Effect of Nanoparticles on Ion Transport in Polymer Electrolytes

Apr 15, 2015 - Using all atom molecular dynamics and trajectory-extending kinetic Monte Carlo simulations, we study the influence of Al2O3 nanoparticl...
3 downloads 10 Views 6MB Size
Download PDF
Article pubs.acs.org/Macromolecules

Effect of Nanoparticles on Ion Transport in Polymer Electrolytes Santosh Mogurampelly and Venkat Ganesan* Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: Using all atom molecular dynamics and trajectory-extending kinetic Monte Carlo simulations, we study the influence of Al2O3 nanoparticles on the transport properties of ions in polymer electrolytes composed of poly(ethylene oxide) (PEO) melt solvated with LiBF4 salt. We observe that the mobility of Li+ cations and BF4− anions and the overall conductivity decrease upon addition of nanoparticles. Our results suggest that the nanoparticles slow the dynamics of polymer segments near their surfaces. Moreover, the preferential interactions of the ions with the nanoparticles are seen to lead to an enhancement of ion concentration near the particle surfaces and a further reduction in the polymer mobilities near the surface. Together, these effects are seen to increase the residence times of Li+ cations near the polymer backbone in the vicinity of the nanoparticles and reduce the overall mobility and conductivity of the electrolyte. Overall, our simulation results suggest that both the nanoparticle-induced changes in polymer dynamical properties and the interactions between the nanoparticles and ions influence the conductivity of the electrolyte.

1. INTRODUCTION As a consequence of their desirable features of safety, compatibility, flexibility, and mechanical strength, solid polymer electrolytes (SPE) are emerging as promising candidates for replacing liquid electrolytes in lithium ion batteries.1−12 However, despite a number of promising advances in this regard, SPEs such as poly(ethylene oxide) (PEO) still possess significantly lower conductivity compared to those of their liquid and conventional polymer electrolyte counterparts. Hence, there is an active interest in strategies that can improve the conductivity of SPEs without having a concurrent detrimental effect upon the characteristics such as the mechanical strength of the material. In the above regard, recently, researchers have reported that adding micron- or nanosized electrochemically inert ceramic filler particles like TiO2, SiO2, and Al2O3 to SPEs may, in certain circumstances, enhance the conductivity of the underlying polymer matrix.5−7,13 Interestingly, in a number of other contexts, addition of nanoparticles has been shown to also have a significant positive influence upon the mechanical strength of polymer materials.5,6,11,14−18 Together, these observations have sparked an interest in the strategy of adding nanofillers as a means to design mechanically strong, conductive polymer electrolytes. While a number of studies have examined the mechanisms underlying the influence of nanoparticles on the mechanical properties of polymer matrices, in contrast, a comparable molecular level understanding of the mechanisms of ion transport in polymer nanocomposites (PNCs) is still lacking. Transport of ions in PNCs is expected to be a complex process which includes effects arising from ion motion, polymer segmental motion, nanoparticle interaction with ions and the © 2015 American Chemical Society

polymer segments, changes in polymer conformations induced by nanoparticles, etc. It was initially believed that the influence of nanoparticles on ion motion arose exclusively from the impact of the nanoparticles upon the degree of amorphicity of the polymer matrix.19 However, a number of subsequent experiments have demonstrated that the influence of nanoparticles can extend to temperatures higher than the melting point of the polymer electrolyte and also that the ordered crystalline phases of polymeric hosts can support significant ionic conductivity. These results have cast doubts on whether the nanoparticle’s influence on the polymer matrix crystallinity constitutes the sole mechanism underlying the conductivity changes in polymer electrolytes. Moreover, recent experiments have indicated that the Lewis acid/base nature of the filler nanoparticle and its interaction with ions and the polymer electrolyte can also significantly influence transport of ions.8−10,16,20−26 In sum, there is a lack of clarity on the influence of the nanoparticles on ion transport and their relationship to the influence of the nanoparticles upon the static and dynamic properties of the underlying polymer matrix. Molecular dynamics (MD) simulations using atomistic and/ or polarizable force fields serve to provide detailed molecular level information on polymer electrolyte systems and has contributed significantly to the understanding of fundamental mechanisms of ion transport in polymer electrolytes. For instance, Muller-Plathe27 studied the transport of Li+ and I− ions and their clustering properties in PEO electrolyte. Neyertz and Brown28 studied the local structure and dynamics of Na+ Received: December 22, 2014 Revised: April 1, 2015 Published: April 15, 2015 2773

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules and I− ions in PEO electrolyte, and Smith et al.29 studied conformational properties of large molecular weight PEO and report good agreement with experiments. These initial MD simulations have provided useful insights into the structural as well as ion transport mechanisms in PEO. Later, Borodin and Smith30−33 developed many-body polarizable force fields for polyelectrolyte systems with various anions providing the fundamental understanding of the various mechanisms affecting Li+ ion transport. Inspired by the above successes, in a recent short communication, we reported the results of MD simulations which studied the transport mechanisms of ions in polymer nanocomposites consisting of TiO2 nanoparticles dispersed in PEO polymer electrolyte.34 We found that the addition of nanoparticles lowered the cationic mobilities relative to the pure PEO matrix. Analysis of our simulation results implicated the influence of nanoparticles upon the polymer mobility and conformations as the main mechanism underlying the changes in ionic mobilities. In this study, we considered a different system comprising of PEO matrix dispersed with basic Al2O3 nanoparticles. Our motivation for this work is fourfold: (i) To examine the degree of universality of our results and mechanisms deduced in our earlier work pertaining TiO2 nanoparticles.34 (ii) To present conclusive evidence for modified polymer segmental dynamics near the nanoparticles and its corresponding impact upon the ion residence times. In addition, we identify clearly the mechanisms underlying modified polymer segmental dynamics near the nanoparticles and identify the role of ion−nanoparticle interactions. (iii) To study an expanded suite of properties, which include the mobility of cations and anions and the overall conductivity, and probe whether other distinct mechanisms can influence properties such as conductivity. (iv) To provide a more detailed description of the results and methodologies which were presented in brief in our earlier short communication. Additionally, while we do not consider the effect of surface polarity of the nanoparticles as a parameter in the present study, we do note that recent experiments have suggested that rendering the Al2O3 nanoparticles acidic in nature can have a profound effect on ionic conductivities.5−7,9,10,12 The results of the present article can serve as a basis for comparison to those arising in simulations of Al2O3 nanoparticles with acidic surface characteristics. In this article, we report the results of atomistic MD simulations studying ion transport in an amorphous poly(ethylene oxide) (PEO) melt containing Al2O3 nanoparticles solvated with lithium tetrafluoroborate (LiBF4) salt. The effect of Al2O3 nanoparticle on ion transport mechanism was investigated using a combination of fully atomistic simulations and trajectory-extending kinetic Monte Carlo (TEKMC) simulations to access the long time ion dynamics (of the order of 100 ns and longer). We calculate the diffusion coefficient of both cation and anions (D) and the conductivity (σ) at two salt concentrations (EO:Li = 15:1 and EO:Li = 8:1) and at various weight percentages of Al2O3 nanoparticles in PEO melt. We also characterize the influence of temperature by studying the ion transport at 350, 425, and 500 K. We probe the polymer dynamics by calculating the polymer segmental relaxation times and changes in the Li+ ion residence times for coordination with polymer segments to understand the connection with ion transport. The static structural changes were analyzed by studying the radial distribution functions of cations and anions with PEO segments and the Al2O3

nanoparticles. Together our results implicate the nanoparticleinduced changes in polymer dynamics arising from nanoparticle−polymer and nanoparticle−ion interactions as the main mechanisms for the influence of nanoparticle on ion dynamics. We note that in an earlier article Kasemägi et al.35,36 did study the issue of Li+ ion mobility when Al2O3 nanoparticles were dispersed in polymer electrolytes. They reported lower Li+ ion mobility when Al2O3 nanoparticles were dispersed in polymer electrolytes solvated with monatomic anions such as Cl−, Br−, and I− but an increased Li+ ion mobility when the nanoparticles were solvated with polyatomic anion tetrafluoroborate (BF4−). They attributed the decrease in Li+ ion mobility in the former case to ion-pairing and attraction of Li+ ions to the nanoparticle. In contrast, the increase in Li+ ion mobility in the latter case was assumed to arise from the increased free Li+ ions compared to the bulk system. However, their simulations were performed for a very short duration of 1 ns, which is usually not sufficient to extract meaningful information on the dynamics of ions and the conductivity of the polymer matrix. In contrast, we use special simulation techniques to study the long-time dynamics of the ions, and the trends seen in our results differ from the results presented in Kasemägi et al.35,36 The organization of the rest paper is as follows: In section 2, we present the simulation details including the interaction potential and force fields used, initial system setup, equilibration, and TEKMC methodology. In section 3.1, we present results and discussion for mean-squared displacement (MSD) of Li+ ions and the diffusivity of both cations and anions. In section 3.2, we discuss the results for the ionic conductivity. In section 3.3, we present results for the equilibrium ion correlations based on the radial distribution functions as well as the coordination numbers. In section 4, we identify the influence of different mechanisms underlying the changes in ion mobilities in PNCs. Explicitly, in section 4.1 we present results for the relaxation of dihedral angle correlations of the polymer backbones and the polymer segmental mobilities and discuss the correlation with the ionic mobilities. In section 4.2, we discuss the influence of nanoparticles on polymer mobilities as well as residence times of Li+ near to EO as a function of the proximity to the nanoparticle surfaces. We also present results for the residence time of Li−EO pairs and demonstrate its connection to polymer segmental mobilities. In section 4.3, we identify the mechanisms underlying polymer mobilities near to the nanoparticles and discuss the effect of interactions between the nanoparticles and the ions. Finally, in section 5 we summarize our results and identify possible directions which emerge from our findings to improve ionic conductivity in PNCs.

2. SIMULATION METHODS 2.1. Interaction Potential and Force Fields. We performed MD simulations of the PEO−LiBF4−Al2O3 system using LAMMPS simulation package (Large-scale Atomic/ Molecular Massively Parallel Simulator)37 using the following general form of the interaction potential between the different components: U (r ) = U bonded(r ) + A e−Br − 2774

qq C D + 12 − 4 6 4π ϵ0r r r

(1)

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules where Ubonded(r) describes the contribution arising from bonds, angles, and torsions in the PEO. Such interactions in PEO were modeled with a harmonic potential kr(r − r0)2 for bonds, angles with kθ(θ − θ0)2 and torsions with the ∑4n=1kϕ(1 − cos(nϕ)) potential. The nonbonded interactions are modeled with second to fifth terms and include the Buckingham potentials (the second and third terms) and the Coulomb potential (fourth term). Borodin and Smith have developed many-body (MB) polarizable force fields and have demonstrated that computationally expensive many-body polarizable interactions can be approximated in some circumstances by effective twobody (TB) interactions30−33 while still preserving an excellent agreement on the static and dynamic properties of PEO systems. Inspired by their work, we use a two-body, mean-field like dipole polarization interaction (last term in eq 1).31,38 Force fields for PEO and LiBF4 are taken from refs 31 and 33, partial atomic charges on PEO are taken from ref 27, and nonbonded interactions for Al2O3 are taken from refs 35 and 36. The static partial atomic charges used for atoms in Al2O3 nanoparticles were identical to those in ref 39. Partial atomic charges on atoms considered in our simulations are shown in Table 1, and nonbonded parameters are given in Table S1 of the Supporting Information.

proximity of the existing nanoparticles. The snapshot at 5 wt % shown in Figure 1 does in fact indicate a close proximity of the

Table 1. Charges on the Atoms Used in Our MD Simulationsa atom

q (e)

atom

q (e)

atom

q (e)

Oet C H

−0.348 0.103 0.0355

Cm Li B

0.0675 1.0 1.072

F Al OAl

−0.518 2.20539 −1.47026

a

Notation: Oet: ether oxygen; Cm: methyl carbon at the terminus; OAl: oxygen in alumina.

2.2. Initial System Setup. The initial conformations of the PEO were constructed as follows: we built a single PEO chain of 55 repeating monomer units having chemical structure H− [CH2−O−CH2]55−H using Materials Studio suite.40 An energy minimization on the PEO chain was performed to obtain desired bond lengths, angles, and torsion angles as defined by the force fields (FF) parameters.30−33 The minimized configuration of single PEO chain was replicated in a simulation box to create a less dense system consisting 40 PEO chains using Packmol41 software. The molecular weight of the pure PEO matrix is 2425 g/mol. Subsequently, this PEO matrix was solvated with appropriate number of Li+ cations and BF4− anions to get a salt concentration of EO:Li = 15:1 and EO:Li = 8:1. The Al2O3 nanoparticles were dispersed in bulk PEO matrix to get PNC systems with a loading of 5, 10, and 20 wt %. Polymer nanocomposites of 5, 10, and 20 wt % contain 2, 5, and 9 nanoparticles, respectively. Alumina nanoparticle was extracted from its rhombohedral crystal form corundum having space group R3̅c (No. 167) and unit cell parameters a = 4.75 Å and c = 12.99 Å. The diameter of the extracted Al2O3 nanoparticle was approximately 14 Å and roughly spherical (115 atoms). We note that Li+, BF4− ions, and Al2O3 nanoparticles were added randomly in PEO matrix with a close contact distance of 2 Å between any two species to avoid overlapping of atoms. However, since we dispersed the nanoparticles one by one after the polymer melt was built during the initial system setup, in some cases we observed a slightly higher probability of inserting a nanoparticle in the

Figure 1. Representative snapshot of equilibrated PEO−LiBF4−Al2O3 polymer nanocomposite at EO:Li = 15:1 and Al2O3 of 5 wt %. The magenta bonds are shown to distinguish dynamic bonds between the anions and cations which represents ion-pairing. Color coding: small red = ether oxygen, cyan = carbon, white = hydrogen, mauve = lithium ion, ice blue = boron, ochre = fluorine, yellow = aluminum, big red = oxygen in alumina nanoparticles. Color and relative sizes of the atoms are guides to the eye.

two nanoparticles dispersed in the polymer melt with the center-of-mass separation of 23 Å. The density of the system in the initialization stage was chosen as low as 0.15 g/cm3 in order to obtain optimal solvation of Li+, BF4−, and Al2O3 solvent particles in the PEO matrix. Such a low density in the model building stage was chosen to ensure that the PEO melt does not suffer from potential energy traps arising from contact and entanglements. 2.3. Equilibration. The conformations of the polymer matrix and the ions were then minimized using the interaction potential given in eq 1 through the steepest-descent method. The force and energy tolerances were set to 10−5 by restraining PEO chains and Al2O3 in the beginning stage to equilibrate Li+/ BF4− ions with respect to PEO matrix and Al2O3 nanoparticle. Following equilibration of Li+/BF4− ions, minimization was performed by releasing restraints on PEO while still restraining Al2O3. Eventually, the entire PNC system was minimized without any restraints. The temperature of the minimized PNC 2775

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules Table 2. Summary of the Simulation Setupa 0 wt %

5 wt %

10 wt %

20 wt %

EO:Li

T (K)

ρ

ρs

ρ

ρs

ρ

ρs

ρ

15:1

350 425 500 350 425 500

1.117(4) 1.073(5) 1.026(5) 1.180(4) 1.140(5) 1.096(5)

1.143(4) 1.092(5) 1.031(6) 1.206(4) 1.168(4) 1.123(5)

1.112(4) 1.066(5) 1.004(6) 1.181(4) 1.141(4) 1.097(5)

1.160(5) 1.148(5) 1.050(7) 1.246(3) 1.209(4) 1.161(5)

1.092(5) 1.081(5) 0.985(7) 1.183(3) 1.147(4) 1.100(5)

1.204(5) 1.176(6) 1.152(6) 1.291(3) 1.261(4) 1.214(5)

1.086(5) 1.060(6) 1.036(6) 1.181(3) 1.153(4) 1.106(5)

8:1

Density (in g/cm3) of the entire system is denoted as ρs which includes PEO−LiBF4 melt and Al2O3 nanoparticles with specified weight percentage whereas the density of PEO−LiBF4 melt is denoted as ρ. Value given in parentheses is the standard deviation (in units of 10−3 g/cm3). The number of atoms in melt of 40 PEO chains is 15480 corresponding to a molecular weight of 2425 g/mol, the number of Li+ ions is 147 (275) at a salt concentration of EO:Li = 15:1 (EO:Li = 8:1), and the number of atoms in Al2O3 nanoparticles is (230, 575, 1035) at wt % of (5, 10, 20), respectively. a

20 ns trajectory, and the structural analysis reported in this paper was performed on the last 15 ns of such trajectories. We note that such a short trajectory obtained from MD simulations is often inadequate to characterize diffusive properties of ions in PNC since the diffusive regime of MSD curve can extend beyond 15 ns and, in general, could strongly depend on the system under consideration. To obtain long-time ion dynamics in PNC, we employ a variant of kinetic Monte Carlo method by using MD trajectory as input and extend the trajectories beyond the reach of MD simulations (typically to the order of 80−100 ns). The method, also termed as the trajectory-extending kinetic Monte Carlo (TEKMC),34,50,51 is based on the assumption that short time equilibrium MD simulations are adequate to sample penetrant percolating paths of diffusive ions within the penetrant regions inside simulation box. To implement this approach, we map our MD simulation box to a fictitious lattice framework with grid size Dgrid and analyze MD configurations separated by time interval tinput. Equilibrium MD trajectory was used to identify all possible transition events of penetrant particles (ions in our case) between penetrant cells (say, i and j) and calculate respective transition rate constants, {kij}, from cell i to cell j. Since the nanoparticles were restrained with a harmonic potential of large force constant, k = 1000 kcal/(mol Å2), they are assumed to be immobile. The transition rates between different cells were averaged over the input atomistic simulation trajectory and are treated as constants during the KMC simulations. Rate constants calculated from equilibrium MD simulations were then used to propagate ions in time using Bartoz−Kalos−Lebowitz (BKL) algorithm,52 thus allowing us to obtain the time evolution of ions and study their dynamic properties. In the past work, we have successfully employed variants of TEKMC to study various systems34,51,53,54 and have established correspondence with the results obtained from atomistic simulations executed for longer times. To implement the TEKMC simulation framework for the present work, we performed an optimization of the Dgrid and tinput and ensured a match of the atomistic MSD curve for diffusive particles for short time scales (see Figure S1 in Supporting Information). This required us to optimize Dgrid at each temperature, salt concentration, and nanoparticle weight percentage. Such an optimized grid was used to obtain the results reported in the next section.

configuration was slowly increased from 0 to 600 K by restraining Al2O3 to prevent ion aggregation. After heating the system to 600 K, another minimization was performed following which 2 ns of NPT simulations were carried out. Individual systems having different salt concentrations and different nanoparticle weight percentages were then annealed slowly to 500, 425, and 350 K. Finally, the entire system was minimized without restraints to account for any stress built due to the nanoparticle restraints during annealing. This minimization protocol ensured appropriate equilibration of respective PNC system prior to long production runs. Figure 1 displays a snapshot of the equilibrated PNC at Al2O3 nanoparticle 5 wt % containing PEO melt, Li+, BF4−, and Al2O3 nanoparticles. During the NPT production runs, we restrained Al2O3 nanoparticle with a harmonic potential of spring constant 1000 kcal/(mol Å2). All bonds involving H atoms were constrained using the SHAKE algorithm42 which enabled us to employ a large simulation time step of 2 fs. Electrostatics were treated with particle mesh Ewald (PME)43 method during the initial equilibration protocol and with particle−particle particle−mesh solver (PPPM)44 during long NPT production runs. Temperature and pressure were controlled using a Nosé− Hoover thermostat and barostat45,46 with damping relaxation times of 0.1 and 1 ps, respectively. Periodic boundary conditions were applied in all three directions with dimensions longer than twice the cutoff distance. A cutoff distance of 10 Å was used for all nonbonded interactions. Translational center of mass of the entire system was removed every time step. A summary of the simulation setup and equilibrium densities is given in Table 2. The density of PEO−LiBF4 melt in particle free (i.e., 0 wt %) system at EO:Li = 15:1(EO:Li = 8:1) was found to be 1.12(1.18), 1.08(1.14), and 1.03(1.10) g/cm3 respectively at 350, 425, and 500 K and are in accord with experimental observations47−49 and other simulations.28,29 We note that the long-range corrections to the pressure and energy arising from the truncation of the Buckingham potential as well as the mean field polarization interaction are not accounted for in our simulations. However, the densities obtained in our simulations for nanoparticle free systems were comparable to other reports,28,29,47−49 and hence we believe that such corrections are not critical for the conditions considered in this article. When nanoparticles were added to PEO melt, we observe negligible change in PEO densities at all the temperatures compared to nanoparticle free system. 2.4. Trajectory-Extending Kinetic Monte Carlo (TEKMC). Each individual system was simulated to obtain a

3. RESULTS AND DISCUSSION 3.1. Mean-Squared Displacement and Diffusion Coefficient. Following the equilibration protocol described 2776

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

the ether oxygens (EO) of PEO chain environment. Thermal agitations assist Li+ ions to escape from such potential well and eventually exhibit transition to diffusive motion at long time scales. The onset of the transition from subdiffusive to diffusive region occurs at a time scale of the order of 100 ps at 500 K but increases to 1000 ps when the temperature is decreased to 350 K at EO:Li = 15:1 salt concentration (see Figure S2 in Supporting Information for MSD results at 350 and 425 K). When the salt concentration is increased to EO:Li = 8:1, the onset to diffusive motion is seen to occur at slightly higher time scales (of the order of an additional 50 ps) compared to that of EO:Li = 15:1. This result can be understood to be a consequence of the higher probability of Li+ ion coordination with EO arising from the higher salt concentration. Moreover, the density of the melt system is higher for larger concentration of salt, which leads to reduced percolating paths for the motion of Li+ ions. The results displayed in Figure 2 suggest that the mobility of Li+ ion decreases with decreasing temperature and increasing nanoparticle loadings. Interestingly, the effects arising from the addition of nanoparticles are seen to mimic the trends resulting from lowering the temperature. Explicitly, we observe that adding nanoparticles leads to a longer regime corresponding to the subdiffusive region which thereby delays the onset of the linear, diffusive region. These results suggest that broadly similar mechanisms may underlie the influence of temperature and nanoparticles upon the mobility of the ions. We explore this hypothesis further in a more quantitative manner in the next section. To quantify the above MSD results, we calculated the diffusion coefficient (D) of Li+ and BF4− ions by using the MSD obtained from the TEKMC method. By employing TEKMC, we are able to obtain MSD curves for long time scales, typically up to 100 ns, where the ion motion is well into the linear diffusive regime. However, at low temperatures and high nanoparticle loading, we rely primarily on MD trajectories of 20 ns to calculate the diffusivities. Figure 3 displays D/D0 for both the cations and anions (D0 is the diffusion coefficient of ions in pure PEO melt at the respective temperature) for salt concentrations EO:Li = 15:1 (filled symbols) and EO:Li = 8:1 (open symbols) at various

in the previous section, we performed NPT simulations at 1 atm pressure and at temperatures 350, 425, and 500 K. The transport characteristics of Li+ and BF4− ions were studied by calculating their mean-squared displacement (MSD = ⟨(R(t) − R(0))2⟩, where R(t) denotes the position of the ions at time t and ⟨···⟩ denotes an ensemble average) at various salt concentrations and weight percentages of Al2O3 nanoparticles. The MSDs were then used to calculate the diffusion coefficient (D) of ions using the Einstein relation 1 D = lim ⟨(R(t ) − R(0))2 ⟩ (2) t →∞ 6t Figure 2 displays MSD curves at 500 K for EO:Li = 15:1 (solid lines) and EO:Li = 8:1 (dashed lines) as a function of the

Figure 2. MSD for Li+ ions obtained from MD simulations as a function of Al2O3 weight percentage at 500 K. Plots of MSD for Li+ ions at other temperatures are give in Figure S2, and plots of MSD for BF4− ions are given in Figure S3.

weight percentage of Al2O3 nanoparticles. As seen therein, the MSD of Li+ ions displays an initial subdiffusive regime followed by a transition to the diffusive regime. In the subdiffusive region, the Li+ ions can be understood to be caged in a potential well arising from the neighboring BF4− anions and/or

Figure 3. Ion diffusion: D/D0 of (a) cations and (b) anions as a function of Al2O3 weight percentage at three different temperatures for EO:Li = 15:1 (filled symbols) and EO:Li = 8:1 (open symbols) salt concentration. The insets in both figures displays corresponding D0 as a function of temperature. Ion diffusivities in a non-normalized representation is shown in Figure S6 of the Supporting Information for a better understanding of the differences between different temperatures and salt concentrations. Lines are a guide to the eye. 2777

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

Figure 4. Ionic conductivity: (a) the average value of the degree of uncorrelated ion motion, α̅ , and (b) the ionic conductivity, σ, as a function of Al2O3 weight percentage at three different temperatures for EO:Li = 15:1 (filled symbols) and EO:Li = 8:1 (open symbols) salt concentration. In the inset, α(t) is shown for short time scales at 425 K for EO:Li = 15:1 on the left panel and EO:Li = 8:1 on the right panel. α(t) at other temperatures are given in Figure S4. Lines are a guide to the eye.

3.2. Ionic Conductivity. While the mobility of the ions is an important property of polymer electrolyte, more relevant to applications is the overall conductivity of the polymer electrolyte. In an infinitely dilute solution of ions, the ionic conductivity of the electrolyte is directly proportional to the sum of the diffusivities of the anions and the cations.27 However, at higher salt concentrations, ionic conductivity becomes influenced by ion correlation effects such as ionpairing and the tendency for ion collisions arising from longrange electrostatic interactions. Hence, in general, ionic conductivity is understood as a collective transport property of the cations and anions and may exhibit trends which differ from that of the individual mobilities. Motivated by such considerations, in this section, we present results which identify the influence of the nanoparticles upon the ionic conductivities of the polymer electrolytes. Specifically, we seek to clarify whether the nanoparticles influence the ionic conductivity by mechanisms beyond the former’s direct influence upon the ionic mobilities. We calculated the ionic conductivity using Green−Kubo formula as the autocorrelation function of the charge flux,27,31,59 j(t) = e∑izivi(t)

Al2O3 weight percentages and temperatures. Overall, we observe that the ion mobility increases with increasing temperature, a result consistent with the expectations of increased polymer segmental mobilities. For instance, in the nanoparticle free system at EO:Li = 15:1, we find D0 of Li+ ions to be 0.37 × 10−7 cm2/s at 350 K, which increases to 6.30 × 10−7 cm2/s when the temperature is increased to 500 K. We also observe a decrease in D when the salt concentration was increased from EO:Li = 15:1 to EO:Li = 8:1. For example, in the nanoparticle free system at 500 K, D0 decreases from 6.30 × 10−7 cm2/s at EO:Li = 15:1 to 3.78 × 10−7 cm2/s when the salt concentration was increased to EO:Li = 8:1. Large concentration of salt results in increased density of the system as shown in Table 2. Moreover, there is expected to be a ion coordination induced reduction in polymer segmental mobilities.31,55,56 Together, these effects lead to the observed decreased diffusion coefficient. Such a result is consistent with earlier studies on the diffusion of the ions in polymer electrolyte systems.57,58 Interestingly, similar qualitative features are also observed for the results of D of BF4− anions with temperature and Al2O3 nanoparticle weight percentages (Figure 3b). However, we observe that D of BF4− ions is larger than that of Li + ions at a specified temperature and Al 2 O 3 nanoparticle loading. The latter observations are also consistent with earlier computer simulation results for pure PEO melt.31 More pertinently, we observe that both the Li+ and BF4− ion mobilities decrease monotonically as we increase the loading of Al2O3 nanoparticles. Interestingly, we observe that the normalized diffusivities for different temperatures exhibit quantitatively similar values for different particle weight fractions. Similarly, despite the differences in the D0 for different salt concentrations, the normalized ion diffusivities are also seen to display very similar values for the two salt concentrations at different particle loading. Together, these observations suggest that the broad mechanisms underlying the influence of particles upon the ionic mobilities are likely to be independent of both the salt concentration and the temperature. Prior to delving deeper into the mechanisms underlying the above results, we discuss the influence of nanoparticles upon the ionic conductivity.

σ=

1 3VkBT

∫0

∞

⟨j(t ) ·j(0)⟩ dt

(3)

where vi(t) and zi are velocity and charge of ion i, e is electronic charge, V is volume of the simulation box, kB is Boltzmann’s constant, T is the absolute temperature, and ⟨···⟩ represents ensemble average. In molecular dynamics simulations, it is often useful to write the above equation in the form of an equivalent Einstein relation, σ = lim

t →∞

e2 6tVkBT

∑ ∑ ⟨zizj(R i(t ) − R i(0))(R j(t ) − R j(0))⟩ i

j

(4)

The above equation can be written as

σ = σuncorr + σcorr

(5)

where σuncorr and σcorr are given by 2778

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

Figure 5. Radial distribution function (gLi−X(r)) and coordination number (nLi−X(r)) for X = EO and B of Li+ ions at 500 K.

Figure 6. Fraction of free ions that are not in the first coordination shell of their counterions at 500 K.

σuncorr = lim

t →∞

σcorr = lim

t →∞

e2 6tVkBT

2e 2 6tVkBT

∑ ⟨zi2(R i(t ) − R i(0))2 ⟩ i

However, instead of calculating σcorr, we use the short-time atomistic statistics to calculate the degree of uncorrelated ion motion α(t).60−63 The time average value of α(t) in the subdiffusive regime, denoted henceforth as α̅ , was then used to calculate the overall conductivity as σ = α̅ σuncorr. Such an approximation relies on the fact that we are able to obtain much more accurate results for σuncorr through our implementation of the TEKMC methodology. Figures 4a and 4b display the degree of uncorrelated ion motion, α̅, and the conductivities, σ, calculated using the abovediscussed procedure. For all the systems examined in this work, the α̅ values were found to be in the range of 0.36−0.58. In contrast, the values of α̅ for ionic liquids have been reported in the range of 0.55−0.75,60 0.63−0.69,61 0.6−0.7,62 and 0.5− 0.863 which suggest a higher degree of ion-pairing in our systems compared to ionic liquids. Our results also show that α̅ is slightly higher for higher salt concentrations, a result which is in agreement with Smith et al.61 Overall, with respect to the influence of nanoparticles, the α̅ are seen to remain almost a constant (within the errors involved in these calculations) and do not display any perceptible trends. However, since the conductivity reflects both the correlated motion and the uncorrelated ionic diffusivities, we observe that σ (Figure 4b) is reduced upon

(6)

∑ ∑ ⟨zizj(R i(t ) − R i(0))(R j(t ) − R j(0))⟩ i

j>i

(7)

The first term σuncorr is the sum of mean-squared displacement of the individual ions and can be directly computed from the self-diffusion coefficient of ionic species. The second term accounts for cross-correlations between the different ions for i ≠ j. The degree of uncorrelated ion motion, defined as α(t) =σ/σuncorr,60−63 is 1.0 for completely uncorrelated ion motion, whereas α = 0.0 for fully correlated ion motion. A value of α = 1.0 corresponds to the situation of maximum conductivity, whereas α = 0.0 corresponds to the case in which cations and anions move together as ion pairs giving rise to zero conductivity. Accurate calculation of σ in the diffusive regime is challenging due to the difficulty in accessing the correlated part σcorr.60−63 To minimize the errors involved in the computation of σ, we adopt a procedure proposed in refs 60−63. Explicitly, we use the long-time statistics and MSDs to calculate accurate values for the uncorrelated component σuncorr. 2779

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

Figure 7. (a) Dihedral autocorrelation function (Cϕϕ(t)) for C−O−C−C dihedral angle ϕ which represents PEO chain backbone for all the systems studied. (b) Segmental mean relaxation time of PEO chains as a function of Al2O3 weight percentage at three different temperatures for EO:Li = 15:1 (filled symbols) and EO:Li = 8:1 (open symbols) salt concentration. The inset displays τ0 as a function of temperature for both salt concentrations. Lines are a guide to the eye.

particles to polymer electrolytes reduce the ionic conductivities of the matrices and that such an effect is directly ascribable to the influence of the nanoparticles upon the ion mobilities. Moreover, our equilibrium structural data confirm that the nanoparticles have no significant effect on the static ion− polymer structure and their coordination.

the addition of nanoparticles. These results indicate that in our system the nanoparticles do not significantly influence the overall extent of ion pairing within our electrolyte and that the influence of nanoparticles upon the conductivity arises from their influence upon the ionic mobilities. 3.3. Equilibrium Ion Correlations. In the preceding section we demonstrated the influence of the nanoparticles upon the ionic conductivities arose mainly from their impact in turn upon the ion diffusivities. In other words, we argued that the nanoparticles do not significantly influence the coordination between cation and anions. However, as we pointed out in such a context, the data underlying Figure 4a relies on short-time statistics to infer the behavior of the correlated component of the conductivity and is hence expected to be influenced by statistical errors. As additional evidence to support the overall conclusions presented therein, in this section we present the results characterizing the influence of nanoparticles upon the equilibrium structural coordination characteristics of the cation and anions. To examine the equilibrium ion coordination characteristics, we calculated various radial distribution functions, gLi−X(r), for X = EO and B and the respective coordination numbers, nLi−X(r) (the coordination of ions with the nanoparticles is presented and discussed in section 4.3). From the results, displayed in Figure 5, we observe no significant changes in either gLi−X(r) or nLi−X(r) for X = EO and B with increase in nanoparticle loading. We also analyzed the number of free ions (Nf) in the system by calculating ions not present in the first coordination shell of their counterions. The fraction of free ions are displayed in Figure 6 at 500 K for all the systems studied. Explicitly, we do not find the fraction of free ions to have explicit dependence on the nanoparticle loading. Radial distribution functions for various other atomic pairs, viz. Li− Li, B−B, C−C, EO−EO, Li−B, Li−C, Li−EO, and C−EO, are shown in the Supporting Information (Figures S10−S17) and are consistent with the same conclusion. This reveals that in our simulations the nanoparticles do not affect any of the static structure and coordination of Li+ ions with polymer and BF4− ions. Taken together, the results presented in the present and preceding section indicate that the addition of Al2O3 nano-

4. MECHANISMS UNDERLYING ION MOBILITIES 4.1. Polymer Segmental Dynamics and Ion Mobilities. A number of experimental and theoretical studies have studied ion transport in polymer electrolytes and have identified a variety of relevant transport mechanisms such as the hopping motion of cations through the formation of a weak coordination shell between Li+ ions and ether oxygens (EO), free ion motion along percolating channels in PEO melt, etc.19,26,27,31,34,55,64 However, among the different controlling factors, the segmental mobility of the polymer backbone has been identified as a key factor underlying the cation and anion mobilities. For instance, Borodin and Smith55 explicitly compared the diffusivities of Li+ cation and TFSI anions with the time scale of motion of EO segments and showed that the translational dynamics of the ions exhibited the same dependence upon the salt concentration as the polymer segmental dynamics. More directly, they modified the dynamics of the polymer segments and demonstrated that the cation and anion diffusivities were also correspondingly modified by the same factor. We also note that in a recent short communication34 we presented results for ion mobilities in PNCs containing TiO2 nanoparticles and demonstrated that the influence of the nanoparticles upon the ion mobilities were strongly correlated to the influence of the former upon the polymer segmental mobilities. Moreover, the results presented in section 3 suggested that the effect of adding nanoparticles had the same qualitative effect as lowering the temperature. Since temperature mainly influences the polymer segmental dynamics, such results suggest probing the nanoparticle induced changes in the polymer dynamics and examining their correlation to the changes in ion mobilities. To probe the segmental dynamics of the PEO chains, we used the autocorrelation function of dihedral angle involving C−O−C−C atoms calculated as32,65−68 2780

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules Cϕϕ(t ) =

⟨ cos ϕ(t ) cos ϕ(0)⟩ − ⟨cos ϕ(0)⟩2 ⟨cos ϕ(0) cos ϕ(0)⟩ − ⟨ cos ϕ(0)⟩2

mobilities. In this and the following section, we clarify the mechanisms underlying the nanoparticles influence upon the polymer segmental dynamics and its resulting impact upon the ion mobilities. A number of previous works have studied the influence of nanoparticles upon the polymer dynamical properties such as rheology, glass transition temperature, etc.16,22,23,69−72 It is generally believed that the effect of nanoparticles primarily arises from their influence upon the conformation and dynamical properties of polymer segments near the nanoparticle surfaces. Indeed, a number of works have pointed to the existence of an “interfacial layer” which has been attributed as the key determinant of the property changes noted in PNCs.30,32,35,36,67,73−79 To probe the applicability of such a mechanism to our model, we divided the polymer segments in our system based on their proximity to the nanoparticle surfaces. For this, we used the PEO configuration at specific instant of time t (a configuration at t = 5 ns was chosen) along the MD simulation trajectory. The minimum distance between the surface of the nanoparticles and the center of mass of the four atoms (C−O−C−C) involved in a dihedral angle was used to classify the dihedral angles as either “near” or “far” from the nanoparticles (a cutoff distance of 8 Å was used to classify the dihedral angles). Subsequently, we probed the relaxation of the autocorrelation function of dihedral angle for these two classes of polymer segments. In Figure 9, we display the plots for Cϕϕ(t) and the corresponding polymer relaxation times for the segments distinguished by their proximity to the nanoparticle surface. It is observed that the relaxation profiles and the relaxation times of the segments which are proximal to the nanoparticles are significantly slower compared to the bulk segments. Interestingly, the relaxation times of segments which are farther from the nanoparticle surfaces are practically unaffected by the nanoparticle loading. In contrast, the relaxation times near the particle surface is seen to increase with increasing particle loading. Such a behavior may be rationalized as a consequence of the overlap of the interfacial zones and an effect of the polymer segments being influenced by multiple nanoparticles. In addition, we also observe that the relative disparity in the time scales increase with decreasing temperatures. Such a trend is indicative of an enthalpic origin for the modified relaxations of the interfacial layer. To understand the influence of the above effects upon ion mobilities, we probed the Li+ ion residence times in the vicinity of EO segments present in the PEO chains. For this purpose, we used the residence time autocorrelation function R(t) for Li+ ions around PEO chains as

(8)

where ϕ(t) is the dihedral angle of C−O−C−C atoms in PEO chain at time t. To quantify the polymer segmental mobilities, we fitted simulation results for Cϕϕ(t) to a Kohlrausch− Williams−Watts (KWW) stretched exponential function of the β form Cϕϕ(t) = e−(t/t*) , where t* and β are the fitting parameters and obtained the mean segmental relaxation time (τ) as τ=

∫0

∞

⎡ ⎛ t ⎞β⎤ ⎛ 1⎞ exp⎢ −⎜ ⎟ ⎥ dt = t *Γ⎜1 + ⎟ ⎝ ⎠ β⎠ ⎝ ⎣ t* ⎦

(9)

where Γ denotes Gamma function. The results for Cϕϕ(t) for different temperatures and nanoparticle loading are displayed in Figure 7a. We observe that at both salt concentrations Cϕϕ(t) decays slower in particle filled PNCs compared to the pure PEO melt. The corresponding mean segmental relaxation times for the PEO chain backbone (normalized by the mean relaxation time in nanoparticle free melt, τ0) are displayed in Figure 7b. Consistent with the results of Figure 7a, the mean relaxation time of PEO chains is seen to increase with the addition of nanoparticles. Together, the results of Figures 7a and 7b indicate that the mobility of the PEO segments is reduced with addition of the nanoparticles. In Figure 8, we display the ionic mobilities obtained from our simulations and the (inverse) polymer segmental mobilities at

Figure 8. Diffusion coefficient of Li+ ions (filled symbols with bold lines) and the inverse of the mean relaxation times of the polymer segments (open symbols with dashed lines) as a function of the temperature at various loading of the nanoparticles at EO:Li = 8:1.

R (t ) =

EO:Li = 8:1 as a function of the temperature (the results for other salt concentrations and anions are shown in Figure S7). It is observed that there is an almost perfect correlation between the changes in the polymer segmental mobilities and the corresponding effects on anionic and cationic mobilities. These results conclusively indicate that the influence of the nanoparticles upon the ionic mobilities can in turn be attributed to the changes in the polymer segmental dynamics arising from the introduction of nanoparticles. 4.2. Influence of Nanoparticles on Polymer Segmental Dynamics and Ionic Mobilities. In the preceding section, we demonstrated that the nanoparticles modifies the segmental dynamics of the polymer and thereby influences the ionic

⟨H(t )H(0)⟩ ⟨H(0)H(0)⟩

(10)

where H(t) is 1 if Li+ ion and EO are in their first coordination shell (taken to be as distance of 3.5 Å) and 0 otherwise. R(t) provides a direct estimate of the time spent by Li+ ion in the coordination shell of EO.25,31,80 To explicitly quantify the influence of nanoparticles, we identified a Li−EO pair to be “near” to the nanoparticle surface if the center of mass of Li− EO pair lies within 8 Å distance from the surface of the nanoparticle. The Li−EO pairs which were farther than such a cutoff were labeled as “far”. Plots of residence time autocorrelation function, R(t) at EO:Li = 15:1 are shown in Figure 10. At both the salt 2781

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

Figure 9. (a) Dihedral autocorrelation function for PEO chains “near” and “far” from Al2O3 nanoparticles at 5 wt %, EO:Li = 15:1 and the (b) mean segmental relaxation times “near” and “far” from Al2O3 nanoparticles for all the systems.

Figure 10. Li+ ion dynamics around polymer segments: (a) residence time autocorrelation function R(t) as a function of temperature at 5 wt %, EO:Li = 15:1 and (b) as a function of wt % at 500 K, EO:Li = 15:1. Plots of R(t) at other temperatures and wt % are given in Figures S8 and S9.

Figure 11. Time scales: (a) Bottom panel displays the mean residence time, τr, of Li+ ions around polymer segments and top panel displays the polymer segmental mean relaxation times, τ, for a comparison. Qualitative features of τr and τ are similar to each other. (b) Mean residence time of Li+ ions near to the Al2O3 nanoparticle (top panel) and away from the Al2O3 nanoparticle (bottom panel). Filled symbols represent EO:Li = 15:1, and open symbols represent EO:Li = 8:1. Lines are a guide to the eye.

2782

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

Figure 12. Comparison of the (a) radial distribution functions and corresponding coordination numbers and the (b) polymer relaxation times, between ion-doped systems and ion-f ree systems at various loading of the nanoparticles at 425 K. Atoms EO and OAl are chosen to represent polymer and nanoparticles.

concentrations investigated, we observe R(t) decays rapidly at higher temperatures, revealing lower mean residence times for ion coordination. From the R(t) shown in Figure 10, we observe two important features: First, R(t) near to the surface (solid lines) decays much slower compared to the R(t) for ions further away from the surface (dashed lines). Second, R(t) far away from nanoparticle surface approaches the results expected for pure PEO systems (dotted lines), indicating that the effect of nanoparticle surface extends to a finite distance (corresponding to the “interfacial zone”) from the surface. We fitted the overall R(t) as well as those corresponding to “near” and “far” from the particles to a stretched exponential function (eq 9) similar to that employed for Cϕϕ(t) and obtained the mean residence time, τr, for Li+ ions. The results for the overall mean residence times, τr, are displayed in Figure 11a (bottom panel). Also displayed for comparison (in Figure 11a) are the overall polymer segmental relaxation times. The qualitative features of τr are seen to be very similar to those exhibited by polymer segmental relaxation times, τ. For instance, the mean residence time, τr, was found to be at least 2 times higher at a given temperature when the salt concentration was increased to EO:Li = 8:1, an observation consistent with the dihedral relaxation times τ. Overall, these results are consistent with the expectation that the ion motion, especially the component which involves the coordination between the polymer segments and ions, is strongly correlated to the polymer segmental relaxation times. More pertinent to the discussion of the present section are the results displayed in Figure 11a which depicts τr based on the proximity of the polymer segments to the nanoparticles. We observe that τr near the Al2O3 nanoparticles is much larger than that of the bulk system as shown in the top panel of Figure 11b. It is also seen that the τr away from the nanoparticles is practically unaffected by the loading of the nanoparticles, indicative of the presence of an “interfacial zone” beyond which the influence of the nanoparticle does not manifest. The above results are consistent with the trends observed in Figure 9 and conclusively identify that the origin of changes in polymer segmental dynamics and the ionic mobilities arise from the presence of an interfacial layer around the nanoparticles. Such an interfacial layer possesses significantly reduced polymer mobility and correspondingly the ion motion associated with

the polymer segments is also lowered. In contrast, the dynamics of both the polymer and ions in the “bulk” region of the PNC is seen to be unaffected by the presence of the nanoparticles. While the previous section showed the broad origins of the changes in the ionic mobilities, the results presented in this section identify the microscopic mechanisms underlying such trends. 4.3. Mechanisms Underlying Polymer Segmental Mobilities in the Interfacial Zone. The preceding section presented evidence for the presence of an interfacial layer around the nanoparticles wherein the polymer mobilities were significantly reduced. As a consequence, the ion mobilities in the layer as the overall ion diffusivities and conductivities were shown to be reduced. The final issue we consider in this work is the identification of the mechanisms underlying the modified polymer dynamics in the interfacial layer. In a number of other works,54,81−83 the influence of nanoparticles on the polymer segmental mobilities have been shown to be a result of the polymer nanoparticles interactions and the resulting polymer conformation/density changes which occur near the surface. Indeed, favorable or attractive polymer particle interactions have been shown to lead to enhanced polymer densities and slower polymer dynamics near the surface. In contrast, repulsive or unfavorable polymer particle interactions have been shown to lead to depletion and enhanced mobilities of polymer segments near surfaces.54,81,83−85 We note that the above-discussed mechanism implicates only the interaction between the polymer and the nanoparticles. Hence, to probe such a feature, we considered an ion-free system at different particle loadings. Shown in Figure 12a are the radial distribution functions and coordination numbers between the polymer components and the nanoparticles. In Figure 11b, we display the corresponding polymer segmental relaxation times for the “near” and “far” segments for the PNC matrix. While the radial distribution functions by themselves are not suggestive of a strong interaction between the polymer segments and the nanoparticles, it is however seen that the polymer segmental relaxation are indeed slowed near the particle surface (relative to the bulk). Together these results indicate that the favorable polymer particle interaction does 2783

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

Figure 13. Radial distribution function (gLi−X(r)) and coordination number (nLi−X(r)) for X = Al and OAl of Li+ ions at 500 K. Radial distribution functions of Li, EO, and B with the atoms of nanoparticles are given in Figures S18−S23.

composites. Nonconductive, quasi-spherical and nanosized Al2O3 particles were well dispersed in PEO melt to create PNC systems at EO:Li = 15:1 and EO:Li = 8:1 salt concentration and Al2O3 nanoparticle wt % of 5, 10, and 20. Ionic mean-squared displacement and mean-squared charge displacement were used to calculate diffusion coefficient and conductivity at temperatures 350, 425, and 500 K for all the systems. We found that the diffusivity of the ions and the conductivity decreased monotonically with decreasing temperature and with the addition of Al2O3 nanoparticles in PEO melt. To unravel the effect of nanoparticle on Li+ ion mobilities, we calculated the segmental mean relaxation times of polymers from the autocorrelation function of backbone C−O−C−C dihedral angle. In addition, residence time autocorrelation functions of EO−Li pairs were calculated in the entire system, “near” and “far” away from Al2O3 nanoparticle. By fitting the dihedral angle and residence time autocorrelation functions to stretched exponential function, we calculated segmental mean relaxation time, τ, and mean residence time, τr, respectively. We found that both the τ and τr were to be increasing with the nanoparticle weight percent, revealing slow dynamics of PEO. By a detailed analysis we implicated the presence of an interfacial region with modified polymer segmental dynamics as being the origin of the influence of nanoparticles. Moreover, by consideration of ion-f ree systems and by comparison to iondoped systems we identified that both the polymer−nanoparticle and ion−nanoparticle interactions are responsible for the presence and the characteristics of the interfacial layer. Overall, our results suggest that nanoparticle induced modifications of conformational and dynamical properties of PEO have a strong effect on lithium ion mobilities in PNC. An interesting future direction is to study the influence of surface chemistry of Al2O3 nanoparticle. Experiments have demonstrated that protonation of Al2O3 nanoparticle can render the surface chemistry acidic in nature20,24,25 and can have a drastic effect upon the conductivities.22,24,25,86 Nanoparticle size and shape are also shown to have significant influence on ion conductivity and hence is another promising question to address.5,14,35,83,87

plays a key role in influencing the presence and the dynamical properties within the interfacial zone. What is the influence of the ions upon the above results? In Figure 12 we compare the results of the ion-f ree with the iondoped system. We observe that the relaxation times of the polymer segments in both the interfacial and the bulk zones in the ion-free system are much faster than the relaxation times for the ion-doped system. However, it is seen that the relative time scales (shown in the inset as τnear normalized by τfar) between the “near” and “far” regions are slightly reduced in the ion-doped system. The above trends can be understood by comparing the radial distribution functions for the ion-f ree and ion-doped systems in Figure 12a. We observe that the coordination numbers for the polymer segments are indicative of a slightly reduced attraction between the nanoparticle and the polymer backbone in the iondoped system. Such results are in-turn rationalized by the results displayed in Figure 13 where it is seen that Li+ ions strongly “bind” to OAl of Al2O3 nanoparticles. Correspondingly, the coordination number nLi−X(r) for X = Al, OAl of Li+ ions shown in Figure 13 also reveals that the number of Li+ ions bound to Al2O3 nanoparticles increases with Al2O3 weight percentage. In sum, the association between the ion and nanoparticles reduces the polymer coordination with the nanoparticles and explains the reduction in the relative time scales between the “near” and “far” segments. The results presented in this section serve to clarify the origins of the modified polymer dynamics in the interfacial region. Explicitly, both nanoparticle−polymer and the nanoparticle−ion interactions were shown to be important. In our system, the Li+ ions are preferentially attracted to and can hydrogen bond with the electron donating basic surface sites present in Al2O3 nanoparticles. Moreover, the basic surface sites can also possibly participate in a weak interaction with methylene (CH2) groups of polymer chains. Together, these Lewis acid−base interactions were shown to form an interfacial region with slowed polymer relaxation times.

5. CONCLUSIONS To summarize, all atom molecular dynamics and trajectoryextending kinetic Monte Carlo simulations were performed to study transport mechanisms of Li+ ions in polymer nano2784

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules

■

(18) Tang, C.; Hackenberg, K.; Fu, Q.; Ajayan, P. M.; Ardebili, H. Nano Lett. 2012, 12, 1152−1156. (19) Berthier, C.; Gorecki, W.; Minier, M.; Armand, M. B.; Chabagno, J. M.; Rigaud, P. Solid State Ionics 1983, 11, 91−95. (20) Wieczorek, W.; Zalewska, A.; Raducha, D.; Florjanczyk, Z.; Stevens, J. R. J. Phys. Chem. B 1998, 102, 352−360. (21) Croce, F.; Persi, L.; Scrosati, B.; Serraino-Fiory, F.; Plichta, E.; Hendrickson, M. Electrochim. Acta 2001, 46, 2457−2461. (22) Ganapatibhotla, L. V. N. R.; Maranas, J. K. Macromolecules 2014, 47, 3625−3634. (23) Johansson, P.; Ratner, M. A.; Shriver, D. F. J. Phys. Chem. B 2001, 105, 9016−9021. (24) Eilmes, A.; Kubisiak, P. J. Phys. Chem. B 2011, 115, 14938− 14946. (25) Wu, H.; Cummings, O. T.; Wick, C. D. J. Phys. Chem. B 2012, 116, 14922−14932. (26) Gadjourova, Z.; Andreev, Y. G.; Tunstall, D. P.; Bruce, P. G. Nature 2001, 412, 520−523. (27) Muller-Plathe, F. Acta Polym. 1994, 45, 259−293. (28) Neyertz, S.; Brown, D. J. Chem. Phys. 1996, 104, 3797−3809. (29) Smith, G. D.; Yoon, D. Y.; Jaffe, R. L.; Colby, R. H.; Krishnamoorti, R.; Fetters, L. J. Macromolecules 1996, 29, 3462−3469. (30) Borodin, O.; Smith, G. D.; Bandyopadhyaya, R.; Redfern, P.; Curtiss, L. A. Model. Simul. Mater. Sci. Eng. 2004, 12, S73−S89. (31) Borodin, O.; Smith, G. D.; Douglas, R. J. Phys. Chem. B 2003, 107, 6824−6837. (32) Borodin, O.; Smith, G. D.; Bandyopadhyaya, R.; Byutner, E. Macromolecules 2003, 36, 7873−7883. (33) Borodin, O.; Smith, G. D. J. Phys. Chem. B 2003, 107, 6801− 6812. (34) Hanson, B.; Pryamitsyn, V.; Ganesan, V. ACS Macro Lett. 2013, 2, 1001−1005. (35) Kasemägi, H.; Klintenberg, M.; Aabloo, A.; Thomas, J. O. J. Mater. Chem. 2001, 11, 3191−3196. (36) Kasemägi, H.; Klintenberg, M.; Aabloo, A.; Thomas, J. O. Solid State Ionics 2002, 147, 367−375. (37) Plimpton, S. J. Comput. Phys. 1995, 117, 1−19. (38) Rittner, E. S. J. Chem. Phys. 1951, 19, 1030−1035. (39) Sarsam, J.; Finnis, M. W.; Tangney, P. J. Chem. Phys. 2013, 139, 204704. (40) Materials Studio v 3.0; Accelrys Software Inc., San Diego, CA. (41) Martinez, L.; Andrade, R.; Birgin, E. G.; Martinez, J. M. J. Comput. Chem. 2009, 30, 2157−2164. (42) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Phys. 1977, 23, 327−341. (43) Darden, T.; York, D.; Pedersen, L. J. Chem. Phys. 1993, 98, 10089−10092. (44) Hockney, R. W.; Eastwood, J. W. Computer Simulation Using Particles; Taylor & Francis: Boca Raton, FL, 1989. (45) Nosé, S. J. Chem. Phys. 1984, 81, 511−519. (46) Hoover, W. G. Phys. Rev. A 1985, 31, 1695−1697. (47) Mark, J. E. Physical Properties of Polymers Handbook; Springer: New York, 2007. (48) Polymer Handbook, 4th ed.; Brandrup, J., Immergut, E. H., Grulke, E. A., Eds.; Wiley-Interscience: New York, 1999. (49) Wunderlich, B. Macromolecular Physics; Academic Press: New York, 1973; Vol. I. (50) Neyertz, S.; Brown, D. Macromolecules 2010, 43, 9210−9214. (51) Hanson, B.; Pryamitsyn, V.; Ganesan, V. J. Phys. Chem. B 2012, 116, 95−103. (52) Bortz, A. B.; Kalos, M. H.; Lebowitz, J. L. J. Comput. Phys. 1975, 17, 10−18. (53) Kipp, D.; Ganesan, V. J. Appl. Phys. 2013, 113, 234502. (54) Pryamitsyn, V.; Hanson, B.; Ganesan, V. Macromolecules 2011, 44, 9839−9851. (55) Borodin, O.; Smith, G. D. Macromolecules 2006, 39, 1620−1629. (56) Fullerton-Shirey, S. K.; Maranas, J. K. Macromolecules 2009, 42, 2142−2156.

ASSOCIATED CONTENT

S Supporting Information *

Nonbonded parameters, comparison of the MSD curve obtained from TEKMC with that of atomistic MD at 425 K and EO:Li = 15:1 for short time scales, MSD curves obtained from MD for Li+ and BF4− ions, α(t) in the subdiffusive regime, transference number for cations, D of ions versus wt %, D and τ−1 versus temperature, residence time autocorrelation functions and radial distribution functions for various atomic pairs, viz., Li−Li, B−B, C−C, EO−EO, Li−B, Li−C, Li−EO, C−EO, Li−Al, Li−OAl, EO−Al, EO−OAl, B−Al, and B−OAl. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (V.G.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing computing resources that have contributed to the research results reported within this paper. This work was supported in part by grants from Robert A. Welch Foundation (Grant F1599) and National Science Foundation (DMR1306844) and the US Army Research Office under Grant W911NF-13-1-0396. The authors acknowledge useful discussions with Drs. Victor Pryamitsyn, Oleg Borodin, and Benjamin Hanson.

■

REFERENCES

(1) Armand, M. Solid State Ionics 1983, 9−10, 745−754. (2) Armand, M. B. Annu. Rev. Mater. Sci. 1986, 16, 245−261. (3) Springer, T. E.; Zawodzinski, T.; Gottesfeld, S. J. Electrochem. Soc. 1991, 138, 2334−2342. (4) Meyer, W. H. Adv. Mater. 1998, 10, 439−448. (5) Croce, F.; Appetecchi, G. B.; Persi, L.; Scrosati, B. Nature 1998, 394, 456−458. (6) Croce, F.; Curini, R.; Martinelli, A.; Persi, L.; Ronci, F.; Scrosati, B.; Caminiti, R. J. Phys. Chem. B 1999, 103, 10632−10638. (7) Tarascon, J. M.; Armand, M. Nature 2001, 414, 359−367. (8) Christie, A. M.; Lilley, S. J.; Staunton, E.; Andreev, Y. G.; Bruce, P. G. Nature 2005, 433, 50−53. (9) Arico, A. S.; Bruce, P. G.; Scrosati, B.; Tarascon, J. M.; van Schalkwijk, W. Nat. Mater. 2005, 4, 366−377. (10) Bruce, P. G.; Scrosati, B.; Tarascon, J.-M. Angew. Chem., Int. Ed. 2008, 47, 2930−2946. (11) Smitha, B.; Sridhar, S.; Khan, A. A. J. Membr. Sci. 2005, 259, 10− 26. (12) Hallinan, D. T., Jr.; Balsara, N. P. Annu. Rev. Mater. Res. 2013, 43, 503−525. (13) Fullerton-Shirey, S. K.; Maranas, J. K. J. Phys. Chem. C 2010, 114, 9196−9206. (14) Nagasubramanian, G.; Attia, A. I.; Halpert, G.; Peled, E. Solid State Ionics 1993, 67, 51−56. (15) Appetecchi, G. B.; Passerini, S. Electrochim. Acta 2000, 45, 2139−2145. (16) Walls, H. J.; Zhou, J.; Yerian, J. A.; Fedkiw, P. S.; Khan, S. A.; Stowe, M. K.; Baker, G. L. J. Power Sources 2000, 89, 156−162. (17) Podsiadlo, P.; Kaushik, A. K.; Arruda, E. M.; Waas, A. M.; Shim, B. S.; Xu, J.; Nandivada, H.; Pumplin, B. G.; Lahann, J.; Ramamoorthy, A.; Kotov, N. A. Science 2007, 318, 80−83. 2785

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786

Article

Macromolecules (57) Ma, Y. P.; Doyle, M.; Fuller, T. F.; Doeff, M. M.; Dejonghe, L. C.; Newman, J. J. Electrochem. Soc. 1995, 142, 1859−1868. (58) Ferry, A.; Oradd, G.; Jacobsson, P. J. Chem. Phys. 1998, 108, 7426−7433. (59) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed.; Academic Press: New York, 2001. (60) Borodin, O.; Gorecki, W.; Smith, G. D.; Armand, M. J. Phys. Chem. B 2010, 114, 6786−6798. (61) Li, Z.; Smith, G. D.; Bedrov, D. J. Phys. Chem. B 2012, 116, 12801−12809. (62) Lesch, V.; Jeremias, S.; Moretti, A.; Passerini, S.; Heuer, A.; Borodin, O. J. Phys. Chem. B 2014, 118, 7367−7375. (63) Picalek, J.; Kolafa, J. J. Mol. Liq. 2007, 134, 29−33. (64) Diddens, D.; Heuer, A. J. Phys. Chem. B 2014, 118, 1113−1125. (65) Ndoro, T. V. M; Boehm, M. C.; Mueller-Plathe, F. Macromolecules 2012, 45, 171−179. (66) Smith, G. D.; Borodin, O.; Bedrov, D.; Paul, W.; Qiu, X. H.; Ediger, M. D. Macromolecules 2001, 34, 5192−5199. (67) Smith, G. D.; Borodin, O.; Paul, W. J. Chem. Phys. 2002, 117, 10350−10359. (68) Ahumada, O.; Theodorou, D. N.; Triolo, A.; Arrighi, V.; Karatasos, C.; Ryckaert, J. P. Macromolecules 2002, 35, 7110−7124. (69) Zhao, J.; Morgan, A.; Harris, J. Polymer 2005, 46, 8641−8660. (70) Paul, D. R.; Robeson, L. M. Polymer 2008, 49, 3187−3204. (71) Fakhraai, Z.; Forrest, J. A. Science 2008, 319, 600−604. (72) Ilton, M.; Qi, D.; Forrest, J. A. Macromolecules 2009, 42, 6851− 6854. (73) Gong, S.; Chen, Q.; Moll, J. F.; Kumar, S. K.; Colby, R. H. ACS Macro Lett. 2014, 3, 773−777. (74) Jouault, N.; Moll, J. F.; Meng, D.; Windsor, K.; Ramcharan, S.; Kearney, C.; Kumar, S. K. ACS Macro Lett. 2013, 2, 371−374. (75) Kumar, S. K.; Jouault, N.; Benicewicz, B.; Neely, T. Macromolecules 2013, 46, 3199−3214. (76) Starr, F. W.; Schroder, T. B.; Glotzer, S. C. Macromolecules 2002, 35, 4481−4492. (77) Kropka, J. M.; Pryamitsyn, V.; Ganesan, V. Phys. Rev. Lett. 2008, 101, 075702. (78) Kropka, J. M.; Putz, K. W.; Pryamitsyn, V.; Ganesan, V.; Green, P. F. Macromolecules 2007, 40, 5424−5432. (79) Ganesan, V.; Ellison, C. J.; Pryamitsyn, V. Soft Matter 2010, 6, 4010−4025. (80) Danckwerts, P. V. Chem. Eng. Sci. 1953, 2, 1−13. (81) Pandey, Y. N.; Doxastakis, M. J. Chem. Phys. 2012, 136, 24706. (82) Rissanou, A. N.; Harmandaris, V. J. Nanopart. Res. 2013, 15, 1589. (83) Ganesan, V.; Jayaraman, A. Soft Matter 2014, 10, 13−38. (84) Krishnamoorti, R. MRS Bull. 2007, 32, 341−347. (85) Ganesan, V. J. Polym. Sci., Part B: Polym. Phys. 2008, 46, 2666− 2671. (86) Croce, F.; Sacchetti, S.; Scrosati, B. J. Power Sources 2006, 161, 560−564. (87) Surve, M.; Pryamitsyn, V.; Ganesan, V. Langmuir 2006, 22, 969−981.

2786

DOI: 10.1021/ma502578s Macromolecules 2015, 48, 2773−2786