The Effect of Chain Bending on the Photophysical Properties of

Article pubs.acs.org/JPCB

The Effect of Chain Bending on the Photophysical Properties of Conjugated Polymers Nicholas J. Hestand and Frank C. Spano* Department of Chemistry, Temple University, Philadelphia, Pennsylvania 19122, United States S Supporting Information *

ABSTRACT: The impact of chain bending on the photophysical properties of emissive conjugated polymers (CPs) is studied theoretically using Holstein-style Hamiltonians which treat vibronic coupling involving the ubiquitous vinyl/ring stretching mode nonadiabatically. The photophysical impact of chain bending is already evident at the level of an effective Frenkel Hamiltonian, where the positive exciton band curvature in CPs translates to negative excitonic coupling between monomeric units, as in J-aggregates. It is shown that the absorption and photoluminescence (PL) spectral line shapes respond very differently to chain bending. The misalignment of monomeric transition dipole moments with bending selectively attenuates the 0−0 PL peak intensity while leaving the 0−1 intensity practically unchanged, a property which is ultimately due to the uniquely coherent nature of the 0−0 peak. Hence, the 0−0/0−1 PL ratio, as well as the radiative decay rate, decrease with chain bending, effects that are more pronounced at lower temperatures where exciton coherence extends over a larger portion of the chain. Increasing temperature and/or static disorder reduces the exciton coherence number, Ncoh, thereby reducing the sensitivity to bending. In marked contrast, the absorption vibronic progression is far less sensitive to morphological changes, even at low temperatures, and is mainly responsive to the exciton bandwidth. The above results also hold when using a more accurate 1D semiconductor Hamiltonian which allows for electron−hole separation along the CP chain. The findings may suggest unique ways of controlling the radiative properties of conjugated polymer chains useful in applications such as organic light emitting diodes (OLEDs) and low-temperature sensors. henes;23,24 circular porphyrin nanorings;25−27 and virtually defect-free, bending-in-plane β-phase polyfluorene molecules.28 With the abundant diversity of possible backbone conformations available to CP chains, the question of the effect of morphology on the optical properties naturally arises. As shown in the pioneering works of Moerner and coworkers, single molecule spectroscopy (SMS) allows for observation and modeling of photophysical properties that are normally washed out in ensemble measurements, thereby providing a more comprehensive picture of the system in question.29 J. L. Skinner has contributed significantly to the field in his investigations of spectral diffusion and single molecule line shape distributions.30,31 SMS has also emerged as an important tool for understanding the complex dynamics and photophysical processes occurring in CP chains.20,32−43 In SMS studies on CP chains, the anisotropy of the polarized excitation and emission has been used to probe conformation.20,28,32,35,41,43 Highly anisotropic excitation spectra point

I. INTRODUCTION Since the discovery of conducting polymers,1−3 a great deal of attention has been devoted to understanding the electronic properties of single conjugated polymer (CP) chains. Although initial theoretical investigations dealt primarily with the properties of excited electrons and charges in ideal “straight” chains, it soon became apparent that polymer chain morphology has a profound effect on transport and photophysical properties, limiting the conjugation and diffusion lengths,4−14 both of which need to be optimized in organic electronic devices such as solar cells15−17 and light emitting diodes18,19 which rely on efficient energy and charge transport. Most CP assemblies assume complex morphologies; random coils, molten globules, toroids, rods, defect-coils, and defectcylinders are all possibilities depending on the chain stiffness, the interactions between chain segments, and the interactions between the chain and the surrounding medium.20 On the other hand, ordered and sometimes exotic CP structures are either known or have been suggested, including straight, practically disorder-free polydiacetylene chains polymerized within their monomer matrix;21 solvent induced, single molecule aggregates of polythiophene with a suggested helical structure;22 an extensive range of macrocyclic oligothio© 2014 American Chemical Society

Special Issue: James L. Skinner Festschrift Received: February 21, 2014 Revised: April 7, 2014 Published: April 29, 2014 8352

dx.doi.org/10.1021/jp501857n | J. Phys. Chem. B 2014, 118, 8352−8363

The Journal of Physical Chemistry B

Article

Figure 1. (a) Schematic representing a bent chain. Each monomer unit is represented by a black arrow. The bend angle θ is the angle between adjacent vectors, which taken together define the polymer backbone. The transition dipole moments (red arrows) make an angle θR with respect to the backbone. (b) A segment of a polythiophene chain in the all cis-conformation showing the transition dipole moments of each monomer. Here θR = 0.

extremely sensitive to chain bending unlike the analogous ratio in the absorption spectrum. The sensitivity is most pronounced at low temperatures, and diminishes with increasing temperature. The simple relationships derived for J-chains are shown to remain essentially unchanged in a more complex model based on direct band gap semiconductor chains with charge-separated (Wannier-like) excitations.

to well-ordered, elongated conformations, whereas strongly reduced anisotropy suggests collapsed, globular structures.20,43 Anisotropic fluorescence spectra do not necessarily convey the same information: fluorescence normally occurs from longer, lower energy segments subsequent to rapid population relaxation. On the basis of a pronounced red-shifted emission frequency, single chromophore emitters have been associated with ordered, linear CP segments or perhaps aggregates thereof. Conversely, multichromophore emission results from incomplete energy relaxation in more disordered polymers and is considerably blue-shifted. Yu and Barbara observed that the red-shifted single molecule emission derives from longer segments with reduced vibronic coupling, as the 0−0 peak is much larger than the 0−1 peak in the steady-state PL spectrum.34 A similar effect in the excitation spectrum was reported by Walter et al. where the 0−0/0−1 ratio was largest for segments probed at the lowest energies.38 In this work, we focus our attention on the impact of chain bending on the steady-state absorption and PL line shapes in CPs. Chain bending can result from defects or stress within the CP chain but can also occur in a more regular fashion, as, for example, within polythiophene hairpins, where all of the sulfur atoms are in the syn position.44 Chain bending in polyfluorene as evidenced by a reduced fluorescence anisotropy coincident with an increased line width was reported by Da Como et al.28 Our group has recently shown that fluorescent CPs behave photophysically like J-aggregates,45−48 a hypothesis supported by observations made by several groups.47−52 Linear Jaggregates exhibit photoluminescence (PL) 0−0/0−1 peak ratios and radiative decay rates which are proportional to the exciton coherence number and inversely proportional to the square root of temperature. This same behavior is consistent with observations by Schott and co-workers on essentially straight, defect-free polydiacetylene chains grown in situ from the monomer crystal.53−55 The similarity between J-aggregates and CPs ultimately follows from the fact that in both systems the curvature of the exciton band is positive at the gamma point, thus establishing the nodeless (κ = 0) exciton as the lowest in energy. We therefore begin by investigating the effects of bending on the photophysics of CP chains modeled as effective Frenkel exciton “J-chains” where the exciton is vibronically coupled to the intramolecular vibrational modes commonly observed in conjugated molecules. It is shown that the PL spectrumin particular, the 0−0/0−1 ratiois

II. MODELING MORPHOLOGY In what follows, we consider a chain of coupled two-level chromophore units which is uniformly “bent” such that a constant bend angle θ is maintained between adjacent units see Figure 1. A nonzero bend angle impacts the chain in two ways: it changes the value of the nearest neighbor electronic coupling Jn,n+1, and it misaligns the chromophore transition dipole moments as shown in the figure. Since, Jn,n+1(θ) is not a function of n, we can approximately describe the chain using periodic boundary conditions as was done earlier for straight chains.56 For a chain containing N chromophores, changing θ from 0 to 2π/N transforms the chain from a linear to a circular conformation. As shown in several of our previous papers, an (emissive) CP chain can be qualitatively described as a J-aggregate. This arises because the curvature of the exciton band at the gamma point is positive, consistent with a negative value for Jn,n+1 (see section V). We should point out, however, that Jn,n+1 in a CP derives mainly from through-bond coupling between adjacent monomeric units, unlike in a conventional J-aggregate, where Jn,n+1 is due to Coulombic (through-space) coupling. In either case, Jn,n+1 < 0, thereby ensuring that the lowest energy exciton is nodeless (κ = 0). The linear J-aggregate model of ref 56 considers an electronically excited state, treated as a Frenkel exciton (bound electron/hole pair), that couples locally to the prominent monomer vibrational mode corresponding to the vinyl-stretching/ring-stretching mode that is ubiquitous among conjugated polymers. The vibrations are treated nonadiabatically using a Holstein-style Hamiltonian. When periodic boundary conditions are invoked, the bent chain Hamiltonian is identical in form to the linear chain Hamiltonian. The Hamiltonian in the subspace containing a single electronic excitation is therefore represented in the k-, q- basis as56−58 8353



Article

with α labeling the states of a given κ and ñq=0 in order of increasing energy. The ket |ñq=0⟩ gives the number of totally symmetric phonons, while |k; nq1, nq2, ..., nqN−1⟩ represents the product state of an exciton with wavevector k having nql accompanying non-totally symmetric phonons with wavevector ql. The summations in eq 6 are restricted so that the total momentum quantum number κ satisfies eq 5. The associated eigenstate energies are given by

† Ĥ = ℏω0 − 0 + ℏωvibbq̃ = 0bq̃ = 0 + ℏωvib ∑ bq†bq q≠0

+ +

ℏωvibλ N

∑ k ,q≠0

{|k⟩⟨k + q|bq†

+ |k⟩⟨k − q|bq

∑ Jk̃ |k⟩⟨k| + (1 − 1/N )ℏωvibλ 2

} (1)

k

ℏω0−0 is the adiabatic electronic transition energy pertaining to a single (isolated) chromophore. |k⟩ represents a delocalized electronic excitation with wave vector k. In the quasi-1D chain, the dimensionless wave vector assumes the values k = 2πl/N, with l = 0, 1, ..., N − 1. q represents the wave vector of a delocalized vibrational mode. All such modes are taken to be dispersionless Einstein phonons based on the vinyl/ring stretching intramolecular mode with energy ℏωvib = 0.17 eV (≈1400 cm−1) . Similar to k, q is dimensionless and assumes the values q = 2πl/N, with l = 0, 1, ..., N − 1. The Hamiltonian in eq 1 shows that the totally symmetric (q = 0) phonon is special in that it decouples from all excitons. The second term represents the energy of the q = 0 mode, while the third term represents the energies of all non-totally symmetric phonons with q ≠ 0. Non-totally symmetric phonons couple linearly to the excitons, |k⟩, through the fourth term in Ĥ , where λ2 is the Huang−Rhys (HR) parameter. Finally, the exciton coupling is diagonal in k-space with Jk̃ in the fifth term given by Jk̃ = 2Jn , n + 1 cos(k)

Eκ , α , nq̃ =0 = ℏω0 − 0 + nq̃ = 0ℏωvib + ℏωκ , α

where ℏωκ,α are eigenvalues of Ĥ excluding the first two terms. As depicted in Figure 1, the bent chains considered here are limited to those in which the chromophore transition dipole moments lie in the (xy) plane. The transition dipole moments are taken to be at an angle θR relative to the backbone tangent. However, the model can be easily generalized by allowing θ to vary with n, or by introducing an n-dependent torsional angle between adjacent units, φn, needed to generate more complex three-dimensional conformations. All of our basic conclusions will remain valid as long as the electronic couplings remain at most weakly dependent on n so that κ remains approximately a good quantum number. In this regard, Beenkin has shown that torsional localization requires morphologies that include severe torsional angles (near 90°) between monomer units.59,60 Hence, we dismiss, for now, large changes in electronic couplings associated with kinks or abrupt and large torsional angles, which can induce significant exciton localization. As we show in section VI, however, we can understand such effects by generalizing the exciton coherence length, which appears in our expressions for spectral observables, to include the localizing influences of static disorder. The transition dipole moments appear in the expressions for spectral observables. The absorption and PL spectra are given by the respective equations

(2)

where Jn,n+1 is the nearest neighbor excitonic coupling assumed to be independent of n. The phonons in eq 1 are expressed in terms of the boson operator bq† =

1 N

∑ eiqnbn†

(3)

n

†

bq̃ = 0 = bq†= 0 +

∑

A(ω) =

where b†q (bn) creates (annihilates) a vibrational quantum on the nth chromophore. bq is the Hermitian conjugate of b†q . The totally symmetric mode is shifted relative to the non-totally symmetric modes by λ/√N λ N

(7)

fκ , α , ñ Γ(ω − ωκ , α − nq̃ = 0ωvib)

κ , α , nq̃ = 0

q=0

(8)

S(ω) = ⟨Iκ0,−αv, tnq̃ =0 Γ(ω − ωκ , α − nq̃ = 0ωvib + vt ωvib)⟩T

∑ ∑ vt

κ , α , nq̃ = 0

(4)

(9)

The vacuum state is defined relative to all unshifted modes. Hence, in the absence of electronic excitations, a single phonon with wave vector q is given by |q⟩ = b†q |vac⟩, where |vac⟩ is the vacuum state in which there are no vibrational quanta on all chromophores (or in all q modes). Translational symmetry dictates that the total quasimomentum

where the homogeneous line shape function, Γ, is taken to be Gaussian with standard deviation σ = 0.07ℏωvib in all calculations to follow. Absorption from the vibrationless ground state, |G⟩ ≡ |g⟩ ⊗ |vac⟩, to the exciton, |Ψκ,α,ñq=0>, is dictated by the line strength61 fκ , α , ñ

=

q=0

κ = k + nq q1 + nq q2 + ... + nq qN − 1 1

(5)

N−1

2

M̂ ≡

k

nq , nq ,..., nq 1

2

N−1

1

2

N−1

1

2

(11)

Here, μn is the transition dipole moment of the nth chromophore (or monomer unit), which generally depends on n. In a chain uniformly bent within the xy-plane, the monomeric transition dipole moment is given by

ckκ;,nαq , nq ,..., nq |k ; nq , nq , ..., nq ⟩

∑

∑ μn {|n⟩⟨g | + |g ⟩⟨n|} n

|Ψκ , α , nq̃ =0⟩ = |nq̃ = 0⟩

∑

(10)

where the chain transition dipole moment operator is

(mod(2π)) is a good quantum number. The number of totally symmetric phonons, ñq=0, also serves as a good quantum number. All eigenstates of Ĥ in eq 1 can therefore be expressed as56

⊗

1 |⟨Ψκ , α , nq̃ =0|M̂ |G⟩|2 μ2

N−1

μn = μ sin(nθ )i + μ cos(nθ )j

(6) 8354

(12)



Article

III. ABSORPTION AND PHOTOLUMINESCENCE IN BENT CHAINS AT LOW TEMPERATURE We now consider the effect of chain bending on the absorption and PL spectral line shapes in J-chains. We first treat systems in which the exciton is coherent across the entire length of the chain, i.e., a hypothetical chain with no disorder at low temperatures. Under these conditions, emission comes from only the lowest energy eigenstate of the system by Kasha’s rule.62 Thermal excitation is negligible when kBT is less than the energetic separation between the two lowest energy excitons. This condition results in the inequality kBT < 4πℏωc/N2,45,56 where ℏωc is the curvature of the lowest-energy exciton band near κ = 0,45 defined by making a parabolic approximation for the lowest-energy exciton band

where i and j are the normal Cartesian unit vectors and the bend angle θ is defined in Figure 1. The chain transition dipole moment operator M̂ is purely electronic. Its matrix elements between the ground electronic state |g⟩ and exciton states |k⟩ in a chain with bend angle θ are given by μ ⟨g |M̂ |k⟩ = ∑ eikn{sin(nθ)i + cos(nθ)j} N n (13) Hence, for a circular chain (θ = 2π/N), the vector matrix elements are Mk =

i N μ(δk ,2π / N − δk , −2π / N )i 2 1 N μ(δk ,2π / N + δk , −2π / N )j + 2

(14)

E(κ ) = E(0) + ℏωcκ 2

where the complex “i” is not to be confused with the unit vector along x. The Mk are finite only when k = ±2π/N. In the limit of a perfectly straight chain along the x-axis, the transition dipole moment matrix elements are Mk =

N μδk,0 i

where E(0) = Eκ=0,α=1,ñq=0=0 and the curvature is ℏωc ≡ (ℏ/2)(d2ωκ,α=1/dκ2), evaluated at κ = 0. For J-chains the curvature is positive so that the lowest energy exciton is nodeless with κ = 0. In our work, all energies are evaluated fully numerically via diagonalization of the Holstein Hamiltonian in eq 1 represented in the one- and two-particle basis set (see the Supporting Information). The band structure is quite complex, resulting in a series of vibronic bands, the lowest of which is described as ñq=0 = 0, α = 1.45 The band curvature for the lowest energy band is obtained by fitting the energies of the (κ, α = 1, ñq=0 = 0) excitons near κ = 0 to eq 20. The curvature is approximately equal to

(15)

and therefore require k = 0. Changing θ results in a redistribution of oscillator strength away from the k = 0 exciton. Matrix elements of the aggregate transition dipole moment 0−vt also govern light emission. The PL line strength Iκ,α,ñ q =0 appearing in S(ω) in eq 9 corresponds to the matrix element of M̂ between the excited state, |Ψκ,α,ñq=0⟩, and the ground electronic state with a total of vt vibrational quanta (phonons). For example, emission to the vibrationless ground state is governed by Iκ0,−α0, nq̃ =0 =

ℏωc = −FJn , n + 1

1 = 2 μ

∑ |⟨g ; nq

l

l

⎛ λ2 ⎞ 0, α = 1 2 F ≡ exp⎜ − ⎟|ckκ==0;0,0,...,0 | ⎝ N⎠

(16)

is a generalized FC factor ranging from exp(−λ ) for weak excitonic coupling (ωc ≪ λ2ωvib) to exp(−λ2/N) in the strong coupling limit (ωc ≫ λ2ωvib).56 The negative sign of Jn,n+1 ensures that the curvature is positive for a J-chain. Figure 2 shows calculated absorption and PL spectra for several bend angles in a J-chain with N = 40 chromophores and a constant nearest neighbor coupling of Jn,n+1 = −200 cm−1. For such calculations, all eigenstates and energies of the Hamiltonian in eq 1 are obtained numerically, from which the spectral observables are constructed. In the calculations of Figure 2, we have limited the impact of bending to the misalignment of monomeric transition dipole moments by neglecting the changes to the nearest neighbor coupling. This approximation is quite good considering that θ increases from zero to only 2π/40 (9°) in going from the linear to circular morphologies of Figure 2. Moreover, even if the coupling changes, it would do so in a uniform manner (independent of n), thereby leaving κ a good quantum number. Figure 2 shows that chain bending has a dramatic impact on the PL spectrum, in marked contrast to the absorption spectrum, which is largely insensitive. The high PL sensitivity is limited to the response of the 0−0 peak, which drops precipitously with bending while the satellite peaks remain essentially unchanged. It has previously been shown that for a perfectly straight J-aggregate at low temperatures (and in the absence of site disorder), the 0−0 peak increases linearly with

2

= 1|M̂ |Ψκ , α , nq̃ =0⟩|

(17)

(18)

and I 0 − 1 ≡ ⟨Iκ0,−α1, nq̃ =0⟩T

(22) 2

In eq 17, |g; nql = 1⟩ represents the electronic ground state with one phonon of wave vector ql with all other modes in the vacuum state. The brackets ⟨...⟩T appearing in S(ω) indicate a thermal (Boltzmann) average over exciton emitters. The thermal average of the line strength−line shape product in eq 9 is very nearly factorable, such that the line strengths in the vibronic progression in eq 9, as determined from spectral areas, are very accurately given by the thermal average of the line strengths in eqs 16 and 17. Hence, we define the thermal average over the 0−0 and 0−1 line strengths in eqs 16 and 17 simply as56 I 0 − 0 ≡ ⟨Iκ0,−α0, nq̃ =0⟩T

(21)

where

1 |⟨G|M̂ |Ψκ , α , nq̃ =0⟩|2 μ2

whereas emission to the ground electronic state with a single phonon is governed by Iκ0,−α1, nq̃ =0

(20)

(19)

Finally, we point out that cubic frequency dependencies in S(ω) have been suppressed in order to focus on a construct dependent solely on line strengths. S(ω) is more properly referred to as a reduced PL spectrum. 8355



Article

Equation 24 reduces to eq 23 in the limit that the bend angle θ approaches zero. With increased bending (θ > 0) eq 24 captures the decrease in SR due to the misalignment of the transition dipole moments. In the limit of a circular chain (θ = 2π/N) eq 24 reduces to SR = 0 reflecting a total cancellation of all transition dipole moments within the κ = 0 exciton. Figure 3 shows how the numerically evaluated line strengths as well as the PL ratio change with θ in a J-chain with N = 40

Figure 2. Absorption and PL spectra of bent J-chains. The 0−1 and 0−2 peaks in the PL spectrum are shown at 10× magnification in the insets. Notice that the spectra look nearly identical regardless of morphology except for the intensity of the 0−0 peak in the PL spectrum. The parameters used for these simulations are N = 40, Jn,n+1 = −200 cm−1, ℏωvib = 1400 cm−1, and λ2 = 1. The chain morphologies are defined by the bend angle, which is θ = 0, 2π/160, 2π/80, and 2π/ 40 in going from I to IV. The PL spectra are calculated at T = 0 K. Figure 3. The 0−0 and 0−1 line strengths and their ratio, SR, as a function of bend angle (in radians) for a J-chain with N = 40 at T = 0 K. The maximum bend angle, 2π/40, corresponds to a circular chain. The open circles are numerical results using Jn,n+1 = −200 cm−1. The red curves correspond to the analytical expressions shown in the insets. The blue curves for the 0−1 line strength and SR correspond to a more accurate expression derived from perturbation theory, eqs S9 and S10 in the Supporting Information. The generalized FC factor, F, is found numerically to be 0.465. Note that F exactly cancels out in the ratio, SR. Vibronic parameters are the same as those reported in Figure 2.

N while the sideband peaks are N-independent. This arises because the 0−0 peak is uniquely sensitive to the exciton’s coherence, and is therefore sensitive to the constructive interference (sum) of the monomeric transition dipole moments. In essence, the 0−0 line strength arises from the square of the sum over monomeric transition dipole moments within the exction coherence length and hence is sensitive to θ. Conversely, the sideband line strengths are due to the sum of the squares of the transition dipole moments, an incoherent construct that is independent of θ (and N). This important distinction was originally noted in ref 63 for herringbone aggregates. In a linear J-aggregate (θ = 0), the constructive interference is maximum at low temperature and in the absence of defects. In this case, the ratio of the 0−0 and 0−1 PL line strengths is given by56

using the same parameters as in Figure 2. The figure shows that the main impact of bending is felt in the 0−0 emission strength, with the 0−1 intensity remaining practically unchanged from its θ = 0 value of Fλ2.56 The expressions shown in the inset are exact for a linear chain (θ = 0) but remain an excellent approximation over all bend angles. The deviations between the full numerical results and the analytical expressions in red do, however, increase with |Jn,n+1| (not shown), but even for Jn,n+1 as high as −1000 cm−1, the deviations are not greater than about 5%. Higher couplings also induce a small θ-dependence to the 0−1 line strengths. For larger couplings, the blue curves shown in the figure and derived in the Supporting Information are more accurate for describing I0−1 and SR. We next consider the effects of bending on the absorption spectrum. For a linear chain, only the totally symmetric κ = 0 exciton states carry oscillator strength from the vibrationless ground state. However, bending the chain redistributes oscillator strength away from the κ = 0 states. We therefore expect there to be a response of the absorption spectrum to changes in morphology, even if Jn,n+1 remains unchanged. The reason we do not observe a noticeable change in the simulated

I 0−0 N kBT < 4π ℏωc /N 2 = 2 (23) I 0−1 λ which is rigorously exact over the whole phase space defined by ωvib, λ2, N, and ωc (as long as periodic boundary conditions are maintained). Furthermore, eq 23 remains accurate with increasing static or thermal disorder when N is replaced by the coherence number Ncoh.56 As we outline in the Supporting Information, an analogous formula to eq 23 can be derived for bent chains. To an excellent approximation, the PL ratio for systems suffering a constant bend is given by SR ≡

SR ≈

csc 2(θ /2) sin 2(Nθ /2) Nλ 2

kBT < 4π ℏωc /N 2 (24) 8356



Article

Hence, Ncoh = N at low temperatures. The effect of increasing temperature is included by averaging the emission over a steady state Boltzmann population of low-energy excitons. Since kBT ≪ ℏωvib, only the lowest vibronic band in the excited state manifold of states contributes to emission; hence, α = 1 and ñq=0 = 0. The contribution of the exciton with total momentum κ to the emission is weighted by the associated Boltzmann population factor

spectra in Figure 2 is due to the magnitude of the exciton coupling and the manner in which a bending morphology distributes the oscillator strength among exciton states. As shown in the Supporting Information, the majority of oscillator strength for a bending chain is distributed between κ = 0 and κ = ±2π/N. Furthermore, the magnitude of the coupling used in our simulations is small so that the energetic splitting between these states (= 4π2ωc/N2) is less than the line width. Therefore, two distinct peaks are not observed. In general, the absorption spectrum depends on the bend angle θ and the coupling Jn,n+1, even if the latter is independent of θ. As alluded to above, if we were to systematically increase the coupling while keeping all other parameters constant, then for bent systems, we would expect to observe a splitting of the 0−0 peak in the absorption spectrum as the energetic separation between the κ = 0 and κ = ±2π/N states increases. In addition, the 0-0/0-1 peak ratio in the absorption spectrum increases with the exciton bandwidth. For linear J-chains the ratio is given by,45,64 0−0 I Ab 0−1 I Ab

⎛ 1 + 0.24W /ℏωvib ⎞2 =⎜ ⎟ ⎝ 1 − 0.073W /ℏωvib ⎠

⎡ ΔE ⎤ Pκ = Z(N , T )−1 exp⎢ − κ ⎥ ⎣ kBT ⎦

(27)

where ΔEκ ≡ Eκ,1,0 − E0,1,0 is the energy difference between the exciton κ and the band bottom. Under the parabolic approximation (see eq 20), we have ΔEκ ≈ ℏωcκ2. Z(N, T) ≡ ∑κ exp[−ΔEκ/kBT] is the partition function. Effects of temperature on the chain structure are omitted, as well as the thermal broadening associated with lattice−phonon coupling. The latter does not impact the vibronic line strengths, I0−n, which are proportional to the spectral areas. Figure 4 shows the temperature profiles of the PL ratio associated with the chain morphologies of Figure 2 for several

λ2 = 1 (25)

where W is the free exciton bandwidth W ≡ 4|Jn , n + 1|

(26)

The 0-0/0-1 enhancement predicted by eq 25 is due to interband coupling between κ = 0 states in different vibronic bands. The equation accurately reproduces the 0-0/0-1 spectral ratio (≈1.35 ) observed in the absorption spectra of Figure 2, independent of bend angle. As long as one computes the ratio using the spectrally integrated vibronic bands, eq 25 with W = W(θ) is valid for all bend angles. Therefore, when considering the absorption spectrum, the exciton bandwidth, W, affects how the oscillator strength is distributed between the various vibronic bands, whereas the bend angle θ affects how the oscillator strength is distributed between states within the same vibronic band (i.e., among different κ states). Essentially, the absorption spectrum is less sensitive to bending compared to the PL spectrum, because the absorption spectrum abides by an oscillator sum rule. Bending redistributes oscillator strength. The PL spectrum lacks a sum rule so that in certain cases the 0-0 peak vanishes entirely.

Figure 4. Left side: The PL line strength ratio as a function of temperature for the N = 40 chains of Figure 2. The magnitude of the exciton coupling increases from top to bottom, while the vibronic parameters remain the same as those used in Figure 2. The numerically calculated ratio using a one- and two-particle basis set is plotted as hollow circles, and the ratio using the cosine modulated coherence function in eq 30 is plotted as crosses. The nearly perfect superposition of hollow circles and crosses attests to the accuracy of eq 30. The solid lines represent eq 31, which is accurate in the thermodynamic limit and therefore diverges from the numerical results at low temperatures where finite size effects dominate. Right side: The PL ratio plotted as a function of 1/T1/2. In linear J-chains (black) with λ2 = 1, Ncoh is simply the PL ratio (see eq 34). When Ncoh is approximately 10 or smaller, i.e., less than N/4, the PL ratio is no longer able to discriminate between morphologies. This is readily observed for the larger exciton couplings. All other parameters are given in Figure 2.

IV. THERMAL EFFECTS ON THE PHOTOLUMINESCENCE RATIO In section III, we considered completely coherent systems by neglecting the effects of thermal fluctuations and energetic disorder. The latter results from a nonuniform environment due primarily to the presence of a variety of defects. As the temperature and/or disorder increases, the exciton remains coherent over a limited number of monomer units, Ncoh, fewer than the total number N in the chain. It has been shown that, for linear J-aggregates, the PL ratio directly scales with Ncoh.56 The coherence number averaged over a Boltzmann distribution of exciton emitters diminishes with temperature, as reflected in an attenuated PL ratio at higher temperatures. In this section, we investigate the analogous situation for bent chains. We anticipate that the ratio should depend on Ncoh, as in straight chains, and also the bend angle θ, through the resulting dipole misalignment. We begin by examining the effects of temperature on the PL ratio in bent J-chains which are otherwise free of static disorder.

Jn,n+1. As can be seen, the ratios are strongly dependent on bend angle at low temperatures (kBT 4π ℏωc /N 2

(32)

For a circular aggregate, we find that Figure 5. The numerically computed thermally averaged coherence function for the N = 40 chains of Figure 2 at various temperatures. All parameters are the same as those reported in Figure 2. Note: the coherence function is the same for all four conformations of Figure 2.

S R (T ) ≈

1 λ2

4π ℏωc −4π 2ℏωc / N2kBT e kBT

kBT > 4π ℏωc /N 2

(33)

This is the same result as for a linear chain but modified by the Boltzmann factor which takes into account the activation energy required to thermally excite the κ = ±2π/N excitons, the only excitons that have oscillator strength to the vibrationless ground state and therefore the only excitons that contribute to 0−0 emission. As a result, the PL ratio is zero at low temperatures where only the κ = 0 exciton is populated. Interestingly, full numerical evaluations of the sideband intensities in a circular chainor a chain with any bend

delocalized over the entire system and Ncoh = N. As the temperature increases, the exciton localizes and Ncoh < N. Though not explicitly considered here, the localization also results when energetic disorder is introduced. Following ref 56, we are able to express the PL ratio for a bent chain in terms of the thermal coherence function as 8358



Article

Figure 6. The coherence function, Ncoh, diminishes with increasing temperature and/or disorder. When the exciton coherence extends over the entire system (a), the PL ratio is most sensitive to chain bending through the associated misalignment of the monomer transition dipole moments. When the exciton is localized (b), it can coherently experience only a portion of the chain, and thus, the PL ratio responds only to the local morphology. The green line represents the coherence function, while the gray lines define C̅ (s) = 0. The spheres represent the monomer units of a Jchain with transition dipole moments oriented in the bending plane.

Hence, when the product, N2kBT, is sufficiently small relative to the band curvature, the impact of chain bending on the PL ratio becomes observable. Using Ncoh(T) ≈ (4πℏωc/kBT)1/2, eq 36 can be rearranged to give

angleshow that, unlike the 0−0 emission, they do not approach zero at T = 0 K. Emission from a k ≠ 0 exciton can satisfy the Δκ = 0 selection rule by terminating on the ground electronic state with q ≠ 0 phonons.46,56 As shown in the Supporting Information, the 0−1 line strength remains steadfast at Fλ2, practically independent of temperature, as was shown previously for linear chains.56 With increasing temperature, the PL ratio in circular chains initially increases via thermal activation of the 0−0 emitting excitons having κ = ±2π/N and peaks when kBT is approximately equal to the energy difference between the κ = ±2π/N excitons and the bandbottom exciton (κ = 0). Equation 29 shows that, unlike the PL ratio, the coherence length is independent of bend angle, a rigorous conclusion as long as the associated changes in Jn,n+1 are small (so that the band curvature is approximately constant). The independence of Ncoh with θ arises because Jn,n+1 remains independent of n upon bending, thereby preserving κ as a quantum number (assuming, of course, periodic boundary conditions). Hence, the temperature dependence of Ncoh in the thermodynamic limit can be taken from the simple straight-chain case (θ = 0) where it was shown that Ncoh(T) ≈ (4πℏωc/kBT)1/2.45,56 This allows eq 31 to be recast into the most useful form S R (T ) ≈

Ncoh(T ) λ2

c(θ , T )

Ncoh > 0.18N

The extreme sensitivity of the PL ratio allows one to ponder the possibility of controlling the radiative properties of the material by controlling its morphology. Consider, for example, the radiative decay rate, krad, of an exciton. When the 0−0 emission is dominant, krad is proportional to the square of the 0−0 transition dipole moment matrix elements. One can readily show that k rad(T ) = FNcoh(T )c(θ , T )k mon

1 Nμ2

(34)

2

∑ e−ℏω κ /k T |Mκ |2 c

B

(35)

κ

The θ dependence arises through the distribution of |Mκ| . For example, as the chain bends, |Mκ|2 changes from Nμ2δκ,0 when θ = 0 to Nμ2(δκ,2π/N + δκ,−2π/N)/2 when θ = 2π/N. Equation 34 shows that the PL ratio is sensitive to the product of the coherence number and a conformation factor, c(θ, T). The latter is unity in a straight chain, where it is independent of temperature, and diminishes with chain bending. For a circular chain at low temperatures, c(θ, T) is zero but increases to almost unity with increasing temperature. Numerical evaluations of Ncoh and insertion into eq 34 yield results for SR that are indistinguishable from eq 30. Figure 4 shows that the PL ratio experiences the largest change in going from an unbent chain where c(θ = 0, T) = 1 to a circular chain. Thus, the quantity 1 − c(θ = 2π/N, T) gives the maximum percentage change in SR at a given temperature. A difference in SR of at least 10% between these two extremes yields the condition 2

N 2kBT < 40π 2ℏωc

(38)

where kmon is the radiative decay rate for a single monomer. Thus, since the 0−0 emission intensity is coherently enhanced, so is the radiative decay rate, a phenomenon known as superradiance.66 Like the 0−0 intensity, krad is also sensitive to chain bending, due to the misalignment of the monomeric transition dipoles occurring within the coherence length. Thus, for example, at low temperatures, superradiance exists in linear chains but is essentially quenched in a circular chain where c(θ, T) = 0. Interestingly, in both cases, the coherence extends over the entire chain length, allowing the transition dipole sum to achieve its maximum value in a linear chain and a value of zero in a circular chain. The intermediate morphologies can be viewed in terms of these extremes. It is interesting to think of the range of applications that could possibly exploit this effect. For instance, one could imagine an OLED where the emission intensity is controlled by the morphology of the material. It is also conceivable to design low temperature sensors based on the PL ratio and the morphology of the material.

where the chain conformation factor is given by c(θ , T ) ≡

(37)

V. EXTENSION TO POLYMER CHAINS MODELED AS ONE-DIMENSIONAL SEMICONDUCTORS Up to this point, we have focused on a Frenkel exciton model and shown how bending affects the vibronic progression in the absorption and PL spectra of J-chains. For CPs, a superior model includes through-bond coupling between monomer units and the formation of Wannier-like or, perhaps more accurately, charge-separated (CS) excitons in which the electron and hole are not bound to the same monomer unit. In ref 46, we generalized the Holstein Hamiltonian in eq 1 to account for through-bond coupling and charge separation, necessitating the introduction of separate HR factors for excited Frenkel-like states (in which the electron and hole reside at the same monomer), cation states (holes) and anion states (electrons). We obtained accurate numerical results for absorption and PL

(36) 8359



Article

spectra using an expanded three-particle basis set, the details of which can be found in ref 46. The main conclusions of refs 46 and 45 is that emissive CPs behave photophysically just like J-aggregates, thereby justifying our J-chain approximation in previous sections. The close analogy arises because of the direct band gap nature of CP semiconductors: the band curvature near the gamma point (κ = 0) is positive, so that the κ = 0 exciton is lowest in energy, as in J-aggregates. This is consistent with the results of perturbation theory, which gives an effective nearest neighbor coupling between monomer units of Jeff = −2teth/(U − V1), where te(th) is the electron (hole) transfer integral between monomer units and U − V1 is the Coulomb energy required to separate the electron and hole over the distance between neighboring monomers. For direct band gap semiconductors, the signs of te and th are the same (using a convention based on translational symmetry) so that Jeff is negative, as in a J-aggregate. Using the Hamiltonian found in eqs 1 and 3 of ref 46 and employing a three-particle basis set, we ran analogous simulations to the J-chain simulations of previous sections; Figures 7 and 8 are analogous to Figures 2 and 3 but for CP Figure 8. The 0−0 and 0−1 line strengths and their ratio, SR, as a function of bend angle (in radians) for a semiconducting (direct band gap) chain with N = 40 at T = 0 K. The open circles are numerical results using the Hamiltonian from ref 46 represented in a threeparticle basis set. The parameters defining the CP chain are te = th = −0.5 eV, U = 2 eV, V1 = 1 eV, ℏωvib = 1400 cm−1, λ2 = 1, and λ+2 = λ−2 = 0.5. The FC factor, F, is found numerically to be 0.528. Note that analytical expressions in red are the same as those for the J-chains in Figure 3 except for an added κ̅ factor in the I0−0 expression. The more accurate expression for I0−1 (blue curve) captures the slight deviations with bending and is given by eq S9 in the Supporting Information, using Jeff = −1600 cm−1.

curvature that is several times greater than that used to describe the J-chains in Figures 2 and 3. This allows the redistribution of oscillator strength with bending to be observed in the CPs of Figure 7: as the chain bends, the oscillator strength moves from the lowest energy κ = 0 exciton (θ = 0) to the second-lowest energy excitons with κ = ±2π/N in the circular aggregate. For the half-circle, the oscillator strength is (approximately) evenly divided between the κ = 0 and κ = ±2π/N excitons, yielding the split pair of peaks. Since the energy separation between the peaks is inversely proportional to N2, the two peaks should merge into a single peak as N increases beyond the value of 10 used in Figure 7. We have previously shown that eq 23 for a J-chain can be modified for 1D CP chains by the simple inclusion of a dimensionless factor (κ)̅ which accounts for the differences between the nuclear shifts in Frenkel-like versus ionic monomer units.46 Indeed, the expression shown in the inset of Figure 8 accurately describes SR over all bend angles using κ̅ = 0.76. This remains despite some obvious deviations in the computed 0−1 line strengths from Fλ2. Finally, and most importantly, the simple decomposition of the PL ratio into the product of the exciton coherence number and a morphology factor also holds for the semiconducting CP chains; the only modification required on eq 34 is the inclusion of the prefactor κ.̅ Hence, we have the general expression

Figure 7. Absorption and PL (0 K) spectra calculated using the Hamiltonian from ref 46, corresponding to the morphologies presented in Figure 2, but for N = 10 due to the large basis set needed for these calculations [i.e., (I) linear θ = 0, (II) quarter circle θ = 2π/40, (III) half circle θ = 2π/20, and (IV) circle θ = 2π/10]. The parameters used for these calculations are te = th = −0.5 eV, U = 2 eV, V1 = 1 eV, ℏωvib = 1400 cm−1, λ2 = 1, and λ+2 = λ−2 = 0.5.

chains modeled as one-dimensional semiconductors. The CP chains behave essentially the same as J-chains, with the PL ratio hypersensitive to bending while the absorption spectrum is only minimally impacted. One interesting difference appears in the absorption spectrum for a half circle (III) in Figure 7, where the main band is split into two closely spaced peaks. The effect is essentially due to the much larger effective coupling (curvature) present in the CP chains. The realistic parameters used to simulate the CPs (reported in the figure captions) yield a

S R (T ) ≈ κ ̅ 8360

Ncoh(T ) λ2

c(θ , T )

(39)



Article

the impact of additional localization (beyond the thermal localization considered here) reflected in a further reduction in the Ncoh factor. This is true since site disorder, which involves a distribution of monomeric transition energies, does not additionally impact the directions of the transition dipole moments. For linear J-aggregates in the presence of site inhomogeneous broadening, it was shown in ref 56 that the PL ratio remains accurately determined by Ncoh/λ2, the limiting form of eq 34 when θ = 0, as long as the width of the inhomogeneous distribution of monomer site energies is not much greater than approximately 0.1 times a vibrational quantumor about 150 cm−1 for the high frequency mode ubiquitous to CPs. If the static disorder results in both a change in the monomer site energies and the transition dipole moment directions, as will occur, for example, in a distribution of torsional angles, then eq 39 will be modified by a reduced Ncoh and a change in the conformation factor. In this case, a more fundamental starting point would be the thermally and configurationally averaged coherence function

which reduces to the expression derived previously for a linear CP found in ref 46, where c(θ, T) = 1. Finally, the radiative decay rate is modified from the J-chain expression in eq 38 though an additional factor of ϕ0, which is the probability that the electron and hole reside on the same monomer unit k rad(T ) = Fϕ0Ncoh(T )c(θ , T )k mon

(40)

Together, eqs 39 and 40 constitute the main results of this work.

VI. DISCUSSION/CONCLUSION In summary, we have investigated theoretically the impact of chain bending on the steady-state absorption and photoluminescence spectra of conjugated polymers. This has been accomplished using a J-aggregate Holstein-style model for polymer chains which treats vibronic coupling nonadiabatically. The model shows that the 0−0 peak in the PL spectrum as well as the radiative decay rate is highly sensitive to chain bending when the exciton coherence length is large enough for the exciton to sample a significant change in the alignment of the monomeric transition dipole moments. In marked contrast, the largely incoherent sideband intensities are practically independent of chain bending, making the PL ratio, I0−0/I0−1, an excellent probe of exciton coherence and chain conformation. The main result of our work is embodied in eq 39: the PL ratio is directly responsive to the product of the exciton coherence number Ncoh and a conformation factor c(θ, T). For straight chains, c(θ,T) is unity independent of temperature, and the PL ratio reduces to Ncoh(T)/λ2, as previously derived in ref 46. As the chain bends c(θ,T) becomes temperature dependent. At low temperatures (kBT < 4πℏωc/N2) c(θ,T) increases from zero in the limit of a circular chain, to unity for a linear chain, even though the exciton remains f ully delocalized with Ncoh = N. However, in the high temperature limit, c(θ, T) rises to unity independent of θ, so that the effect of chain bending disappears, and the PL ratio scales as the inverse square-root of temperature, typical of a linear J-aggregate.45,46,56 Very recent experiments on cycloparaphenylenes with varying numbers of constituent benzene rings (N) showed a clear increase of the PL ratio with N at room temperature.67 Within our formalism, this observation can be explained by an increase in just the 0−0 intensity due to a lowering of the thermal activation energy, 4π2ωc/N2 with N. Nishihara et al. also showed that the radiative decay rate scales with T−1/2, in accordance with the present theory in the thermodynamic limit. Though to the best of our knowledge the PL ratio as a function of temperature has not been measured in these systems, they may prove ideal for testing our theory. A very interesting case arises for a chain with irregular bending in which θ becomes n-dependent. If the chain remains in the xy-plane and the bending is not too extreme so that Ncoh = N, then the PL ratio is directly proportional to the square of the distance between the first and last chromophore. This is readily appreciated in the simple limits of a linear chain and a circular chain. Hence, the PL ratiowhich also tracks the radiative decay ratecould, in principle, be used to probe conjugated polymer structure, and elicit information about exciton coherence in devices such as OLEDs and photovoltaic devices. Although we have not explicitly considered other means of exciton localizationfor example, the presence of static or site energetic disorderwe expect eq 39 to remain accurate, with

CM(s) ≡

1 ⟨∑ ⟨Ψ(α)|Bn†Bn + s |Ψ(α)⟩μn ·μn + s ⟩T , C N n

(41) 2 −1

from which the PL ratio is found to be SR ≈ [CM(s = 0)λ ] ∑s CM(s). The approach presented here provides a spring-board for investigations into more exotic morphologies such as hairpin folds, suspected to occur in high molecular weight CP aggregates44 as well as spiral shaped “slinky” aggregates.22 The latter are particularly interesting, as the successive rings will interact electronically, forming “intramolecular aggregates” with H-like and J-like interactions. We are currently exploring such morphologies in greater detail with the goal of acquiring a more fundamental understanding necessary for applications in organic electronics.

■

ASSOCIATED CONTENT

S Supporting Information *

A more detailed description of the multiparticle basis set, derivations of analytical results, a demonstration of the insensitivity of the 0-1 PL peak to temperature, and the impact of open boundary conditions are presented. This material is available free of charge via the Internet at http://pubs.acs.org.

■ ■ ■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS F.C.S. is supported by the National Science Foundation, Grant No. DMR-1203811. REFERENCES

(1) Heeger, A. J.; Kivelson, S.; Schrieffer, J. R.; Su, W. P. Solitons in Conducting Polymers. Rev. Mod. Phys. 1988, 60, 781−850. (2) Patil, A. O.; Heeger, A. J.; Wudl, F. Optical Properties of Conducting Polymers. Chem. Rev. 1988, 88, 183−200. (3) Shirakawa, H.; Louis, E. J.; Macdiarmid, A. G.; Chiang, C. K.; Heeger, A. J. Synthesis of Electrically Conducting Organic Polymers Halogen Derivatives of Polyacetylene, (CH)X. J. Chem. Soc., Chem. Commun. 1977, 578−580. (4) Barford, W.; Lidzey, D. G.; Makhov, D. V.; Meijer, A. J. H. Exciton localization in disordered poly(3-hexylthiophene). J. Chem. Phys. 2010, 133, 044504.

8361



Article

(26) Hoffman, M.; Kärnbratt, J.; Chang, M.; Herz, L. M.; Albinsson, B.; Anderson, H. L. Enhanced π Conjugation around a Porphyrin[6] Nanoring. Agnew. Chem., Int. Ed. 2008, 47, 4993−4996. (27) Sprafke, J. K.; Kondratuk, D. V.; Wykes, M.; Thompson, A. L.; Hoffman, M.; Drevinskas, R.; Chen, W.; Yong, C. K.; Kärnbratt, J.; Bullock, J. E.; Malfois, M.; Wasielewski, M. R.; Albinsson, B.; Herz, L. M.; Zigmantas, D.; Beljonne, D.; Anderson, H. L. Belt-Shaped πSystems: Relating Geometry to Electronic Structure in a Six-Porphyrin Nanoring. J. Am. Chem. Soc. 2011, 133, 17262−17273. (28) Da Como, E.; Becker, K.; Feldmann, J.; Lupton, J. M. How Strain Controls Electronic Linewidth in Single β-Phase Polyfluorene Nanowires. Nano Lett. 2007, 7, 2993−2998. (29) Moerner, W. E. A Dozen Years of Single-Molecule Spectroscopy in Physics, Chemistry, and Biophysics. J. Phys. Chem. B 2002, 106, 910−927. (30) Reilly, P. D.; Skinner, J. L. Spectral Diffusion of Single Molecule Fluorescence: A Probe of Low-Frequency Localized Excitations in Disordered Crystals. Phys. Rev. Lett. 1993, 71, 4257−4260. (31) Geva, E.; Skinner, J. L. Theory of Single-Molecule Optical LineShape Distributions in Low-Temperature Glasses. J. Phys. Chem. B 1997, 101, 8920−8932. (32) Barbara, P. F.; Gesquiere, A. J.; Park, S.-J.; Lee, Y. J. SingleMolecule Spectroscopy of Conjugated Polymers. Acc. Chem. Res. 2005, 38, 602−610. (33) Kim, D. Y.; Grey, J. K.; Barbara, P. F. A detailed single molecule spectroscopy study of the vibronic states and energy transfer pathways of the conjugated polymer MEH-PPV. Synt. Met. 2006, 156, 336−345. (34) Yu, Z. H.; Barbara, P. F. Low-temperature single-molecule spectroscopy of MEH-PPV conjugated polymer molecules. J. Phys. Chem. B 2004, 108, 11321−11326. (35) Lupton, J. M. Single-Molecule Spectroscopy for Plastic Electronics: Materials Analysis from the Bottom-Up. Adv. Mater. 2010, 22, 1689−1721. (36) Da Como, E.; Borys, N. J.; Strohriegl, P.; Walter, M. J.; Lupton, J. M. Formation of a Defect-Free pi-Electron System in Single betaPhase Polyfluorene Chains. J. Am. Chem. Soc. 2011, 133, 3690−3692. (37) Lupton, J. M.; Becker, K. Dual Species Emission from Single Polyfluorene Molecules: Signatures of Stress-Induced Planarization of Single Polymer Chains. J. Am. Chem. Soc. 2005, 127, 7306−7307. (38) Walter, M. J.; Lupton, J. M. Unraveling the Inhomogeneously Broadened Absorption Spectrum of Conjugated Polymers by SingleMolecule Light-Harvesting Action Spectroscopy. Phys. Rev. Lett. 2009, 103, 167401. (39) Mirzov, O.; Scheblykin, I. G. Photoluminescence spectra of a conjugated polymer: from films and solutions to single molecules. Phys. Chem. Chem. Phys. 2006, 8, 5569−5576. (40) Lin, H. Z.; Hania, R. P.; Bloem, R.; Mirzov, O.; Thomsson, D.; Scheblykin, I. G. Single chain versus single aggregate spectroscopy of conjugated polymers. Where is the border? Phys. Chem. Chem. Phys. 2010, 12, 11770−11777. (41) Bolinger, J. C.; Traub, M. C.; Brazard, J.; Adachi, T.; Barbara, P. F.; Vanden Bout, D. A. Conformation and Energy Transfer in Single Conjugated Polymers. Acc. Chem. Res. 2012, 45, 1992−2001. (42) Adachi, T.; Brazard, J.; Ono, R. J.; Hanson, B.; Traub, M. C.; Wu, Z. Q.; Li, Z. C.; Bolinger, J. C.; Ganesan, V.; Bielawski, C. W.; Bout, D. A. V.; Barbara, P. F. Regioregularity and Single Polythiophene Chain Conformation. J. Phys. Chem. Lett. 2011, 2, 1400−1404. (43) Huser, T.; Yan, M.; Rothberg, L. J. Single chain spectroscopy of conformational dependence of conjugated polymer photophysics. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 11187−11191. (44) Noriega, R.; Rivnay, J.; Vandewal, K.; Koch, F. P. V.; Stingelin, N.; Smith, P.; Toney, M. F.; Salleo, A. A general relationship between disorder, aggregation and charge transport in conjugated polymers. Nat. Mater. 2013, 12, 1038−1044. (45) Yamagata, H.; Spano, F. C. Strong Photophysical Similarities between Conjugated Polymers and J-aggregates. J. Phys. Chem. Lett. 2014, 5, 622−632. (46) Yamagata, H.; Spano, F. C. Vibronic coupling in quantum wires: Applications to polydiacetylene. J. Chem. Phys. 2011, 135, 054906.

(5) Tozer, O. R.; Barford, W. Exciton Dynamics in Disordered Poly(p-phenylenevinylene). 1. Ultrafast Interconversion and Dynamical Localization. J. Phys. Chem. A 2012, 116, 10310−10318. (6) Dykstra, T. E.; Hennebicq, E.; Beljonne, D.; Gierschner, J.; Claudio, G.; Bittner, E. R.; Knoester, J.; Scholes, G. D. Conformational Disorder and Ultrafast Exciton Relaxation in PPV-family Conjugated Polymers. J. Phys. Chem. B 2009, 113, 656−667. (7) Van Averbeke, B.; Beljonne, D. Conformational Effects on Excitation Transport along Conjugated Polymer Chains. J. Phys. Chem. A 2009, 113, 2677−2682. (8) Collini, E.; Scholes, G. D. Coherent Intrachain Energy Migration in a Conjugated Polymer at Room Temperature. Science 2009, 323, 369−373. (9) Hwang, I.; Scholes, G. D. Electronic Energy Transfer and Quantum-Coherence in pi-Conjugated Polymers. Chem. Mater. 2011, 23, 610−620. (10) Rossi, G.; Chance, R. R.; Silbey, R. Conformational Disorder in Conjugated Polymers. J. Chem. Phys. 1989, 90, 7594−7601. (11) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R. Conformational dynamics of photoexcited conjugated molecules. Phys. Rev. Lett. 2002, 89, 097402. (12) Schwartz, B. J. Conjugated polymers as molecular materials: How chain conformation and film morphology influence energy transfer and interchain interactions. Annu. Rev. Phys. Chem. 2003, 54, 141−172. (13) Westenhoff, S.; Daniel, C.; Friend, R. H.; Silva, C.; Sundstrom, V.; Yartsev, A. Exciton migration in a polythiophene: Probing the spatial and energy domain by line-dipole Forster-type energy transfer. J. Chem. Phys. 2005, 122, 094903. (14) Bjorklund, T. G.; Lim, S.-H.; Bardeen, C. J. The optical spectroscopy of poly(p-phenylene vinylene)/polyvinyl alcohol blends: from aggregates to isolated chromophores. Syn. Met. 2004, 142, 195− 200. (15) Gunes, S.; Neugebauer, H.; Sariciftci, N. S. Conjugated polymerbased organic solar cells. Chem. Rev. 2007, 107, 1324−1338. (16) Brabec, C. J.; Gowrisanker, S.; Halls, J. J. M.; Laird, D.; Jia, S. J.; Williams, S. P. Polymer-Fullerene Bulk-Heterojunction Solar Cells. Adv. Mater. 2010, 22, 3839−3856. (17) Bredas, J.-L.; Norton, J. E.; Cornil, J.; Coropceanu, V. Molecular Understanding of Organic Solar Cells: The Challenges. Acc. Chem. Res. 2009, 42, 1691−1699. (18) Burroughes, J. H.; Bradley, D. D. C.; Brown, A. R.; Marks, R. N.; Mackay, K.; Friend, R. H.; Burns, P. L.; Holmes, A. B. Light-Emitting Diodes Based on Conjugated Polymers. Nature 1990, 347, 539−541. (19) Sirringhaus, H.; Tessler, N.; Friend, R. H. Integrated optoelectronic devices based on conjugated polymers. Science 1998, 280, 1741−1744. (20) Hu, D. H.; Yu, J.; Wong, K.; Bagchi, B.; Rossky, P. J.; Barbara, P. F. Collapse of stiff conjugated polymers with chemical defects into ordered, cylindrical conformations. Nature 2000, 405, 1030−1033. (21) Lécuiller, R.; Berréhar, J.; Lapersonne-Meyer, C.; Schott, M. Dual Resonance Fluorescence of Polydiacetylene Chains Isolated in their Crystalline Monomer Matrix. Phys. Rev. Lett. 1998, 80, 4068− 4071. (22) Kiriy, N.; Jahne, E.; Adler, H. J.; Schneider, M.; Kiriy, A.; Gorodyska, G.; Minko, S.; Jehnichen, D.; Simon, P.; Fokin, A. A.; Stamm, M. One-dimensional aggregation of regioregular polyalkylthiophenes. Nano Lett. 2003, 3, 707−712. (23) Iyoda, M. Syntheses, Structures, and Supramolecular Properties of Giant π-Expanded Macrocyclic Oligothiophenes. Heteroat. Chem. 2007, 18, 460−466. (24) Donehue, J. E.; Varnavski, O. P.; Cemborski, R.; Iyoda, M.; Goodson, T., III. Probing Coherence in Synthetic Cyclic LightHarvesting Pigments. J. Am. Chem. Soc. 2011, 133, 4819−4828. (25) Chang, M.; Hoffmann, M.; Anderson, H. L.; Herz, L. M. Dynamics of Excited-State Conformational Relaxation and Electronic Delocalization in Conjugated Porphyrin Oligomers. J. Am. Chem. Soc. 2008, 130, 10171−10178. 8362



Article

(47) Paquin, F.; Yamagata, H.; Hestand, N. J.; Sakowicz, M.; Bérubé, N.; Côté, M.; Reynolds, L. X.; Haque, S. A.; Stingelin, N.; Spano, F. C.; Silva, C. Two-dimensional spatial coherence of excitons in semicrystalline polymeric semiconductors: Effect of molecular weight. Phys. Rev. B 2013, 88, 155202. (48) Yamagata, H.; Hestand, N. J.; Spano, F. C.; Köhler, A.; Scharsich, C.; Hoffmann, S. T.; Bässler, H. The red-phase of poly[2methoxy-5-(2-ethylhexyloxy)-1,4-phenylenevinylene] (MEH-PPV): A disordered HJ-aggregate. J. Chem. Phys. 2013, 139, 114903. (49) Niles, E. T.; Roehling, J. D.; Yamagata, H.; Wise, A. J.; Spano, F. C.; Moule, A. J.; Grey, J. K. J-aggregate behavior in poly-3hexylthiophene nanofibers. J. Phys. Chem. Lett. 2012, 3, 259−263. (50) Kohler, A.; Hoffmann, S. T.; Bassler, H. An Order-Disorder Transition in the Conjugated Polymer MEH-PPV. J. Am. Chem. Soc. 2012, 134, 11594−11601. (51) Baghgar, M.; Labastide, J.; Bokel, F.; Dujovne, I.; McKenna, A.; Barnes, A. M.; Pentzer, E.; Emrick, T.; Hayward, R.; Barnes, M. D. Probing Inter- and Intrachain Exciton Coupling in Isolated Poly(3hexylthiophene) Nanofibers: Effect of Solvation and Regioregularity. J. Phys. Chem. Lett. 2012, 3, 1674−1679. (52) Baghgar, M.; Pentzer, E.; Wise, A. J.; Labastide, J. A.; Emrick, T.; Barnes, M. D. Cross-Linked Functionalized Poly(3-hexylthiophene) Nanofibers with Tunable Excitonic Coupling. ACS Nano 2013, 7, 8917−8923. (53) Lécuiller, R.; Berréhar, J.; Ganiére, J. D.; Lapersonne-Meyer, C.; Lavallard, P.; Schott, M. Fluorescence yield and lifetime of isolated polydiacetylene chains: Evidence for a one-dimensional exciton band in a conjugated polymer. Phys. Rev. B 2002, 66, 125205. (54) Dubin, F.; Melet, R.; Barisien, T.; Grousson, R.; Legrand, L.; Schott, M.; Voliotist, V. Macroscopic coherence of a single exciton state in an organic quantum wire. Nat. Phys. 2006, 2, 32−35. (55) Schott, M. Optical properties of single conjugated polymer chains (Polydiacetylenes). In Photophysics of molecular materials: from single molecules to single crystals; Lanzani, G., Ed.; Wiley-VCH: Weinheim, Germany, 2006; pp 49−145. (56) Spano, F. C.; Yamagata, H. Vibronic Coupling in J-Aggregates and Beyond: A Direct Means of Determining the Exciton Coherence Length from the Photoluminescence Spectrum. J. Phys. Chem. B 2011, 115, 5133−5143. (57) Holstein, T. Polaron motion. I. Molecular-crystal model. Ann. Phys. 1959, 8, 325−342. (58) Scherer, P. O. J.; Fischer, S. F. On the theory of vibronic structure of linear aggregates. Application to pseudoisocyanin (PIC). Chem. Phys. 1984, 86, 269−283. (59) Beenken, W. J. D.; Pullerits, T. Spectroscopic Units in Conjugated Polymers: A Quantum Chemically Founded Concept? J. Phys. Chem. B 2004, 108, 6164−6169. (60) Beenken, W. J. D. Excitons in conjugated polymers: Do we need a paradigma change? Phys. Status Solidi A 2009, 206, 2750−2756. (61) Here the linear frequency dependence in the oscillator strength is omitted, since the vibrational frequency is much smaller than the optical transition frequency. (62) Kasha, M. Characterization of Electronic Transitions in Complex Molecules. Discuss. Faraday Soc. 1950, 9, 14−19. (63) Spano, F. C. The fundamental photophysics of conjugated oligomer herringbone aggregates. J. Chem. Phys. 2003, 118, 981−994. (64) Spano, F. C. The Spectral Signatures of Frenkel Polarons in Hand J-Aggregates. Acc. Chem. Res. 2010, 43, 429−439. (65) Although eq 31 is valid at high temperatures, kT cannot exceed a vibrational quantum. This condition is, however, easy to satisfy, since a vibrational quantum is about 7 times kT at room temperature. (66) Potma, E. O.; Wiersma, D. A. Exciton superradiance in aggregates: The effect of disorder, higher order exciton-phonon coupling and dimensionality. J. Chem. Phys. 1998, 108, 4894−4903. (67) Nishihara, T.; Segawa, Y.; Itam, K.; Kanemitsu, Y. Exciton recombination dynamics in nanoring cycloparaphenylenes. Chem. Sci. 2014, in press.

8363


The Effect of Chain Bending on the Photophysical Properties of

Recommend Documents