Relationship between Molecular Structure and Electron Transfer in a

Article pubs.acs.org/JPCC

Relationship between Molecular Structure and Electron Transfer in a Polymeric Nitroxyl-Radical Energy Storage Material Travis W. Kemper,*,† Ross E. Larsen,*,† and Thomas Gennett‡ †

Computational Science Center and ‡Chemical and Materials Sciences Center, National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, Colorado 80401, United States S Supporting Information *

ABSTRACT: In recent years, stable organic radical functional groups have been incorporated into a variety of polymeric materials for use as electrodes within energy storage devices, for example, batteries and capacitors. With the complex nature of the charge-transfer processes in a polymer matrix, the morphologies of the polymer films can have a significant impact on the redox behavior of the organic-based radical. To elucidate possible effects of packing on electron-transport mechanisms, theoretical modeling of the wellcharacterized cathode material poly(2,2,6,6-tetramethylpiperidinyloxy methacrylate) (PTMA) was conducted. Polymer morphologies were modeled using classical molecular dynamics simulations, and subsequently, the electroniccoupling matrix element between each radical site was calculated. Building on a previously derived treatment of diffusion in inhomogeneous materials, an expression for an effective electron diffusion length and an effective electron diffusion rate was derived in terms of an electronic-coupling-weighted radial distribution function. Two primary distances were found to contribute to the effective electron transfer length of 5.5 Å with a majority of the electron transfer, nearly 85%, occurring between radical sites on different polymer chains. Finally, we point out that this analysis of charge transfer using an electroniccoupling-weighted radial distribution function has application beyond the specific system addressed here and that it may prove useful more generally for simulating electron-transfer processes in disordered molecular materials.

1. INTRODUCTION Developing new energy storage capabilities relevant to current and future energy supplies is a challenging issue with a diverse range of requirements. Applications can range from large-scale grid storage to electric vehicles to flexible electronic devices.1−4 Each of these applications requires different energy-storage device properties; consequently, different underlying architectures and materials are being used in these domains. In all cases, the next generation of energy-storage materials for these applications needs to be part of a sustainable cycle, including benign composition and recyclability.5 One technology, which fits this sustainability profile, is the organic radical electrode material introduced by Nakahara et al. in 2002.6 This new material offers a potentially low-cost and mechanically flexible platform that may be able to satisfy one or more of the current energy storage needs. However, the underlying fundamental chemical and physical processes that dominate the operation and performance of the polymeric organic radical electrode materials remain unclear. For example: How do electrons move through a radical organic polymer? Are solvent penetration and ion mobility intrinsically connected or could ions move through the solid organic matrix without a solvent sphere or solvent incorporation into the matrix, that is, swelling? This paper is the first in a series of theoretical and experimental reports that focuses on answering these questions. Here, the issue of electrons moving through a radical organic © 2014 American Chemical Society

polymer will be addressed by simulating the polymer morphology, then calculating electron coupling in the morphology, and finally by deriving an expression for the average electron transfer length and rate in the disordered polymer matrix. An organic radical polymer-based electrode incorporates stable radicals such as the neutral redox active nitroxide radical 2,2,6,6-tetramethyl-1-piperidinyloxy (TEMPO), polymerized into poly(2,2,6,6-tetramethylpiperidinyloxy methacrylate) (PTMA), Figure 1.7,8 During charging cycles, electrons are removed from the electrode leading to the oxidation of adjacent neutral radicals; electrons may then transfer from (neutral) radical sites to oxidized radical (cation) sites, while at the same time, counteranions diffuse through the polymer to form complexes with the oxidized sites. During discharging, the process is reversed and the cations are reduced to form neutral radicals and the anions diffuse back into the electrolyte. What makes this technology so exciting is that the PTMA films operate at nearly 100% of their theoretical capacity, have rapid rates of charge/discharge, and display great stability.9,10 The lack of degradation means that devices using this new technology are largely immune to failure from complicated Received: February 14, 2014 Revised: June 18, 2014 Published: June 25, 2014 17213

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incorporate the distance dependence of the electronic-coupling matrix elements and the distribution of intersite separations to derive an expression for the average electron transfer length in the system. Overall, our work allowed us to assess morphological features associated with electron-transfer pathways in PTMA and to compare interpolymer chain and intrapolymer chain contributions to electron transport.

2. COMPUTATIONAL AND THEORETICAL METHODS 2.1. Classical Molecular Dynamics. 2.1.1. Force-Field Parameters. To perform accurate MD simulations of a PTMA film, force-field (FF) parameters and atomic charges appropriate to PTMA polymers are necessary. Recently, a set of FF parameters (bonds, bond angles, and dihedrals) based on the AMBER formalism was published for TEMPO,17,18 and we have applied these parameters to the PTMA monomer. The presence of the ester group opposite the nitroxyl group on the TEMPO ring modifies the dipole of the monomer from that of ordinary TEMPO, and so we modified the atomic charges for the FF by calculating atomic partial charges using an electrostatic-potential (ESP) fit as implemented in Gaussian09.19−21 The calculations were performed on a methylterminated version of the monomer unit shown in Figure 5a, which is a viable model considering it will capture the charge distribution and dipole moment of the entire monomer unit. The ESP fit was performed on the optimized geometry of the neutral gas-phase monomer with the electronic structure computed using density functional theory (DFT) with a three-parameter Becke-style hybrid exchange correlation functional, B3LYP, and a 6-311++G(d,p) basis. The resulting monomer charges, given in the Supporting Information (SI), were then adjusted slightly so as to produce a neutral oligomer when monomers have methyls removed so they can be concatenated to form oligomers. 2.1.2. Molecular Dynamics Simulations. Using the AMBER force fields and our ESP based charges, MD simulations were run with the software package GROMACS.22 Simulations were conducted at constant temperature and pressure with a time step of 0.5 fs using the leapfrog integration scheme. A particle mesh Ewald summation was used to describe Coulombic interactions with a cutoff of 15 Å, and the same cutoff was also applied to the Lennard-Jones interactions. The temperature was controlled with a Nose−Hoover thermostat, and the pressure was controlled using a Parrinello−Rahman barostat scheme, both with time constants of 0.5 ps. Syndiotactic PTMA oligomers 24 monomer units long were created, as shown in Figure 2. One hundred one oligomers were randomly placed into a cubic simulation cell with a box length of 20 nm to give a total of 100 192 atoms and 2424

Figure 1. PTMA monomer unit.

chemistry associated with the charge trapping and lithium depletion that limits metal-oxide performance in standard Liion batteries (LIB). In recent years, Nakahara and co-workers in Japan have done extensive work with PTMA-based cathodes for LIBs, demonstrating the favorable properties of organic radicalbased technology.6,7,10−14 PTMA cathodes have achieved voltages of 3.5 V versus Li metal, have discharge capacities of 110 mAhg−1, have retained 89% of the initial capacity after 1000 cycles, have tolerated high C-rates, and have exhibited promising lifetime results.13,14 Further development of radical polymers for use in energystorage systems requires improving the capacity by increasing the density of radicals in a working cathode while maintaining cyclability and good lifetime performance.12,15,16 This requires establishing a fundamental understanding of the factors that limit charge transport and ion diffusion in polymeric radical cathodes. Establishing the mechanisms of mass and charge transport during charging and discharging cycles will facilitate future advances in organic radical-based energy-storage devices. The degree to which electron and ionic transport are coupled and how such coupling affects the impedance of the organic radical electrode under charge and discharge cycling remains an open question. In this paper, we report on initial steps toward understanding charge-transport processes in organic radical electrodes with a focus on electron transport within PTMA films. For this work, the approach included: classical molecular dynamics (MD) simulations to generate morphologies to understand the structure of an amorphous PTMA film, quantum mechanical electronic structure methods to calculate the electronic-coupling matrix element between radical sites in the morphology generated with the classical MD simulation, and modification of a previously derived formalism for understanding diffusion in inhomogeneous materials to

Figure 2. Syndiotactic PTMA 24-mer used in the molecular dynamics simulations. Gray spheres indicate carbon atoms, white spheres indicate hydrogen atoms, blue spheres indicate nitrogen atoms, and red spheres indicate oxygen atoms. 17214

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was hydrogen terminated by replacing the methacrylate oxygen bonded to the ring with a hydrogen with the appropriate bond length of 1.09 Å; the shift of the ring from its original position during minimization is corrected in the final global minimization. For each pair of sites, the TEMPO rings of i and j were removed from the polymer matrix and hydrogen terminated with the same procedure as the minimization, and Vij was calculated with the corresponding orbital transformation method using the electron-transfer routines in the NWCHEM software package, and the cation geometry was approximated with the neutral geometry, as the charge state has a minimal effect on the geometry, see section 3.2.25,26 In cases where the cation and neutral geometries differ significantly, the electronic coupling should be computed for the two fragments in the geometry at which the hopping event is expected to occur, midway between the cation and neutral geometries.27 In practice, this geometry can be approximated by averaging the internal coordinates of the neutral and cation states. In this implementation, the calculations are performed at the Hartree− Fock level, and the 6-31G Pople-type basis set was used. 2.2.2. Electron Diffusion with a Distribution of Transfer Rates and Distances. It would be desirable to relate the microscopic information about rates that depend on Vij and lengths that depend on atomic separations to a macroscopic diffusion constant, D, rather than modeling the diffusion directly, for example, with a Monte Carlo simulation.28 For the case of a single hopping length, l, and hopping rate, s, the diffusion constant is simply,29

TEMPO groups. The initially random configuration was equilibrated by increasing the temperature (at a pressure of 1 atm) up to 1200 K for 13 ns to reduce the volume, then by cooling the melt to 300 K over 1 ns, and then by equilibrating the system for an additional 9 ns, followed by 10 ns of data collection. This procedure resulted in a cubic periodic simulation cell of 9.7 nm with a density of 1.06 g cm−3, which is in reasonable agreement with the experimentally measured density of PTMA, 1.15 g cm−3.6 To test that our specific choice of a 24-mer does not affect the resulting film morphologies, we also simulated systems based on 12-mers and on 48-mers. As shown in the SI, the film structure was found to be independent of the oligomer lengths used. 2.2. Electron Transport. 2.2.1. Quantum Mechanical Modeling of Electron Transfer. Because PTMA is an electronic insulator,16 electron transfer (ET) is expected to be described by discrete transfer events (hops) rather than by bandlike conductivity seen in materials with delocalized states near the band edges. Hence, we can consider the species referred to in the redox scheme for ET shown below to be the individual TEMPO moieties, consistent with localized charges. At this level of description, the ET process occurs when an electron initially sitting on a neutral radical at site i• is transferred to a neighboring oxidized radical at site in the cation state, j+ i • + j + → i + + j•

(1)

This is considered a self-exchange oxidation/reduction (redox) reaction, as the radical is oxidized and the cation is reduced and each site is chemically equivalent. The rate at which an electron transfers from site i to site j is defined within the Marcus formalism as ij kET

1/2 ⎡ (λ + ΔG°)2 ⎤ Vij2 ⎡ π ⎤ = ⎥ ⎥ exp⎢ − ⎢ 4λkBT ⎦ ℏ ⎣ kBTλ ⎦ ⎣

D = sl 2/6

(3)

and the mobility, μ, can be related to the diffusion constant with the Einstein relation,

(2)

μ = eD/kBT

where T is temperature, Vij is the electronic-coupling matrix element between the sites i and j, λ is the reorganization energy, ΔGo is the free energy associated with the charge-transfer reaction, and ℏ and kB are Planck’s constant over 2π and Boltzmann’s constant, respectively.23,24 For a self-exchange reaction in the absence of an electric field or environmentinduced disorder, ΔGo is zero. The reorganization energy (λ) is the energy associated with geometric and dielectric relaxation related to the ET reaction and will be considered constant in the following discussion. The prediction of the absolute magnitude of the electron-transfer rate is beyond the scope of this work; we are interested here in how the packing in the PTMA film controls electronic coupling between sites i and j, (Vij), so we defer calculation of inner-sphere and outer-sphere contributions to λ to future work. The value of Vij between sites is dependent on how the polymer chains are arranged in the film because this packing will affect the overlap of the molecular orbitals relevant to electron transfer. We calculate Vij for each pair within a 10 Å radius from a single representative snapshot taken from the MD simulation. Because the single snapshot does not allow for a time average over thermal vibrations, a multistep minimization process based on the classical FF was used to remove thermal vibrations. First, the energy of each TEMPO ring in isolation was minimized to remove any intramolecular vibrations, and second, a global minimization was conducted to remove intermolecular vibrations that occur in the MD simulation. Each TEMPO ring was removed from the polymer matrix and

(4)

where e is the electron charge.30 It would be useful to estimate an effective length, leff, and effective rate, seff, in terms of the electronic-coupling matrix element and internitroxyl distances found during an MD simulation. Reducing the understanding of electron diffusion in a complex polymer film to just a few parameters also has the advantage that one can look at how the parameters vary with temperature, pressure, chemical composition, or solvent. Liemant has given a very general derivation of the driftdiffusion equation in inhomogeneous materials31 that shows how to predict an effective length and effective rate in the case of diffusion when there exists a distribution of rates and distances. Unfortunately for our needs here, this derivation implicitly assumed that the distribution of intersite separations was uniform (i.e., that the radial distribution function for the system is unity everywhere); thus, the formulas derived there are not directly applicable to the case of molecular materials having nontrivial correlations between atomic positions, such as in PTMA. It is possible, however, to modify Liemant’s derivation to account for interatomic structure and to apply it to electron diffusion in a disordered molecular material. Liemant demonstrated that a general diffusion rate, that we will take as the ET rate kET in eq 2, from an occupied site at position r, which has energy G, to a site at position r′ with energy G′, has the form kET(r, G ; r′, G′) = f (|r − r′|)s(r, G ; r′, G′) 17215

(5)

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and the division of kET into two pieces is made so that the function f is normalized under three-dimensional integration over all space

∫ drf (r) = 1

where 2 leff

(6)

where r is the magnitude of the vector r. The derivation proceeds by defining the probability distribution, pt(r), of site occupancies at time t, where the energy dependence is implied from the dependence on r for the rest of the derivation. Writing down an ET rate equation by counting electron hops between sites and expanding the resulting rate equation to second order enables us to arrive at a drift-diffusion equation. This gives an average rate of change in the occupancy at site r, ∂pt (r) ∂t

= −pt (r)

seff =

∫ ρkET(r; r′)pt (r′)dr′

(7)

where ρ is the number density (the number of radical sites per unit volume). When the site positions are correlated, the only change one needs to make in this formula for ET away from site i is to include the probability of finding a site at rj given that there is a site at ri. In terms of a particular site, i, we have ∂pt (ri) ∂t

ij = −pt (ri) ∑ kET (ri; rj)(1 − pt (rj)) j≠i ij + (1 − pt (ri)) ∑ kET (ri; rj)pt (rj)

(8)

j≠i

and upon averaging over site positions i and taking a statistical average we find ∂pt (r) ∂t

= −pt (r)

∫ ρgV (|r − r′|)s(r; r′)(1 − pt (r′))dr′

+ (1 − pt (r))

∫ ρgV (|r − r′|)s(r; r′)pt (r′)dr′ (9)

where gV is the squared-electronic-coupling-weighted intersite radial distribution function 1 N

ρgV (r) =

N

N

∑ ∑ (Vij)2 δ(r − (r − r′)) i

(10)

j≠i

where the brackets denote an equilibrium average over atomic positions. In terms of Liemant’s formulation, under the assumption that all sites have equivalent energy (ΔG0 = 0), if we split the Marcus-like formula for kET as follows f (|r − r′|) =

s(r; r′) =

1 ℏ

ρgV (|r − r′|) ρ ∫ gV (r )dr

⎡ π λ ⎤ exp⎢ − ⎥ kBTλ ⎣ 4kBT ⎦

(11)

∫ ρgV (r)dr

1 2 seff leff 6

⎡ π λ ⎤ exp⎢ − ⎥ kBTλ ⎣ 4kBT ⎦

∫ ρgV (r)dr

(15)

3. RESULTS AND DISCUSSION 3.1. Intra- and Intermolecular Structure of PTMA Films. To begin to understand how TEMPO groups and the polymer backbones pack in solid films, we have computed radial distribution functions (RDFs) from atomic positions taken from snapshots sampled every 0.5 ps from our 10 ns production run with the results shown in Figure 3. The interchain RDF for carbons on the polymer backbone shows a

(12)

then the rest of Liemant’s derivation proceeds unchanged. In the limit that the concentration of charges is low,

D=

1 ℏ

(14)

For simplicity, we have canceled the number density in the numerator and denominator in the expression for leff. The mobility is related to the diffusion constant by the aforementioned Einstein relation. The simplified formulation above assumes no site energy disorder, and it is worth commenting on the validity of this approximation. To estimate accurate electron or hole transfer rates, as opposed to just examining the distribution of hopping distances, one also needs the site energy distribution (which would then enter as an additional term in the average that defines gV above) and the reorganization energy associated with charge transfer. For organics, the site energies typically vary on the order of hundreds of meV,28 and gas-phase DFT calculations suggest that the reorganization energy of an isolated TEMPO molecule is much larger, about 1 eV. The estimate of the reorganization energy from the isolated molecule result can be misleading because dielectric response and so-called tunneling corrections must be included, and this can dramatically alter values of λ.32 Because our focus in this paper is on the distribution of intersite couplings via gV, we leave including energy disorder (ΔGo ≠ 0) and estimating infilm reorganization energies to future work, Thus, we have derived an expression for the macroscopic electron diffusion constant in a disordered molecular material on the basis of a general expression for diffusion in inhomogeneous materials. This expression is dependent on an effective length, leff, and an effective rate, seff, which depend on the values of the electron transfer matrix element Vij and intersite separation rij, which can be captured by the squaredelectronic-coupling-weighted intersite radial distribution function, gV. As such, eqs 10, 14, and 15 point to exactly what quantities one can calculate from simulation in order to assess the electron diffusion constant’s dependence on polymer structure and morphology. We are not aware of previous uses of gV to understand electron diffusion in disordered materials with correlations between site positions. We have limited ourselves to a particularly simple limit of low concentrations and constant site energies for the PTMA model system. For cases where there may be an energy dependence in the ET rate or where the concentration of charges is large, the general formulas derived by Liemant31 for diffusion constants and mobilities should also apply with the appropriate redefinitions of the functions f and s defined above for the simplified limit. We defer such considerations to future work.

∫ ρkET(r; r′)(1 − pt (r′))dr′

+ (1 − pt (r))

∫ r 2gV (r )dr = ∫ r f (r )dr = ∫ gV (r )dr 2

(13) 17216

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Figure 3. Simulated radial distribution functions for (a) nitroxyl nitrogen−nitrogen, (b) nitroxyl oxygen−oxygen, (c) nitroxyl nitrogen−oxygen, and (d) polymer backbone−polymer backbone separations. All three nitroxyl panels show the total (black, solid curves), interchain (red, dash-dotted), and intrachain (blue, dotted) RDFs. The nitroxyl nitrogen−oxygen bond peak about 1.2 Å was removed for clarity.

rise at 12 Å, indicating that the backbones are well spaced from one another. This is in contrast to what we observed in a simulated poly(methyl methacrylate) film (not shown) in which the interchain separation is much smaller, with the rise being at roughly 2−3 Å, so we conclude that the large backbone−backbone spacing observed for PTMA is due to the bulky nitroxyl radical groups being well distributed around the backbones and forcing the chains apart. Intersite electron transfer depends on Vij, which is controlled by how the nitroxyl groups on the TEMPO rings are positioned relative to each other. These relative positions can be understood by examining the RDFs associated with internitroxyl nitrogen−nitrogen, oxygen−oxygen, and nitrogen− oxygen separations, shown in Figure 3. The initial shoulder of the oxygen−oxygen RDF occurs at shorter distances than the shoulder of the nitrogen−nitrogen RDF, indicating that the TEMPO groups are aligned with nitrogen−oxygen bonds in a “head-on” configuration, that is, oriented with the oxygens facing each other. One example of this configuration is shown in Figure 4a, where two of the methyl groups of one TEMPO align between the methyl groups of the other TEMPO allowing a close contact of 4.1 Å and 4.5 Å between the internitroxyl nitrogens and internitroxyl oxygens, respectively. The first peak in the oxygen−oxygen RDF occurs at the same separation as the first peak in the nitrogen−nitrogen RDF at approximately 6.5 Å. This is consistent with the axes of the two oxygen− nitrogen bonds being parallel and being perpendicularly bisected by a single axis. While any rotation of a bond around the bisecting axis will result in equal oxygen−oxygen and nitrogen−nitrogen separations, the first peak in the nitrogen− oxygen RDF being at a shorter separation of 5.5 Å indicates that the nitrogen−oxygen bonds are pointing in opposite directions as depicted in Figure 4b where the methyl of one TEMPO group is in between two of the methyls of another causing the internitroxyl nitrogen and internitroxyl oxygen atoms to be separated by 6.5 Å. At larger inter-TEMPO separations, the oxygen−oxygen second peak is positioned at a greater distance than the second nitrogen−nitrogen peak; the second peak can be attributed to the stacking of the TEMPO

Figure 4. Molecular structure taken from a minimized molecular dynamics simulation snapshot, displaying typical packing motifs: (a) head-on, (b) parallel nitrogen−oxygen bond stack, and (c) stacked rings with the same color scheme as in Figure 2.

rings themselves with the nitrogen−oxygen bond forced to greater separation by repulsive interactions among the methyl groups from each TEMPO ring, as depicted for the intrachain case in Figure 4c. In each case, the internitroxyl distance is dominated by the manner in which the protective methyl groups sterically interact. In addition to the atom-type based RDFs, we have decomposed the RDFs into interchain and intrachain components in Figure 3, and the results indicate that interTEMPO separations below 7 Å are mostly between rings on different polymer chains. Evidently, even the fairly flexible backbone of PTMA cannot easily distort to allow close approaches by neighboring TEMPO rings on a single chain. In contrast, the decomposition of the RDFs shows that the second peak is associated with almost equal part TEMPO groups on the same and different chains. Rings can get to within ∼8 Å of each other regardless of whether they are on the same chain or reside on different polymers. The results of our simulations of the film structure suggest that in a PTMA film the inter-radical coupling will be dominated by interchain interactions because of the closer approach between interchain TEMPO nitrogen−oxygen bonds 17217

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Figure 5. Lowest energy conformations of a hydrogen-terminated PTMA monomer in (a) neutral and (b) cation geometries with respective (c) SOMO and (d) LUMO iso-surfaces drawn at contour values of ±0.002 au. The dihedral angle made by the atoms highlighted in yellow is 68° in the neutral geometry and 91° in the cation geometry.

than between intrachain TEMPO nitrogen−oxygen bonds. Hence, largely three-dimensional electron transport is anticipated in PTMA because intergroup ET will not be dominated by transfers along the polymer backbone. These structural results also have implications for how ions and solvent molecules move into the films upon charging in an electrode configuration. We defer detailed study of ion and solvent motions in PTMA films to a subsequent theoretical and experimental work, but we speculate that the fact that the TEMPO groups are so bulky creates enough void volume for ions to efficiently move within the polymer during charging and discharging events. 3.2. Electronic Transfer in PTMA Films. To justify our localized-charge model for electron transfer in the PTMA film, the position of the radical-based electron on the monomer unit was investigated using quantum chemical methods. The singly occupied molecular orbital (SOMO) of the radical and the lowest unoccupied molecular orbital (LUMO) of the cation shown in Figure 5 can be seen to primarily occupy an antibonding π orbital on the oxygen−nitrogen bond. This indicates that when an electron is transferred from the SOMO of the neutral radical to the LUMO of the cation, the electron hops from the vicinity of a nitrogen−oxygen bond of the radical to the vicinity of the nitrogen−oxygen bond of the cation, which is consistent with previously published results.11 When a neutral PTMA monomer is changed to a cation, the primary changes in geometry predicted by the DFT calculations are that the methyl carbon−carbon−nitrogen−oxygen dihedral increases from 68° to 91°, highlighted in Figure 5a and b, and that the oxygen−nitrogen bond length decreases from 1.28 to 1.19 Å. Similar results have been reported for the TEMPO radical that show, as we have found for the PTMA monomer, that the geometry of the ring away from the nitroxyl group is not affected by the oxidation state.32 The intersite coupling Vij was calculated for each of the 12 026 pairs of TEMPO groups in the film according to the method described in section 2. The average values of V2ij as a function of nitrogen separation are shown in Figure 6b. At

Figure 6. (a) Radial distribution function of all nitrogen−nitrogen, interchain nitrogen−nitrogen, and intrachain nitrogen−nitrogen separations in PTMA films, (b) average electron coupling matrix elements V2ij for pairs of TEMPO groups from a single snapshot of the MD trajectory for the simulated PTMA film, (c) the couplingweighted radial distribution function ρgV, and (d) the radially weighted coupling-weighted radial distribution function 4πr2ρgV. In all panels, the total quantity is the solid black curve; interchain quantities are red, dash-dotted curves; and intrachain quantities are blue, dotted curves. The bins for the distributions are taken to be 0.1 Å wide. Error bars are given by the standard deviation of the mean with the appropriate twotailed Student’s t-distribution prefactor to give a 90% confidence interval, and error bars are not included for distances with a single data point (i.e., small r).

short distances corresponding to the shoulder of the RDF, interchain nearest neighbors come into close head-on contact, and the average V2ij value falls exponentially in accordance with exponential decay expected for the electron density around the molecule. At the first peak in the RDF at 6.5 Å, the exponential decay in the average V2ij value plateaus because of the stacking of nitrogen−oxygen bonds, which causes the overlap of the SOMO wave function of the radical neutral and the LUMO of 17218

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4. CONCLUSION In this paper, we have used atomistic molecular dynamics simulations to explore the structure and morphology of polymer films and to examine the interplay between intermolecular packing and charge transport in PTMA films. The radical electron was found to primarily occupy the oxygen−nitrogen bond of the TEMPO ring of PTMA. The simulated film structures revealed that methyl-methyl interactions control the internitroxyl distances such that there are three primary packing motifs associated with the inter-TEMPO radical position, consisting of close contact of the TEMPO oxygen atoms in a head-on arrangement, oxygen−nitrogen bond stacking, and TEMPO ring stacking. When the electronic coupling between TEMPO rings was calculated, it was found that the strongest inter-TEMPO ring coupling was between rings on different chains with interchain rings able to arrange with the head-on and oxygen−nitrogen bond stacking motifs associated with the initial shoulder and first peak of the interTEMPO nitrogen and oxygen RDF. Because the oxygen− nitrogen bonds stack parallel to each other at moderate separations, this allows for a favorable overlap of the molecular orbitals associated with ET, producing a plateau in the magnitude of inter-ring coupling of roughly 6−7 Å. This favorable stacking motif allows for ET events to occur over much greater distances than would be anticipated if transport were dominated entirely by short hops between the most strongly coupled pairs of TEMPO rings. Our simulations also showed that stacked rings resulted in somewhat parallel nitrogen−oxygen bonds that were slightly farther apart, approximately 8 Å, because of the repulsions between the methyl groups when the TEMPO rings were stacked; this larger radical site separation can be associated with rings either on the same chain or on different chains. ET over this distance was found to contribute little to the overall electron transport mechanism, but one might anticipate that modifications to the TEMPO structure to allow slightly closer inter-ring stacking (e.g., by replacing methyl groups with smaller groups that still stabilize the radical) could lead to large increases in electron mobility because of an increase in prevalence of longer ET events. We also introduced an electronic-coupling-weighted radial distribution function, gV(r), that allowed us to draw conclusions about the relative importance of interchain versus intrachain electron transfer. This distribution is intuitively appealing, insofar as it clearly shows the interplay between intersite structure and intersite coupling that determines both how far and how rapidly charges can move in the TEMPO matrix. Most significantly, our derivation of gV(r) as the relevant quantity for predicting electron diffusion was not specific to only organic radical polymer systems. The coupling-weighted radial distribution function falls naturally out of the extension of Liemant’s very general treatment31 of diffusion processes in inhomogeneous materials. Hence, this quantity will prove useful for understanding charge transport in other systems for which charges are localized on sites that have nontrivial intersite organization because of intermolecular packing, such as for organic photovoltaics and organic light emitting diode systems. In light of the fact that the coupling-weighted radial distribution function can be simulated directly in a straightforward manner, we suggest that this quantity should be utilized in concert with atomistic simulations to understand how changes in molecular structure lead to changes in electronic conductivity of organic

the cation to be less affected by separation and to vary weakly when the parallel bonds are displaced parallel to each other. We also calculated the average coupling, denoted ⟨V2ij⟩, as a function of nitrogen−nitrogen separation for both intrachain and interchain pairs of TEMPO groups. As seen in Figure 6b, for both intrachain and interchain interactions, the average values are nearly the same, indicating that the inter-TEMPO group alignment is similar regardless of whether TEMPO rings are on the same or different chains. Hence, rather than there being preferential electron transfer along a chain because of favorable electronic coupling, interchain ET would be favored because (according to the RDFs) the nitroxyl group of TEMPO has greater probability of being in closer proximity with a nitroxyl group from another chain than with a nitroxyl group on the same chain. To make this conclusion quantitative, we have calculated the coupling-weighted radial distribution function, ρgV(r), as a function of internitrogen separation from eq 10, and this is displayed in Figure 6c. The two peaks in ρgV indicate that two effective electron diffusion lengths and rates dominate electron diffusion in the PTMA system. To highlight that at large separations there are more available sites than at small separations, we also plot 4πr2ρgV in Figure 6d. The effective electron diffusion length, l2eff, found from eq 15 is 5.5 Å. This distance is actually in between the two peaks of ρgV; the average distance associated with electron diffusion is one at which few electron hops actually occur. We also decomposed ρgV into intrachain and interchain contributions, and inserting these two versions of ρgV into eq 15 gives an effective electron diffusion length of 6.7 and 5.4 Å, respectively. While the effective electron diffusion length associated solely with intrachain ET is larger than that associated solely with interchain ET, most of the ET events must occur between nitroxyl groups on different chains, as indicated by the different magnitudes of the curves in the bottom panel of Figure 6. A quantitative measure of the relative importance of intrachain and interchain charge transfer can be made by observing that the diffusion constant for charges in the system is given by D = (seffl2eff)6. Hence, the diffusion constant in eq 13 can be written as D=

1 6ℏ

⎡ π λ ⎤ exp⎢ − ⎥ρ kBTλ ⎣ 4kBT ⎦

∫ gV (r)r 2dr

(16)

Thus, diffusion constants are proportional to the second moment of gV. Moreover, because the full electron-couplingweighted distribution (gV) can be expressed as the sum of purely interchain and intrachain pieces, we can write Dtotal = Dintra + Dinter, where Dintra is the intrachain diffusion constant and Dinter is the interchain diffusion constant. By calculating the integral in eq 16 using gV and the intrachain and interchain pieces and then by forming the ratios Dintra/Dtotal and Dinter/ Dtotal, we find that the total electron diffusion constant comes from about 84% interchain ET events and 16% intrachain ET events. At first glance, even 16% percent coming from intrachain motion may be surprising in view of how small the intrachain ρgV is in Figure 6c; however, the additional r2 weighting in eq 16 and the tendency of intrachain distances between nitroxyl groups to be large enhance the intrachain contribution to electron diffusion. 17219

dx.doi.org/10.1021/jp501628z | J. Phys. Chem. C 2014, 118, 17213−17220

The Journal of Physical Chemistry C

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materials generally. Thus, the overall methodology of calculating gV may serve as a standard tactic for evaluating electron transfer in disordered organic systems via a relatively straightforward analysis of molecular dynamics simulated data.

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ASSOCIATED CONTENT

S Supporting Information *

Details of the force-field parameters and molecular dynamics simulations. This material is available free of charge via the Internet at http://pubs.acs.org

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]; phone: (303) 275-3738. *E-mail: [email protected]; phone: (303) 275-4422. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under contract DE-AC36-08GO28308. The research was performed using resources sponsored by the Department of Energy’s Office of Energy Efficiency and Renewable Energy, located at the National Renewable Energy Laboratory.

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