fractal, for which f ¼ 4: The general topology on which we focus is displayed in Fig. 2, which shows schematically in 2d the f ¼ 3 structure
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.10.012
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the RHFs are fractals, whereas (due to their exponential growth) the (theoretical) fractal dimension of dendrimers is infinite [16].
2. Dynamical properties and energy transfer
Fig. 1. Classical dendrimer with functionality f ¼ 3 at the generation g ¼ 4: The open circles show the new sites in the first generation of the dendrimer, the origin being the zeroth generation.
[16–18]. One starts from the object of generation g ¼ 1; consisting here of f þ 1 ¼ 4 beads arranged in star-wise pattern, the central chromophore (bead) having three neighbors. To this object one attaches at the next generation, through f bonds, in star-wise fashion, f identical copies of itself. Hence the next stage object ðg ¼ 2Þ consists of 16 beads. One continues then the construction iteratively. Fig. 2 presents the finite f ¼ 3 regular hyperbranched fractal (RHF) for g ¼ 3: Comparing now dendrimers and RHFs, one notes that in dendrimers the number of branches increases exponentially, so that their chemical synthesis must stop rather soon; usually one attains five or six generations [1]. On the other hand, the RHFs have a much more ramified structure; based on the regular pattern of Fig. 2, one sees readily [16,17] that the fractal dimension d r of RHFs depends on f dr ¼
lnð f þ 1Þ : ln 3
(1)
We note that hyperbranched polymers are not limited in their growth and that they are much easier to synthesize than dendrimers [9,10]. Furthermore,
Many dynamical properties of connected structures (such as the vibrational spectra, the relaxation modes, but also random-walks over them) depend on the spectrum of the eigenvalues of their connectivity matrix A ¼ ðAij Þ [6,7,16,17,19–21]. For a system consisting of N beads, A is an N N matrix. The off-diagonal elements Aij are 1 if beads i and j are connected by a bond, and 0, otherwise; furthermore the Aii elements obey Aii ¼ PN 0 j¼1 Aij ; where the prime excludes the case j ¼ i from the sum. Remarkably, the matrix A is also fundamental in describing the dynamics of energy transfer over a system of chromophores. For this we assume that the energy is resonantly exchanged only between nearest neighbors. Given that for dipolar interactions the Fo¨rster transfer rate obeys wðrÞ ¼ C=r6 ; where r is the relative distance between the chromophores involved, in the geometry of Fig. 2 nearest-neighbor transfer is by far dominant. Moreover, as we have discussed earlier [22], transfer among chromophores whose dipolar moments are oriented randomly but fixed in time (frozen) depolarizes very efficiently both in 3d and also in 2d. Thus, measuring the polarization of the spontaneously emitted fluorescence allows to determine the probability that the originally excited donor is (still or again) excited at time t. We stop to note that the problem is related to that of energy transfer and trapping in dendrimers, studied in Refs. [23–25]. As we proceed to discuss, the depolarization problem studied here admits a closed-form expression. Now, the energy transfer among the chromophores (beads) obeys the master equation ! N N X dPi ðtÞ X0 0 ¼ T ij Pj ðtÞ T ji Pi ðtÞ; (2) dt j¼1 j¼1 where Pi ðtÞ is the probability for bead i to be excited at time t and T ij denotes the transfer rate
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Fig. 2. Regular hyperbranched fractal with functionality f ¼ 3 at generation g ¼ 3: The open circles show the new sites in the first generation of the fractal, the origin being the zeroth generation.
from bead j to bead i. We separate, as usual, the radiative decay (equal for all chromophores) from the transfer problem. The radiative decay leads, in fact, only to the multiplication of all the Pi ðtÞ by expðt=tR Þ; where t1 R is the corresponding decay rate. Assuming now all microscopic rates to nearest-neighbors to be equal, say k, Eq. (2) takes the form N
X0 dPi ðtÞ ¼ k Aij Pj ðtÞ ðkAii ÞPi ðtÞ; dt j¼1
(3)
with A being defined as above. This relation can be written in a simple matrix form d PðtÞ ¼ kAPðtÞ (4) dt by introducing PðtÞ ¼ ðP1 ðtÞ; . . . ; PN ðtÞÞT ; where T denotes the transposed vector.
The matrix A is a real symmetric matrix, which was studied in many fields. The transformation K ¼ Q1 AQ;
(5)
where the matrix Q consists of a complete, orthonormal set of eigenvectors of A; brings A to diagonal form; the matrix K has then on its diagonal the eigenvalues li ði ¼ 1; . . . ; NÞ: The li are all real and nonnegative. For the connected structures studied in the following, only one single eigenvalue (we call it l1 ) vanishes, l1 ¼ 0; while li 40 for 2pipN (see, for instance, Ref. [6]). Eq. (4) is now readily solved by using Eq. (5). One has namely, by setting SðtÞ ¼ Q1 PðtÞ d d 1 SðtÞ ¼ Q PðtÞ ¼ kQ1 AQQ1 PðtÞ dt dt ¼ kKSðtÞ;
ð6Þ
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which admits an immediate integration. In matrix form the solution reads PðtÞ ¼ Q expðktKÞQ1 Pð0Þ:
(7)
If the excitation is initially on site i, Pj ð0Þ ¼ dij ; one has at later times, based on Eq. (7) Pn ðtÞ ¼
N X
Qnm expðktlm ÞðQ1 Þmi ;
(8)
m¼1
and the conditional probability of being at the initial site i is Pi ðtÞ ¼
N X
Qim ðQ1 Þmi expðktlm Þ:
(9)
m¼1
A considerable simplification occurs now by averaging over all sites, a procedure fully justified when the dipolar orientations are independent of the bead’s position in the system. Then, under a homogeneous weak exciting field, the observable is hPðtÞi
N 1 X Pi ðtÞ N i¼1
N N X 1X ¼ ðQ1 Þmi Qim N m¼1 i¼1
3. Evaluation of eigenvalues ! expðktlm Þ: ð10Þ
Moreover, given that Q1 Q ¼ 1 holds, it follows that (see also for instance Ref. [26]) hPðtÞi ¼
N 1 X expðklj tÞ; N j¼1
matrix A: Of course, there are a lot of procedures that can perform this task numerically, as long as the number N of chromophores (beads) of the structure is not very large [29,30]. On the other hand, in special cases one can do much better than this, by making use in a judicious way of the symmetries of the systems considered. Based on other work (mostly from the theory of polymers), we know that in special cases one can determine the eigenvalues lj in semi-analytical fashion, either by succeeding in separating the characteristic polynomial of A in a product of polynomials of much lower degree and computing their roots or, in exceptional cases, by obtaining the roots of such polynomials in an iterative fashion. In fact, the first method can be used for dendrimers [3,8,31,32] while the second method was successfully developed for RHFs [16,17,20].
(11)
an expression which depends on all the lj ; but not on the Q: In other physico-chemical contexts, several forms close to Eq. (11) have been established [6,8,19,27,28]. Such forms turned out to be extremely helpful in relating dynamical properties to the underlying geometry. Thus in polymer science forms akin to Eq. (11) describe mechanical, dielectric and magnetic relaxation phenomena [19]. In the following, we will make use of Eq. (11) to determine for dendrimers and RHFs the probability that an excitation, originally found on one site, will be (still or again) at that site at a later time t. As is clear from Eq. (11), all that is technically necessary is to diagonalize the pertinent
We now turn our attention to the determination of the eigenvalues for finite dendrimers and for finite RHFs, eigenvalues which will be used in the following in order to evaluate hPðtÞi according to Eq. (11). For dendrimers we follow the procedure developed by Cai and Chen and used in several works dealing with dendrimers, dendritic wedges, and networks built out from them [3,8,18,32]. The basic idea is that one can divide all the eigenvectors (denoted by k in the following) into two classes: in the first class the component Q1k of the eigenvector k corresponding to the central bead i ¼ 1; is nonvanishing, Q1k a0; in the second class of eigenvectors one has Q1k ¼ 0: In the first class of eigenvectors the eigenvalues are nondegenerate and can be obtained from the roots of the implicit transcendental equation [3,32] pffiffiffiffiffiffiffiffiffiffiffi f 1 sin ðg þ 1Þck ¼ sin gck ; (12) so that lk is given by pffiffiffiffiffiffiffiffiffiffiffi lk ¼ f 2 f 1 cos ck :
(13)
Eq. (12) leads (for f X3) to g distinct eigenvalues. To these, nondegenerate eigenvalues of the first class is added the eigenvalue l1 ¼ 0:
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In the case that Q1k ¼ 0 a similar method applies [32]. One proceeds from the center sequentially, generation after generation, by denoting with n the last generation in which the eigenvector k under scrutiny is such that the components Qjk of all the beads j belonging to generation n vanish, but where at least one component Qik related to a bead i of the next generation (n þ 1Þ does not vanish. For 0pnpg 2 the equation to be solved reads pffiffiffiffiffiffiffiffiffiffiffi sinðg þ 1 nÞck ¼ f 1 sinðg nÞck ; (14) with lk being, as before, given by Eq. (13). Here, however, one may possibly find only ðg n 1Þ roots instead of ðg nÞ: If this is the case, an additional root is obtained from the implicit equation pffiffiffiffiffiffiffiffiffiffiffi sinhðg n þ 1Þc ¼ f 1 sinhðg nÞc; (15) with the eigenvalue L being given by pffiffiffiffiffiffiffiffiffiffiffi L ¼ f 2 f 1 cosh c:
(16)
Here each of these eigenvalues is ð f 2Þ-fold degenerate for n ¼ 0 and f ð f 2Þð f 1Þn1 -fold degenerate otherwise. For n ¼ g 1 one has as eigenvalue lk ¼ 1; which is f ð f 2Þð f 1Þg2 fold degenerate [32]. For finite RHFs the early work has centered on the case f ¼ 4 (Vicsek-fractals); in this case Jayanthi and Wu [12–14] succeeded in determining the eigenvalues of A by computing the zeros of iteratively determined polynomials. In [16,17,20] the problem was solved for general f; there it was shown that the RHF-eigenvalues can be obtained easily, for arbitrary f and g, through an algebraic iterative procedure, which involves the Cardanosolution for cubic equations [33]. The determination of the eigenvalues lk is based on the realization that the RHFs rescale under a realspace renormalization transformation, see for instance Ref. [17] for details. Fundamental in this respect is the polynomial PðlÞ ¼ lðl 3Þðl f 1Þ:
(17)
One finds namely that a part of the eigenvalues of the RHF at generation (g þ 1Þ are connected to
331
the RHF-eigenvalues at generation g through the relation Þ ¼ lðgÞ Pðlðgþ1Þ i i :
(18)
More specifically, each eigenvalue at generation g (apart from l1 ¼ 0Þ gives rise, based on Eqs. (18) and (19), to three new ones at generation ðg þ 1Þ: Note that through this iterative procedure the degeneracy is transferred to the new generation. Clearly, at every generation one also has the nondegenerate eigenvalue l1 ¼ 0: Furthermore, one finds in each generation one additional nondegenerate eigenvalue with value ( f þ 1) and Dg new degenerate eigenvalues of value 1 each, where Dg is given by Dg ¼ ð f 2Þð f þ 1Þg1 þ 1:
(19)
As shown in Ref. [17], in this way the total number of eigenvalues in each RHF generation equals the number of beads, as indeed it should be. Furthermore, Eqs. (17) and (18) can be used to compute the lðgÞ iteratively, based on the roots of the i polynomial [16–18,20] x3 ð f þ 4Þx2 þ 3ð f þ 1Þx a ¼ 0:
(20)
Introducing 1 p ¼ ½ f ð f 1Þ þ 7 ; 3
q¼
1 ð5 f Þð f þ 4Þð2f 1Þ; 27
(21)
(22)
and r ¼ jp=3j3=2
(23)
the roots of this polynominal are given by the Cardano-solution, see Ref. [33] xn ¼ ð f þ 4Þ=3 þ 2r1=3 cosððf þ 2pnÞ=3Þ; with n 2 f1; 2; 3g;
ð24Þ
where f ¼ arccosðða qÞ=2rÞ:
(25)
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As discussed in Refs. [16,17], from Eq. (20) one also obtains the spectral dimension of the RHF 2 lnð f þ 1Þ d~ ¼ : lnð3f þ 3Þ
(26)
Thus, in this way we succeed in obtaining both for dendrimers and for RHFs for arbitrary values of f (in the following we use f ¼ 3 and 4) and g, all the eigenvalues. It is clear that for large g the standard numerical diagonalization methods are of no avail, since increasing g lets N increase exponentially, which quickly exceeds the computer capacity and speed which are available today; in fact the usual limit for diagonalizing full matrices and obtaining all their eigenvalues [30] is around N ¼ 5000: Needless to say, we have verified the correctness of our semi-analytical approach in many instances, by using smaller matrices and comparing our theoretical methods with the results obtained from numerical diagonalization [16,17,20,34]. We always obtained perfect agreement, both with respect to the numerical accuracy of each eigenvalue, as well as to its degeneracy.
4. Fluorescence depolarization In this way we can now turn to the evaluation of Eq. (11). In Fig. 3 we present for different RHFs 0
log10
-2
-4
-6
-8
-10 -2
0
2
4
6 8 log10t
10
12
14
16
Fig. 3. The average probability hPðtÞi; Eq. (11), shown in dimensionless units for regular hyperbranched fractals with f ¼ 3; and g ¼ 4; 7; 10; and 13 from above, hence N ¼ 44 ; 47 ; 410 ; and 413 : The scales are double logarithmic to basis 10.
the average probability hPðtÞi that an initially excited chromophore is excited at time t and display the results in double-logarithmic scales. In Fig. 3 we focus on RHFs with f ¼ 3 and let g vary from g ¼ 4 to g ¼ 13 which corresponds to N ¼ 44 and to N ¼ 413 ; respectively. Plotted are the results for g ¼ 4; 7; 10; and 13. What is immediately clear from Fig. 3 is that in an intermediate time domain (which grows with increasing g) the decay is linear in the chosen, double-logarithmic scales, i.e. that hPðtÞi scales. Formally this means that hPðtÞi is well represented by a power-law hPðtÞi ta : We like to recall that this is the form of the hPðtÞidependence introduced by Alexander and Orbach and used by them to define the fracton (spectral) dimension in their classical work [35]. In Fig. 3 a ranges from a ¼ 0:539 for g ¼ 4 to a ¼ 0:557 for g ¼ 13: This behavior can be very well understood e based on the theoretical expectation [35] a ¼ d=2; e ¼ 0:55788 where de is given by Eq. (26), so that d=2 in our case.In addition to this scaling behavior, in Fig. 3 one can also observe some superimposed waviness, which stems from the underlying hierarchical structure. Furthermore, the decay laws show at long times a plateau; this behavior is due (in the absence of radiative decay) to the equipartition of the energy over all the RHF-beads, so that each bead has a probability of 1=N of harboring the excitation. In Fig. 4 we display the situation which obtains for finite RHFs with f ¼ 4; again by considering g ¼ 4; 7; 10; and 13. The expectation that the slope e ¼ 0:59431 is well-fulfilled; we be now close to d=2 find from the figure a ¼ 0:580 for g ¼ 4 and a ¼ 0:594 for g ¼ 13: Here the waviness superimposed on the quasilinear slope is even more pronounced than in Fig. 3; as in other models involving fractals [17,20,21] the waviness grows with growing f. A different situation emerges when the structure under consideration is a dendrimer. In Fig. 5 we present hPðtÞi for dendrimers with f ¼ 3 and where g varies from g ¼ 4 up to g ¼ 19: These values correspond to dendrimers whose number of beads varies between N ¼ 46 and N ¼ 1572862: Plotted are the curves for the values of g ¼ 4; 7; 10; 13; 16; and 19. In the double logarithmic scales of Fig. 5 differences from the RHFs situation previously discussed are evident: Whereas the
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0
log10
-2
-4
-6
-8
-10
-2
0
2
4
6 8 log10t
10
12
14
16
Fig. 4. The average probability hPðtÞi; Eq. (11), shown in dimensionless units for regular hyperbranched fractals with f ¼ 4; and g ¼ 4; 7; 10; and 13 from above, hence N ¼ 54 ; 57 ; 510 ; and 513 : The scales are double logarithmic to basis 10.
0 -1
333
approximately) straight lines. Indeed, this feature may allow to differentiate readily, in a straightforward fashion, between RHFs and dendrimers. At this point we would like to stress that similar results were observed for the dynamical relaxation of polymeric structures which are dendrimers [6,7] or RHFs [16,17,20]. Although the physical situation was very different (here the nearest neighbor distances are kept constant, whereas for polymers they vary) and the measurements are different (here the depolarization decay is monitored, there the mechanical and dielectric relaxation were considered), in all cases one observes scaling in the decay for the RHFs and no scaling for dendrimers; this assertion pertains, of course, to the intermediate time and frequency domains. The deep reason for these findings is the fact that the dynamics is, in all cases, dominated by the topological symmetries of the structures investigated and that hPðtÞi and the other dynamical observables (such as the mechanical storage and loss moduli and also the dielectric function) are all related to each other through simple mathematical operations, such as integrations and Laplace-transformations [19,22].
log10
-2
5. Summary
-3 -4 -5 -6 -7
-2 -1 0
1
2
3
4 5 log10t
6
7
8
9 10
Fig. 5. The average probability hPðtÞi; Eq. (11), shown in dimensionless units for classical dendrimers with f ¼ 3 and g ¼ 4; 7; 10; 13; 16; and 19 from above, hence N ¼ 46; 382; 3070; 24574; 196606; and 1572862. The scales are double logarithmic to basis 10.
overall behavior at very short and very long times is similar, a large difference occurs in the domain which corresponds to intermediate times. Here no scaling is to be seen; by this we mean that the shape of the curves does not follow (not even very
In summary, in this work we have focused on the polarization of fluorescence by energy transfer and especially on the way in which to study the underlying structures by monitoring observables related to the probability hPðtÞi of a random walker to be (still or again) at its point of departure. As we have exemplified using RHFs and dendrimers, such depolarization measurements allow an easy differentiation between the underlying structures. Of utmost importance is here, as in all problems related to fractals and fractional calculus (see also Ref. [36]), the basic idea of scaling. Indeed, the polarization decay shows for fractals a scaling behavior and allows in the case of dendrimers to see their exponential growth with increasing generation. We are confident that such macroscopic measurements, such as monitoring the depolarization, may reveal (as in the case of mechanical and dielectric relaxation) much about the underlying microscopic structures.
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Acknowledgements The support of the Deutsche Forschungsgemeinschaft is gratefully acknowledged. A.B. acknowledges also the help of the Fonds der Chemischen Industrie.
References [1] J. Roovers, in: M.K. Mishra, S. Kobayashi (Eds.), Star and Hyperbranched Polymers, Marcel Dekker, New York, 1999, p. 285. [2] R. La Ferla, J. Chem. Phys. 106 (1997) 688. [3] C. Cai, Z.Y. Chen, Macromolecules 30 (1997) 5104. [4] Z.Y. Chen, C. Cai, Macromolecules 32 (1999) 5423. [5] F. Ganazzoli, R. La Ferla, G. Raffaini, Macromolecules 34 (2001) 4222. [6] P. Biswas, R. Kant, A. Blumen, Macromol. Theory Simul. 9 (2000) 56. [7] P. Biswas, R. Kant, A. Blumen, J. Chem. Phys. 114 (2001) 2430. [8] A.A. Gurtovenko, Yu.Ya. Gotlib, A. Blumen, Macromolecules 35 (2002) 7481. [9] A. Sunder, R. Hanselmann, H. Frey, R. Mu¨lhaupt, Macromolecules 32 (1999) 4240. [10] A. Sunder, J. Heinemann, H. Frey, Chem. Eur. J. 6 (2000) 2499. [11] C.S. Jayanthi, S.Y. Wu, J. Cocks, Phys. Rev. Lett. 69 (1992) 1955. [12] C.S. Jayanthi, S.Y. Wu, Phys. Rev. B 48 (1993) 10188. [13] C.S. Jayanthi, S.Y. Wu, Phys. Rev. B 48 (1993) 10199. [14] C.S. Jayanthi, S.Y. Wu, Phys. Rev. B 50 (1994) 897. [15] W.A. Schwalm, M.K. Schwalm, M. Giona, Phys. Rev. E 55 (1997) 6741. [16] A. Blumen, A. Jurjiu, Th. Koslowski, Ch. von Ferber, Phys. Rev. E 67 (2003) 061103.
[17] A. Blumen, Ch. von Ferber, A. Jurjiu, Th. Koslowski, Macromolecules 37 (2004) 638. [18] Th. Koslowski, A. Jurjiu, A. Blumen, J. Phys. Chem. B 108 (2004) 3283. [19] A.A. Gurtovenko, A. Blumen, Macromolecules 35 (2002) 3288. [20] A. Jurjiu, Th. Koslowski, Ch. von Ferber, A. Blumen, Chem. Phys. 294 (2003) 187. [21] A. Jurjiu, Ch. Friedrich, A. Blumen, Chem. Phys. 284 (2002) 221. [22] H. Kellerer, A. Blumen, Biophys. J. 46 (1984) 1. [23] R. Kopelman, M. Shortreed, Z.-Y. Shi, W. Tan, Z. Xu, J.S. Moore, A. Bar-Haim, J. Klafter, Phys. Rev. Lett. 78 (1997) 1239. [24] A. Bar-Haim, J. Klafter, R. Kopelman, J. Am. Chem. Soc. 119 (1997) 6197. [25] A. Bar-Haim, J. Klafter, J. Phys. Chem. B 102 (1998) 1662. [26] S. Jespersen, A. Blumen, Phys. Rev. E 62 (2000) 6270. [27] H. Schiessel, Phys. Rev. E 57 (1998) R5775. [28] H. Schiessel, Ch. Friedrich, A. Blumen, in: R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000, p. 331. [29] C. Lanczos, J. Res. NBS B 45 (1950) 225. [30] J.K. Cullum, R. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Problems, vol. I: Theory; vol. II: Programs, Birkha¨user, Boston, Basel, Stuttgart, 1985. [31] Yu.Ya. Gotlib, D.A. Markelov, Polym. Sci. 44 (2002) 1341. [32] A.A. Gurtovenko, D.A. Markelov, Yu.Ya. Gotlib, A. Blumen, J. Chem. Phys. 119 (2003) 7579. [33] I.N. Bronstein, K.A. Semendjajev, Taschenbuch der Mathematik (Handbook of Mathematics), Nauka and Teubner, Moscow and Leipzig, 1985 (Chapter 2.4.2). [34] C. Satmarel, A.A. Gurtovenko, A. Blumen, Macromol. Theory Simul. 13 (2004) 487. [35] S. Alexander, R. Orbach, J. Physique Lett. (Paris) 43 (1982) L625. [36] I.M. Sokolov, J. Klafter, A. Blumen, Physics Today 55 (11) (2002) 48.
Journal of Luminescence 111 (2005) 327–334 www.elsevier.com/locate/jlumin
Monitoring energy transfer in hyperbranched macromolecules through fluorescence depolarization A. Blumena,, A. Voltaa, A. Jurjiua, Th. Koslowskib a
Theoretische Polymerphysik, Universita¨t Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany b Institut fu¨r Physikalische Chemie, Universita¨t Freiburg, Albertstr. 23a, D-79104 Freiburg, Germany Available online 4 January 2005
Abstract Fluorescence depolarization under dipolar quasiresonant energy transfer allows to monitor the presence of the excitation on the originally excited chromophore. Moreover, a simple model relates the arrangement of the chromophores to the depolarization decay via the eigenvalues of the corresponding connectivity matrix. Now, for important classes of polymers (such as dendrimers and fractal hyperbranched macromolecules) these eigenvalues can be obtained without the need of diagonalizing the corresponding connectivity matrices, a fact which allows to study very large systems also. We determine the fluorescence depolarization behavior of hyperbranched systems and emphasize the possibilities offered by such measurements in differentiating between the underlying geometries. r 2004 Elsevier B.V. All rights reserved. PACS: 36.20.r; 33.50.j; 05.40.Fb; 61.43.hv Keywords: Dendrimers; Hyperbranched macromolecules; Dynamics; Energy transfer; Random walks; Scaling
1. Introduction The last years have seen, besides a keen interest in dendrimers, a considerable increase in the investigations of general hyperbranched macromolecules [1–10]. In topological terms all these structures are trees, since they do not have loops. Now the Corresponding author. Tel.: +49 761 203 5905; +49 761 203 5906. E-mail address: [email protected] (A. Blumen).
fax:
dendrimers are particular, since they are regular subsets of the infinite Cayley tree [1–10], see Fig. 1. From a central point f branches exit, which in their next generation have at their ends ( f 1) new branches each, where f is the coordination number (number of nearest neighbors). Besides the dendrimers, of much importance are hyperbranched polymers, some of which may be regular fractals [11–15]. A specific example for these is the classical Vicsek fractal, for which f ¼ 4: The general topology on which we focus is displayed in Fig. 2, which shows schematically in 2d the f ¼ 3 structure
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.10.012
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328
the RHFs are fractals, whereas (due to their exponential growth) the (theoretical) fractal dimension of dendrimers is infinite [16].
2. Dynamical properties and energy transfer
Fig. 1. Classical dendrimer with functionality f ¼ 3 at the generation g ¼ 4: The open circles show the new sites in the first generation of the dendrimer, the origin being the zeroth generation.
[16–18]. One starts from the object of generation g ¼ 1; consisting here of f þ 1 ¼ 4 beads arranged in star-wise pattern, the central chromophore (bead) having three neighbors. To this object one attaches at the next generation, through f bonds, in star-wise fashion, f identical copies of itself. Hence the next stage object ðg ¼ 2Þ consists of 16 beads. One continues then the construction iteratively. Fig. 2 presents the finite f ¼ 3 regular hyperbranched fractal (RHF) for g ¼ 3: Comparing now dendrimers and RHFs, one notes that in dendrimers the number of branches increases exponentially, so that their chemical synthesis must stop rather soon; usually one attains five or six generations [1]. On the other hand, the RHFs have a much more ramified structure; based on the regular pattern of Fig. 2, one sees readily [16,17] that the fractal dimension d r of RHFs depends on f dr ¼
lnð f þ 1Þ : ln 3
(1)
We note that hyperbranched polymers are not limited in their growth and that they are much easier to synthesize than dendrimers [9,10]. Furthermore,
Many dynamical properties of connected structures (such as the vibrational spectra, the relaxation modes, but also random-walks over them) depend on the spectrum of the eigenvalues of their connectivity matrix A ¼ ðAij Þ [6,7,16,17,19–21]. For a system consisting of N beads, A is an N N matrix. The off-diagonal elements Aij are 1 if beads i and j are connected by a bond, and 0, otherwise; furthermore the Aii elements obey Aii ¼ PN 0 j¼1 Aij ; where the prime excludes the case j ¼ i from the sum. Remarkably, the matrix A is also fundamental in describing the dynamics of energy transfer over a system of chromophores. For this we assume that the energy is resonantly exchanged only between nearest neighbors. Given that for dipolar interactions the Fo¨rster transfer rate obeys wðrÞ ¼ C=r6 ; where r is the relative distance between the chromophores involved, in the geometry of Fig. 2 nearest-neighbor transfer is by far dominant. Moreover, as we have discussed earlier [22], transfer among chromophores whose dipolar moments are oriented randomly but fixed in time (frozen) depolarizes very efficiently both in 3d and also in 2d. Thus, measuring the polarization of the spontaneously emitted fluorescence allows to determine the probability that the originally excited donor is (still or again) excited at time t. We stop to note that the problem is related to that of energy transfer and trapping in dendrimers, studied in Refs. [23–25]. As we proceed to discuss, the depolarization problem studied here admits a closed-form expression. Now, the energy transfer among the chromophores (beads) obeys the master equation ! N N X dPi ðtÞ X0 0 ¼ T ij Pj ðtÞ T ji Pi ðtÞ; (2) dt j¼1 j¼1 where Pi ðtÞ is the probability for bead i to be excited at time t and T ij denotes the transfer rate
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Fig. 2. Regular hyperbranched fractal with functionality f ¼ 3 at generation g ¼ 3: The open circles show the new sites in the first generation of the fractal, the origin being the zeroth generation.
from bead j to bead i. We separate, as usual, the radiative decay (equal for all chromophores) from the transfer problem. The radiative decay leads, in fact, only to the multiplication of all the Pi ðtÞ by expðt=tR Þ; where t1 R is the corresponding decay rate. Assuming now all microscopic rates to nearest-neighbors to be equal, say k, Eq. (2) takes the form N
X0 dPi ðtÞ ¼ k Aij Pj ðtÞ ðkAii ÞPi ðtÞ; dt j¼1
(3)
with A being defined as above. This relation can be written in a simple matrix form d PðtÞ ¼ kAPðtÞ (4) dt by introducing PðtÞ ¼ ðP1 ðtÞ; . . . ; PN ðtÞÞT ; where T denotes the transposed vector.
The matrix A is a real symmetric matrix, which was studied in many fields. The transformation K ¼ Q1 AQ;
(5)
where the matrix Q consists of a complete, orthonormal set of eigenvectors of A; brings A to diagonal form; the matrix K has then on its diagonal the eigenvalues li ði ¼ 1; . . . ; NÞ: The li are all real and nonnegative. For the connected structures studied in the following, only one single eigenvalue (we call it l1 ) vanishes, l1 ¼ 0; while li 40 for 2pipN (see, for instance, Ref. [6]). Eq. (4) is now readily solved by using Eq. (5). One has namely, by setting SðtÞ ¼ Q1 PðtÞ d d 1 SðtÞ ¼ Q PðtÞ ¼ kQ1 AQQ1 PðtÞ dt dt ¼ kKSðtÞ;
ð6Þ
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which admits an immediate integration. In matrix form the solution reads PðtÞ ¼ Q expðktKÞQ1 Pð0Þ:
(7)
If the excitation is initially on site i, Pj ð0Þ ¼ dij ; one has at later times, based on Eq. (7) Pn ðtÞ ¼
N X
Qnm expðktlm ÞðQ1 Þmi ;
(8)
m¼1
and the conditional probability of being at the initial site i is Pi ðtÞ ¼
N X
Qim ðQ1 Þmi expðktlm Þ:
(9)
m¼1
A considerable simplification occurs now by averaging over all sites, a procedure fully justified when the dipolar orientations are independent of the bead’s position in the system. Then, under a homogeneous weak exciting field, the observable is hPðtÞi
N 1 X Pi ðtÞ N i¼1
N N X 1X ¼ ðQ1 Þmi Qim N m¼1 i¼1
3. Evaluation of eigenvalues ! expðktlm Þ: ð10Þ
Moreover, given that Q1 Q ¼ 1 holds, it follows that (see also for instance Ref. [26]) hPðtÞi ¼
N 1 X expðklj tÞ; N j¼1
matrix A: Of course, there are a lot of procedures that can perform this task numerically, as long as the number N of chromophores (beads) of the structure is not very large [29,30]. On the other hand, in special cases one can do much better than this, by making use in a judicious way of the symmetries of the systems considered. Based on other work (mostly from the theory of polymers), we know that in special cases one can determine the eigenvalues lj in semi-analytical fashion, either by succeeding in separating the characteristic polynomial of A in a product of polynomials of much lower degree and computing their roots or, in exceptional cases, by obtaining the roots of such polynomials in an iterative fashion. In fact, the first method can be used for dendrimers [3,8,31,32] while the second method was successfully developed for RHFs [16,17,20].
(11)
an expression which depends on all the lj ; but not on the Q: In other physico-chemical contexts, several forms close to Eq. (11) have been established [6,8,19,27,28]. Such forms turned out to be extremely helpful in relating dynamical properties to the underlying geometry. Thus in polymer science forms akin to Eq. (11) describe mechanical, dielectric and magnetic relaxation phenomena [19]. In the following, we will make use of Eq. (11) to determine for dendrimers and RHFs the probability that an excitation, originally found on one site, will be (still or again) at that site at a later time t. As is clear from Eq. (11), all that is technically necessary is to diagonalize the pertinent
We now turn our attention to the determination of the eigenvalues for finite dendrimers and for finite RHFs, eigenvalues which will be used in the following in order to evaluate hPðtÞi according to Eq. (11). For dendrimers we follow the procedure developed by Cai and Chen and used in several works dealing with dendrimers, dendritic wedges, and networks built out from them [3,8,18,32]. The basic idea is that one can divide all the eigenvectors (denoted by k in the following) into two classes: in the first class the component Q1k of the eigenvector k corresponding to the central bead i ¼ 1; is nonvanishing, Q1k a0; in the second class of eigenvectors one has Q1k ¼ 0: In the first class of eigenvectors the eigenvalues are nondegenerate and can be obtained from the roots of the implicit transcendental equation [3,32] pffiffiffiffiffiffiffiffiffiffiffi f 1 sin ðg þ 1Þck ¼ sin gck ; (12) so that lk is given by pffiffiffiffiffiffiffiffiffiffiffi lk ¼ f 2 f 1 cos ck :
(13)
Eq. (12) leads (for f X3) to g distinct eigenvalues. To these, nondegenerate eigenvalues of the first class is added the eigenvalue l1 ¼ 0:
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In the case that Q1k ¼ 0 a similar method applies [32]. One proceeds from the center sequentially, generation after generation, by denoting with n the last generation in which the eigenvector k under scrutiny is such that the components Qjk of all the beads j belonging to generation n vanish, but where at least one component Qik related to a bead i of the next generation (n þ 1Þ does not vanish. For 0pnpg 2 the equation to be solved reads pffiffiffiffiffiffiffiffiffiffiffi sinðg þ 1 nÞck ¼ f 1 sinðg nÞck ; (14) with lk being, as before, given by Eq. (13). Here, however, one may possibly find only ðg n 1Þ roots instead of ðg nÞ: If this is the case, an additional root is obtained from the implicit equation pffiffiffiffiffiffiffiffiffiffiffi sinhðg n þ 1Þc ¼ f 1 sinhðg nÞc; (15) with the eigenvalue L being given by pffiffiffiffiffiffiffiffiffiffiffi L ¼ f 2 f 1 cosh c:
(16)
Here each of these eigenvalues is ð f 2Þ-fold degenerate for n ¼ 0 and f ð f 2Þð f 1Þn1 -fold degenerate otherwise. For n ¼ g 1 one has as eigenvalue lk ¼ 1; which is f ð f 2Þð f 1Þg2 fold degenerate [32]. For finite RHFs the early work has centered on the case f ¼ 4 (Vicsek-fractals); in this case Jayanthi and Wu [12–14] succeeded in determining the eigenvalues of A by computing the zeros of iteratively determined polynomials. In [16,17,20] the problem was solved for general f; there it was shown that the RHF-eigenvalues can be obtained easily, for arbitrary f and g, through an algebraic iterative procedure, which involves the Cardanosolution for cubic equations [33]. The determination of the eigenvalues lk is based on the realization that the RHFs rescale under a realspace renormalization transformation, see for instance Ref. [17] for details. Fundamental in this respect is the polynomial PðlÞ ¼ lðl 3Þðl f 1Þ:
(17)
One finds namely that a part of the eigenvalues of the RHF at generation (g þ 1Þ are connected to
331
the RHF-eigenvalues at generation g through the relation Þ ¼ lðgÞ Pðlðgþ1Þ i i :
(18)
More specifically, each eigenvalue at generation g (apart from l1 ¼ 0Þ gives rise, based on Eqs. (18) and (19), to three new ones at generation ðg þ 1Þ: Note that through this iterative procedure the degeneracy is transferred to the new generation. Clearly, at every generation one also has the nondegenerate eigenvalue l1 ¼ 0: Furthermore, one finds in each generation one additional nondegenerate eigenvalue with value ( f þ 1) and Dg new degenerate eigenvalues of value 1 each, where Dg is given by Dg ¼ ð f 2Þð f þ 1Þg1 þ 1:
(19)
As shown in Ref. [17], in this way the total number of eigenvalues in each RHF generation equals the number of beads, as indeed it should be. Furthermore, Eqs. (17) and (18) can be used to compute the lðgÞ iteratively, based on the roots of the i polynomial [16–18,20] x3 ð f þ 4Þx2 þ 3ð f þ 1Þx a ¼ 0:
(20)
Introducing 1 p ¼ ½ f ð f 1Þ þ 7 ; 3
q¼
1 ð5 f Þð f þ 4Þð2f 1Þ; 27
(21)
(22)
and r ¼ jp=3j3=2
(23)
the roots of this polynominal are given by the Cardano-solution, see Ref. [33] xn ¼ ð f þ 4Þ=3 þ 2r1=3 cosððf þ 2pnÞ=3Þ; with n 2 f1; 2; 3g;
ð24Þ
where f ¼ arccosðða qÞ=2rÞ:
(25)
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As discussed in Refs. [16,17], from Eq. (20) one also obtains the spectral dimension of the RHF 2 lnð f þ 1Þ d~ ¼ : lnð3f þ 3Þ
(26)
Thus, in this way we succeed in obtaining both for dendrimers and for RHFs for arbitrary values of f (in the following we use f ¼ 3 and 4) and g, all the eigenvalues. It is clear that for large g the standard numerical diagonalization methods are of no avail, since increasing g lets N increase exponentially, which quickly exceeds the computer capacity and speed which are available today; in fact the usual limit for diagonalizing full matrices and obtaining all their eigenvalues [30] is around N ¼ 5000: Needless to say, we have verified the correctness of our semi-analytical approach in many instances, by using smaller matrices and comparing our theoretical methods with the results obtained from numerical diagonalization [16,17,20,34]. We always obtained perfect agreement, both with respect to the numerical accuracy of each eigenvalue, as well as to its degeneracy.
4. Fluorescence depolarization In this way we can now turn to the evaluation of Eq. (11). In Fig. 3 we present for different RHFs 0
log10
-2
-4
-6
-8
-10 -2
0
2
4
6 8 log10t
10
12
14
16
Fig. 3. The average probability hPðtÞi; Eq. (11), shown in dimensionless units for regular hyperbranched fractals with f ¼ 3; and g ¼ 4; 7; 10; and 13 from above, hence N ¼ 44 ; 47 ; 410 ; and 413 : The scales are double logarithmic to basis 10.
the average probability hPðtÞi that an initially excited chromophore is excited at time t and display the results in double-logarithmic scales. In Fig. 3 we focus on RHFs with f ¼ 3 and let g vary from g ¼ 4 to g ¼ 13 which corresponds to N ¼ 44 and to N ¼ 413 ; respectively. Plotted are the results for g ¼ 4; 7; 10; and 13. What is immediately clear from Fig. 3 is that in an intermediate time domain (which grows with increasing g) the decay is linear in the chosen, double-logarithmic scales, i.e. that hPðtÞi scales. Formally this means that hPðtÞi is well represented by a power-law hPðtÞi ta : We like to recall that this is the form of the hPðtÞidependence introduced by Alexander and Orbach and used by them to define the fracton (spectral) dimension in their classical work [35]. In Fig. 3 a ranges from a ¼ 0:539 for g ¼ 4 to a ¼ 0:557 for g ¼ 13: This behavior can be very well understood e based on the theoretical expectation [35] a ¼ d=2; e ¼ 0:55788 where de is given by Eq. (26), so that d=2 in our case.In addition to this scaling behavior, in Fig. 3 one can also observe some superimposed waviness, which stems from the underlying hierarchical structure. Furthermore, the decay laws show at long times a plateau; this behavior is due (in the absence of radiative decay) to the equipartition of the energy over all the RHF-beads, so that each bead has a probability of 1=N of harboring the excitation. In Fig. 4 we display the situation which obtains for finite RHFs with f ¼ 4; again by considering g ¼ 4; 7; 10; and 13. The expectation that the slope e ¼ 0:59431 is well-fulfilled; we be now close to d=2 find from the figure a ¼ 0:580 for g ¼ 4 and a ¼ 0:594 for g ¼ 13: Here the waviness superimposed on the quasilinear slope is even more pronounced than in Fig. 3; as in other models involving fractals [17,20,21] the waviness grows with growing f. A different situation emerges when the structure under consideration is a dendrimer. In Fig. 5 we present hPðtÞi for dendrimers with f ¼ 3 and where g varies from g ¼ 4 up to g ¼ 19: These values correspond to dendrimers whose number of beads varies between N ¼ 46 and N ¼ 1572862: Plotted are the curves for the values of g ¼ 4; 7; 10; 13; 16; and 19. In the double logarithmic scales of Fig. 5 differences from the RHFs situation previously discussed are evident: Whereas the
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0
log10
-2
-4
-6
-8
-10
-2
0
2
4
6 8 log10t
10
12
14
16
Fig. 4. The average probability hPðtÞi; Eq. (11), shown in dimensionless units for regular hyperbranched fractals with f ¼ 4; and g ¼ 4; 7; 10; and 13 from above, hence N ¼ 54 ; 57 ; 510 ; and 513 : The scales are double logarithmic to basis 10.
0 -1
333
approximately) straight lines. Indeed, this feature may allow to differentiate readily, in a straightforward fashion, between RHFs and dendrimers. At this point we would like to stress that similar results were observed for the dynamical relaxation of polymeric structures which are dendrimers [6,7] or RHFs [16,17,20]. Although the physical situation was very different (here the nearest neighbor distances are kept constant, whereas for polymers they vary) and the measurements are different (here the depolarization decay is monitored, there the mechanical and dielectric relaxation were considered), in all cases one observes scaling in the decay for the RHFs and no scaling for dendrimers; this assertion pertains, of course, to the intermediate time and frequency domains. The deep reason for these findings is the fact that the dynamics is, in all cases, dominated by the topological symmetries of the structures investigated and that hPðtÞi and the other dynamical observables (such as the mechanical storage and loss moduli and also the dielectric function) are all related to each other through simple mathematical operations, such as integrations and Laplace-transformations [19,22].
log10
-2
5. Summary
-3 -4 -5 -6 -7
-2 -1 0
1
2
3
4 5 log10t
6
7
8
9 10
Fig. 5. The average probability hPðtÞi; Eq. (11), shown in dimensionless units for classical dendrimers with f ¼ 3 and g ¼ 4; 7; 10; 13; 16; and 19 from above, hence N ¼ 46; 382; 3070; 24574; 196606; and 1572862. The scales are double logarithmic to basis 10.
overall behavior at very short and very long times is similar, a large difference occurs in the domain which corresponds to intermediate times. Here no scaling is to be seen; by this we mean that the shape of the curves does not follow (not even very
In summary, in this work we have focused on the polarization of fluorescence by energy transfer and especially on the way in which to study the underlying structures by monitoring observables related to the probability hPðtÞi of a random walker to be (still or again) at its point of departure. As we have exemplified using RHFs and dendrimers, such depolarization measurements allow an easy differentiation between the underlying structures. Of utmost importance is here, as in all problems related to fractals and fractional calculus (see also Ref. [36]), the basic idea of scaling. Indeed, the polarization decay shows for fractals a scaling behavior and allows in the case of dendrimers to see their exponential growth with increasing generation. We are confident that such macroscopic measurements, such as monitoring the depolarization, may reveal (as in the case of mechanical and dielectric relaxation) much about the underlying microscopic structures.
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Acknowledgements The support of the Deutsche Forschungsgemeinschaft is gratefully acknowledged. A.B. acknowledges also the help of the Fonds der Chemischen Industrie.
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