Dependence of Ion Dynamics on the Polymer Chain Length in Poly

Article pubs.acs.org/JPCB

Dependence of Ion Dynamics on the Polymer Chain Length in Poly(ethylene oxide)-Based Polymer Electrolytes Joyjit Chattoraj, Marisa Knappe, and Andreas Heuer* Institut für Physikalische Chemie, Westfälische Wilhelms-Universität Münster, Corrensstrasse 28/30, D-48149 Münster, Germany ABSTRACT: It is known from experiments that in the polymer electrolyte system, which contains poly(ethylene oxide) chains (PEO), lithium-cations (Li + ), and bis(trifluoromethanesulfonyl)imide-anions (TFSI−), the cation and the anion diffusion and the ionic conductivity exhibit a similar chain-length dependence: with increasing chain length, they start dropping steadily, and later, they saturate to constant values. These results are surprising because Li-cations are strongly correlated with the polymer chains, whereas TFSIanions do not have such bonding. To understand this phenomenon, we perform molecular dynamics simulations of this system for four different polymer chain lengths. The diffusion results obtained from our simulations display excellent agreement with the experimental data. The cation transport model based on the Rouse dynamics can successfully quantify the Lidiffusion results, which correlates Li diffusion with the polymer center-of-mass motion and the polymer segmental motion. The ionic conductivity as a function of the chain length is then estimated based on the chain-length-dependent ion diffusion, which shows a temperature-dependent deviation for short chain lengths. We argue that in the first regime, counterion correlations modify the conductivity, whereas for the long chains, the system behaves as a strong electrolyte.

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In addition, Shi and Vincent4 assume that the anion diffusion is M-independent. Interestingly, an M-dependent anion diffusion has been reported in the experiments.12,13 Thus, the prediction of σ as a function of M from the ion diffusion, which is proposed in ref 4, is too simple, and a closer understanding of the individual contributions to ionic conductivity is required. In this work, we investigate the effects of the polymer chain length on the diffusion of ions and the ionic conductivity via molecular dynamics (MD) simulation. This has been proved to be an essential tool to capture the microscopic mechanisms for such systems5−11 (see also the references cited by these articles). We predict the cation diffusion as a function of the number of monomers per chain N (proportional to the molecular weight) based on the transport model. This is the first time we verify the model for the same polymer electrolyte systems and in the same temperature regime at which experimental data are available. A model for σ is developed using the transport model where we assume that the system behaves as a strong electrolyte (i.e., no correlation takes place between ions). This assumption is then critically discussed.

INTRODUCTION Over the last few decades, an extensive amount of research1,2 has been conducted on polymer electrolytes, which are a class of materials where polymers dissociate salts into their cations and anions in absence of any other solvent. This feature is advantageous for many energy storage devices (e.g., Li-metal batteries). In such systems, the molecular weight of the polymers significantly modifies the ion dynamics and hence the ionic conductivity σ. A crossover from a strong molecularweight dependence to a very weak dependence has been reported by Teran et al.3 for a system where poly(ethylene) oxide PEO is used as the host polymers and lithium bis(trifluoromethanesulfonyl)imide Li-TFSI is used as the salt. This type of crossover had already been predicted by Shi and Vincent4 based on the dependence of the cation diffusion on the polymer dynamics. They essentially decompose the cation diffusion into two parts: (i) the molecular weight (M)dependent center-of-mass diffusion of chains Dcom(M) ∝ 1/M and (ii) M-independent cation diffusion, which is due to the polymer segmental relaxation processes. The second part of the cation diffusion, which was initially assumed to be M-independent by the authors, has been further rationalized by the Rouse-based transport model.5−8 The model predicts a strong-to-weak crossover behavior with M for this diffusion defined as DM. Overall, the transport model yields a similar characteristic behavior of the cation diffusion with M incorporating both Dcom and DM.7,8 A systematic simulation study of the cation diffusion as a function of chain length has so far not been conducted to verify the model prediction and also to compare with the corresponding experimental data. © XXXX American Chemical Society

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MOLECULAR DYNAMICS SIMULATIONS Our study focuses on the polymer electrolyte systems containing PEO chains, Li-cations, and TFSI-anions. We prepare four polymer electrolyte systems with different polymer sizes, where the numbers of monomers per chain N are 5, 10, 20, and 54. The numbers of Li-cations (27), TFSI-anions (27), Received: December 21, 2014 Revised: March 27, 2015

A

DOI: 10.1021/jp512734g J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B and the ratio between ether oxygen atoms and Li-cations EO/ Li (20:1) are kept always constant for all systems. We use the Amber MD code, where the many body polarizable force field is included. This has been especially developed for the PEO-LiTFSI system.6,14 We prepare a system in gas phase at temperature T = 423 K (well above the melting point). Then we allow the system to shrink under normal pressure (1 atm) for a few tens of nanoseconds via MD simulations keeping T fixed. For example, we shrink the system containing our longest polymer chains (54 monomers) for 80 ns, which is roughly two times longer than the chain relaxation time (or the Rouse time) at this temperature. Thus, the chain relaxation time, determining, for example, the relaxation of the end-to-enddistance, is by far smaller than our equilibration time. In the equilibrated state the average system density becomes 1.127 ± 0.008 g/cm3. After that, we start the production run under constant number of atoms, volume, and temperature. We discard the first few nanoseconds of the production run for our data analysis to ensure the equilibration of the system at its fixed box volume. During all MD simulations, the smallest time step of 1 fs is considered for the time integration. The temperature is kept constant using the Berendsen thermostat with a time constant of 0.1 ps. The pressure is kept constant using a similar barostat with a time constant of 1 ps. The atomic bonds involving hydrogen atoms are constrained using the Shake algorithm. Long-range electrostatic interactions for periodic images are handled using the particle mesh Ewald method. MD simulations at lower temperatures are started each time by taking a well-equilibrated configuration from the previous higher T. The system is then re-equilibrated under constant number of atoms, normal pressure, and temperature for a few nanoseconds, and the production run under constant number of atoms, volume, and temperature is started afterward. We accumulate data for temperatures of 423, 393, 363, 333, and 303 K. The duration of a production run depends on reaching the diffusive regime of Li-cations, which again depends on N and T (e.g., for N = 5, T = 423 K and N = 54, T = 303 K we have 40 and 800 ns MD trajectories, respectively).

Figure 1. ⟨Δn2(t)⟩ versus t for T = 423 K and all our N values.

n2(τ3) ⟩ = 2D1τ3,15 where τ3 is the average residence time of a cation on its host chain. The values of τ1 as a function of T for N = 54 are given in Table 1. Table 1. Simulation Data with Length in Å, Time in ns, and T in Ka

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CATION DIFFUSION AND THE TRANSPORT MODEL The Li-dynamics in the polymer electrolyte systems is strongly correlated with the polymer dynamics. A cation is always found to be coordinated with 4−6 EO atoms either of a single chain or two chains. Note that for very short chains, the number of chains coordinated with a cation tends to be higher. The cation moves through the host polymer via continuous bond (with EOs) breaking and making, which is analogous to a onedimensional random walk. In the transport model,7 this motion is characterized as the intrachain motion (M1), and it can be quantified by measuring the mean square change of the monomers ⟨Δ n2(t)⟩, where Δn(t) is the number of monomers of a chain traversed by a cation over time t. By assuming that this dynamics is diffusive (i.e., ⟨Δ n2(t)⟩ = 2 D1t) a time τ1 = (N−1)2/π2D1 can be defined as the typical time taken by a cation to traverse a polymer chain. We find that ⟨Δ n2(t)⟩ for all our N values nearly collapse (see Figure 1). Of course, later these curves saturate at different plateaus. They reflect the effect of the finite chain length. The collapse implies that D1 is independent of N and thus, τ1(N) ∝ (N−1)2. Strictly speaking, the motion along the chain is slightly subdiffusive (i.e., ⟨Δ n2(t)⟩∝ t0.8). Using the data for N = 54, we define D1 via ⟨Δ

τ1

N

T

DLi

Dcom

DTFSI

54

423 393 363 333 303

3.2 1.9 0.96 0.32 0.08

1.2 0.81 0.41 0.17 0.05

20

423 393 363 333

9.5 3.7 2.0 0.91

8.2 3.6 1.6 0.89

10

423 393 363 333

23 16 7.0 2.1

32 18 9.1 3.0

5

423 393 363 333 303

53 42 29 12 6.6

96 66 49 19 11

15 100 11 165 6.6 357 3 930 0.56 3241 R2E= 1600 24 − 14 − 7 − 3 − R2E= 515 40 − 26 − 14 − 5 − R2E= 226 72 − 57 − 40 − 17 − 9.7 − R2E= 94

τ2

τ3

ρ1

122 155 245 750 3200

13 28 74 239 1393

0.51 0.46 0.40 0.36 0.32

9 15 54 104

7.2 15 40 98

0.46 0.50 0.41 0.41

1.4 2.1 4.6 18

3.8 7.3 17 35

0.41 0.36 0.25 0.22

0.32 0.57 0.91 1.8 2.9

1.6 3.1 4.8 10 17

0.29 0.19 0.12 0.07 0.06

a

The maximum error does not exceed 10% of the actual value. Note that τ1 for other chain lengths except N = 54 is not calculated explicitly as we have shown that D1 is independent of N.

The EO atoms of a single chain which are bound to a cation are moving themselves. This segmental motion (M2) of bound EO atoms, relaxing with time τ2, thus, also contributes to the cation diffusion. In the transport model this is quantified using the Rouse dynamics16 of a linear polymer chain via gbo(t ) =

2RE2 π

2

N−1

∑ p=1

1 − exp[−p2 t /τ2] p2

(1)

where the mean-square displacement (MSD) of bound EO atoms gbo(t) with respect to the polymer center-of-mass frame B

DOI: 10.1021/jp512734g J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B is expressed as a sum over normal modes p. R2E is the mean square end-to-end distance of polymer chains. R2E and τ2 can be obtained fitting the simulation data of gbo(t) by the above equation. We already showed for N = 5415 that the fitting works reasonably well. It is known from the Rouse dynamics that R2E ∝ N and τ2 ∝ N2. Simulation values of R2E, which are also found to be independent of T, verify the relation as reported in Table 1. The prescribed scaling for τ2, however, does not hold for our two smallest chain lengths. Moreover, for high temperatures (≥363 K) a higher exponent (approximately 30%) fits τ2 data very well for all N values. The transport model incorporates the two mechanisms M1 and M2, and finds a similar expression for the MSD of a cation coordinated with a single polymer chain, which is gLiM12(t ) =

2RE2 π2

N−1

∑

1 − exp[−p2 t /τ12] p2

p=1

(2)

with τ12 = [1/τ1 + 1/τ2]−1. The MSD becomes uncorrelated when the cation undergoes an interchain jump (M3) from its host chain to another, which sets a residence time τ317 for the cation on its host chain. We find an N-dependence on τ3, which becomes weaker with increasing N (see Table 1). For short chains a cation can easily leave a chain due to its intrachain motion as it lies close to the chain ends. This effect quickly fades out with higher N and τ3 is expected to become Nindependent. In general, τ3 follows the probability distribution: ρ(t,τ3) = exp[−t/τ3]/τ3 with some exceptions at small τ3.8 Thus, the time average of gM12 Li (t) for all interchain jumps (i.e., ∫ 0∞ρ (t,τ3) gM12 Li (t) dt) becomes ⟨gLiM12(τ3)⟩M3 =

2RE2 π

2

N−1

∑ p=1

⎤ 1⎡ 1 ⎢1 − ⎥ 2 2 p ⎣ 1 + p τ3/τ12 ⎦

Figure 2. (a) The transport model prediction of Li diffusion DM for T = 333 K as a function of N, when a Li-cation is bound to single chain or two chains and both. (b) Dcom as a function of N for T = 333 K: experiments12,13 (open symbols), our simulation data (filled symbols), and the solid line is the fit. (c) The transport model prediction of Dcom, DM and Dcom + DM for T = 333 K. (d) Dcom(N) +DM(N) (solid lines) is compared with DLi for two Ts 363 and 333 K: experiments4,12 (open symbols), and simulations (filled symbols). Here, for Dcom(N), we use the three scaling behaviors described in panel (c); DM(N) is obtained from eq 4 after plugging in R2E ∝ N, τ1 ∝ N2, τ2 ∝ N2, τ3 ∝ N0 and ρ1 ∝ N0 based on values for N = 54.

(3)

For a cation bound to two chains, the intrachain motion is absent.15 For such cases, the MSD can be quantified as ⟨gM12 Li (τ3; τ1→ ∞) ⟩ from eq 3. A diffusion coefficient DM can be defined as DM =

The center-of-mass diffusion Dcom of the polymers directly adds to the cation diffusion as the polymers carry the cations. Hence, the diffusion of Li-cations can be expressed as DLi = DM + Dcom (5)

1 [ρ ⟨g M12(τ3)⟩ + (1 − ρ1)⟨gLiM12(τ3; τ1 → ∞)⟩] 6τ3 1 Li (4)

Figure 2b displays Dcom in the range of short chains: both the experiment12 and our simulation results (agree quite well) suggest a stronger scaling ∝N−1.85 than the Rouse scaling ∝N−1 for linear chains. For N ≳ 225, chains become entangled. Here the reptation model16 predicts Dcom ∝ N−2. In the intermediate regime, roughly N ∈ [75, 225], another scaling Dcom ∝ N−3 is reported in the literature.18 However, the contribution of Dcom to DLi, as compared to DM, has already started to become minor, as shown in Figure 2c. It is clear from the transport model that DLi ≈ Dcom for N ≲ 30, and DLi ≈DM for N ≳ 100. This finding is very well complemented by both experiments and simulations, as shown in Figure 2d. Interestingly, for very short chains, the cation dynamics does not strictly follow the above picture. First, we find that DLi for N = 5 is significantly (∼40%) smaller than Dcom (see Table 1) because the latter diffusion includes both the motion of bound PEO chains hosting cations and unbound PEO chains. The bound chains move much slower than the unbound chains. We checked, and indeed, the MSD of cations and the MSD of bound chains collapse. Later, for larger N, no chain remains

where single chain and two chain effects are involved. ρ1 is the probability that a cation is bound to single chain. Previously we showed15 that the polymer segmental dynamics for N = 54 and for this temperature regime follows the Rouse dynamics. Thus, assuming the ideal scaling R2E ∝ N, τ1 ∝ N2, τ2 ∝ N2 and τ3 ∝ N0, we estimate DM as a function of N based on the values of the four quantities at N = 54 and using eq 4. DM for single chain (i.e., ρ1 = 1) and for two chains (i.e., ρ1 = 0) is shown in Figure 2a. For short chains, the effect of cation coordination with number of chains is negligible on DM; here, DM rapidly grows with increasing N and finally, it saturates at around N = 100. For large N, the number of coordinated chains significantly modifies DM. Thus, we consider both the single chain and the double chain effects to estimate DM for any N and T using the value of ρ1 at N = 54 and the corresponding T. In general, ρ1 increases weakly with increasing T and with increasing N, except the values of ρ1 for N = 5, which are approximately 40−60% smaller than the values for other chain lengths (see Table 1). C

DOI: 10.1021/jp512734g J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B DTFSI = A 0(T ) + A1(T )/N1.22

unbound, and DLi is close to Dcom (see Figure 3). Second, the N-dependent τ3 values (obtained from our simulations) would

(6)

Figure 3. MSD of Li-cations, bound PEOs, and all PEOs for N = 5, 20 at T = 393 K.

increase DM few times in case of N = 5, 10, as compared to the values obtained by using τ3 ∝ N0. Unfortunately, DM cannot be calculated directly from simulation data to verify this finding. In any event, this potential increase in DM would hardly have any relevance for the observed Li-diffusivity because DM ≪ Dcom in the regime of small N; see Figure 2c. We also checked the system size effect on diffusion. The mean square displacements of PEO and Li-cations for two system sizes are shown in Figure 4. The larger system, which is

Figure 5. (a) DTFSI versus N for T = 363, 333, 303 K: experiments12,13 (open symbols), our simulation data (filled symbols). Solid lines are corresponding fits of the experimental data. (b) The radial distribution function (RDF) between TFSI-anions and ethylene oxide monomers for N = 5, 54 and for T = 363 K. (c) The probability of the number of ethylene oxide monomers and (d) the probability of the number of PEO chains surrounding a TFSI-anion shown for all four chain lengths.

In the strong N-dependent regime the drop of DTFSI with N is weaker than Dcom (the two exponents are 1.22 and 1.85, respectively). This suggests that a weak correlation is present between the polymer and anion dynamics. We cannot figure out the source of such correlation through our data analysis; for example, the radial distribution function between TFSI-anions and ethylene oxide monomers show no N-dependence (Figure 5b), and the squared-displacement of an anion as a function of the distance between the anion and a polymer chain displays no distance-dependent behavior. Our simulation snapshots indicate that the polymer topology plays a key role here. We calculate the probability of the number of ethylene oxide monomers lying inside the first neighbor shell (∼8.4 Å) of a TFSI-anion (Figure 5c) and find that 28−30 monomers are most likely (slight deviation for N = 5) to surround an anion. Similarly, we calculate the probability of the number of PEO chains contributing to these monomers (Figure 5d). The number of PEO chains decreases with increasing N. This suggests that for long chains, polymer segmental dynamics might effect the anion diffusion as an anion is surrounded by many monomers from the same chain. This might explain why the diffusivity of TFSI-anions increases with decreasing N.

Figure 4. MSDs of PEOs (N = 54) and Li-cations for two system sizes at T = 393 K.

twice the size of our working system, contains 80 PEO chains. The short time MSD increases slightly with the system size, whereas in the diffusive regime no significant system size effect is visible.

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ANION DIFFUSION In contrast to Li-cations, TFSI-anions are weakly bound to polymer chains. Nevertheless, the diffusion coefficient of TFSIanions DTFSI varies similarly with N as Li-cations, as displayed in Figure 5a. DTFSI values for all temperatures can be fitted by

IONIC CONDUCTIVITY The ionic conductivity σ as a function of N can be estimated considering the N-dependence of DLi (eq 5) and DTFSI (eq 6) via D

DOI: 10.1021/jp512734g J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B σ≈

e 2n [DM + Dcom + DTFSI] kBTV

estimated σ collapses with both the experimental results except for 363 K, where the estimation is approximately 25% too low. The collapse indicates that systems with long polymer chains behave as strong electrolytes. In the strongly N-dependent regime, the deviation between our estimated values and experiments is large for 363 and 333 K. Interestingly, at 333 K, the two sets of experimental results also significantly vary among themselves. We speculate that the conductivity data of Teran et al. reflect stronger cation−anion correlations as their systems contain a larger fraction of ions (EO/Li = 20:1.7) compared to our system (20:1) and Hayamizu et al. (20:1,12,13). At 303 K, the deviation between the estimated and the experimental values becomes small, which suggests weaker correlations between counterions with lowering temperature. This phenomenon (i.e., the decrease of correlation between counterions with increasing N and for decreasing T) can be deduced from the radial distribution function between Li-cations and oxygen atoms of TFSI-anions. Here, the first peak drops both with increasing N and decreasing T (see Figure 6c), which implies that the number of TFSI-anions by which a Li-cation is surrounded inside its first neighboring shell decreases. Stolwijk et al.22 have reported a similar effect of temperature (i.e., the ion pair formation between Li-TFSI ions becomes negligible with decreasing T).

(7)

Here, e is the electronic charge, n is the total number of ions (54), kB is the Boltzmann constant, V is the system volume. Note that σ, estimated by the above equation, can be directly compared with the experimental results (measured using some specific electrochemical techniques19,20) only for the strong electrolyte systems where the dynamics of the different ionic species is uncorrelated. If an ion is coordinated with its counterion they transport mass and not charge. Therefore, σ estimated from eq 7 will be higher than the experimental values. It is obvious from eq 7 that σ has a similar N-dependence as DLi and DTFSI. We estimate the individual contribution of the three diffusion terms in eq 7 to σ (reported here as σM, σcom, and σTFSI) shown in Figure 6a. σ strongly drops with N until

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CONCLUSIONS In this study we have shown a crossover behavior of the cation and anion diffusion. Both strongly drop with increasing chain length and saturate around N = 100. The cation diffusion can be characterized by the Rouse-based transport model, which expresses DLi as a sum of DM and Dcom. In the strong Ndependence regime DLi ≈ Dcom, which varies as N−1.85, and in the N-independent regime DLi ≈ DM, which, for large N, can be expressed by the three time scales: τ1 (∝N2), τ2 (∝N2), and τ3 (∝N0). Both the simulation and experimental diffusion data validate the model prediction for different temperatures. However, these scaling relations are strongly violated for small N. Because DM does not matter for the cation-diffusivity in this regime, these deviations do not matter in practice. DTFSI data follows a fitting equation A0 + A1N−1.22, although TFSIanions are considered as free ions for such systems. Possible sources of correlation have been indicated above. The ionic conductivity as a function of N, which is estimated from the N-dependent cation and anion diffusion functions, shows a similar crossover behavior as expected, where TFSI diffusion has the major contribution. However, the direct comparison with the experimental data reveals that in the strong N-dependent regime our prediction is too high. This suggests a strong anion−cation correlation in order to reduce the conductivity. These correlations become weaker with increasing N and decreasing T as clearly seen from the structural data. As a consequence, the assumption of a strong electrolyte works best for low temperatures and large chain lengths. In summary, we have extended our previous work to the limit of short chains and included the mobility of the anions. This allows us to obtain a complete picture of diffusivity and conductivity for all chain lengths in a broad temperature regime.

Figure 6. (a) σ versus N estimated using eq 7 along with the contributions of Li-cations and TFSI-anions for T = 363 K. (b) σ versus N for different Ts: our estimation (dot-dashed lines), the fitting equation of Teran et al.3 (solid lines), experiments12,13 (open symbols). (c) Radial distribution functions (RDF) between Li-cations and oxygen atoms of TFSI-anions for N = 5 (solid lines) and N = 54 (dashed lines).

σTFSI starts to saturate at around N = 100; here σcom and σTFSI both contribute to σ. When N ≳ 100, the effect of N on σ becomes extremely weak. In this regime, σM contributes by 20% to the total conductivity. Overall σTFSI always dominates except for extremely small N values. For Li-metal batteries, the cation transport is of major interest and it is often measured as the transference number,21 which is DLi/(DLi + DTFSI) . We find that the transference number drops from 40% to 15% when comparing N = 5 and N = 54, respectively. Naturally, the conductivity decreases with deceasing temperature. Teran et al.3 suggested the fit σ(T) = 590exp[−4660/ T]+ 23(0.0073−2.3/T) /N, which can describe their experimental results for T ∈ [330, 368]K. In Figure 6b we present σ versus N for T = 363 K, 333 K calculated using the fitting equation (solid lines) and for T = 363 K, 333 K, 303 K estimated using eq 7 (dot-dashed lines). In addition, we show the experimental data for T = 333 K, 303 K (open symbols) reported in.12,13 In the nearly N-independent regime our

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. E

DOI: 10.1021/jp512734g J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Notes

(20) Joost, M.; Kunze, M.; Jeong, S.; Schönhoff, M.; Winter, M.; Passerini, S. Ionic mobility in ternary polymer electrolytes for lithiumion batteries. Electrochim. Acta 2012, 30, 330−338. (21) Hayamizu, K.; Aihara, Y.; Nakagawa, H.; Nukuda, T.; Price, W. S. Ionic Conduction and Ion Diffusion in Binary Room-Temperature Ionic Liquids Composed of [emim][BF4] and LiBF4. J. Phys. Chem. B 2004, 108, 19527−19532. (22) Stolwijk, N. A.; Wiencierz, M.; Heddier, C.; Kösters, J. What Can We Learn from Ionic Conductivity Measurements in Polymer Electrolytes? A Case Study on Poly(ethylene oxide) (PEO)-NaI and PEO-LiTFSI. J. Phys. Chem. B 2012, 116, 3065−3074.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Diddo Diddens for setting up the initial simulations using Amber package and Nitash Balsara, and Volker Lesch for their helpful insights. This work is conducted in connection with HI-MS (Helmholtz-Institute Münster).

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REFERENCES

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DOI: 10.1021/jp512734g J. Phys. Chem. B XXXX, XXX, XXX−XXX