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Keywords:

  • computer modeling;
  • conjugated polymers;
  • simulations;
  • solar cells;
  • theory

Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES

Organic materials, in particular conjugated polymers, have recently become the subject of extensive research for photovoltaic device applications. This increase of interest is primarily the result of their potentially low manufacturing cost, compatibility with flexible substrates, diverse chemical tunability, scalability, and ease of processing currently available for suitable bulk heterojunction (BHJ) construction. However, to date, these materials have not been able to exceed power conversion efficiencies (PCE) beyond 5–9%, values short of those considered commercially viable. The deficit in PCE appears to derive from a combination of physicochemical and device complexities associated with inadequate hole transport mobility, solubility and miscibility with an appropriate acceptor, narrow electronic band gap for efficient solar light harvesting, appropriate highest occupied molecular orbital (HOMO) and lowest unoccopied molecular orbital (LUMO) energies to maximize the open-circuit voltage (Voc) and electron transfer to the acceptor, and in particular the control of the multidimensional problem of BHJ morphology. In this review article, we provide an overview of some of the recent progress toward implementing theory, modeling, and simulation approaches in combination with results from precision synthesis, characterization, and device fabrication as a mean to overcome/understand the inherent issues that limit practical applications of organic photovoltaics. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012


INTRODUCTION AND GENERAL OVERVIEW

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES

The bulk heterojunction (BHJ) design1 is one promising strategy to access large-scale organic photovoltaics (OPVs) to address some of the world's energy harvesting needs.2 OPVs based on semiconducting π-conjugated materials are intensely pursued because of their potentially low manufacturing cost, compatibility with flexible substrates, tunability, and ease of processing.3 BHJ designs blend a conjugated polymer (CP) or conjugated oligomer (CO; see Fig. 1) as the electron donor with an electron-accepting material [e.g., a fullerene like [6,6]-phenyl-C61-butyric acid methyl ester (PCBM)] to form an interpenetrating active layer of a solar cell device.4–8 The CO/CP must boast high hole transport mobility, good solubility, miscibility with the acceptor, a small band gap for efficient solar light harvesting [often facilitated by a donor–acceptor (D–A) arrangement in the main chain; Fig. 1(a,b)], and appropriate HOMO and LUMO energies to maximize the open-circuit voltage (Voc) and electron transfer to the acceptor, respectively. The control of BHJ morphology is also important. Clearly, this multidimensional problem, when considered in the context of the other requirements outlined above, constitutes one of the core challenges in OPV optimization. Despite intense efforts, organic materials currently available for BHJ construction are still challenged to exceed power conversion efficiencies (PCE) beyond 5–9%, which unfortunately falls short of those considered to be commercially viable (however, one should note that <10% PCE and <10 year lifetime could still potentially allow for the generation of electricity at a price equal or inferior to that of grid power9). There is no clear path to improve such relatively poor efficiencies in spite of the fact that the community has become particularly adept at designing (e.g., using electronic structure calculations10, 11) and preparing, through organic synthesis, π-conjugated constructs with desirable optoelectronic and stability characteristics.12 The lack of progress can be explained in part by the fact that the optoelectronic properties that have been tailored at the molecular level are modulated in, often, unpredictable ways by multicomponent interactions in solution,13 in bulk,14 and more significantly at interfaces.15 As such, tailoring the morphology of the active molecules in a specific supramolecular environment has quickly become an emphasis in the quest to optimize charge injection/separation/transport and to achieve acceptable device performance. However, a majority of the research efforts have been conducted using a trial-and-error methodology, and to date, have generated a large amount of mainly limited information in regard to the material morphologies needed to improve the PCE. Briefly, based on a recent report by Dang et al.,16 the extensive literature on organic solar cells included 1953 publications in the year 2010 alone. A similar literature search using the key words “organic solar cells” for the period of January 2011–January 2012 returns 2074 publications, showing that the prolific trend in reported studies continues. In the literature, the dominant organic system studied is composed of a blend of poly(3-hexylthiophene) (P3HT) and PCBM for which measured PCEs are found to reach values of up to about 4–5%. Figure 2 summarizes the results reported for PCE and clearly illustrates the variability obtained for a single organic system. From a statistical point of view, Figure 2 has a very large variance with a mean value of ∼3%. Clearly, the BHJ design of P3HT/PCBM involves a large level of complexity that is highly sensitive to materials preparation and device fabrication details. It follows that it is difficult to make substantial improvements and that a brute force trial-and-error approach is unlikely to achieve immediate success (however, one can note a good incremental progress; see refs.17–21).

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Figure 1. Generic donor (D)–acceptor (A) polymer (a) used in OPVs and D–A oligomer arrangements (b).

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Figure 2. Power conversion efficiencies of P3HT-/PCBM-based solar cells reported in 579 publications from a screened survey. (Adapted from ref.16, with permission from Wiley-VCH Verlag.)

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Concerning materials characterization, neutron scattering could be of potential help for better elucidation of the “nonequilibrium” structure of the active OPV layer. Neutrons have an important advantage in that fullerenes and hydrogenous polymers show a strong neutron contrast, allowing a structural determination by reflectometry of which component is enriched at the top and bottom interfaces.21 Additionally, combined with mesoscale simulation and modeling of neutron reflectometry for polymers may offer key insight. In particular, the generalized field theory (FT) approach (see “Field Theory” section for more details on FT) can be applied to model polymer aggregation in solution and on a surface. Environmental conditions of solvent quality, concentration, composition, and nanoconfinement of thin films or brushes can be modeled, and the computed density profiles can be used to simulate the neutron reflectivity traces for a direct comparison with the experiments on similar systems. Another important advantage of neutrons is that the solvent can be deuterated, or solvent mixtures can be used to show a blend component to more selectively track the migration of one component during evaporation. Given the intimate connection between structure and performance in OPV devices, an improved understanding of how to manipulate structure through processing or chemical modification to yield selective interactions with the appropriate surface could yield a substantial payoff.

Why have polythiophenes (PTs) emerged as one of the principal materials of choice for use in BHJ solar cells (Fig. 2)? The answer is multifaceted: in part it is due to the ability of PTs to absorb light across the entire visible spectrum. It is also due to the inherent high carrier (hole) mobility when compared with other conjugated soft materials.22–25 Also critical is the ability to synthetically control molecular weight (recent work on thiophene-based polymers has demonstrated how molecular weight has drastic consequences on molecular orientation and subsequently PCE; see ref.26) and to tailor the PT backbone with alkyl substitutents that induce aggregation into crystalline π-stacked domains on the size scale of tens of nanometers.27 Additionally, some of the desirable photophysical properties likely originate from the existence of low-lying triplet states that are accessible from singlet fission, provided there is proper electronic alignment.28–32 Triplet electronic states generate considerably long-lived excitonic states that can subsequently increase PCE.33 Although this is a potentially exploitable aspect for improving PCE, it is also one aspect that may add further complexity to understanding how to optimize P3HT-based materials. In fact, to date, it has been difficult to satisfactorily map out accurate structure–property–transport relationships for this class of materials in the thin film morphologies of BHJs. To move beyond the sample preparation and environmentally and time-sensitive photophysical properties of P3HT-based BHJs, one key aspect is a detailed understanding of the inherent thin-film molecular structure of P3HT and a blend of P3HT/PCBM so that the underlying energy landscape that controls the time-dependent evolution and interfacial structures can be fully understood. In such a material, interfacial interactions between the donor and acceptor domains become critical as even small changes in the orientation of a P3HT chain relative to an acceptor like PCBM can drastically decrease charge transfer (it is desirable to have charge rectification at the interface to reduce the possibility of recombination). As has been recently experimentally verified,34 charge transport properties are extremely sensitive to the polymer aggregation states (crystalline-like domains) and also to the spatial location in the film. In this regard, molecular structure factors such as P3HT/PCBM loadings, P3HT side chain size, molecular weight, and polydispersity index (PDI) all strongly affect P3HT domain aggregation. Comparison of one P3HT/PCBM system to another may indeed be difficult if the polydispersity (P3HT/PCBM is not a binary system) is not the same. Even small variances in PDI can cause broad morphology changes35, 36 as well as introduce other local modifications that can lead to complex phenomena like jamming and non-Fickean diffusion.37 Altogether, a thin film of P3HT/PCBM presents an inherent heterogeneous multicomponent material structure.

To facilitate progress toward untangling all of these inherent complexities, it is important for theory, modeling, and simulation approaches to be implemented in synergy with experiments. Computational models can provide a number of details relevant to the explicit nanoscale self-assembly of P3HT and P3HT/PCBM blends and as such probe how moving away from a perfect PDI would alter the process. Ideally, one would like to be able to perform a rational design of the material to optimize its OPV performance. Although this is currently not fully possible, active progress is being pursued toward that end.

Below we give a brief overview of some of the primary computational modeling tools that have been deployed to study physicochemical properties/processes in OPVs. Our discussion in this regard is not exhaustive, as we have restricted attention toward aspects of electronic structure methods, classical particle-based methods, and mesoscale field theoretic approaches. Although there are various other approaches, including the use of continuum-based models,38–40 these are not explicitly addressed.

BACKGROUND ON SIMULATION APPROACHES

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES

The generation and decay of excitonic states play a critical role in both optoelectronic and photovoltaic systems. These quasi-particles, generated primarily by photoexcitation, are held together by Coulombic interactions.41 At the molecular level, excitonic states can correspond to localized or delocalized electron-hole densities that extend over the underlying material. Appropriate electronic coupling to charge-transfer states42 can give rise to efficient exciton dissociation,43 which is one key to enhancing OPV efficiency (see the schematic in Fig. 3 showing the formation of a charge-transfer exciton/exciplex). For semiconductor heterojunctions, exciton decay toward a charge-separated state is strongly influenced/controlled by molecular-level electronic interactions at the interface. In addition to the important energetic criteria of the band-offset (Fig. 3), fast charge transfer is another critical component to the overall PV efficiency. Reducing the probability of exciton recombination in the donor phase by strong wavefunction overlap to excited states can open channels for fast adiabatic electron transfer.44–46 By engineering the molecular chemistry at the surface, the electronic structure can be modified to promote charge-transfer bridging states44 that can facilitate transfer47 as well as significantly modulate the open-circuit voltage (Voc).48 Similarly, chemical modification of a semiconducting polymer backbone can potentially be exploited to manipulate the underlying electronic structure.49

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Figure 3. A schematic diagram depicting the dissociation of a singlet exciton to form a charge-transfer exciton/exciplex (dashed oval) across a donor–acceptor interface. EEX is the single exciton binding energy and ECT is the charge-transfer exciton binding energy. (Adapted from ref.42, with permission from American Physical Society Publications.)

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Experimentally measureable quantities that directly relate to device performance metrics such as the PCE include50 short-circuit current (Jsc), open-circuit voltage (Voc), and fill factor (FF). The product of these three quantities divided by the total solar input power yields the PCE. Jsc depends on the external quantum efficiency associated with fundamental molecular and nanoscale processes, whereas Voc is still not well understood. Many authors equate it to the energetic difference between the ionization potential (HOMO) of the donor and the electron affinity (LUMO of the acceptor).12 Because such information is a direct outcome of the underlying electronic structure, quantum chemistry and electronic structure calculations can provide a direct means for accessing them and making appropriate correlations.10, 51–53

Theoretical and computational chemical physics and materials sciences have reached a status that is ideally positioned to offer considerable aid in the quest to understand materials structure–property–transport relationships. This status is fundamentally promising because structure and properties of molecules, solids, and liquids are direct expressions of the underlying quantum motion of their electrons. Additionally, the rapid progression of computational hardware (1000× performance in the last 5 years) with the power to numerically solve the key underlying differential equations for large system size often means that realistic system sizes in relevant environments can now be simulated.

On the algorithm and theory side, the advent of semilocal and especially nonlocal descriptions of exchange and correlation effects within the density functional theory (DFT) now allows describing bonding in molecules and solids with a sufficient accuracy to compare quantitatively to experiments for many classes of systems. It is therefore becoming possible to develop a semiquantitative description of a large number of systems and processes relevant to PV development. As such, we briefly review DFT and its various extensions to include nonlocal terms that are important for long-range dispersion interactions that dominate many self-assembly processes, molecular surface adsorption processes, solution processes, and biological and polymeric materials. Additionally, as computational simulation of many physicochemical processes, especially those related to fabrication of materials, often requires using direct molecular dynamics (MD) simulations or methods that allow treatment of systems under extreme environments, we also review new quantum dynamics methodologies. We note that considerable progress is still needed in designing efficient electronic structure algorithms that can accurately treat excited state processes and that can speed up the required evaluation of energy and forces (see “Prospects on Improving Computational Approaches” section). Some relatively short-term hope may arise from new computer architectures and programming paradigms such as general programming of graphical processing units (GPGPU).

Electronic Structure: DFT and DFT-Based Methods

DFT proposes to solve electronic structure problems using the electron charge density, ρ(r), as the fundamental variable,54 which is formally based on the Hohenberg and Kohn theorems.55 In practice, DFT is applied using the Kohn–Sham (KS) ansatz, using a mean-field approach.56 The KS method represents the density as a linear combination of the inner products of independent spin-orbital functions and the energy as a functional of ρ(r) as

  • equation image(1)

where the first and second terms are the kinetic energy of independent particles, T0[ρ], and the Coulomb interaction energy, and J[ρ] = ½ ∫∫dr′drρ(r′)ρ(r)/|r′ − r|. The term υext(r) is the external potential generated by the nuclei and experienced by the electrons, and VNN is the nuclear repulsion energy for a fixed nuclear configuration. In eq 1, the contribution Exc[ρ] is the exchange-correlation energy, which includes the electron exchange interaction as well as the many-body contributions to the kinetic and electron–electron repulsion potentials (i.e., part of Vee[ρ]) that are not included in T0[ρ] or J[ρ], that is, Exc[ρ] = Vee[ρ] + T[ρ] − J[ρ] − To[ρ]. The explicit expression of Exc remains unknown; however, there are many approaches that have shown satisfactory results using standard approximations such as general gradient approximations and hybrid functionals. Such approaches are grouped according to their treatment of the density into “generations” or “ladder's rungs.”57 The most common are based on the local density approximation or generalized gradient approximation (GGA).58 Although some functionals have shown impressive results, those are not always transferable for every problem and can often fail for the description of long-range interactions and excited states. The origins of these difficulties are attributed to the incorrect cancellation of electron self-interaction,59 incorrect treatment of dynamic correlation, and so forth. There are many strategies to avoid these problems, some of which involve the inclusion of explicit terms from wavefunction theories (hybrid functionals), treatments with optimized effective potentials, an adjustment to the asymptotic correction exchange correlation potentials, and the addition of empirical energy terms.

The pervasiveness of hydrogen bonding and London forces among biological60–62 and supramolecular63–65 entities has motivated persistent efforts to develop reliable computational methodologies for noncovalent interactions.66, 67 Extensive work has established the importance of high levels of electron correlation, as embodied by coupled cluster through perturbative triples, CCSD(T),68 for the proper characterization of dispersion attractions. Yet, achieving this “gold standard” of chemical accuracy69 presents a formal cost of O(N7), where N is proportional to system size. Meanwhile, the efficient DFT [formally, O(N3) − O(N4), or O(N5) for double hybrids) can falter for even qualitative descriptions.70–73 The introduction of the DFT + dispersion (DFT-D) and exchange-hole dipole moment (XDM)74–79 methods and the crafting of several long-range functionals80–89 have attempted to improve the quantum chemical techniques available for nonbonded systems by introducing an largely empirical dispersion term. These are computationally efficient and promising approaches that have emerged from a very active field that strives to improve performance for classes of noncovalent interactions governed by disparate intermolecular forces and spanning broad-length scales.

Exchange-correlation functionals developed for covalent systems are often successful in treating hydrogen bonding and other electrostatically dominated noncovalent interactions that act over short-length scales (<2 Å).90 In contrast, the dispersion attraction, which arises from correlated motions of electrons, is primarily prominent over longer distances [∼2–5 Å (medium range) and >5 Å (long range)], and none of the typical components of a functional, the local electron density (ρσ), its gradient (▿ρσ, present in semilocal GGA functionals), or its kinetic energy (τσ, present in nonlocal meta-GGA functionals), is fully capable of acting over a suitable span of length scales. The original introduction of hybrid functionals included long-range behavior (nonlocality) through Hartree–Fock exchange; however, there are still issues with local correlation and, therefore, many such functionals are also unable to correctly describe the R−6 asymptotic distance dependence of dispersion forces. A number or research groups have developed successive families of functionals of parameterized hybrid meta-GGAs for main-group thermochemistry and noncovalent interactions.82, 84 By including systems with noncovalent interactions into the fitting set and incorporating double the amount of nonlocal exchange, the functionals strive to capture medium-range electron correlation, for reasonable results up to ∼5 Å separation of complexes.83, 84 This area of research and development continues to introduce improved functionals that are providing very encouraging results.91

The challenges to adapt efficient scaling wavefunction and DFT methods for treatment of dispersion have been the subject of several recent reviews in the literature.84, 92–96 Among the empirical treatments developed are dispersion-corrected atom-centered potentials97 and DFT-D.80, 98–100 For common implementations of the former, a pseudo-potential optimized to replicate reference interaction energies is placed over each atom, thereby incorporating medium-range correlation but not capturing the correct asymptotic form, whereas for the latter, a damped atom–atom dispersion term is added to the total DFT energy, thereby accurately encoding the long-range attraction, yet necessitating a carefully tuned damping function to evenhandedly address double counting of the correlation energy across all system configurations and types of interaction. The exchange-hole dipole moment (XDM) is constructed similarly to DFT-D as a correction to the base electronic energy but, by modeling the instantaneous dipole that arises between an electron and its exchange hole, it generates dispersion coefficients that incorporate the effect of the chemical environment. There are also efforts to include medium-range dispersion in conventional semilocal DFT or to incorporate correlation components from wavefunction theory.86–89 Both approaches can often address long-range effects through an additional DFT-D-like term. A more complex task is the development of explicitly nonlocal correlation functionals from first principles. Currently at the forefront are vdW-DF101 and VV09,102 which show very good promise but do not necessarily exceed the accuracy of some of the more empirical methods.

The local or semilocal character of conventional density functionals necessarily leads to underestimate the long-range correlation interactions that capture attractive van der Waals forces. Initially proposed by Wu and Yang98 for ab initio DFT after its first introduction into the tight binding DFT approach by Elstner et al.103 and further developed by Grimme,80, 99, 100 the DFT + dispersion (DFT-D) method appends to the base functional a scaled, damped, and fitted leading term to the well-known dispersion energy series,

  • equation image(2)

Many calculations follow the DFT-D2 variant, where the correction takes the explicit form:

  • equation image(3)

The dispersion coefficients, Cmath image, are obtained from the geometric mean of tabulated elemental values, are summed over interatomic distances, Rij, modulated by a damping function, fdamp(Rij), that gradually activates the dispersion correction over a distance characterized by the sum of the two atomic vdW radii, whereas an overall scaling term, s6, is optimized to be unique to each DFT exchange functional. The scaling factors used are sB2PLYP6 = 0.55, sPBE06 = 0.6, sPBE6 = 0.75, sB9706 = 0.7564, sBP866 = 1.05, sB3LYP6 = 1.05, and sB976 = 1.25. Grimme et al.100 recently presented a refined method, DFT-D3, that incorporates an additional R−8 term in the dispersion series and adjusts the Cmath image combination formula and damping function. The individual atomic Cmath image are interpolated from several reference values based on coordination numbers extracted from the molecular structure, rather than assigned solely by atomic identity as in DFT-D2, and thereby incorporate some awareness of the chemical environment into an otherwise largely heuristic correction. The −D3 dispersion has the following form:

  • equation image(4)

where sr and s8 are the customary nonunity parameters fitted for individual functionals. These parameters are tabulated in ref.100.

A strategy to place dispersion corrections on a somewhat less-empirical footing than DFT-D is the “semiclassical” exchange-hole dipole moment model first formulated in 2005 by Becke and Johnson74 (later placed on firmer theoretical foundations76–78). This approach is based on the concept that the nonzero dipole moment of an exchange hole104 can induce an instantaneous dipole moment and, thus, dispersion interactions between nonoverlapping systems. The method uses first-principles calculation of the dispersion coefficients for the classical dispersion force and forms a fully self-consistent and first-principles treatment of dispersion effects (the form of the dispersion interaction is not based on the first-principles calculation, but the coefficients are based on this calculation).

Recently, one of the present authors helped perform a systematic study of techniques for treating noncovalent interactions within the computationally efficient DFT framework through comparison to benchmark-quality evaluations of binding strength compiled for molecular complexes of diverse size and nature.105 Through comparison with available CCSD(T)/CBS benchmarks, the performance of DFT methods was evaluated both overall and for hydrogen-bonded, mixed-influence, and dispersion-bound subsets. Complete evaluations of test sets were conducted with aug-cc-pVDZ and aug-cc-pVTZ, and closer analysis with families of Dunning double-ζ and triple-ζ basis sets have permitted appraisal of these bases as well as the utility of the counterpoise correction. The efficiency and accuracy of functionals deliberately crafted to encompass long-range forces, a posteriori DFT + dispersion corrections (DFT-D2 and DFT-D3), and exchange-hole dipole moment (XDM) theory was assessed against a large collection of 469 energy points. For most results, M05-2X, B97-D3, and B970-D2 yield superior values in conjunction with aug-cc-pVDZ, for a mean absolute deviation of 0.41–0.49 kcal/mol, and B3LYP-D3, B97-D3, ωB97X-D, and B2PLYP-D3 dominate with aug-cc-pVTZ, affording, together with XYG3/6-311+G(3df,2p), a mean absolute deviation of 0.33–0.38 kcal/mol. These results provide a benchmark for many of the different possible approaches for including dispersion into DFT and highlight the fact that indeed chemical accuracy can be achieved for a nonbiased set of test systems. This is important as it allows the computational efficiency of DFT to be maintained while enabling accurate calculation for a much broader class of molecular materials. However, one should be cautious and note that for anisotropic systems with a high polarizabilty, nonadditive cross-polarization effects can yield a substantially altered van der Waals attraction strength that may cause inaccuracies, and thus, care is still required when using such methods.106, 107 More exact approaches such as the “vdW Density Functional”108–110 may offer an improved general capability for the latter case; however, this is still an active research area.

An extension to the standard DFT approach, time-dependent DFT (TDDFT), uses the correspondence between a time-dependent external potential and the time-dependent one-particle density to enable computation of excited states properties. However, as with standard DFT, difficulties with the current exchange-correlation functionals still lead to problems with electron self-interaction, which means that TDDFT will generally overestimate wavefunction delocalization. These errors tend to shift the vertical excitation energies (absorption energy) when compared with experiment, and as such a quantitative comparison to experiment is unlikely. A recent review presents an overview of the current concerns for polymeric materials.111

Quantum Dynamics-Based Methods

A complete description of molecular systems requires solving the full time-dependent Schrödinger equation for the constituent electrons and nuclei. This task is currently out of reach of state-of-the-art computational approaches and resources. To simplify the problem, it is usually assumed that the motion of electrons (fast) and nuclei (slow) can be separated and that Born-Oppenheimer approximation is valid. As a consequence, the time-dependent equations are replaced by (a) classical Newton's equations for nuclei and (b) time-independent equation for electrons. The resulting method is referred to as Born-Oppenheimer MD (BOMD)112–115 or its extended Lagrangian variants116, 117 in which nuclei move classically on the ground-state potential energy surface (or on its approximation) obtained on-the-fly self-consistently (or are propagated classically) from the stationary electronic structure.

The high computational demand of the electronic structure calculation renders BOMD only practical in implementations based on an independent particle model (Hartree–Fock, tight-binding, and density functional theories). In these methods, all nuclei move on the potential energy surface generated by the electronic structure, and the total “electronic” energy of the system can be expressed as a sum over occupied orbitals

  • equation image(5)
  • equation image(6)

or equivalently

  • equation image(7)

in which the electronic structure is represented by the density matrix P. VNN represents the interaction between nuclei and does not depend on the electronic structure. The second, so-called one-electron term, Hcore, depends linearly on P and represents the kinetic energy of electrons and interaction of electrons with nuclei. The last term, G(P), describes the electron–electron repulsion and depends quadratically on P. The last equation can be used regardless of whether one solves a time-independent electronic structure (BOMD) problem or quantum-dynamical time-dependent Schrödinger equation. It is orbital free (but not basis set free) and is more general. The choice of molecular orbital is arbitrary as the electronic structure represented by density matrix P may be arbitrarily described by many different choices of molecular orbitals.

In the Born Oppenheimer case, the electronic structure is determined through minimization of the energy according to the variational principle. This is usually achieved by diagonalization of the total matrix Hamiltonian [F = Hcore + G(P)]. Alternatives to diagonalization methods exist (e.g., a Fermi operator expansion or direct density matrix minimization) and are described elsewhere (see refs.118–120 and references therein). Once the electronic structure and corresponding nuclear forces are known, the molecular motion can be determined and related properties can be described.

Alternatively to time-independent solutions of the electronic structure problem in dynamics, one can integrate a time-dependent Schrödinger equation along with Newton equations of motion for the nuclei. Another option includes quantum nuclear effects into dynamics for selected nuclei. Other possible approaches include quantum wavepacket dynamics, path integral, and Bohmian trajectory approaches. Notably, quantum dynamical treatment plays an especially important role in proton and electron transfer processes.118–120

Classical Mechanics-Based Approaches

Although many of the intrinsic processes governing OPV operation are quantum mechanical in nature (e.g., exciton generation/recombination, charge carrier mobility, and charge transfer), structural and dynamical processes of long-chain molecules typically used in OPVs are better described within classical or statistical mechanics. In this case, particle-based methods such as classical MD, Monte Carlo (MC), molecular mechanics (MM), and various coarse-grained versions can be of great utility. These are well-tested methods with an established track record of success within the field of polymer physics, and therefore, we only briefly introduce them (e.g., see refs.121 and122 for more details).

For molecular-based materials, these methods require the specification of potential energy functions (usually known as force fields) to describe the various relevant many-body interactions. As one of the main interests here is to understand the structure and bulk morphology in solution-cast thin films, the use of potential functions with a proven record of accurate prediction of structure and vibrational spectra for conjugated organic molecules are required. Such potentials are typically composed of harmonic or Morse oscillators for the bond-stretching and angle-bending terms (both in plane and out of plane), truncated Fourier series for the torsion interactions (regular dihedral and improper), and Lennard-Jones 6-12 plus Coulomb potentials for the nonbonded interactions. Several standard force fields fall into this category, such as the MM2, MM3, MM4, Dreiding, UFF, MMFF, CHARMM, AMBER, GROMOS, TRIPOS, and OPLS models.123

Molecular Dynamics

MD simulations (for details see refs.124–126) essentially consist of numerically integrating Hamilton's, Newton's, or Lagrange's equations of motion using small integration time steps. Although these equations are valid for any set of conjugate positions and momenta, the use of Cartesian coordinates greatly simplifies the kinetic energy term. The resulting coupled first-order, ordinary differential equations can be solved using numerical integrators.127 Despite the simplicity of MD methods, the simulation of millions of atoms over time-scales up to several seconds is not possible without substantial reduction of the number of equations of motion, a feat typically achieved by course-graining groups of atoms into one element (see “Course-Grained Models” section). This approach not only reduces the number of equations, but when proper separation of the time scales for motion can be achieved, it also allows the use of considerably larger integration time steps and thus helps to bridge the large gap between simulation and experimental time scales.

Molecular Mechanics

Molecular mechanics (MM) methods (for details see ref.128) apply the laws of classical physics to predict structures and properties of molecules by optimizing the positions of atoms based on the energy derived from an empirical force field describing the interactions between all nuclei (electrons are not treated explicitly). As such, MM can determine the equilibrium geometry in a much more computationally efficient manner than ab initio quantum chemistry methods, yet the results for many systems are often comparable, at least qualitatively. However, as MM treats molecular systems as an array of atoms governed by a set of potential energy functions, the model nature of this approach should always be noted.

Course-Grained Models

Course-grained models (CGMs) consist of replacing an atomistic description of a molecule with a lower resolution model that averages away some of the fine details of the interactions. Numerous CGMs (differing in the metric used to obtain the course-grained potential, such as the fitting to the forces or to structural features, and in the definition of the course grain) have been developed to investigate the longer time- and length-scale dynamics that are critical to many long-chain molecular processes, such as polymer, lipid membranes, and proteins. Coarse graining can also refer to the removal of certain degrees of freedom (e.g., vibrational modes between two atoms) by freezing the bonds, bends, or torsional degrees of freedom, but more typically it implies that two or more atoms are collapsed into a single particle representation (the so-called united atom model was one of the first popular CGMs). Fundamentally, the level to which a system may be coarse grained is bound by the accuracy in the dynamics and structural properties desired from a simulation. We refer to several review articles for more details.129–131

Field Theory

FT representations can be effectively obtained from CGMs and offer an advantage of significantly reducing the number of differential equations for the particles to a diffusion equation that can be numerically solved using a self-consistent algorithm.132–134 Another advantage of the FT approach is that it ensures thermodynamic equilibrium of the final morphology and thus provides access to modeling much larger length and time scales. Such improvement is difficult to obtain with the particle-based approaches (atomistic or course grained) due to the existence of a very high-dimensional potential energy landscape plagued with multiple local minima. The field theoretic approach can also incorporate realistic polymer models that are often difficult for macroscopic, continuum simulations. In fact, the numerical self-consistent FT (SCFT) for dense polymer melts has been proven to be highly successful in describing the complex morphologies in multiblock copolymers and even for nanoparticles in polymers.132, 133, 135, 136 SCFT simulations for a dense melt of copolymers in bulk use chemical specificity of blocks through the value of the Flory parameter χ, which controls the strength of the chain segregation that drives the formation of microphase segregated morphologies. The course-grained Hamiltonian of a dense melt of n polymer chains containing distinct monomers A and B can be written as follows:

  • equation image(8)

The first term in eq 8 gives the free-energy contribution from the so-called Gaussian thread model. The position vector of a bead's s contour steps along the α-th chain is given by the space curve equation image. The second term is the free-energy contribution in a model where dissimilar monomers in contact have a higher free energy than like monomers in contact. The tendency for chemically dissimilar monomers to segregate is described by the χ value and is a function of the specific chemical species A and B as well as temperature. The model given by eq 8 corresponds to a linear AB diblock copolymer. However, it can be easily generalized to a larger number of blocks to include architectural asymmetry such as branched copolymer architectures, surface/substrate corrugation, and hybrids with nanoparticles.

The coarse-grained Hamiltonian is used to form the partition function of a dense melt of n copolymer chains. The monomer density operators may be recast as continuous monomer density fields by application of a Hubbard-Stratonovich transformation that introduces chemical potential fields w(r) conjugate to the monomer density fields.137 The partition function has the following form:

  • equation image(9)

where ρA and ρB are the density fields and ωA and ωB are the conjugate chemical potential fields that are introduced through the FT transformations. The p field is introduced through the constraint of overall monomer incompressibility. The integrals are over all possible realizations of the fields with F given by

  • equation image(10)

where Q is the single-chain partition function and is a functional of the chemical potential fields and a function of the chain length N. In the mean-field approximation, the extremum of F is assumed to make the largest contribution to the overall sum. In that case, the resulting system of equations can be solved self-consistently using an iterative algorithm to solve for the ρ(r) and ω(r) fields that satisfy the SCFT mean-field equations.

Monte Carlo Methods

MC methods (for details see ref.138) used in macromolecular science generally begin by constructing a Markov Chain generated by the Metropolis algorithm (i.e., sampling of states according to their thermal importance: Boltzmann distribution for the ensemble under consideration, usually the canonical ensemble). As the chain length of a simulated system becomes longer, it quickly becomes necessary to introduce a series of biased moves in which additional information about the system is incorporated into the MC selection process in such a way as to maintain a detailed balance. The most commonly used biased sampling techniques are the continuum configuration bias and concerted rotation moves.139 These modern algorithms or slightly modified versions can efficiently generate dense fluid polymer systems for chain lengths of 30–100 monomers. Longer polymer chain lengths pose additional convergence problems and often require the use of other types of biased moves, in particular the double-bridging moves.140, 141 Addition of solvent via continuum models (implicit solvation models)142 or explicit atoms can also be implemented. A primary interest in using the MC method is to ensure adequate equilibration of longer chain polymers, that is, those with hundreds of monomers. MM and MD methods can also be used but generally require considerably longer times to equilibrate, as they often spend long times in local minima.

A useful extension of the MC method is to introduce a stochastic time variable into the simulation to allow modeling of the rates of specified processes. This approach is referred to as kinetic MC (KMC) and is often used to examine aspects of crystal-growth dynamics143 and to estimate charge transport.144 The method requires input from other computational or experimental methods to provide the energetics for the processes; however, it has been used to obtain critical information for a number of relevant cases. For example, Liu et al.145 used KMC to examine the growth of a C60 phase on graphite. Extensions of KMC to treat nucleation and growth in solutions can also be achieved.146

Multiscale Methods

Multiscale methods generally involve using two or more of the above computational approaches in order to treat a broader set of processes that may span multiple length and time scales (e.g., see refs.147–149). Typically, the so-called multiscale modeling is actually more of a multistep approach (a loose integration), whereby a needed parameter/information at one length or time scale is obtained using an appropriate computational approach that is subsequently used to start another type of simulation for a different scale. For example, aligned nanorod inclusions in polymeric materials were recently studied using this type of approach.150 First, the self-assembly of a mixture of A-coated and B-coated rods in an AB phase-separating blend was examined with dissipative particle dynamics151 simulations. This showed that the steric repulsion between ligands causes the coated rods to preferentially align end to end within the minority phase of the binary blend. The information was then used to formulate a coarse-grained approach, which combined a Cahn–Hilliard152 model for the polymer blend with a Brownian dynamics153, 154 simulation for the rods. Interestingly, the authors found that a small volume fraction of B rods in the majority B phase promotes the percolation of A-like rods within the A phase so that the percolation threshold for the A rods is significantly lowered. However, if the number of B nanorods in the B phase exceeds a particular volume fraction, the B particles inhibit the percolation of the A rods. The output from the morphological studies was then used as input to the lattice spring model155 to characterize the mechanical behavior of the composite revealing that nanorods oriented along the tensile direction contribute to the enhancement of the macroscopic mechanical properties.

Which Computational Approach?

To gain the benefits and apply the power of computational modeling to the characterization and engineering of OPV materials, a proper selection of a theoretical/computational model requires the consideration of the following issues:

  • 1
    If the character of the processes is quantum mechanical in its nature, accurate modeling will require using electronic structure methods. In addition, if the nature of the internal interactions has significant contributions that are nonlocal or involve very strong electron correlation, one may need to consider many-body quantum mechanical approaches, including self-interaction corrections, random phase approximation156–162 DFT,156 or modified DFT functionals.
  • 2
    If the system size is composed of a few hundreds of atoms or more, a rigorous quantum approach will be extremely difficult. The chemistry of reactive bulk materials is intrinsically different from the chemistry of single molecules and thus requires different theoretical approaches. In the latter case, the electronic state can be controlled with spectroscopic accuracy and the long-term decay leads to a well-defined state (i.e., an eigenstate of the system's Hamiltonian). In the former case (bulk phase), the coupling between electronic and nuclei and electrons is nonadiabatic and leads to constant energy exchange between these degrees of freedom where the electronic state of the system is likely a superposition of the electronic eigenstates.
  • 3
    Materials are often produced under harsh experimental conditions, including high temperature, pressure, and external electric or shear fields. Such conditions induce a high probability of significant transfer of energy between electronic and nuclear degrees of freedom. The chemistry of such systems needs to involve electronically excited states and is intrinsically nonadiabatic.
  • 4
    Experimental observations are statistical in nature, thus statistics is generally required for modeling/simulation to reproduce the experimental observation. For example, single trajectory simulations may not lead to qualitatively correct results. Performing an ensemble of trajectory simulations with statistically varied initial conditions to match an experimental setup is generally required to be representative of the experimental observations.
  • 5
    Electronic structure-based evaluation of forces and energy is far more expensive than any classical force field-based or semiempirical calculation. The electronic structure methods used for the evaluation of energy and forces must be very cheap to enable affordable MD simulations. As a rule of thumb, computational cost of one electronic structure calculation of energy and forces (one time step of MD) has to be on the order of 1 min (or less). Two minutes spent for one evaluation of the energy and forces translate into 2 weeks spent for 10,000 steps of MD. Thus, a balance between accuracy and efficiency is often required to address relevant time and length scales appropriate to experiments.

Prospects on Improving Computational Approaches

To fully understand the properties of excitons at the nanoscale, examination of exciton binding, relaxation, and dissociation are required. This implies accurate characterization and optimization of the electronic states where electrons and holes are bound instead of those which act as free carriers. Simple models based on using a subset of selected electronic and vibrational states can be useful to develop some preliminary understanding. However, such reduced models necessitate an intimate knowledge of the electronic structure and dynamics that requires using multiconfiguration time-dependent quantum calculations. We must note that there is still considerable work needed to accurately describe the ground- and excited-state potential energy surfaces in complex environments. For this reason, novel approaches based on DFT and TDDFT, multireference quantum chemistry calculation, GW/Bethe-Salpeter, and quantum MC are under development.163 These new approaches will need to provide better abilities toward treating the following factors:

  • 1
    Charge-transfer states and oxidation-reduction reactions: The problem of electron self-interaction in many standard DFT exchange-correlation functionals is that they can over-delocalize electrons leading to difficulties in fully describing different oxidation states on the same ion or charge transfer between conducting polymers, where holes or electrons can incorrectly delocalize along one chain.
  • 2
    Electron-transfer and proton-coupled electron-transfer processes: A wide variety of processes and reactions in electrochemistry, molecular electronics and optoelectronics, and biochemistry involve nonadiabatic electron transfer processes from a donor to an acceptor via a “bridge.” The excited state of an electron transferring from one complex to the other needs to be calculated correctly for many thousands of configurations to obtain the free-energy diabatic surfaces for the studied reactions.
  • 3
    Excited-state potential energy surfaces and dynamics: In any photoinduced or photocatalytic process, optical excitations force the system to evolve on a higher potential energy surface, with a competition between fast nonradiative decays and the radiative re-emission of photons.
  • 4
    Realistic descriptions of large-scale systems in complex electrostatic and solvation environments: Accurate representations of the physicochemical environment of a materials is essential for modeling the response behavior, and thus, it is important to embed quantum calculations in the correct electrostatic environment.

OVERVIEW OF RECENT PROGRESS IN MODELING OPVS

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES

The largest body of literature devoted to the theoretical modeling of OPVs can be found for specific applications of computational approaches to optoelectronic materials. In this case, there are a number of recent reviews that summarize these efforts.10, 164 Similar types of detailed applications for specific OPVs are considerably less abundant, which is partly due to the overall complexity of the materials (large molecules with relatively slow dynamics) and the underlying physical processes (excited states and charge transfer). Nonetheless, there have been a number of studies that have attempted to bridge the inherent complexities (photon adsorption, exciton generation, diffusion and dissociation, carrier transport, and charge collection) to provide some level of fundamental understanding (e.g., see refs.11 and52). Most research papers fall into loose categories of:

  • a
    modeling of the fundamental electronic structure (see review by Risko et al.11);
  • b
    modeling of the active layer self-assembly process;
  • c
    modeling of the nanostructured morphology;
  • d
    modeling of charge mobility and transfer at interfaces;
  • e
    modeling of exciton formation, diffusion, or recombination; and
  • f
    modeling of two or more of (a)–(e), often using a multistep or multiscale modeling approach.

The most extensive literature can be found for category (a). In that regard, Risko et al.11 have recently provided an extensive review. Categories (b) and (c) on self-assembly and morphology165 are beginning to receive some attention, which rationally follows as this is clearly an important variable for OPV performance,166, 167 and one that clearly possesses a complicated high dimensionality. In general, morphology influences BHJ performance due to a number of factors: the domain size of the donor phase (the size needs to be small enough that excitons diffuse to acceptor regions without recombining), interface-to-volume ratio of the blend that needs to be as high as possible to allow efficient exciton dissociation, adequate percolation between the donor and acceptor to ensure charge carrier transport and ultimately collection at the electrodes, and proper donor molecular alignment to optimize the carrier mobility of the donor phase. Unfortunately, morphology appears to be strongly affected by almost every quantifiable device fabrication parameter, including thin-film deposition spinning speed,168–170 polymer molecular weight,26, 171 thermal treatment temperature,172–175 solvent choice,176–181 blend composition ratio,182 drying time of the thin films,183 and polydispersity. At the same time, experimental characterization of donor–acceptor interfaces or crystalline-like CP aggregates on the tens of nanometer size scale is a formable challenge.172, 184–190 In fact, most recent evidence is beginning to point toward a three-phase morphology (P3HT crystalline-like domains, PCBM, and a mixture of amorphous P3HT/PCBM) instead of the predominately believed two-phase one,191 independent of thermal annealing. This implies that the impact of molecular miscibility192, 193 on nanoscale morphology and device performance needs considerably more examination. Numerous studies have already demonstrated results that elude to the importance of such a three-phase morphology (in particular with 20% amorphous P3HT) and how the “amorphous glue” is required to generate adequate pathways for holes, a hole-carrying network, and thus an improved PCE176 (also see refs.194–197 and more recently ref.198). Some insight into the structure of fullerene and P3HT blends can be obtained from coarse-grained computational simulations199, 200 and for conjugated homopolymers.201 A model representation of a P3HT amorphous phase showing substantial fullerene concentration is shown in Figure 4.202 Notable from this particular study is the use of a multistep simulation where MC simulations provided an initial “bulk-like” geometry based on a DFT determined molecular structure for the P3HT and C60 components making up the MC simulation, and a semiempirical Pariser-Parr-Pople Hamiltonian203 with configuration interaction single excitations to investigate the excited states. From that study, optical absorption spectra, charge transfer states, and exciton bands were elucidated, showing that the absorption spectra are dominated by intramolecular contributions while low-lying charge transfer states are predicted in the optical gap. Interestingly, depending on the shape and overlap of the exciton bands, the ratio of energy relaxation to hopping rates and the presence or absence of low-energy radiative processes may be reduced up to 40%. This is interesting in regard to the so-called Shockley-Queisser limit to PV efficiency,204 in that it suggests that nonabsorbing “dark” states could be as important for OPVs as the initial absorption of light. A detailed analysis of the importance of balancing loss mechanisms for OPV BHJs is given in ref.205.

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Figure 4. Final structure of a simulated fullerene (green)–oliothiophene system (red). This snapshot shows a region that could be considered as a heterogeneous amorphous phase. (Adapted from ref.202, with permission from Royal Society of Chemistry Publishing.)

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Another interesting study on elucidating the dependence on the bulk morphology on blend ratios between P3HT and PCBM was recently provided by using coarse-grained simulations.206 The simulated blend films clarified to some extent why a blend ratio of ∼1:1 (P3HT:PCBM) provides an optimal morphology for OPV performance. Figure 5 shows the thermally annealed and equilibrated coarse-grained structures obtained for different blend ratios, illustrating that a 1:1 blend (a detailed analysis was done) has the highest interface-to-volume ratio and an adequately percolated network that should provide for a more balanced charge carrier transport. Interestingly, charge mobility and transport behavior in ordered and disordered states of P3HT have been shown to exhibit hole mobility that are an order of magnitude larger for intrachain versus the π–π interchain route.207–211 Intrachain disorder is largely due to conformational changes209 within a chain and can lead to exciton localization. The initial exciton relaxation via intramolecular pathways is also not a primary stochastic process but is driving by exciton–phonon coupling to torsional motions.212 Indeed, the wavefunction of the charge carrier (hole) for P3HT has not only been shown to be localized in correspondence to long-lived traps but also revealed that these states are present in the crystalline phase.213 The electronic structure in the spectral region around the band gap is determined by the conformation of the main thiophene chains with only minor contributions from side chains.214 The authors of ref.207 suggests that charge mobility in disordered domains where P3HT chains intersect/cross one another or form bridging points would play a key role in the overall measured (average) mobility. Thus, in the presence of both ordered and disordered P3HT, the disordered domains may represent the rate-limiting part for charge transport.207 However, one should note that it has been found that having some disordered/amorphous P3HT is needed to achieve maximum PCE (see discussion above). This is likely due to the macroscopic conductivity being controlled by small number of crucial percolation paths.215

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Figure 5. Results from coarse-grained molecular dynamics simulations showing equilibrated bulk structures for different P3HT (red): PCBM (blue) ratios. (Adapted from ref.206, with permission from Royal Society of Chemistry Publishing.)

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Modeling and simulation should also be able to guide the analysis and to develop a better understanding of the molecular chain order within the P3HT aggregate domains and their composition (pure P3HT, partial blend of P3HT/PCBM, mixture of P3HT/PCBM, and amorphous P3HT). The work in ref.206 used a reverse mapping from the coarse-grained results to gain insight into the molecular packing and to provide atomistic details of the underlying structures. Indeed, the estimated radial distribution functions showed ordering within P3HT fibrils with peaks associated to distances between chains optimal for π–π interactions. Rather such types of reverse mappings can indeed provide a rigorous and unique representation, as there are multiple possibilities, is often a pertinent question. However, the results are intriguing and qualitatively in line with experiments. In principle, such simulations or more appropriately, fully atomistic or ab initio methods could ultimately be used toward designing improved ternary systems to enable enhanced OPV performance. In practice, this type of noble achievement has been quite elusive although it is a well-recognized possibility in the broader context of “materials by design.” This type of computationally enabled capabilities is one topic that is addressed in the so-called materials genomics initiative.216

A recently proposed step toward understanding the self-assembly of P3HT chains is based on using a combination of quantum DFT and MD simulations.217 This study found that P3HT domains are composed of ordered P3HT chains stacked into a staggered geometry that leads to two-dimensional hydrophobic foil-like structures that subsequently assemble into zigzag bulk structures. The self-assembly process is altered in the presence of surface interactions where the P3HT chains favor alignment instead of stacking. An extensive DFT study of the crystal packing structure of P3HT also found that the lowest energy structure corresponded to a tilted structure that is induced by the alkyl side chains.218 A very strong dependence of the electronic structure and subsequent hole effective mass was found as a function of the staking structure.

P3HT self-assembly in the presence of carbon nanotubes is another example that demonstrates the importance of interfacial structure in charge transfer.219, 220 Overall, these results suggest that tuning of the substrate, possibly by using a surface-segregated monolayer, could facilitate chain alignment via modification of the strength and direction of the surface dipole moment.48 A detailed account of how surface electrostatics dictate the dependence of molecular orientation was recently given by Heimel et al.221 By elucidating the energies of the frontier electronic states in organic thin films, they showed a clear dependence on the orientation of the constituent molecules with respect to the device-relevant interfaces. Additionally, DFT calculations for supramolecular P3HT/fullerene structures have demonstrated that P3HT and fullerene wavefunctions can strongly overlap at an interface (see Fig. 6) and produce a set of intragap charge-transfer states that induce charge transfer in the ground state.45

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Figure 6. Ground-state HOMO and LUMO orbitals for a P3HT/C60 supramolecule with the pentagonal face of the C60 interacting with a five-monomer P3HT chain. (a) HOMO and (b) LUMO at the optimized distance. (c) HOMO and (d) LUMO for separation distance of 6 Å. (Adapted from ref.45, with permission from Elsevier.)

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The interplay between this effect and the presence of an intrinsic dipole on the polymer creates a supramolecular–interfacial dipole pointing from the fullerene to the polymer.222 The strength of the dipole is strongly dependent on the orientation and distance between P3HT and C60, and the dipole modulates a potential energy barrier, which is due to being immersed in the electric field of the germinate electron–hole pair, that hinders electron–hole recombination (rectification). Further support to this effect was recently given by McMahon et al.46 where they used large-scale atomistic models combined with DFT and semiempirical quantum calculations (ZINDO)223, 224 to show how electrons and holes of the P3HT are repelled by the interface due to an increased bandgap (see Fig. 7) that is caused by larger disorder of the P3HT chains close to the PCBM.46, 225 The actual dynamics of electron transfer at a P3HT/PCBM interface has been examined using ab initio nonadiabatic MD simulations,226 revealing an ultrafast, ∼70 femtoseconds, transfer which is dominated by an adiabatic process. However, the reverse transfer (a recombination) was found to occur on a much longer time scale (nanoseconds) thus showing rectification as mentioned earlier. The analysis of the electron dynamics demonstrated that it is determined by the energy evolution driven by the coupled electron-ion dynamics.

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Figure 7. Electronic density of states (DOS) as a function of distance from the P3HT/PCBM interface (schematic: middle frame; snapshot: right frame). (Adapted from ref.46, with permission from American Physical Society Publications.)

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The ability to use the substrate as an “ordering” parameter could be a very useful concept.227, 228 Recently, MD simulation229 results corroborated experimental work230, 231 to demonstrate how functionalized substrates can effectively pattern the ordering of the P3HT chains; disordered substrates with smoothly varying potential energy landscapes lead to edge-on P3HT, whereas ordered substrates can promote a reduction of edge-on orientation. These results suggest possibilities for using a chemically mixed substrate monolayer to introduce packing irregularities that could encourage optimal P3HT chain alignment for optoelectronic applications.

Multistep or multiscale simulations (see “Multiscale methods” section) may offer enhanced information on the combined effects of disorder on carrier mobility and exciton separation. This is primarily due to the fact that in order to model the range of length and dynamic scales involved in typical BHJ processes, one must consider time scales from femtoseconds (molecular vibrations) to nanoseconds and microseconds (charge-carrier transients across a device) along with length scales of angstroms at the interface and for intermolecular chain–chain interactions to tens of microns for the device. No single method from “Background on Simulation Approaches” section is capable of treating all of these length and time scales, and as such, the typical approach is to break the problem into a multistep calculation where the appropriate method to treat the particular aspect of the problem is used to gain information that is subsequently used for another lower resolution method. A recent account on such a multiscale modeling of charge transfer in OPV materials was given by Nelson et al.144 This article provides details on the successful prediction of carrier mobilities for a given molecular packing disorder, including systems with various side chains and molecular weights (see Fig. 8). Such capabilities suggest that quantitative prediction is now possible for charge transport in macromolecular systems with anisotropic heterogeneity and multicomponents. These authors have also successfully demonstrated temperature-dependent growth of C60 films on substrates (Fig. 9) that provide some insight into thermal annealing affects, showing distinctly larger crystalline domains and subsequent increase in carrier mobility for higher temperature (larger grain size). Adequate percolation of C60 to enable electron propagation on exciton dissociation is fundamentally important toward the development of PCE with increased efficiency. Recent theoretical work has provided information on optimal orientation for fullerene-based materials, showing that a fullerene–fullerene face-to-face interaction is a dominate factor in optimizing the charge transport properties.232

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Figure 8. Simulated low-field limit time-of-flight mobility (filled stars) and measured FET mobilites (open triangles). (Adapted from ref.144, with permission from American Chemical Society Publications.)

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Figure 9. FET mobility as a function of grain size (left) and temperature effects on the size of C60 crystalline domains (right). (Adapted from ref.144, with permission from American Chemical Society Publications.)

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Another recent example233 of this type of approach used Marcus equation,234 DFT, ZINDO,223, 224 and MC simulations51, 233 to compute 1D and 3D charge transport properties in ordered and orientationally disordered CPs showed a decrease in carrier mobility for the disordered systems that agreed with experimental results.

Clearly, all of these works have shown that exciton generation (internal quantum efficiency is near 100%) and exciton separation can be very efficient for a P3HT/PCBM material, provided a proper interface and multiple nanophases are formed. Some concerns with these results are in regard to the sensitivities of exciton dissociation to the orientation and distance of the donor–acceptor interface as the donor–acceptor OPV active layer is a “plastic” material. Such nanocomposites have an enormously high-dimensional local minimum energy landscape, which means that the system is in a pseudo nonequilibrium state and will relax over time (aging). Relaxation will upset the delicate balance of the interface and cause detrimental effects to the overall PCE. Coaxing a polymer nanocomposite to form an optimal nanostructured interface,235–238 while giving improved results in terms of measured properties, may not fair well over time in the harsh environment of solar irradiation.239 As such it will be important to validate or produce accelerated laboratory aging protocols to fully access performance.240

In this regard, a novel perspective241 in the OPV field is based on a concept of using polymer nanoparticle superlattices (Fig. 10) to overcome some of the shortfalls of BHJs. The authors propose the possibility of decoupling nanoscale morphology by constructing discrete nanoparticle superlattices. A host of intriguing new photophysics has emerged from preliminary studies. However, the concept is challenged by adequately achieving an appropriate crystalline nanoparticle ordering as that would require a rather narrow distribution of particle sizes for each component (donor and acceptor), which in turn places stringent demands on synthetic control. Additionally, obtaining good interfacial interactions between donor (P3HT) and acceptor (PCBM) nanoparticles in the prepared nanoparticle superlattice geometry will be difficult. However, this new perspective direction for OPVs offers a refreshing step back from the relentless pursuit of improving BHJs.

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Figure 10. Illustration of the crystal structure of AlB2 showing the alternating layered structure (left) and the packing of discrete nanoparticles into a similar lattice structure (right). The continuous phases of the hole conductor (blue) and electron conductor (orange) in principle should allow very efficient charge transport. (Adapted from ref.241, with permission from American Chemical Society Publications.)

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SUMMARY AND CONCLUSIONS

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES

In an effort to balance energy usage, environmental preservation, human health, and economic prosperity, chemistry and physics will continue to be key sciences involved in addressing the associated formidable challenges. In regard to energy, present requirements that need to be attained for a viable alternative or supplementary energy source are that the source must:

  • 1
    be abundant and last a long time (i.e., renewable);
  • 2
    introduce no long-term environmental or health issues (i.e., safe and produce no side products that cause detrimental environmental effects); and
  • 3
    be energy efficient and cost effective.

Solar energy can in principle satisfy all these requirements simultaneously.242 However, the successful “harvesting” of solar energy as a viable contributor to our current energy needs still requires considerable advances in appropriate energy materials. Although organic-based materials are touted to be one promising possibility, the path forward is often clouded by a complex multivariable physicochemical and device performance optimization problem. Traditional trial-and-error approaches for such a complex space of variables is limited toward achieving a serendipitous breakthrough and as such a rational approach is highly desired/needed. Modern computational modeling and simulation combined with precision synthesis, characterization, and device fabrication is beginning to make some inroads toward better understanding of all of the underlying processes and such progress should help to enable future improvement.

From a computational and theoretical standpoint, although the current computer architectural shift toward multicore heterogeneous computing (CPU/GPGPU) poses challenges to the existing simulation paradigms, it opens new opportunity to unify scalable mathematical algorithms, electronic structure methods, and advanced computing tools so as to enable efficiently simulating material systems and physicochemical processes at scale.243 Although present simulations are often restricted in the number of atoms (size scale) and time scales that can be examined (e.g., most DFT-based electronic structure can treat only up to thousands of atoms, atomistic MD up to million atoms for a few nanoseconds, and coarse-grained MD in the tens of millions for hundreds of nanoseconds), multiscale methods and approaches may begin to bridge the gaps. Predictive simulations of entire device structures from first-principle calculations that are critical to advancing the understanding of structure–property–transport relationships are needed and are becoming increasingly possible. Such simulation capabilities could potentially promote the virtual design of molecular and nanoscale assemblies while providing new discovery tools for rapid prototyping. Combined with precision synthesis, characterization, and device fabrication, these advances should lead to the improved capability for discovery/understanding of novel phenomena and for providing principles for the design of efficient energy paradigms.

Acknowledgements

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES

The authors acknowledge the support from the Polymer-Based Materials for Harvesting Solar Energy (PHaSE) Center, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001087. B. G. Sumpter also acknowledges the support from the Center for Nanophase Materials Sciences (CNMS), which is sponsored at Oak Ridge National Laboratory by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Some of the calculations used resources of the Oak Ridge Leadership Facility and the National Center for Computational Sciences. The authors also thank K. Xiao and C. Rouleau for useful discussions and for assistance with the Table of Content graphics.

Biographical Information

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES
Thumbnail image of

Vincent Meunier received his Bachelor of Science (1996), Master of Science (1998), and Ph.D. (1999) in Physics all from Namur University, Belgium. Following postdoctoral studies at the North Carolina State University in 2000–2002, he joined the Computer Science and Mathematics Division at Oak Ridge National Laboratory in the Computational Chemical Sciences group and later the Nanomaterials Theory Institute. He now holds the Kodosky Constellation Chair at the department at Renssealer Polytechnic Institute where he is an associate professor of physics and materials science. Dr Meunier's research is focused on the fundamental understanding of electronic processes at the nanoscale, including simulation of experiments and the development of scalable software for computing electronic transport properties in nano structures.

Biographical Information

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES
Thumbnail image of

Bobby G. Sumpter received his Bachelor of Science in Chemistry from Southwestern Oklahoma State University (1983) and a Ph.D. in Physical Chemistry from Oklahoma State University in 1986. Following postdoctoral studies in Chemical Physics at Cornell University 1987–1988 and in Polymer Chemistry at the University of Tennessee, Bobby joined the Chemistry Division at Oak Ridge National Laboratory in the Polymer Science group. He is currently the group leader for the Computational Chemical and Materials Sciences group and the Nanomaterials Theory Institute. Dr Sumpters' research is focused on the fundamental understanding of nanoscale self-assembly processes, interactions at interfaces, the structure and dynamics of molecular-based materials including multi-component polymers and composites, and the physical, mechanical and electronic properties of nanoscale materials.

REFERENCES AND NOTES

  1. Top of page
  2. Abstract
  3. INTRODUCTION AND GENERAL OVERVIEW
  4. BACKGROUND ON SIMULATION APPROACHES
  5. OVERVIEW OF RECENT PROGRESS IN MODELING OPVS
  6. SUMMARY AND CONCLUSIONS
  7. Acknowledgements
  8. Biographical Information
  9. Biographical Information
  10. REFERENCES AND NOTES