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Title Modeling organic electronic materials: bridging length and time scales

Permalink https://escholarship.org/uc/item/6jv1c5zk

Journal Molecular Simulation, 43(10-11)

ISSN 0892-7022

Authors Harrelson, TF Moulé, AJ Faller, R

Publication Date 2017-07-03

DOI 10.1080/08927022.2016.1273526

Peer reviewed

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Modeling organic electronic materials: bridging length and time scales

Thomas F. Harrelson, Adam J. Moulé & Roland Faller

To cite this article: Thomas F. Harrelson, Adam J. Moulé & Roland Faller (2017): Modeling organic electronic materials: bridging length and time scales, Molecular Simulation To link to this article: http://dx.doi.org/10.1080/08927022.2016.1273526

Published online: 02 Mar 2017.

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Download by: [The UC Davis Libraries] Date: 02 March 2017, At: 08:51 MOLECULAR SIMULATION, 2017 http://dx.doi.org/10.1080/08927022.2016.1273526

ENERGY APPLICATIONS Modeling organic electronic materials: bridging length and time scales

Thomas F. Harrelson, Adam J. Moulé, Roland Faller

Chemical Engineering, UC Davis, Davis, USA

ABSTRACT ARTICLE HISTORY Organic electronics is a popular and rapidly growing field of research. The optical, electrical and mechanical Received 2 October 2016 properties of organic molecules and materials can be tailored using increasingly well controlled synthetic Accepted 10 December 2016 methods. The challenge and fascination with this field of research is derived from the fact that not only KEYWORDS the chemical identity, but also the spatial arrangement of the molecules critically affects the performance Coarse graining; organic of the material. Thus synthetic, fabrication, characterisation and computational scientists need to work electronics; molecular closely to relate a materials performance in a device to the molecular details that cause and optimise that dynamics performance. For computational scientists in particular, the need to relate macroscopic device performance to details of molecular electronic structure brings challenges in methodology due to the need to bridge many orders of time and length scales. This article provides a survey of computational methods applied to multiple-length and time scale problems in organic electronic materials. Here we seek to highlight a few particular approaches that expand the simulation toolbox.

1. Introduction Organic electronic devices and materials have become an increasingly important research and commercial area with a Organic molecules have been recognised for their potential to predicted market value of ∼$70 billion by 2026 [9–11]. Organic harvest and emit light for device applications for decades [1–3]. components are particularly valuable because molecular design Since these humble beginnings, organic light emitting diodes allows an almost infinite variety of structures and functions to be (OLED) have become commercially available for lighting and synthesised [12]. Ideally, it would be possible to conduct mod- are widely used for display applications [4,5]; organic photo- eling experiments to design organic molecules for electronic voltaics (OPV) have achieved over 10% power efficiency [6,7] applications and to characterise the function of the structures and both small molecule and polymer semiconductors have using computers. However this task is currently impossible achieved over 1 cm2/Vs mobility [8]. The progress represented because organic electronic molecules could be crystalline or by these astonishing technological achievements in the use of amorphous with molecular weight between 16 and 108 g/mol. organic materials for electronic applications has come about as They could be liquids, liquid solutions, network solids, gels, or a result of simultaneous major advances in organic materials insoluble materials that are evaporated into place. They could synthesis, capabilities in organic materials characterisation, and be pure hydrocarbons, organo-metallics, metal organic frame- the development of tools for organic materials modeling. This works, organic/inorganic nano-hybrids, biological solids, pure research effort is so vast that it alone could fill a library. In carbon solids and a variety of other forms. In other words, the this review, we will highlight a subset of the research on the variety of forms of organic molecules that must be described use of computational modeling tools used to describe the struc- − using simulations is vast. In addition, length scales from 10 11 ture, dynamics and energetics of organic electronic materials. to 100 m need to be considered to cover all relevant questions The highlighted studies focus on modeling the semiconducting from atomic arrangement to fully fabricated devices. Also time polymer poly-3-hexylthiophene (P3HT) (Figure 1(a)). While − scales from 10 15 to 109 s must be considered to cover from P3HT is not the highest performing polymer for any application, exciton formation to the lifetime of an OLED or PV device. Here it has been more studied than any other electronic polymer we consider the subset of modeling techniques that describes and so there is a wealth of verification data to test simulated structure over length scales from 0.1 to 1000 nm and dynamics results against. Therefore, P3HT is an excellent test subject for over time scales from fs to µs. organic electronic model development. Our goal is to examine One of the most important advantages of organic electronic the process of using multiple different simulation methods to materials is that they can be deposited from solution, which simulate structure and properties. In most simulation studies, potentially makes coating over large areas very inexpensive multiple modeling methods must be used in order to extract [13–16]. The drawback of solution deposition is that the or- meaningful data because a single method is not able to bridge ganic species must self-assemble into the desired molecular length and time scales. We will discuss limitations that can be configuration [17,18]. It is difficult to simulate self-assembly addressed using improved modeling and verification methods. processes because these processes often involve phase changes, This should justify the need for further research in developing reactions, complex interactions with the solvent and changing modeling tools for organic electronic materials.

CONTACT Roland Faller [email protected] © 2017 Informa UK Limited, trading as Taylor & Francis Group 2 T.F. HARRELSON ET AL. concentrations and temperature changes. For organic field effect consistent way, while simultaneously producing a molecular transistor materials the desired self-assembly is large molecular picture of the system. As with experimental characterisation crystals with few defects and a particular molecular orientation techniques, there are a number of modeling tools for different [19]. For OLED materials, typically amorphous materials are length and time scales [34]. In general, molecular models can desired to prevent exciton quenching. Also mixtures with low be broken into three categories. First there are electronic or volume percentages of well spaced emitters are often desired [4]. quantum mechanical models that explicitly treat the electron For solution processed OPV active layers, a mixture of donor position or density separately from the nuclear position. Second and acceptor materials is preferred [20,21]. These materials there are molecular dynamics or classical models that calculate should phase separate on a length scale that maximises exciton the position and forces on atoms or groups of atoms using separation at donor acceptor interfaces while at the same time classical force fields. Third there are continuum models that providing charge transport for holes through the donor material do not explicitly account for atoms, but rather keep track of to the anode and for electrons through the acceptor material to densities (mass, charge etc.) and/or change in densities within the cathode [22,23]. a volume as a function of time. At all length scales, the meth- The discovery of state-of-the-art materials used for elec- ods can be used to determine a static structure or an explicit tronic devices is achieved through combined synthesis, char- time dependence can be added to determine how a change in acterisation and simulation techniques. This is necessary to conditions (temperature, pressure, electric field etc.) affects the understand the morphology of the donor–acceptor mixture molecular/electronic structure. because we need information over different length scales, time Modelling of organic electronic materials almost always re- scales, and with different contrast. For a simulation scientist, quires the use of more than one length scale because electronic validation of a molecular model using experimental data is materials properties depend sensitively on the atomic/molecular often the most difficult challenge because the real sample is structure over length scales that are not accessible using elec- almost always more disordered, larger, and less well defined tronic simulations. Alternatively, to determine structure using than the simulation sample. It is often impossible to make classical methods, electronic simulations are needed to deter- ‘apples-to-apples’ comparisons. Realistic models that match mine the partial charge distribution. Figure 1(b) shows a work sample conditions can however be used to make predictions flow chart that shows the interplay of different molecular sim- for new physics. The social challenge is develop close work- ulation methods for bridging length and time scales (as will ing relationships between simulation and characterisation sci- be discussed in much greater detail below). Here we show how entists that enable development of new modeling tools and simulations at each length scale need to be validated by measure-

validation using appropriate samples and methods. Here is a ments at the appropriate length/time scale. For example, a CG brief list of references for characterisation techniques for or- model may be validated using GISAXS or electron microscopy ganic electronic materials. Neutron scattering methods mea- data. However, to explain molecular details, the CG model sure the structure and dynamics of hydrogen atoms in organic would need to be fine grained to the atomistic MD scale and then electronics through coherent scattering (small angle neutron the molecular structure should be validated using a different scattering) and/or energy loss measurement (inelastic neutron technique. GIWAXS or NMR could be used to show that the MD scattering) [24]. Electron microscopy and tomography create model is consistent with measured results at the atomistic scale. 2D and 3D images of OPV device layers with nanometer res- If the goal of the study is to determine details of the electron olution [25,26]. Similar to neutron methods, electromagnetic transport in the simulated data a second fine graining to an radiation techniques study both the structure and dynamics electronic model would be necessary to impart moving electrons of non-hydrogen atoms in organic electronics, as the atomic to the nuclear positions determined using MD and within the electromagnetic cross-section increases with atomic number. morphology developed by CG modeling. Once again, validation X-ray techniques such as grazing incidence wide angle scattering of the electronic model using, e.g. pulsed spectroscopy, UPS or (GIWAXS) and small angle scattering (GISAXS) are the most electronic measurements would be needed to determine that useful tools to determine the spacing of molecules in ordered the model describes a physical reality. This process of multiple domains and the spacing of domains of separated materials, modeling and validation steps is enormously time consuming respectively [27]. UV/Vis spectroscopy provides information and requires many different skill sets. As a result there are very for the accessible excited electronic states of organic electron- few molecular electronic samples that have been thoroughly ics. Infrared and Raman spectroscopy give vibrational dynamic simulated using multi-scale models. We posit here that the de- information about the material, but selection rules prevent ob- velopment of improved methods to bridge modeling techniques servation of all vibrational modes [28].Electronspinresonance coupled with optimised experimental validation could greatly and nuclear magnetic resonance (NMR) spectroscopies provide reduce the required effort and increase the predictive power detailed information about the local magnetic environment of delivered by computational methods. electrons and nuclei, which are used to determine molecular structure and the relaxation rates of dynamic molecular mo- 2. Electronic methods tion [29]. Detailed reviews of characterisation techniques used for OPV device and materials characterisation are listed here Electronic structure methods are invaluable to understanding [30–33]. the charge transport in organic semiconductors (OSCs). The The experimental techniques provide a patchwork under- electronic structure in OSCs is locally dependent on the molec- standing of the molecular system. Simulation methods are ular structure and globally dependent on the electronic state needed to stitch together the experimental information in a distribution caused by local variations in molecular structure. MOLECULAR SIMULATION 3

Figure 1. (Colour online) (a) The chemical structure of poly-3-hexylthiophene (P3HT), the polymer mainly discussed in this review. (b) Flow chart that shows the connection between different length/time scale simulations (left column) and the type of information gained (right column) integrated with the experiments.

Evaluating the electronic structure of a molecular system is a single orbital, φi, which is formed by an effective single body extremely computationally demanding, meaning that only small potential (Vs) containing all the repulsive forces from the other systems (usually limited to 100s of atoms) can be treated. The charges. Equation (2) defines the electron density, n,asthe electronic structure for the limit volume can only be determined sum of the individual orbital densities. The effective single body for one particular conformer at one time. For highly crystalline potential, Vs, contains contributions from three terms: the ex- materials, a small set of atoms is usually sufficient to characterise ternal potential (V), the electron–electron repulsion and the a material in silico using electronic structure methods and the exchange–correlation potential (VXC). VXC has no classical assumption that the structure has no variations over a large meaning, and is a consequence of mapping a quantum mechan- sample volume is a good one [35,36]. This is called the infinite ical many-body problem onto an effective single body system. crystal approximation. Most high-performing OSCs are locally The exchange–correlation potential is the functional derivative disordered [37], where the disorder introduces a large variety of of the exchange–correlation (XC) functional with respect to molecular conformers that cannot be simulated using symmet- electron density, which means the functional is invariant to ric boundary conditions. Thus the length scale or number of changes in the system configuration. Thus, in order to solve conformers is much larger than can be feasibly simulated. Elec- the equations for DFT for any system, we must specify the form tronic structure methods are still useful because they provide of the exchange–correlation functional. Finding the exact form molecular scale electronic information to parameterise larger of the exchange–correlation functional would be solving the full models. When materials are disordered, these simulations fail to many body problem, but several approximate functionals exist, account for the structural heterogeneity found in most samples which provide many flavours of DFT that are optimised for such that a sufficient number of possible conformers need to different problems [38–40]. be calculated. Thus a clever choice of sample configurations is needed to extract meaningful electronic simulation results. −2 ∇2 + Vs(r) φi(r) = iφi(r) (1) Electronic structure methods can be grouped into two main 2m categories: ab initio and semi-empirical methods. Ab initio N methods do not require external parameters other than atomic 2 n(r) = |φi(r)| (2) positions to complete the calculation, whereas semi-empirical i methods have fitting parameters. Within these two categories, e2n(r) V (r) = V(r) + r + V (r) we focus on density functional theory (DFT) and tight-binding s d XC Hamiltonian methods. r − r (3) 2.1. Density functional theory The information contained in the electronic structure pro- DFT presents a computationally efficient way to approximately vides insight into the structure (and in the case of time depen- solve the many-body Schrödinger equation, actually the dent DFT or Car–Parinello MD) dynamics of the nuclei. The Kohn–Sham equation. The efficiency of DFT comes from the Hellman–Feynman theorem provides a connection between the collapse of a high-dimensional many body problem (positions inter-atomic forces and DFT equations allowing the computa- of every electrons represented as variable) into a simpler prob- tion of equilibrium positions or nuclear dynamics using lem of a self-consistent electron density equation. The set of time-dependent simulations. Combining the computation of self-consistent equations solved in DFT are below (Equations inter-atomic forces and a optimisation scheme such as steepest (1)–(3)). Equation (1) is an effective single body equation for descent, produces an optimised geometry in a local energetic 4 T.F. HARRELSON ET AL. minimum. Geometry optimisations have been applied to OSCs to find static equilibrium states of organic electronic oligomers and/or dopants [41]. If coupled with a molecular dynamics scheme, DFT can reasonably accurately simulate the dynam- ics of molecules, including non-equilibrium geometries [42]. However, the computational expense required for computing inter-atomic forces using DFT limits the ability of these schemes to properly sample phase space in all but the simplest systems. Technically the calculations need a basis to the vector space of orbitals (or electron density distributions). The simulation can be tailored to specific molecular systems through the choice of an appropriate basis set. Gaussian basis sets are used to describe molecule(s) in the gas phase. If the dielectric of the background is adjusted, the conditions mimic an implicit sol- Figure 2. (Colour online) B3LYP-D/6-31+G(d)-optimised geometries of the vent, which extends the use of Gaussian basis sets to the liquid F4TCNQ/4T complex at three energy minima (top view, left; side view, right). phase. Deficiencies of this approach include the absence of Interatomic distances between two molecules (in Å) and the amount of charge transfer, CT, are shown. For clarity, hydrogen atoms are not shown in the side neighbour interactions, and the computational expense of DFT views. Reprinted with permission from [41]. Copyright (2011) American Chemical limits polymer simulations to oligomers (as one can feasibly Society. simulate 100s of atoms). Plane wave basis sets are used to model extended crystalline solids due to the periodicity of the basis functions [35,36], but again computational cost limits the unit cell size. However, neighbour interactions are included and infinitely long crystalline polymers can be simulated. However, disordered systems remain difficult to model due to the inability to include multiple conformers within one unit cell and the small size of the unit cell mandates an artificial symmetry. Rigorous experimental validation of DFT methods is cru- cial due to the number of assumptions. Common experiments

used to validate electronic simulations include ultraviolet pho- toelectron spectroscopy (UPS) [43,44], ultraviolet–visible spec- troscopy (UV–vis) [45–47] and infrared spectroscopy (IR) [46,48,49]. Each experiment validates a different observable of the DFT calculation. UPS provides the energy levels of the valence electrons in the ground state, which can be accurately calculated using DFT. UV–vis measures the energy differences between excited electronic states and the ground state. This is difficult to reproduce with DFT; it is difficult to assign meaning to excited orbitals in DFT, as the theorems of DFT hold for the ground state only [50]. IR spectroscopy measures symmetry allowed molecular vibrations, which can be accurately simulated by finding the eigenvalues of the a matrix of interatomic force constants [48,49]. In all cases, the ease of validation is determined by the com- plexity of the material. Heterogeneous materials, such as semicrystalline P3HT, have intermixed crystalline and amor- Figure 3. (Colour online) Top panel: Schematic of a one-dimensional acceptor lattice phous domains. Crystalline domains simulated using DFT must model. Central panel: corresponding on-site energy i (L = 13 Å and /0 = 3.5). be perfectly periodic, meaning that only single crystal systems The arrows indicate short time quantum diffusion and long time relaxation toward are well suited for DFT simulations. If the system is com- the lowest energy site. Bottom panel: Schematics of time evolution of charge density along acceptor sites. Used with permission from [55]. Copyright (2015) posed of randomly ordered crystallites, plane wave DFT will American Chemical Society. over-estimate alignment of dipoles, quadrupoles, etc. This is problematic because most OSCs have a significant quadrupolar moment [51]. Thus, theorists are presented with the options and assumed to provide meaning to the sample in general. Only of underestimating (Gaussian DFT) or overestimating (PW- a wise choice of characteristic simulations makes the simulation DFT) environmental electrostatic effects. Reducing the effect meaningful for the sample. of these assumptions in crystalline domain OSC simulations DFT with Gaussian basis sets has been used to study charge- represents an ongoing challenge for DFT method development transfer states between dopants and OSCs providing a link [52]. As discussed above, to simulate amorphous domains a between polymer length and the amount of charge transferred huge number of different molecular configurations would need from the OSC molecule to the dopant [41,53]. Figure 2(left) to be addressed. Instead particular configurations are simulated demonstrates the effect of molecular geometry on charge MOLECULAR SIMULATION 5

N transfer between quatro-thiophene and the dopant F4TCNQ. Hˆ = λ(k)u(k) |ii| Different charge-transfer configurations display changes in amount el-ph i of fractional charge transfer. At experimental temperatures, k i we expect multiple instances of each configuration present in (k) (k) (k) + α ui − uj |i j (6) Figure 2(1), which demonstrates the importance of sampling j=i±1 multiple configurations when relating DFT simulation results to N OSC properties. In addition to charge-transfer properties, Gaus- ˆ 1 (k) (k) 2 1 (k) (k) (k) 2 Hph = m u˙ + m ω u (7) sian based DFT can be used to parameterise classical molec- i i k i 2 2 ular dynamics models [54] and model Hamiltonian based schemes [55]. Molecular dynamics simulations require an es- timation of partial atomic charges, and the potential arising In the context of OSCs, an electronic state (a single wave- from the nuclei and the electron density (determined from function) rests on a molecular ‘site’, but the actual wave-function DFT simulations of a particular conformer) is fit to an effec- is never computed. Coarse graining a set of nuclear positions to tive electrostatic potential originating from non-integer charges represent an electronic site limits the specificity of the model for centred at the nuclear positions [56]. Since the electron density a particular sample. When applied to OSCs, model Hamiltoni- is unique to the nuclear positions, the effective partial charges ans typically set the on-site energies () and various coupling may change upon nuclear rearrangement, which is problematic energies (τ) between electronic sites to be constant, which may for molecular dynamics as the partial charges are fixed for be parameterised using DFT. The coupling constants allow the all configurations. This means that the choice of molecular electrons to hop from one site to another, and are given by the conformer used for the DFT simulation can have a large effect overlap of the wave-functions between sites. However, the static on the partial charges and accuracy of the molecular dynamics and dynamic disorder of OSCs requires the on-site energies and model. coupling constants to vary to account for energetic variation between sites due to different molecular environments and/or molecular vibrational motion (phonons). To approximate dis- 2.2. Model Hamiltonian methods order, the on-site energies and coupling constants have been randomised around a mean value [47,57–59]. Recently, cou- DFT based methods are cumbersome when studying the elec- pling to vibrational modes and a thermal bath have been added tron dynamics of complex molecular systems. Poor computa- to model Hamiltonian approaches to test the effects of dynamic tional scaling of DFT severely limits system size, which means and static disorder on electronic dynamics [55]. electron dynamics can only be studied on the length scale of Lee et. al. provide an example of a model Hamiltonian ap- nanometers. proach applied to OSCs to explain the charge separation through To reduce the computational expense, many groups turn quantum diffusion [55]. Figure 3 shows a schematic of what to model tight-binding Hamiltonians to simulate the quan- the model Hamiltonian represents, which is accompanied by tum dynamics of electrons without explicitly accounting for some relevant results. Initially an electron is placed at position the positions of the nuclei. A tight-binding Hamiltonian is a 2, which is separated from the positive charge by 26 Å. At matrix of interactions between neighbouring states (commonly short times (∼ 100 fs), the electron quickly separates from the referred to as coupling between states). An example of a model charge before settling into the lowest energy state at longer Hamiltonian is provided in Equations (4)–(7)[55]. This set times (∼ 10 ps). The important result is that a portion of the of equations modifies the typical tight-binding Hamiltonian electron density separates quickly from the positive charge, (Equation (5)) to include coupling to a bath of phonons. In this  τ while the remainder of the density moves toward the positive example, is the on-site energy of the electron, allows hopping interface. This charge density bifurcation can be explained by between adjacent sites and λ/α allow local/nonlocal transfer of |i conservation of average energy within the electronic Hamil- electron energy to vibrational modes. represents an electronic tonian; the system lowers its energy by moving the electron state, whose form is not clearly defined, but it can be assumed density closer to the positive interface, but this must be balanced that the set of all states are orthonormal. Equation (7)represents by a portion of density moving away from the interface to the energy from classical harmonic oscillators represented in conserve energy. Without electron–phonon coupling, this bi- the phonon modes of the system. The expansion of a com- furcation is effectively coherent in opposite directions. Inclusion plex Hamiltonian into a series of interaction potentials/energies of electron–phonon coupling and vibrational energy into the allows one to identify parameters responsible for changes in Hamiltonian reduces this outward motion, as the inclusion electron dynamics. These parameters can be attributed to the of other energy types destroys the conservation of electronic chemical structure of the underlying molecules in the material, energy. High energy electronic eigenstates couple to low energy which provides insight into potential molecular design rules. states through phonons, allowing electronic relaxation, which u(k) m(k) Here i are the position of the nuclei, are the masses, and decohere the electronic density over time. The strength of the ω(k) are the respective frequencies. electron–phonon coupling governs this decoherence, and re- duces the charge separation efficiency. The authors assume only Hˆ = Hˆ + Hˆ + Hˆ total el ⎡ el-ph ph ⎤ (4) one high and one low energy mode, and electron–phonon cou- N pling only exists between the high energy mode, which prevents ˆ ⎣ ⎦ Hel =  |ii| + τ |i j (5) quantitative conclusions. In all cases, the authors were able to i j=i±1 show fast charge separation, and slow relaxation to thermal 6 T.F. HARRELSON ET AL. equilibrium, leading them to conclude that the yield of free possible conformers in a sample. For example, a fully planar charges is governed by the interplay of those two competing conjugated chain may be used to determine the partial charge processes. distribution, but only a fraction of the chains in the sample may The accuracy of model Hamiltonian approaches critically have this fully planar configuration. Also as a partial charge depend on the accurate parametrisation of the energies and does not relate to a operator in quantum-mechanical sense coupling constants. This parametrisation is non-trivial with the there is ambiguity in the way to map the electron distribution inclusion of vibrational coupling, which, in principle, intro- onto partial charges. Since the partial charges remain fixed duces ∼3N parameters for each type of vibrational coupling throughout the duration of the MD simulation, the chosen DFT considered, where N is the number of atoms in the solid. The configuration will affect the structures and dynamics sampled parametrisation can be simplified as many vibrational modes using the MD model. To complicate this issue, organic elec- are roughly equivalent. However, the smallest possible unit cell tronic molecules can become polarised or interacting donor– of crystalline P3HT contains 100 atoms (294 vibrational modes), acceptor pairs can transfer charge at heterojunctions. These which makes parametrisation difficult and time consuming. Ad- intermolecular electronic interactions dynamically change the ditionally, the lack of information on atomic positions precludes partial charges in the system, limiting the applicability of classi- the sampling of different disordered configurations typically cal MD methods. Also, in films of pure P3HT, we see differences seen in polymeric OSCs. To properly understand disordered in the UV–vis spectrum as a function of dihedral stiffness, configurations present in most OSCs, a method accounting indicating that partial charge may also be a function of the for atomic degrees of freedom is required. To mitigate the intermonomer dihedral angle [61]. computational scaling limitations of electronic modeling, one Poelking and Andrienko studied the stability of the crys- must use molecular dynamics methods. talline phase of P3HT using MD [51] to address the solid/solid  phase transition from the metastable I polymorph to the more 3. Classical simulation methods thermodynamically stable form I of P3HT [49,62]. Figure 4  shows sample configurations for polymorph I (a–c) and poly- 3.1. Molecular dynamics morph I (d–f). The MD simulations show that increasing tem-  Molecular dynamics is a method that tracks the classical mo- perature causes an irreversible transition from polymorph I to I, tion of a particle based system following Hamilton’s equations indicating that polymorph I is thermodynamically favoured according to a (semi-)empirical force field. The position of and should be present in P3HT films at relevant temperatures. every interaction site (in the case of an atomistically detailed Experimentally, it is difficult to distinguish between the two model an atom) at every time step is known, providing complete structures [63–65], due to the significant structural disorder at information about the system. The use of a semi-empirical force the atomic scale and similar spacing between parallel polymer field in a classical scheme improves computational performance backbones. In this example, MD is used to provide structural due to the elimination of electronic degrees of freedom, allowing and thermodynamic information that is unobtainable from ex- the simulation of 105 atoms or more for up to microseconds. periments. The downside is that the results are only as reliable The massive amount of information is used to generate sta- as the assumptions made in the MD model. tistical mechanical ensembles and from there the macroscopic After finding the molecular trajectories with MD, Poelking properties of the system, which provide a clear path to compare and Andrienko fed the simulated morphologies into a semiclas- simulation to experimental results. In the context of OSCs, elec- sical Marcus hopping scheme allowing the estimation of elec- trical properties are of primary interest, which are not directly tronic transport behaviour. The electronic transport behaviour available from MD simulations because electron dynamics is provided estimates for charge mobilities that showed excellent not included in the simulation. Thus, MD simulation results are agreement with experiments in P3HT nanofibers. Specifically, commonly combined with electron dynamics methods, such they show how a few defects in side-chain attachment (90% as semiclassical Marcus theory, to create electronic properties regioregular P3HT) lead to an order of magnitude drop in from the structural results of MD. In this section, we discuss the charge-carrier mobility [66]. This study shows how classical fundamentals of MD, and some relevant examples of how MD molecular dynamics can be coupled with an electronic dynamics provides significant insight into the understanding of electrical scheme (semiclassical Marcus theory) to provide electron dy- properties of OSCs. namics at longer length scales than is possible using DFT alone. A good comparison between experiment and theory requires This approach is limited by the assumptions made to simulate a well parameterised force field. The force field typically con- electron dynamics at this length scale; the system is effectively tains the partial charges, dispersion force parameters, and bond an infinitely large crystal with chain ends occurring every 20 constants for stretching, angle bending, and dihedral motion. monomers. Films of P3HT have randomly oriented crystallites The partial charges and some bonded parameters are often embedded within amorphous domains. The long chains can parameterised from DFT methods, whereas the dispersion pa- wrap into several crystalline domains or the same domain sev- rameters are optimised from empirical data although DFT can eral times, meaning that a single chain has both amorphous also been used [60]. Care must be taken when parameterising and crystalline components. This structural heterogeneity was partial charges from DFT, as long range electrostatic effects (not not considered in this study due to the increased computational accounted for in DFT) have a strong effect on the electronic expense that simulation of the more disordered system would structure in highly ordered organic electronics [51]. Also, the entail. partial charge distribution can be skewed by choosing a single In a similar study, Alexiadis et al. used atomistic MD to molecular conformer in DFT that does not represent all of the study self-organisation and structure in (semi)crystalline MOLECULAR SIMULATION 7

Figure 4. (Colour online) Regioregular poly(3-hexylthiophene) lamellar crystals of polymorphs I (a–c) and I (d–f) as obtained from molecular dynamics simulations. Projections are constructed along the (a) and (d) [0 0 1], (b) and (e) [0 1 0], (c) [1 1 0] and (f) [2 1 0] crystal direction. Note the degree of interlamellar correlation mediated by the hexyl side chains even in their disordered phase. Reprinted with permission from [51].

Figure 5. (Colour online) Snapshots of the simulation configuration of the system with P3HT:C60 = 1.27:1 (w/w) with Nmono = 48 at t = 0 ns (left), t = 30 ns (centre) and t = 135 ns (right). The C60 molecules in the largest cluster are highlighted in blue and all other particles in the system are shown as dots. The length of each side of the simulation box is roughly 25 nm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Reprinted with permission from [68].

rr-P3HT by addressing the amorphous and crystalline parts sep- highlights the importance in carefully choosing/parameterising arately. Within the crystalline region, a unit cell that looks sim- the force field from DFT and validating with measured data.  ilar to polymorph I is the dominant structural moiety at room Atomistic molecular dynamics of P3HT and PBTTT-C12 temperature [67]. The reason for the differences in reported (Poly[2,5-bis(3-dodecylthiophen-2-yl)thieno[3,2-b]thiophene]) structure from Poelking et al. is the different force field, which were compared by our group to clarify why nanoscale structural 8 T.F. HARRELSON ET AL. properties lead to higher measured hole mobility in PBTTT orders of magnitude faster to allow the study of morphology of vs. P3HT [54]. We found that the dihedral angle between the polymer/fullerene bulk heterojunctions at length and time scales two thiophenes in PBTTT is more planar than that in P3HT, relevant to OPV devices. These models although optimised at while the thienothiophene dihedral in PBTTT displayed similar only one state-point each correctly represent the structure of variation to P3HT. The bulkier side chain in PBTTT increases the systems in a sizable range of state space although that is not antiparallel order between intra-chain monomers relative to guaranteed a priori [70]. Figure 5 shows the growth of fullerene P3HT, which produces a structure with fewer defects. In this clusters in a P3HT/fullerene mixture using an IBI developed case we did not directly calculate how the dihedral angles of the system [68]. polymer backbone effect the charge mobility but rather inferred A variation of the model using the same IBI methodology this data from the structure. The increased order and the net was used by the Huang group to study the aggregation of P3HT increase in backbone planarity are the likely causes for higher into high-aspect-ratio nanofibers, nanowires, or nanoribbons electrical mobilities in PBTTT. Our simulation also allowed us in implicit weak solvents such as anisole. The CG model was to predict that the variation in the thienothiophene–thiophene adapted to the local structure and dynamics of an atomistic dihedral angle can be reduced with attachment of a small side model with explicit solvent. The simulations match the ex- chain to increase rotational inertia making deformation less perimental phase behaviour of P3HT in anisole as they yield likely. The advantage to this approach is reduced computational aggregation below ≈320 K but not above. At room temperature expense because we did not need to calculate transfer integrals. hairpins and helices of single chains are predicted. These sin- The disadvantage is lack of quantitative data and an inability to gle chain conformations are the building blocks for the larger validate to electronic data. scale structures. These simulations are particularly interesting As mentioned above, classical MD does not allow dynamic because they, for the first time, used polymers with molecular change of partial charges which means that the charge cannot weight similar to experiment and also showed hairpin folding react to its local environment or other charges. As a result on the same length scale as seen in experiment. Further the helix it is not possible to simulate fully the dynamic processes for structure has not been experimentally detected, as so is a theory charged sites, like the diffusion of a dopant molecule or reaction driven prediction. In addition to providing insight into this dynamics like formation of a cross linked bond. Future MD will mechanisms of fiber formation, the simulations can also resolve need to include polarisable or even reactive modeling to expand details of the molecular-level organisation in the fibers [71]. the modeling tool kit. A second challenge is to model structure So this approach shows several advantages in moving to longer formation which is currently limited for classical MD in con- length and time scales, allowing formation of realistic molecular

densed matter due to both lack of structure sensitive force fields structures that are found in P3HT anisole and toluene solutions or access to long time and length scales. MD is limited to at best [61]. But these CG models are devoid of fine structure or elec- microseconds, which is not long enough to observe structure tronic information. Similar to polystyrene [72] CG models of formation, particularly in polymers. To sample longer times, we P3HT validated in either in dilute solution or polymer melts must reduce degrees of freedom, which requires coarse-grained can be used to show structure formation with a high degree of methods. accuracy. This observation shows that CG models can be used outside of the original state-point. In this case the P3HT was coarse grained from a melt but folded to make a crystalline solid 3.2. Coarse grained modeling in a dilute solution. As described above, MD modeling allows simulation of molec- In further work the Huang group used the dihedral angle ular structure over longer length and time scales larger than distribution in the CG P3HT clusters to predict exciton hopping electronic models make possible, but at the cost of force fields under the assumption that the longest straight segments would that are not sensitive to molecular conformation. Box sizes with represent the lowest energy configuration [47]. Although the tens of nm can be simulated for up to microseconds using MD. atomistic details were missing in the CG model, the modelled However many processes occur over longer length scales and exciton hopping kinetics were similar to data acquired using take much longer. One example is the formation of a crystalline ultra-fast laser techniques in clusters of P3HT in solution, vali- domain from a polymer melt. In order to describe and model dating the use of a CG model to extract electronic information. these slow processes, it is necessary to perform molecular dy- In a second extension of the CG model, the Huang and Groves namics on groups of atoms that are connected and have similar developed mixtures of P3HT at different MW averages and function. For example, a benzene ring can be modelled as a distributions [73]. In this study, nanocrystal formation in a single coarse grained (CG) ‘superatom’. Thus a CG-MD model melt was predicted from different MW distributions. The CG has fewer particles with less chemical specificity, but can be used morphology was fine grained to the atomistic scale and then to model larger volumes for longer time periods. the atomistic geometries were used with a kinetic Monte Carlo A few years ago our group developed a model for P3HT and method to determine charge transport characteristics of the P3HT/C60 mixtures based on systematic coarse-graining using semicrystalline P3HT morphologies. The crystalline structure the Iterative Boltzmann Inversion (IBI) Method where a few was used to validate the simulation at the atomistic scale. This heavy atoms – typically 4–6 – together with their hydrogens study completed the entire work flow diagram shown in Figure are subsumed into a ‘superatom’ [69]. The interactions between 1(b). The P3HT was modelled from DFT, the partial charges these superatoms are optimised that local structure (as defined were used in atomistic MD, the atomistic MD was then coarse by pair correlation functions, superbond distances etc.) is cor- grained and high MW morphologies were obtained over long rectly represented and at the same time simulations are several length and time scales. The morphology was then fine grained MOLECULAR SIMULATION 9 to the atomistic scale and finally the nuclear positions were used In a separate publication, they determined a phase diagram for charge transport calculations. At each step, over six years the for melts of thiophene oligomers with different side chain at- models were validated. tachments. Depending on chemistry, disordered systems, lamel- A more generic coarse-grained model can also be developed lae, perforated lamellae, cylinders and ribbons could be formed. for describing the liquid crystalline like order in P3HT on the Oligomer architecture affects also the ODT because backbone mesoscale [74](seeFigure6). The bonded interactions were also beads are the species with the strongest enthalpic driving force here obtained by Boltzmann-inversion from the atomistic scale for aggregation, and the different architectures permit different under -solvent conditions which produces the same single amounts of backbone bead exposure [76]. This clearly shows chain statistics like a melt. The non-bonded interactions are the strength of experimentally validated CG simulations as the anisotropic and soft. They stem from a combination of π − π design rules determined here can be used by polymer synthesis interactions and the entropic repulsion between side chains. groups to predict the effect of chemistry on morphology forma- This model obtains uniaxial and biaxial nematic mesophases, tion of complex polymer systems. where molecular weight effects on phase behaviour are correctly Root et al. use coarse-grained molecular dynamics simula- represented. Conjugation defects tend to localise near chain tions to predict a number of mechanical properties – tensile ends [74]. Such a model is not as chemically accurate as a sys- modulus, Poisson ratio – for P3HT and its blend with C60. tematically coarse-grained model. However, due to the generic It turns out that the degree of coarse-graining has a strong nature such models allow to control fundamental physical or effect on the predicted properties as a one to one mapping chemical properties in the model and therefore study their (one 3-hexylthiophene per bead) leads to inaccurate density influence. and modulus values. A three-site model like proposed by Huang et al. [69] leads to values which are in reasonable agreement with experiments. For longer chains where entanglements play a role 3.2.1. Chemistry vs. morphology a decrease of the entanglement density with increased blending Jayamaran and her group have performed a series of large scale concentration of fullerene is observed which might explain the CG simulations to study the morphology of P3HT-like systems experimental embrittlement [78]. with different chemical identity using a simplified version of Du et al. use dissipative particle dynamics (DPD) with a our IBI derived model [75,76]. They present a high-throughput rather simple potential of the standard P3HT/PCBM system to coarse-grained simulation study that links molecular-level pa- determine the 3D morphology of BHJ solar cells [79]. Then, they rameters to large scale morphological features in pure polymers estimate the performance using graph theory. They find that a and donor–acceptor blends. For the polymers they vary the ar- volume fraction of about 40–50% PCBM is optimal and leads rangement of the sidegroup as they study systems with isotactic, to a bi-continuous morphology with about 6 nm domain sizes. syndiotactic and double-syndiotactic (switching sides every two They also study the effect of processing conditions on the mor- groups) arrangement of side groups. This is supposed to mimic phology where they find that 413 K is the optimal temperature P3HT, PBTTT and PDHTB (poly(l,4-di(2-(hexylthienyl)) ben- and that the degree of phase separation between polymer and zene)). Figure 7 shows simulated morphologies for P3HT and fullerene increases during evaporation, i.e. the solvent weakens PDHTB along with simulated GIWAXS data and measured the phase separation. In spite of these seemingly predictive GIWAXS data. This is an excellent example to show how sim- results, the CG model in this study does not produce a morphol- ulation data can be validated against measured data. In these ogy consistent with P3HT/PCBM. P3HT is a semicrystalline examples, the backbone and side chain of the polymer phase polymer and the crystalline domains at immiscible with PCBM separate more than would be expected in a measured sample. while the amorphous domains mix with PCBM. So the mixed The simulated spectra are lower resolution because the volume morphology is either two or three phased depending on whether of the simulated data is much lower than the volume of the there is enough PCBM to fill the amorphous P3HT volume experimentally measured sample [26,80,81]. The cited DPD model generates a two phase mor- For the blends of the simulated polymers with fullerenes, phology at all mixing ratios and thus produces morphologies they vary the miscibility of the acceptor by changing the non- that are bi-continuous but not physically relevant to P3HT. This bonded parameters of the acceptors with the donors as well is an example in which a model was insufficiently validated by as other acceptors. They are able to obtain a variety of mor- published data. phologies including lamellae, hexagonally packed cylinders and Clearly the extension of simulation techniques over multiple acceptor intercalation among donor side chains. There is also length scales leads to increased prediction power. However the an order disorder transition (ODT) below which short chains cost in scientific and computational time is large. For a CG self-assemble into lamellae or close packed cylinders. Based on simulation, first an electronic model of the polymer must be the simulated results, design parameters can be made for how developed and validated. Then the partial charges are used to to tailor blend morphologies by changing the local chemistry. create a molecular scale MD model, which must also be validated The layer spacings in their morphologies generally agree with by experiment. Then IBI (or another technique like force match- experimental data. The average center-to-center distance in ing [82]) is used to coarse grain the MD model into super atoms. hexagonal morphology for the PDHTB variant is smaller than Finally after multiple length scales of work, the CG model can be experimentally observed [77]. But the model is able to explain used to make structural predictions. Because of these multiple the existence or non-existence of the cylindrical morphology steps and the need for validation, very few polymer models have for the different structures as alternating side-chain orientations been coarse grained. Another set of challenges (not shown here) inhibit cylindrical packing. would be to finegrain the structures determine to form by coarse 10 T.F. HARRELSON ET AL.

Figure 6. (Colour online) (a) Chemical structure of poly(3-hexylthiophene) (P3HT). Atomistic (b) and coarse-grained (c) representations of the P3HT chain. In the coarse- grained model each repeat unit is a single interacting site placed at the intersection of two imaginary lines placed along the bonds connecting the thiophene rings. This choice improves the transferability of the coarse-grained potential. Reprinted with permission from [74].

Figure 7. (Colour online) (a) Imperfect lamellar P1 (left), diffraction pattern calculated from simulation data (centre) and GIWAXS data from annealed P3HT (right). (b) P3 in hexagonally packed cylinders at T∗=2.25 (left), diffraction pattern calculated from simulation data (centre) and GIWAXS data from annealed PDHBT (right). Reprinted with permission from [75]. Copyright (2013) American Chemical Society.

graining into an atomistic MD model. We did discuss several electronic behaviour, but are limited when model systems in- methods to extract electronic information from MD determined crease in size and/or disorder. To simulate larger length and structures, but many of these methods require reintroduction time scales, classical molecular dynamics methods are neces- of DFT or other electronic simulation methods. This discussion sary. Detailed electronic information has to be averaged out clearly shows that a reduction in the time and effort required to to create effective partial charges centred at each atom, which move between simulation methods would yield a huge increase can be a significant source of error in donor–acceptor systems. in the use and predictive power of the methods. Molecular dynamics can simulate processes on length and time scales up to 10s of nanometers and microseconds, respectively. This is ideal to study the local arrangements of atoms and local 4. Conclusions and recommendations neighbourhoods but it is not enough to study self-assembly pro- Understanding structure/property relationships of OSCs re- cesses or morphology formation. To achieve necessary length quires knowledge of length scales ranging from Angstroms to and time scales for that, many groups coarse-grain the atomistic (at least) microns, and time scales from femtoseconds to mil- molecular dynamics by grouping atoms into super-atoms, and liseconds and beyond. The need for advanced simulation tech- also often implicitly accounting for solvent molecules. The IBI niques increases as the experimental toolbox to probe these method is commonly used to transition between atomistic and length and time scales grows. We have shown that DFT and coarse-grained length scales to ensure proper linkage of the model Hamiltonian based methods give detailed information on larger system to the underlying atomistic system. MOLECULAR SIMULATION 11

Despite the success of each of these methods individually, the potential design criteria in the context of molecular design is key issue moving forward is integrating each of these methods paramount for the future of molecular simulation of organic into a consistent multi-scale model and workflow to explain electronics. OSCs. The biggest challenge is the transition from quantum chemical to classical atomistic simulations, as there are many examples of OSC systems that do not behave classically even at Disclosure statement large length scales. Even structural determination is limited in conjugated samples that have differing partial charge distribu- No potential conflict of interest was reported by the authors. tions depending on the dihedral angle between conjugated rings. Further development in polarisable and/or reactive force fields is needed to properly transition from the electronic to dynamically References adaptive atomistic length scales. 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