Exciton Percolation in Molecular Crystals and Aggregates large. What do you mean by large?
F.ADAR.Yes. G . W. CANTERS.Did you calculate the magnitude of this splitting or was it found from the experiment?
F. ~
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A RYou . how that the splitting is possible because the crystal
field has lower symmetry than that of free space. You know that the 5d orbitals split into two levels with two- and threefold degeneracies. Then you can calculate which set is low in energy.
G. W. CANTERS.But you mentioned that the splitting would be
F. ADAR.Large enough to observe a diamagnetic species. E. R. BERNSTEIN. Do you have any other information besidedine width on the lifetimes of the states you observe?
F. ADAR.We ran the absorption spectra at liquid nitrogen temperature from which we estimated the line widths. We found that the emission was consistent with quantum yields that’ the line widths would predict.
Exciton Percolation in Mixed Molecular Crystals and Aggregates: From Naphthalene to Photosynthesis Raoul Kopelman Department of Chemistry, The University of Michigan, Ann Arbor, Michigan 48 109 (Received February 2, 1976)
Dynamic exciton percolation in a ternary solid is formulated in terms of the static percolation theory of a binary solid for the simple, but common, limit where it is independent of the excitation lifetime, jump time, coherence time, and trapping time. Experiments on the SI exciton in the naphthalene-dshaphthalene-hs/ @-methylnaphthalenesystem are interpreted within this limit. This limit is also appropriate for the description of the exciton transfer in the photosystems of higher plants. It predicts correctly the range of chlorophyll a:chlorophyll b concentration ratios. The energy transport in the ternary naphthalene system appears to mimic the natural process. The general case, where the coherence time is an important factor, can also be utilized to experimentally determine coherence times and lengths as a function of temperature. Preliminary results on naphthalene are reported (for 2 K).
Introduction The traditional approach to the long-range energy transport problem has been molded by the examples of the resonance energy transport (exciton motion) in ordered molecular solids (nearly perfect crystals, usually a t low temperature). The traditional ideas are summarized as follows. Perfect resonance transfer can only occur among molecules that are not only chemically identical but also in physically indistinguishable sites. These conditions are guaranteed by the translationall and interchange2 symmetry of the crystal, i.e., its perfect order. While energy can be transferred between two different molecules (or chromophore^)^,^ in a nonresonance process, this usually leads to fast degradation of the energy, and generally this process is also irreversible in its direction. Thus, for a multistep process, the majority of the energytransfer steps have to occur inside highly ordered domains, arranged like perfect crystals. Any significant portion of defects or impurities, i.e., disorder, will either trap or block the energy transport. This traditional picture given above hampered the investigation of energy (exciton) transport in biological aggregates in two ways. First, it lowered the expectations concerning the occurrence of exciton transfer because of the low probability for “perfect” crystallinity in biomolecular aggregates. Second, even in the case where exciton energy transfer is fairly well established,4g5it led to some confusion as well as to the belief
that the appropriate molecular aggregates have to be organized with a very high degree of order, leaving little room for “sloppiness” or damage and thus highly constraining one’s models of reality. A very similar situation existed a few years ago in solid-state physics concerning the electrical conduction of semiconductors. A crystal of germanium or silicon had to be extremely pure and perfect before it could serve as a “solid-state device”. Also, a little radiation damage would affect it badly. Theo-, retically it was hard to believe in the mere existence of amorphous (glassy, disordered) semiconductor devices. The fact is that they do exist6 and that the new theoretical concepts advanced to explain these facts turned out to be useful in additional situations, such as dense gaseous metals and metal-ammonia s o l ~ t i o n s . ~The ~ s key concept is charge “percolation”.6 Our studies of excitations in mixed molecular crystals9 have led us to the concept of exciton percolation. 4-12 This approach allows a significant amount of lattice disorder to exist in a molecular condensate without the elimination of long-range exciton transport. It also defines new kinds of constraints for the persistence of such energy transport. While it is still true that complete order is required for strictly one-dimensional systems, this is not so for other topologies, not even for a linear but (‘stranded” system. One can define a ((phasetransition” in a molecular condensate, where the change is from an “exciton conductor” to an (‘exciton insulator”, this change ocThe Journal of Physical Chemistry, Vol. 80, No. 20, 1976
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P
C
rav
Figure 1. Reduced average cluster size (see eq 2) vs. the A component concentration C,for square, triangular, and square (1.2) lattices. The square (1,2) topology is a square lattice with both nearest and next nearest (“diagonal”) interactionsewhile the other two topologies include Only nearest-neighbor interactions. The locations of the sharp maxima give the critical (percolation)concentrations for the A component while the peak height has no meaning (due to statistical fluctuations). Each computer lattice had 200 X 200 sites.
curring as a function of an order parameter (Le., concentration). We have been able to put together a simple three-component mixed molecular crystal system (substitutionally disordered) which is amenable to experimental investigations under the most controlled conditions. This system mimics the essential features of the primary process of photosynthesis. Moreover, the “mimicking” fails completely13 for the very values of the disorder parameter that do not appear in natural systems (aggregates with such parameters have presumably been eliminated in the evolutionary process). We also give here a theoretical synopsis of exciton percolation and a model of “percolation through a semirandom lattice” for the energy transport in the primary process of photosynthesis. Synopsis of Theory Exciton percolation--the migration of excitons in disordered materials-is based on the availability of a disordered exciton conducting quasi-lattice (A). Simultaneously there exists a (disordered) exciton insulating quasi-lattice (B). The condition of exciton percolation, i.e., efficient migration of the excitation, requires an effectively connected A quasi-lattice. The simplest case of exciton percolation is found in a binary mixed crystal with random substitutional disorder and where the B quasi-lattice is energetically inaccessible to the A excitons. Dynamic percolation, Le., the migration of excitons throughout a quasi-lattice, is described, formally, in terms of the static percolation of this quasi-lattice. The static percolation, in turn, is a well-defined mathematical concept. In the simple case of a binary, substitutionally random lattice the essence of the site percolation problem is the question: What is the probability for any site A to belong to the “infinite A cluster” (maxicluster)? This probability ( p )is zero when the fraction of A is zero and unity when the fraction of A is unity. However, the significant fact is that this probability 8)is zero over a wide range of concentration. For instance, in a square lattice it is zero for concentrationsg up to 0.59. The critical concentration (C,) is here 0.59 (0.50 for a triangular lattice, 0.31 for a simple cubic lattice etc.). Above C, the probability P rises sharply and approaches unity at concentrations quite lower than unity. The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
0
0.2
0.6
0.4
0.0
CG
Flgure 2. Experimental and theoretical dynamical exciton percolation: first singlet excitation of naphthalene. The experimental points are integrated fluorescence intensity ratios /s/ltotal where ls is the sensor (&methylnaphthalene) emission intensity and ltotal = Is In the combined sensor and exciton conductor (naphthalenehe) emission intensity (see eq 5 and 6). Here Cs % (mole fraction) so that C % CA.The low concentration data (C 0.1) actually include contributions from a “dark current” (see text) and are therefore excluded from a comparison to theory. The theoretical curves give the dynamical percolation Paccording to eq 3 (using the more precise eq 14 for the C, regions) for the three topologies described in the Figure 1 caption.
+
kT, T = 300 K
Amalgamated >hT, T = 2 K >kT,T=2K
Experimentally, we used the ternary system perdeuterionaphthalene (B)-naphthalene (A)-fl-methylnaphthalene (S).The experimental details have been given.12J4The results compare very well with the theoretically derived percolation probability P (see Figure a), provided that the exciton interaction topology is roughly between triangular and square. While the nearest neighbor interactions9J5 create a squarelattice topology, the next nearest neighbors (in the ab plane, at least) give a triangular-like t ~ p ~ l ~ gWe y .notice ~ ~ Jthat ~ our actual experimental setup introduced some complications, such as direct excitation into the sensor (bypassing the A lattice). At low A concentration (C < O.l), that would create14 an effective “dark current” (i.e., emission from the sensor). Thus we ignore the very low C region points (Figure 2). Also, the intrinsic intensity and trapping efficiency ratios (A:S) appear to be small and opposing and thus roughly cancel.12 We have seen therefore both an experimental and a theoretical verification of the concept of exciton percolation, or at least its existence above the critical concentration (CJ. Next, we look at some of its uses and applications. The exciton percolation experiments give us a way of determining the degree of the exciton motional coherence in the pure crystal. This is the first method we are aware of for measuring the coherence properties of any singlet excitons (we discuss triplet excitons elsewhere).11J4 Our method is to compare experiments a t low C s with theoretical predictions based on eq 4, with the coherence time as a parameter. Agreement is achieved14J6for a value of about 10 ps, at 1.8K. This agrees well with the highest available lower limit of about 5 ps, based on spectral line width measurements.16J7 An important general conclusion, based on our computations,16 is the unexpected effect of exciton coherence on exciton transport in disordered aggregates. For perfect lattices it is well-known that the longer the coherence time, the more efficient the energy transport. It turns out that quite the opposite is true for exciton percolation over a wide concentration domain above the percolation point.16 Thus for an increase in temperature, which increases exciton phonon coupling, the concomitant decrease in coherence time t , actually improves the exciton transport! This may have interesting biological implications (see below). Simulation of Photosynthesis? There is an amazing similarity between our “synthetic” experimental ternary system discussed above and the photosynthetic systems of the higher plants. This similarity is shown, in part, in Table I. We list for both chlorophyll a5 and naphthalene9J2 a number of important parameters: the nearest-neighbor exciton pairwise interaction ( M ) ;the exciton jump time, Tj = (4cM)-l, where c is the speed of light; the exciton lifetime r; the important ratio r/T?,equivalent to the average number of exciton steps; the exciton trapping effiThe Journal of Physical Chemistry, Vol. SO,No. 20, 1976
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ciency y (at the sensor S, Le., P700 or @-methylnaphthalene); the nature of the sensor-based phonons;13J4 and the energy denominators between the exciton “conductor” A and the exciton “insulator” B (chlorophyll b or CloDa), as well as between the exciton conductor A and the exciton sensor S. In the primary process of photosynthesis, the transfer of excitation from the “antenna” pigment to the reactive centers is of major importance, especially in the green plants. In our quasi-random model, most of the light-absorbing pigments are distributed in a substitutionally nearly random fashion. The light absorbed by the higher energy pigments (B) quickly cascades down to the nearby lowest energy pigment (A), and then this A excitation migrates in the A quasi-lattice, until it reaches a reactive center (S).The same migration occurs for light absorbed directly by A. The topology of the chromophore aggregates in the photosystems is not known but is probably made of a few layers, with a trigonal, square, or triangular topology of nearestneighbor interaction^.^ This puts the critical concentration (roughly, compare Figures 1and 2 and Figures 6 and 7 of ref 14) a t 0.35 < P,
< 0.6
(7) or the ratio of exciton “insulator” (B).to “conductor” (A) chromophore concentrations is [B]/[A] < 1 f 0.5 This prediction of the quasi-random model is certainly amenable to experimental test. We note that the traditional funnel model (i.e., adjacent layers of B and A), involves no such restrictions on the ratio [B]/[A]. We also notice that the present experimental information5J8 gives the region of ratios 0.1 5 [B]/[A] 5 1.0
(9)
Actually there may exist much lower ratios than 0.1, but there is a difficulty in measuring such low ratios.18 We note that the above considerations apply mainly to the “higher” (green) plants, containing large numbers of chromophores, with a nonzero B/A ratio as discussed, for instance, in ref 18, Le., where both chl-a and chl-b are represented. However, in a way our considerations could apply even to cases where only one chromophore (i.e., chl-a) is present, but with different enough e n e r g i e ~ ,due ~ ~to ? ~different ~ environments of the chromophore sites (i.e., state of aggregation, see below). On the other hand, once there is a large enough number of such species (i.e,, chl-a), differing in excitation energy by less than k T , then such species should all be treated practically as one (see Table I). We have treated this problem in much more detail elsewhere.13J4 It suffices to stress here that in order for our approach to be meaningful it is required that
E B - E A ?. k T
(10)
E A - E s 2 kT
(11)
and
as well as a demand for a large number of (statistically meaningful) chromophores. Needless to say, we tried here to present the simplest model which still retains the essential characteristics of exciton percolation. One problem under active consideration is a “cooperative” percolation-thermalization model, related to both mixed crystals and photosynthesis systems for which the constraints of eq 10 and 11 apply only marginally.21 The Journal of Physical Chemistry, Vol. 80, No. 20, 1976
Finally, we note that in our “simulation experiments” involving naphthalene, the sensor S is kept a t a small concentration so as to avoid sensor clusters and other complications. The analogous low sensor (Le,, P700) concentration in the photosystems5J8 may be related to the same reason or may just be dictated by spatial access considerations. Even so, the analogy between our synthetic ternary system and the photosynthetic systems seems to go beyond coincidence. , , We also notice that a sensor of or definitely falls within the limit of very efficient transport14 and therefore the limit of validity of eq 3. Actually, at sensor concentrations below the efficiency of the exciton transport (to the sensor) suffers significantly,2* whether this transport is coherent or i n ~ 0 h e r e n t .We l ~ thus should not be surprised that the ratio of antenna-to-sensor chromophores in photosynthetical systems has not exceeded 10-3. However, our conclusion about the predicted ratios (see (8))is independent of this consideration. Actually, with eq 4 the ratios [B]/[A] tend to be slightly lower than the given limit (unity) of (8),but this might be compensated for by some partial ordering, such as a preference for A-A aggregation (or, alternatively, A-B a g g r e g a t i ~ n )In . ~ conclusion, we demonstrated that exciton percolation may play an important role in biomolecular aggregates, since it relaxes the hitherto assumed requirement of a highly ordered aggregate so as to permit efficient, longrange exciton transport.
.
Acknowledgment. The help of my collaborators, Dr. J. Hoshen and E. M. Monberg, with the computation and graphical representation is gratefully acknowledged. As pointed out in the text, the experimental data were taken from work with P. Argyrakis, E. M. Monberg, and F. W. Ochs. This research was supported by NSF Grant GH-32578X and NIH Grant NS08116. References and Notes (1) A. S. Davydov, “Theory of Molecular Excitons”, Plenum Press, New York, N.Y., 1971. (2) R. Kopelman, J. Chem. Phys., 47,2631 (167). (3) Th. W. Foerster, Ann. Phys. (Leipzig), 2, 55 (1948). (4) R. S.Knox in “Bioenergetlcs of Photosynthesis”, Govindjee, Ed., Academic Press, New York, N.Y., 1975. (5) K. Sauer, ref 4. (6) S.Kirkpatrick, Rev. Mod. Phys., 45,574 (1973), and references therein. (7) T. P. Eggarter and M. H. Cohen, Phys. Rev. Lett., 25,807 (1970). (8) J. Jortner and M. H. Cohen, J. Chem. Phys., 58, 5170 (1973). (9) R. Kopelman, “Excited States”, Vol. II, E. C. Lim, Ed., Academic Press, New York, N.Y., 1975. (IO) H.-K. Hong and R. Kopelman, J. Chem. Phys., 55,5380 (1971). (11) R. Kopelman, E. M. Monberg, F. W. Ochs, and P. N. Prasad, J. Chern. Phys., 82, 292 (1975). (12) R. Kopelman,E. M. Monberg, F. W. Ochs, and P. N. Prasad, Phys. Rev. Lett., 34, 1506 (1975): (13) R. Kopelman, J. Lumin., 12,775 (1976). (14) R. Kopelman in “Non-Radiative Relaxation Processes in Molecules, Solids and Photosynthesis”, Voi. 15 of Applied Physics Series, F. K. Fong, Ed., Springer-Verlag, Berlin, in press. (15) D. M. Hanson, J. Chem. Phys., 52,3409 (1970). (16) P. Argyrakis and R..Kopeiman, unpublished work. (17) F.W. Ochs, P. N. Prasad, and R. Kopelman, Chem. Phys., 8,253 (1974); J. Small, private communication, Sept 1975. (18) C. Arntzen, private communication, July 1975; C. J. Arntzen and J-M. Briantls in ref 4. (19) C. S.French, Proc. Natl. Acad. Sci. U.S.A., 68,289 (1971). (20) Govindjee, “Bioenergetics of Photosynthesis”, Govindjee, Ed., Academic Press, New York, N.Y., 1975. (21) F. W. Ochs, E. M. Monberg, and R. Kopelman, Unpublishedresults. (22) J. Hoshen and R. Kopeiman, J. Chem. Phys., in press.
Discussion
M.W. WINDSOR.If it is the chlorophyll a quasi lattice that initiates exciton migration in green plant photosynthesis, then why has Nature provided so much chlorophyll b? What is the role of the chlorophyll b? Does it, in your opinion, function solely as an accessory pigment?
Exciton Percolation in Molecular Crystals and Aggregates
R. KOPELMAN.First of all, the question should really be asked the other way around. Why hasn’t Nature provided us with about 10 pigments to cover the spectrum very efficiently? And the only answer I know is that you wouldn’t be able to have excess of one pigment, if you divided yourself between 10 different pigments. M. W. WINDSOR.But it does provide many accessory pigments.
R. KOPELMAN.Yes, but accessory does not mean substitutional in the same kind of lattice. With addition of the (b), one can reduce the concentration of (a) down to about 60% almost without losing any efficiency of exciton transfer. The bottleneck doesn’t start until about 50-50. So why not have another pigment to absorb about twice as much light?
R. M. HOCHSTRASSER.What assumptions are being made in regard to long-range vs. short-range energy transfer in your models? Won’t chlomphylls exhibit long-range transfer? R. KOPELMAN.Basically I have assumed short-range interactions. Now this does not mean the nearest neighbor only. Like in naphthalene we talk about nearest neighbor, next nearest, next, next, and so on. Even if one has dipole-dipole interactions it is limited to a plane and that is not very long range. As long as the interactions are not terribly long, then this model is going to work. D. MAUZERALL. For red and brown algae, which do cover the whole absorption spectrum, Nature seems to have chosen the funnel model. The accessory pigments are in the form of clusters (called phycobilisomes) which plug into the photosynthetic membranes. There is even some evidence that the phycoerythrin and phycocyanin may be arranged in layers with the phycocyanin closer to the chlorophyll.
R. KOPELMAN.In certain algae Nature has built a completely different situation where a lot of pigments cover the spectrum much more efficiently. My model is not made for those algae because there is a whole photosynthetic unit; it is very small, and the whole statistical approach would break down anyhow. I am not saying that our
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model will work for those algae. Probably for the most developed systems, which have the largest antenna, our kind of mechanism can work. It certainly does not work for all classes of systems. M. KASHA.Why work with 40 000 if you have only say 400 ih a photosynthetic natural system?
R. KOPELMAN.The answer is, we wanted to check ourselves, so we overkilled the problem by going to large systems. However, one of my slides~showedthat qualitatively there is very little difference between 300 or 500 units and an infinite number of them. There is really little quantitative difference between 400 units and 40 000 or even up to 400 OOO units. We were pushing the problem from the solid state point of view and wanted to make sure that in the limit the computer gave us the analytical solution. Obviously, we also imply ensembles of the 400 size systems. H. SELIGER.What is the applicability of the model for the random formation of chlorophyll aggregates to facilitate energy percolation in the natural photosynthetic system? In the latter there is a formal structure to the lamelles of the chloroplasts where there are special places for the electron transport sensitizer chlorophyll and some regular spacing in the membrane phospholipids for the inclusion of antenna chlorophyll. In fact the chlorophyll appears to be necessary to the formation of the lamellae themselves.
R. KOPELMAN.There is some question as to the validity of the structure of the lamellar system which has been assumed. I was at the Gordon Photosynthesis Conference last summer-and I am obviously not prepared to quote a lot of information-but my overall impression was that for large harvesting systems in green plants it really isn’t that obvious how much structure one has. And my own feeling is that people invented a lot of structure because they were stuck with a model that demanded structure. Now I’m not saying that the system has to be prfectly random, in fact calculations on partially random systems produce different numbers, but qualitatively the situation is the same. One has to have very good order, and nearly perfect structure, before this model breaks down completely.
The Journal of Physical Chemistry, Vol. 80,No. 20, 1976
