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J. Phys. Chem. B 2008, 112, 12386–12393
Photoluminescence Spectra of Self-Assembling Helical Supramolecular Assemblies: A Theoretical Study Leon van Dijk,*,† Sander P. Kersten,† Pascal Jonkheijm,‡ Paul van der Schoot,† and Peter A. Bobbert† Group Polymer Physics, Department of Applied Physics and EindhoVen Polymer Laboratories, Technische UniVersiteit EindhoVen, P.O. Box 513, 5600 MB EindhoVen, The Netherlands, and Molecular Nanofabrication Group, UniVersity of Twente and MESA+ Institute for Nanotechnology, 7500 AE Enschede, The Netherlands ReceiVed: March 28, 2008; ReVised Manuscript ReceiVed: July 15, 2008
The reversible assembly of helical supramolecular polymers of chiral molecular building blocks is known to be governed by the interplay between mass action and the competition between weakly and strongly bound states of these building blocks. The highly co-operative transition from free monomers at high temperatures to long helical aggregates at low temperatures can be monitored by photoluminescence spectroscopy that probes the energetically lowest-lying optical excitations in the assemblies. In order to provide the interpretation of obtained spectroscopic data with a firm theoretical basis, we present a comprehensive model that combines a statistical theory of the equilibrium polymerization with a quantum-mechanical theory that not only accounts for the conformational properties of the assemblies but also describes the impact of correlated energetic disorder stemming from deformations within the chromophores and their interaction with solvent molecules. The theoretical predictions are compared to fluorescence spectra of chiral oligo(p-phenylene-vinylene) molecules in the solvent dodecane and we find them to qualitatively describe the red-shift of the main fluorescence peak and its decreasing intensity upon aggregation. 1. Introduction Biomimetic self-assembly has proven to be a very useful tool for constructing well-organized nanomaterials with widespread applications. Examples include nanotube architectures formed by self-assembly of amphiphilic molecules with applications in bionanotechnology,1 molecules that form responsive gels through self-organization in three-dimensional networks of fibers,2 and supramolecular polymeric (wire-like) assemblies of π-conjugated systems.3 For π-conjugated “nanowires”, selfassembly provides a way to combine the advantages of easy solution-processing, one-dimensional structural order and rigidity. The central tenet is that in selective solvents and under suitable conditions specifically tailored π-conjugated molecules self-assemble into helical fibers and that the overlap of the regularly positioned π-orbitals allows for the transport of excitations (electrons, holes and excitons) along the stacking direction. It has been suggested that these nanowires have potential applications in electronic devices in the 5-100 nm size range.4 Recently, helical supramolecular assemblies have been studied that consist of chiral oligo(p-phenylene-vinylene) derivates (OPV-4, see Figure 1). These are capped on one end by a tridodecyloxybenzene and on the other end by an ureidotriazine unit, which is able to engage in four hydrogen bonds with those of another oligomer.5 If dissolved in an apolar solvent at high temperatures, the OPV oligomers dimerize as a result of the quadruple hydrogen bonding. Upon lowering the temperature, the dimers associate into linear aggregates due to the π-π * Author to whom correspondence should be addressed. E-mail:
[email protected]. † Group Polymer Physics, Department of Applied Physics, and Eindhoven Polymer Laboratories, Technische Universiteit Eindhoven. ‡ Molecular Nanofabrication Group, University of Twente, and MESA+ Institute for Nanotechnology.
Figure 1. Molecular structure of the OPV-4 derivative and a schematic representation of the self-assembly. The blue and red blocks represent the OPV-4 backbone and the hydrogen-bonding end group, respectively.
interactions of the phenylene-vinylene backbones. The formation and growth of these aggregates, which attain a helical conformation below some critical temperature, can be monitored by spectroscopic techniques, in particular ultraviolet and visible (UV/vis) absorption, fluorescence and fluorescence decay as well as circular dichroism (CD) measurements.5-7 The self-assembly of OPVs is not only hierarchical but also highly co-operative; see Figure 1. Indeed, at high temperatures (or very low concentrations) the OPVs form dimers that on cooling down (or increasing the concentration) first self-organize into short, imperfectly ordered stacks of molecules, the properties of which are characterized by isodesmic mass action. The transformation of these irregularly stacked assemblies into highly
10.1021/jp802693s CCC: $40.75 2008 American Chemical Society Published on Web 09/10/2008
Helical Supramolecular Assemblies ordered (helical) ones occurs at a well-defined temperature. It is this transformation that is so co-operative because below this temperature the short ordered helical stacks experience strong elongation resulting in long helical aggregates. The combination of supramolecular polymerization and a conformational transition reminiscent of but not quite identical to the helix-coil transition in biopolymers, has been studied theoretically7-11 and the predictions agree well with CD and UV measurements on OPVs dissolved in alkanes.7 By combining theory and experiment Jonkheijm and co-workers discovered that the solvent plays a crucial role in the ordering transition of the assemblies.7 While spectroscopy is routinely used to study co-operativity in self-assembly, there is no firm theoretical basis for linking measured absorption or fluorescence intensities to the fraction of molecules in assemblies or to some average aggregation number. Indeed, results seem to depend (if only fairly weakly so) on the choice of spectroscopic technique, the wavelength, the normalization of the intensity, and so on. In other words, the interpretation of spectroscopic data by their fitting to statistical mechanical models cannot yet be considered an exact science, because these models alone cannot account properly for absorption and fluorescence intensities. Several studies have been made to come to a quantummechanical description of the problem in hand.12-14 Spano and co-workers, for example, recently reported on a study of the photoexcitations (excitons) in helical OPV-4 aggregates in dodecane solution at T ) 278 K.12 They calculated circularly polarized absorption and emission spectra using a Holstein Hamiltonian with correlated disorder in the excitation energies of the stacked chromophores, the so-called on-site energies, taking into account phenomenologically the impact of coupling of low-frequency phonons to the exciton and the presence of solvent. By comparison to experimental spectra they were able to characterize the excited-state and they found that disorder and exciton-phonon coupling act synergistically in localizing the vibrationally dressed exciton over only a few molecules. While quite advanced, their quantum-mechanical model ignores any influence of mass action and conformational transitions, and their results are applicable only at temperatures around 278 K, far below the helical transition temperature. In that case almost all oligomers are in the aggregated state and the aggregates are very long. Moreover, the aggregates are then almost perfectly helical and conformational transitions do not play a role. At higher temperatures one should take into account the fact that there is a broad distribution of aggregate sizes and that the conformational state of a single aggregate is not fixed because on approach of the helical transition temperature strong fluctuations in the helical order of the assemblies do take place. Both effects potentially affect the absorption and photoluminescence spectra. In fact, Figure 2 shows measured photoluminescence spectra taken at different temperatures,5 together with the calculated (number-averaged) mean aggregate size as a function of temperature (see section 2). Upon lowering the temperature, the main photoluminescence peak red-shifts and the intensity of the spectrum decreases. It is quite natural to presume that these significant changes in the photoluminescence spectra observed with a lowering of the temperature must somehow be related to the concomitant increase in mean aggregate size. Hence, a theory of photoluminescence spectra of supramolecular wires should at least in principle account for the temperature (and concentration) dependence of both the size distribution and the conformational state of the assemblies.
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Figure 2. Photoluminescence spectra of OPV-4 in dodecane solution for different temperatures (taken from ref 5). The inset displays the calculated (number-averaged) mean aggregate size, given as the number of stacked OPV-4 dimers, as a function of temperature (see section 2).
Here, we indeed go one step further by taking into account the influence of mass action and conformational transitions to obtain an understanding of photoluminescence spectra in relation to supramolecular polymerization. We present a quantummechanical model that describes the delocalization of the photoexcitations on the aggregates and combine that with a theory that describes their helical self-assembly. To our knowledge, we are the first to follow such an approach and the combined model is a first step in providing measured fluorescence intensities with a firm theoretical basis. As temperaturedependent fluorescence measurements are used to monitor all aspects of the self-assembly, unlike CD measurements that provide only information about the helical assemblies, a full understanding of how helical self-assembly affects fluorescence spectra would lead to a deeper insight in it. Note that in principle our combined quantum and statistical mechanical approach can be used to obtain temperature-dependent absorption spectra as well. In this work we apply the self-assembled Ising model of helical supramolecular polymerization 8-11 to calculate the size distribution of the aggregates and the distribution of their conformations as a function of temperature (and concentration). For every aggregate of given length and conformation, we calculate within a tight-binding model the energy of the energetically lowest-lying optical excitation, presumed to be the state from which radiative decay to the ground-state takes place. The disorder in the on-site energies stemming from deformations within the chromophores and their interaction with the solvent molecules is taken into account phenomenologically in the same way as done by Spano et al.12 By averaging over the distributions of aggregate lengths and conformations, and by accounting for optical matrix elements that take into account the transition intensities, we finally obtain the luminescence spectrum as a function of temperature (and concentration). In what follows, we will show that by accounting for the influence of mass action and conformational transitions, the redshift of the main photoluminescence peak can be understood qualitatively. We shall see that by comparing theory and experiment we can estimate the mean exciton migration length, the average distance that photoexcitations overcome before decaying radiatively. Furthermore, we shall see that our model predicts the decrease in intensity of the spectrum upon lowering the temperature reasonably well, and that the disorder in the
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on-site energies is predominantly caused by the interaction of the chromophores with the solvent molecules. The remainder of this paper is set up as follows. First, in section 2, we present a brief overview of helical aggregation theory and refer for details to the work of van Gestel and van der Schoot.8-11 In section 3, we discuss our tight-binding model for describing the photoexcitations on a single molecular aggregate of given length. Next, in section 4, we describe how photoluminescence spectra are calculated by combining the two theories. We apply our model to the specific case of OPV-4 aggregates in dodecane in section 5, and conclude in section 6 with a summary, a discussion, and the main conclusions. 2. Assembly Model The helical equilibrium polymerization of chiral molecular units in solution is governed by the interplay of mass action and the competition between high-energy/low-entropy and lowenergy/high-entropy bonded states of neighboring molecules within an aggregate.11 In the high-energy/low-entropy bonded state two neighboring molecules are locked in a helical arrangement, where the helical sense is fixed by the chiral nature of the molecules involved. In the low-energy/high-entropy bonded state the molecules do not have fixed orientations relative to each other. For simplicity we will refer to the former as the helical bond, and to the latter as the non-helical bond. It seems plausible to presume that different conformers are required for each type of bonding. In thermodynamic equilibrium the law of mass action holds. In dilute solution it has the familiar form
F(N) ) Q(N)F(1)
N
∑
Q(N) )
N-1
∑
1 1 H ) - R (sjsj+1 - 1) + P (sj + 1) + (N - 1)M 2 j)1 2 j)1 (2) with sj ) -1 if the bond following the jth monomer unit is non-helical and sj ) +1 if it is helical. For N ) 1, there are no
∑ e-H
(3)
{si}
where the summation over {si} ≡ {s1,...,sN-1} involves all possible conformations. This conformational sum is straightforwardly evaluated using the transfer-matrix method, albeit that boundary values have to be made explicit.15 In principle one would presume free boundary conditions to hold but this cannot be reconciled with the non-isodesmic character of the helical assembly of OPV-4 in alkane solvents.7 Hence, we have used boundary conditions where the first bond is non-helical and the last one free. An asymmetric boundary condition is consistent with the chiral nature of the monomer units (the OPV dimers), and allows for nonisodesmic, nucleated assembly. For every N the partition function in eq 3 can be calculated analytically and closed expressions can be found in ref 10. In order to relate the theoretical parameters M and P to experimentally accessible quantities, we follow ref 8 and make the temperature dependence of these parameters explicit through a first-order Taylor expansion around two reference temperatures ∞ T /0 and T // defined below:
( )
M(T) ≈ M(T 0/) + 1 -
(1)
where F(N) represents a dimensionless concentration of aggregates of size N (actually a mole fraction) and Q(N) the partition function of a single aggregate of size N that in this context may be seen as an equilibrium constant.11 The singleaggregate partition function Q(N) is calculated by applying a model that can be mapped onto the well-known one-dimensional Ising model. The Ising model is a two-state lattice model and in this case the state represents the type of bond, helical or nonhelical. In the model we do not explicitly consider any contributions from flexing, stretching, or writhing of the assemblies, nor of their interaction with the solvent. These contributions should be extensive in the length of the assemblies and may be absorbed in the free energy scales that we present next. The free energy that is associated with the formation of a non-helical bond we denote by M < 0, and from now on we tacitly presume all free energies to be expressed in units of the thermal energy kBT, where kB denotes Boltzmann’s constant. The free-energy difference between a helical and a nonhelical bond we denote by P. A helical bond is more favorable than a non-helical one if P < 0 while for P > 0 the opposite is true. If a molecule is involved in two types of bonding with its direct neighbors, it must be conformationally frustrated and a free energy penalty R g 0 is introduced for those molecules. Accordingly, the dimensionless Hamiltonian H for an aggregate of N > 3 chain units is given by N-2
bonds and we put H ≡ 0, while for N ) 2 and N ) 3 the Hamiltonian depends on the boundary conditions we impose on the first and last bond, s1 and sN-1. For instance, for free boundary conditions, H(2) ) 1/2P(s1 + 1) + M and H(3) ) 1/ P(s + 1) + 1/ P(s + 1) - 1/ R(s s - 1) + 2M. 2 1 2 2 2 1 2 The single-aggregate partition function Q(N) is given by
T hn T 0/ kBT 0/
(4)
and
(
P(T) ≈ 1 -
)
hh T ∞ ∞ T // kBT //
(5)
Here, M(T /0) ≡ ln φ/(2 - 2) is the binding free energy associated with the nonhelical bond at the polymerization transition temperature T /0 where half of the monomer units would have assembled if there were no helical transition. It depends on the overall mole fraction φ of the molecular building blocks. If we set M(T 0/) ) m/kBT 0/ and presume that m is purely enthalpic, this quantity can be fixed given that T /0 is known for one concentration φ. This then gives a prediction of T 0/ for other concentrations. Finally, the quantities hn and hh represent the (now dimension-bearing) enthalpy of the formation of a nonhelical bond and the enthalpy difference between a helical a ∞ non-helical bond, and T // is the helical transition temperature for an infinitely long aggregate. Even if we presume the free-energy penalty R to be temperature independent, the model has five free parameters that (ideally) need to be fixed independently from each other.10 This is, of course, no trivial matter. For the material described in ref 10 this proved possible because of the large body of experimental data available for a range of temperatures and concentrations. For OPV-4 we do have at our disposal sufficient data to fix all model parameters, but unfortunately not to the extent to determine all of them independently. The enthalpy of the formation of a non-helical bond and the polymerization transition temperature we determined by fitting the fraction of monomer units in the aggregate state η ≡ 1 F(1)φ-1 to a normalized UV/vis absorption intensity at the wavelength λ ) 490 nm, and measured at a concentration of 1.1 × 10-5 M, corresponding to a molar fraction φ ) 2.5 × 10-6. The enthalpy of the formation of a helical bond, the helical
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TABLE 1: Parameters Used in the Calculations for OPV-4. calculation parameters
values
hh hn R T /0 ∞ T // φ
-15 kJ/mol -184 kJ/mol 2.2 kBT 342 K 334.4 K 2.5 × 10-6
The linear aggregates under consideration consist of π-stacked chromophores. Each chromophore in the aggregate is taken to be a two-level system with an electronic ground-state |g〉 and an energetically lowest-lying photoexcited state |e〉. We describe the delocalization of photoexcitations on the aggregate in a tightbinding approximation with the quantum-mechanical Hamiltonian,
∑
n)1
N-1
εn|n 〉 〈n| +
N
1
√2π √det C
[
× N
∑
]
1 (C-1)m,nδεnδεm (7) 2 n,m)1
where (C-1)n, m are the elements of the inverse of the covariance matrix C. The elements of this matrix are given by the covariances of the deviations δεn and δεm,
3. Tight-Binding Model
N
[ ] 1
exp -
transition temperature, and the parameter R we estimated by comparing normalized CD intensities at λ ) 466 nm, obtained at the same concentration, to the theoretical weight-averaged fraction of helical bonds.9 Table 1 shows the values which have been used in our calculations for OPV-4. For these values agreement between theory and the spectroscopic data is quantitative, except at low temperatures where OPV-4 is thought to form bundles.7 Given the values for the parameters R, P, and M, we are now in a position to explicitly evaluate the partition function Q(N) in eq 3 and predict the density of aggregates of length N using eq 1. We can also predict the probability that an aggregate of certain length attains any given bonded conformation, as this is proportional to e-H. Both the density distribution and the configurational statistics of the aggregates we need as input in our quantum-mechanical calculations outlined next.
H)
P(δε1, ... , δεN) )
∑ Vn,n+1{|n + 1 〉 〈n| + |n 〉 〈n + 1|}
n)1
(6) representing the contribution of the on-site excitation energies and the transfer of excitations from one chromophore to the next. As before, N stands for the number of stacked chromophores forming the aggregate while |n〉 represents the electronic state in which the nth chromophore is excited and the other N-1 chromophores are in the electronic ground state, |n〉 ≡ |en〉Πm+n|gm〉. This means that we consider low excitation densities, i.e., each aggregate contains at most one photoexcitation. This assumption holds due to the low laser intensity and density used in experiment.16-18 In our simplified model, we also neglect any coupling of the excitation to the conformational degrees of freedom of the chromophores (exciton-phonon coupling). This means that we concentrate on the 0-0 peaks in the luminescence spectra (implying no phonons in either the electronic ground-state or the excited state). Because of thermally induced deformations within each chromophore, e.g., by bending and twisting, statistical conformations of the polymeric side groups, and the presence of some degree of randomness in the direct environment of each chromophore on account of the presence of solvent, the energies εn will fluctuate around a mean on-site excitation energy ε0. As in ref 12, we draw the deviations from the mean, δεn, from a joint Gaussian distribution
Cn,m ) 〈δεnδεm 〉 ) σ2e-|n-m|/l0
(8)
where 〈 · · · 〉 denotes an average over the distribution in eq 7 and σ is a measure for the amount of disorder, defined as σ ≡ 〈δεn2〉. As in ref 12, we assume that the on-site energies are exponentially correlated and l0 is the spatial correlation length in dimensionless units of the distance between two bound chromophores. There is no spatial correlation in the case l0 ) 0 and all on-site energies are identical in the limit where l0 f ∞. Besides depending on the (correlated) disorder in the on-site energies, the energies of the photoexcited states also depend on the conformation of the aggregate because the effective (see below) transfer integral Vn, n+1 of a helical bond differs from that of a non-helical bond. The reason is that two chromophores are much more loosely bound in the non-helical bonded state than they are in the helical state, and presumably are in that case able to rotate freely with respect to each other. Therefore, we expect the transfer integral of two loosely bound chromophores to be much smaller than when they are in the more strongly bound helical state, and for simplicity we set it equal to zero. This assumption is supported by experimental investigations of the transfer rate of photoexcitations along supramolecular OPV assemblies for two different packing geometries. Indeed, in well-defined helical stacks the transfer of excitations is efficient, whereas in disordered assemblies a slow transfer of excitations along the stacks is observed that has been ascribed to the much weaker electronic coupling.19 Exciton-phonon coupling is neglected in our tight-binding model. However, in case of weak excitonic coupling, excitonphonon coupling is known to renormalize the exciton bandwidth with a factor e-λ2, where λ2 is the Huang-Rhys factor. Therefore, we can approximately take into account the influence of exciton-phonon coupling on the 0-0 emission by using an effective transfer integral Vn, n+1 ) e-λ2Jn, n+1, where Jn, n+1 is the transfer integral without exciton-phonon coupling. For an aggregate of given length and conformation, the tightbinding model allows for calculating the energies of the photoexcited states from which radiative decay takes place. Photoluminescence spectra can be calculated by averaging over the distribution of aggregate lengths and the distribution of conformations. This is done by combining the assembly model with the tight-binding model, as we explain in the next section. 4. Calculation of the Spectra Here, we outline the procedure we follow to calculate photoluminescence spectra, which is schematically shown in Figure 3. Before considering the photoluminescence spectrum of an entire distribution of aggregate lengths in solution, we first concentrate on that of a single aggregate of given length N. In our photoluminescence gedanken experiment only one of the chromophores in the aggregate absorbs a high-energy photon, thereby creating a high-energy photoexcitation. We assume that this photoexcitation rapidly relaxes to the energetically lowest-
12390 J. Phys. Chem. B, Vol. 112, No. 39, 2008
van Dijk et al. of the aggregate, which is given by
b) M
N
∑ fµn
(11)
n)1
Taking the z-axis along the stacking direction, the transition dipole moment operator b µn at site n is defined as f µn ) µ0(cos(φn)eˆx + sin(φn)eˆy)(|n 〉 〈gn| + |gn 〉 〈n|)
(12)
where µ0 is the transition dipole moment of a single chromophore, φn the angle between the x-axis and the long axis of the oligomer at site n, along which the dipole moment of OPV-4 is oriented, and êx and êy are unit vectors in the x- and y-directions. Accordingly, the intensity of the luminescence of an aggregate of length N and conformation i becomes Figure 3. Schematic representation of the procedure for calculating the photoluminescence spectra by combining the assembly model and the tight-binding model. The meaning of the different symbols is explained in the main text.
lying excited-state on the aggregate and then decays radiatively to re-establish ground state. This relaxation is possible due to Fo¨rster transfer20 of the excitation along the aggregate, accompanied by the excitation of low-energy phonons. Experiments on aggregates of OPV-3 (containing only one dialkoxybenzene unit) with a small amount of OPV-4 mixed in demonstrate how efficient this process is: almost all luminescence comes from the OPV-4 molecules, which have an energetically lower-lying excited-state than the OPV-3 molecules.21 At low temperatures the aggregates become very long and the assumption that radiative decay takes place from the lowest excited-state of the aggregate breaks down due to the finite lifetime of the photoexcitations. In that case, the photoexcitations can migrate only a finite distance along the stacks and radiative decay will occur before the photoexcitations reach the lowest-lying excited state. To account for this in our calculations, we use an effective maximum aggregate length to calculate the lowest-lying excited state. The energetically lowest-lying excited-state of the aggregate is denoted by ΨN, i ≡ Σnci, n|n〉 and has an energy Ei, where the subscript i refers to a particular conformational state of the aggregate. The coefficients ci, n and the energy Ei can be found by diagonalization of the Hamiltonian in eq 6. In order to obtain the luminescence spectrum SN(E) of all aggregates of length N, we have to perform an average over the conformational states of aggregates of length N,
SN(E) ) 〈IN,iδ(E - Ei)〉
(9)
with IN, i being the intensity of the luminescence of an aggregate of length N and conformation i and 〈 · · · 〉 a conformational average. To determine IN, i we have to take into account non-radiative processes and other possible non-fluorescent decay channels, such as intersystem crossing, which leads to phosphorescence. For simplicity we assume that the rate of non-fluorescent decay knf is independent of the length and conformational state of the aggregate. The rate of fluorescent decay kf,N,i is proportional to the optical matrix element of the lowest-lying excited-state and the Franck-Condon factor of the 0-0 transition
kf,N,i )
k˜f µ02
b |〈ΨN,i|M
|gn 〉|2e-λ ∏ n
2
(10)
Here k˜f is a constant, assumed to be independent of the aggregate b represents the transition dipole moment operator length, and M
IN,i )
∏
b n |gn 〉|2e-λ2/µ02 kf,N,i |〈ΨN,i|M ) kf,N,i + knf |〈Ψ |M b n |gn 〉|2e-λ2/µ02 + knf/k˜f N,i
∏
(13) The ratio knf/k˜f can be estimated from the experimental spectra, as we will show in the next section. To obtain the total luminescence spectrum we have to take an additional average over the aggregate length N. In performing this average, we have to take into account the fact that the absorption cross section of an aggregate is proportional to its length, since each chromophore in the aggregate can absorb a photon. This means that each spectrum of aggregates of length N should be weighed by the probability distribution ∞
P(N) ) NF(N)/
∑ NF(N)
(14)
N)1
with F(N) the number density of such an aggregate, given by eq 1. Hence, the total luminescence spectrum is then obtained as ∞
S(E) )
∑ P(N)SN(E)
(15)
N)1
In the numerical evaluation of the sum in eq 15, we introduced a cutoff Nmax, fixed by P(Nmax)/max(P(N)) ≈ 0.001. For OPV4, this leads to cutoff values between Nmax ) 4 at T ) 353 K and Nmax ) 1430 at T ) 293 K. We take a large number (up to about 106) of aggregate lengths N according to the distribution P(N) of eq 14 and for each of these aggregates we take random on-site energies from the joint Gaussian energy distribution eq 7. For each aggregate we then perform the conformational average in eq 9 by a Monte-Carlo procedure based on single spin-flip sampling. Starting from a random configuration, an equilibration time of 40N Monte-Carlo steps proved sufficient for all temperatures studied, after which data were gathered during 100N steps. With this procedure we obtain a spectrum for each temperature with an accuracy of about 2% around the peak position. 5. Results and Comparison with Experiment Figure 4 shows the experimental spectra (black solid curves, taken from ref 5) together with the theoretical spectra (red dashed curves and blue dash-dotted curves) for OPV-4 in the solvent dodecane, obtained with the parameters taken from Table 1. The theoretical spectra are normalized such that the peak height of the theoretical spectrum at T ) 353 K equals that of the corresponding experimental spectrum. To obtain the theo-
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Figure 4. (a-g) Experimental spectra (black solid curves) and theoretical spectra (red dashed curves and blue dash-dotted curves, corresponding to effective maximum aggregate lengths of 60 and 13, respectively) of OPV-4 aggregates at different temperatures. The spectra are normalized such that the peak height of the theoretical spectrum at T ) 353 K equals that of the corresponding experimental spectrum. The vertical dotted line indicates the position of the experimental 0-0 peak. Note that the graphs have different y scales. (h) Position of the 0-0 emission peak versus temperature. (i) Relative integrated intensity versus temperature. In parts h and i, the vertical dotted line indicates the helical transition temperature as observed from CD measurements.7
retical curves, we put in a theoretical estimate for the transfer integrals. The transfer integral Jn, n+1 for OPV-4 was calculated in ref 22 for two superimposed chromophores, and found to have a value of 57 and 46 meV for rotation angles of 6° and 12°, respectively. In this work we take the median angle 9° and accordingly insert Jn, n+1 ) 50 meV for the transfer integral for the case of a helical bonded state between consecutive chromophores. The effective transfer integral is then given by Vn, n+1 ) e-λ2Jn, n+1, and we have used λ2 ) 1.2 for the Huang-Rhys factor, as in ref 12. As we have pointed out in section 3, in case of a non-helical bonded state we expect the transfer integral to be much smaller and for simplicity we set it equal to zero. We have determined the experimental 0-0 peak positions by carrying out a four-Gaussians fitting procedure to the experimental spectra. The average on-site excitation energy ε0 and the width of the energy distribution σ are then determined by the Gaussian corresponding to the 0-0 peak of the emission spectrum at T ) 353 K, a temperature at which the solution consists mainly of single oligomer pairs. The optimal values
we find are ε0 ) 2.471 eV and σ ) 77 meV. The latter value is quite close to the value of 85 meV obtained in ref 12 deep in the helical polymerized regime. This suggests that the disorder in the on-site energies cannot be associated with the conformational properties of the assemblies, a point we discuss further in the next section. Furthermore, Spano and co-workers found that the on-site energies of the chromophores in the aggregated state are correlated. They determined the correlation length by fitting the ratio between the intensities of the 0-1 and the 0-0 peak, and found l0 ) 4.5.12 In our calculations we have used the same correlation length, as we will clarify further on. As argued in the previous section photoexcitations cannot reach the lowest-lying excited-state when aggregates are too long. In fact, calculations of the migration of photoexcitations along OPV-4 stacks predict a root-mean-squared displacement of 20-25 nm, corresponding to 53 to 67 stacked oligomer pairs, in the first 100 ps after excitation.22 As indicated by timeresolved photoluminescence experiments, more than 50% of the photoexcitations decays during this time,23 suggesting that photoexcitations on very long aggregates experience an effective
12392 J. Phys. Chem. B, Vol. 112, No. 39, 2008 maximum aggregate length. For that reason, we use an effective maximum aggregate length of 60 to calculate the lowest-lying excited-state for all longer aggregates. To take into account non-fluorescent decay processes we have estimated the ratio knf/k˜f in eq 13. We have determined numerically that on average the optical matrix element for an aggregate of length 60 is equal to 0.22µ02. Because the average aggregate length at temperatures below 293 K is much larger than 60, we find from eq 13 that the intensity of luminescence at these low temperatures is proportional to 0.22e-1.2/(0.22e-1.2 + knf/k˜f). At T ) 353 K the solution contains almost exclusively single oligomer pairs. Since the optical matrix element of a single oligomer pair is equal to µ0 the intensity of luminescence at T ) 353 K is proportional to e-1.2/(e-1.2 + knf/k˜f). From the fact that the integrated experimental fluorescence intensity for T e 293 K is about 26% of the integrated intensity at T ) 353 K (see Figures 2 and 4i) we estimate that knf = 1.2k˜f, and this is the value we will use in our calculations for an effective maximum aggregate of length of 60. We now concentrate on the behavior of the 0-0 peak as a function of temperature. Experimentally, it red-shifts and reduces in intensity with decreasing temperature, as can be seen in Figures 2 and 4a-h. The theoretically predicted red-shift of the spectra, the red dashed curve in Figure 4, follows the experimental red-shift reasonably well at high temperatures. To calculate these spectra we have used a correlation length of l0 ) 4.5. In preliminary calculations we have observed that smaller correlation lengths lead to a clear overestimation of the redshifts at these high temperatures. On the other hand, we have observed that the calculated spectra do not change upon increasing the correlation length if l0 g 4, and we have chosen to use the same correlation length as in ref 12. The theoretically predicted red-shift starts to deviate substantially from the experimental results below T ) 333 K. In the spectrum at the lowest temperature shown in Figure 4g, where the aggregates are on average much longer than the effective maximum aggregate length of 60 (see Figure 2), the theoretical red-shift is about 0.07 eV too large. When an effective maximum aggregate length of 13 is used, as shown by the blue dash-dotted curves in Figure 4, the theoretical red-shift matches the experimental red-shift at temperatures lower than 293 K. Again, to take into account nonfluorescent decay processes, we have estimated the ratio knf/k˜f. On average, the optical matrix element for an aggregate of length 13 is equal to 0.24µ02, resulting in knf = 3.8k˜f. The effective maximum aggregate length of 13 can be interpreted as an estimate for the mean exciton migration length, as was argued in the previous section. Our theoretical predictions for the intensity of the 0-0 peak as a function of temperature follow the experimental results reasonably well, see Figure 4i. Below 323 K, the decrease in intensity becomes very substantial, which is connected with the formation of helical aggregates. If perfect cofacial stacking would occur without energetic disorder, the aggregates would be so-called H-aggregates and the 0-0 emission would be strictly forbidden by symmetry. The presence of helicity and disorder leads to a symmetry breaking, causing a weakly allowed 0-0 emission. The onset of the decrease as predicted by theory occurs at lower temperatures than as measured. In the next section, we discuss possible explanations for this discrepancy. 6. Summary, Discussion, and Conclusions We have presented a statistical-mechanical theory for the selfassembly of quasi one-dimensional helical aggregates in solution
van Dijk et al. and combined that with a quantum-mechanical tight-binding model to describe the photoexcitations on these aggregates. The combined theory allows us to calculate photoluminescence spectra in systems of supramolecular aggregates that not only adjust their size distribution to the ambient conditions but in addition can undergo a helix-coil type conformational transition. The assembly of this kind of supramolecular polymer can be highly co-operative and non isodesmic. We have applied the theory to OPV-4 aggregates that are thought to belong to this class of supramolecular polymers.7 The theory is able to qualitatively describe the initial red-shift of the 0-0 (zero phonon) peak in the photoluminescence spectrum with decreasing temperature. However, the theoretical red-shift is substantially larger than the experimental one at further decrease of the temperature. Below a temperature of 293 K, the theory overestimates the red-shift by about 0.07 eV when an effective maximum aggregate length of 60 is used to account for the mean exciton migration length. The decrease in intensity of the spectrum with decreasing temperature is reasonably predicted by the theory, albeit that it occurs at a too low temperature (see Figure 4h-i). Furthermore, Figure 4h and Figure 4i show different behavior: the theoretical red-shift starts to deviate from experiment below the helical transition temperature, whereas the disagreement between theory and experiment for the integrated intensity is most substantial above the helical transition temperature. When an effective maximum aggregate length of 13 oligomer pairs is used, the theoretical red-shift equals the experimentally observed red-shift below T ) 293 K. The fact that this length is relatively short is probably due to the fact that we consider the oligomer pairs to be single chromophores. However, due to disorder, the two chromophore energies of the dimer might not be equal. Therefore, our estimate of 2.471 eV for the average chromophore excitation energy might be too low. With a higher value of this excitation energy a higher effective maximum aggregate length will be found. In fact, Spano and collaborators obtain good agreement between their calculated spectrum and the experimental spectrum at a low temperature of T ) 278 K by using aggregates consisting of 20 oligomer pairs. We also note that the oligomer pairs are actually J-aggregates,12 and therefore their emission intensity is much higher than that of the long helical stacks, which are H-aggregates. Therefore, the oligomer pairs will dominate the emission spectra until almost all of them are in the aggregated state and this could explain why the theoretical red-shift starts to deviate from experiment below the helical transition temperature. The fact that in our theory the integrated intensity of the spectrum drops at a temperature somewhat below the temperature at which the experimental integrated intensity drops may be due to our assumption that the transfer integral between nonhelical bonds is zero. A nonzero but still small transfer integral could lead to some delocalization of the exciton and consequently to a reduction of the spectrum in the temperature region where predominantly nonhelical aggregates are formed. Also, transfer of excitons along the aggregates to quenching sites may occur by Fo¨rster transfer even without such a transfer integral. In our theoretical description we took a width of σ ) 77 meV for the energetic disorder distribution, which was extracted from the spectrum of single OPV-4 pairs. One may wonder if this value also applies to aggregates. Remarkably, this value is very close to the value σ ) 85 meV obtained by Spano and coworkers for their fit to all-helical aggregates.12 This indicates that the disorder in the on-site energies for the aggregates is
Helical Supramolecular Assemblies the same as for the single pairs, which suggests that the origin of the disorder is not associated with any property of the aggregated state. We put forward that this can only be explained by presuming that the fluctuations in the on-site energies of the chromophores are caused by the interaction of the chromophores with the solvent molecules. Note that in the aggregated state the side chains of the OPVs are strongly sterically hindered so one would expect to see a significant change in σ upon aggregation. Furthermore, the intermolecular separation along the stacks is found to be 0.375 nm,22 and accordingly the correlation length of l0 ) 4.5 found by Spano and co-workers12 corresponds to a distance of about 1.7 nm along the stacks, which is comparable to the bulk correlation length of the solvent. In our opinion, this highlights once again the important role of the solvent in supramolecular polymerization.7 Finally, an obvious improvement of the quantum-mechanical part of the theory would be the inclusion of exciton-phonon coupling because this would give the entire spectrum instead of only the 0-0 emission line. Including exciton-phonon coupling would, however, complicate our calculations significantly. The reason is that our methodology requires a sampling over the lowest-lying excited states of many aggregates of different length, disorder configurations and bond conformations. Because the numerical diagonalization of a tight-binding Hamiltonian is so easy, such a sampling is feasible with this Hamiltonian. Inclusion of exciton-phonon coupling leads to much more complicated Hamiltonians and excessively long sampling times. On the other hand, our calculations show that the helical transition in OPV-4 aggregates occurs over a relatively small temperature interval. At temperatures below this interval, the aggregates are almost perfectly helical and the sampling over the different conformational states could then possibly be avoided. In future work, we intend to investigate this possibility. Acknowledgment. This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”. The authors thank Prof. Frank Spano, Dr. Albert Schenning, and Dr. Stefan Meskers for valuable discussions.
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