Charge Transport in Nanostructured PS–PEO–PS Triblock Copolymer

Apr 10, 2014 - Smith , T. W.; Abkowitz , M. A.; Conway , G. C.; Luca , D. J.; Serpico , J. M.; Wnek , G. E. Macromolecules 1996, 29, 5042– 5045. [AC...
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Charge Transport in Nanostructured PS−PEO−PS Triblock Copolymer Electrolytes R. Bouchet,*,† T. N. T. Phan,‡ E. Beaudoin,‡,∥ D. Devaux,§,‡ P. Davidson,∥ D. Bertin,‡ and R. Denoyel§ †

Laboratoire d’Electrochimie et de Physico-chimie des Matériaux et des Interfaces (LEPMI) UMR CNRS 5279, Grenoble Universités, 1130 rue de la piscine, 38402 St. Martin d’Hères, France ‡ Institut de Chimie Radicalaire - UMR 7273, Chimie Radicalaire Organique et Polymères de Spécialité, Aix-Marseille Université, Campus Saint Jérôme, Case 542, 13397 Marseille, Cedex 20, France § MADIREL - UMR 7246, Matériaux divisés, interfaces, réactivité, électrochimie, Aix-Marseille Université, Campus saint Jérôme, Bât. MADIREL, 13397 Marseille, Cedex 20, France ∥ Laboratoire de Physique des Solides, UMR 8502 CNRS, Université Paris-Sud, Bâtiment 510, 91405 Orsay, Cedex, France S Supporting Information *

ABSTRACT: Nanostructured block copolymer electrolytes (BCE) are currently attracting widespread interest for applications in rechargeable lithium batteries. In order to investigate the influence of the composition, and therefore that of confinement, on the conductivity, a series of triblock PS− PEO−PS copolymers, with three linear PEO blocks of molecular weights 9, 10, or 35 kg mol−1 and with PEO weight fractions varying from 36% to 75%, were synthesized and doped with LiTFSI. Measurements by impedance spectroscopy of the conductivity show that it increases with PEO molecular weight, which is quite counterintuitive. To explain this phenomenon, the conductivity of the BCE has been modeled using three factors: (1) the conductivity of bulk PEO, (2) the topology of the PEO percolating network, described by the tortuosity parameter, and (3) the influence on the tortuosity of the effective volume fraction of the PEO phase useful for the conduction, taking into account a “dead zone” excluded from ionic transport, at the PS/PEO interface. In this approach, by analogy with porous materials and in contrast with previous work, the tortuosity is not kept constant but depends on the PEO volume fraction effectively useful for charge transport. The thickness of the dead zone, 4−5 EO units (∼1.6 nm), is the same as that of the exclusion zone for crystallization previously reported. This value does not depend either on the PEO molecular weight (from 9 to 35 kg mol−1) or on the EO/Li ratio (from 20 to 30). The absence of both conduction and crystallization in the excluded region could be due to the low mobility of the PEO chains in this zone. Consequently, the conductivity of BCE increases with PEO molecular weight because the proportion of the excluded zone becomes smaller as the PEO molecular weight increases. This model therefore provides a fairly precise description of the ionic conductivity of BCE used in lithium battery applications.

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INTRODUCTION

poly[oligo(oxyethylene) methacrylate] as central block and polystyrene (PS) as end block, exhibits a high ionic conductivity of 10−4 S cm−1 at room temperature for PEO volume fractions above 85%. However, its mechanical properties were poor. These two groups actually worked on fairly similar materials, but Mayes et al. studied diblock copolymers whereas Kanamura et al. studied triblock copolymers. Other experimental12,13 and theoretical14,15 studies also show that ionic conductivity in polymer electrolytes depends on chain mobility and should therefore decrease with increasing molecular weight. However, unexpectedly, Balsara et al. recently observed the opposite trend in a series of nearly symmetrical lamellar PS−PEO diblock copolymers doped with

The use of poly(ethylene oxide) (PEO)-based polymer materials with microsegregated structures is presently being investigated worldwide for applications as solid electrolytes in rechargeable lithium batteries. PEO is particularly appealing in this respect because its ether coordination sites dissociate lithium salts and because the flexibility of PEO chains assists the ionic transport. The design of these microsegregated electrolyte materials has been based on different architectures where PEO can be found as a linear block1−5 or as oligomeric side chains in block copolymers6−9 or grafted to a macromolecular backbone.10,11 On the basis of a variety of copolymers, Mayes and co-workers8−11 showed that block copolymer electrolytes (BCE) with enhanced conductivity are obtained when the conducting block is attached to a soft rubbery block rather than to a hard glassy block. Nevertheless, Kanamura et al.6,7 recently reported that a BAB triblock copolymer, consisting of © 2014 American Chemical Society

Received: February 25, 2014 Revised: April 2, 2014 Published: April 10, 2014 2659

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Table 1. Molecular Weights and Compositions of the PS-b-PEO-b-PS Block Copolymers sample

Mn,PEOa (kg mol−1)

Mn,PSb (kg mol−1)

Mw,totc (kg mol−1)

PDIc

PEO:PS (wt %)

εd (%)

SEO9S_74 SEO9S_56 SEO9S_37 SEO10S_68 SEO10S_62 SEO10S_42 SEO35S_77

9

3.2 7.1 15 4.5 5.8 13 10.7

12.2 16.1 24 14.5 15.8 23 45.7

1.10 1.13 1.15 1.19 1.17 1.56

74:26 56:44 37:63 69:31 63:37 43:57 77:23

76 59 40 71 65 46 79

10

35

a

Number molecular weight of PEO from the suppliers. bNumber molecular weight of PS from 1H NMR. cTotal number molecular weight of the triblock copolymers and PDI from SEC in THF (the PDI of SEO35S_77 could not be measured). dVolume fraction of the conducting domain (PEO + LiTFSI) of the BCE. 10 kg mol−1 were prepared by the NMP method; the synthesis process is shortly described in the Supporting Information. Different triblock copolymers were synthesized, and their characteristics are listed in Table 1. The PS-b-PEO-b-PS macromolecules were labeled “SEOxS_Y”, where x corresponds to the number-average molecular weight, Mn, of PEO in kg mol−1 and Y corresponds to the wt % of PEO in the copolymer. Preparation of Block Copolymer Electrolytes (BCE). The BCE films were prepared by the solvent casting method. The block copolymers listed in Table 1 and the LiTFSI salt were mixed with molar ratio EO/Li = 20 or 30 using dichloromethane/acetonitrile (50% v/v) solvent to form a homogeneous solution of 10% w/w BCE solution. This solution was cast on a Teflon Petri dish, and the solvent was allowed to evaporate slowly at 20 °C for 24 h. The as-formed film was first annealed in vacuum for 24 h at 50 °C and then placed in a glovebox filled with argon (H2O < 1 ppm, Jacomex) for 1 week. The final thickness of the BCE films was about 150 μm. The relevant parameter for the conductivity analysis is the volume fraction, ε, of PEO laden with LiTFSI. This parameter was calculated assuming an ideal mixture between PEO and LiTFSI and using the PEO, LiTFSI, and PS densities (1.11, 2.15, and 1.05 g cm−3, respectively).26 Physical Studies. Structural Studies of the BCE. The morphology of the BCE was examined by small-angle X-ray scattering (SAXS). All SAXS experiments were performed at room temperature at the bending magnet beamline BM02 (D2AM) of the European Synchrotron Radiation Facility (Grenoble, France) with X-ray energy set at 11 keV.27 The distance from sample to detector was 1.30 m. Acquisition times typically ranged from 5 to 15 s. The data were acquired using a CCD Peltier-cooled camera (SCX90-1300, from Princeton Instruments Inc., Trenton, NJ) with 1340 × 1300 pixels. Data preprocessing (dark current subtraction, flat field correction, radial regrouping, and normalization) was performed using the “bm2img” software developed at the beamline. Since all the SAXS patterns obtained were isotropic, they were azimuthally averaged in order to obtain curves of scattered intensity versus scattering vector modulus q (q = (4π sin θ)/λ where 2θ is the scattering angle and λ the wavelength). Electrical Conductivity Measurements. Symmetrical cells were assembled in a glovebox using platinum electrodes to measure the conductivity of the BCE. The active surface was precisely defined by inserting a polyethylene thin film (8 μm thick) with a circular hole between one electrode and the BCE.28 Cell assembly consisted of successive laminations at 80 °C under a roll pressure of 4 bar. The thickness of the electrolytes was checked throughout the lamination process as well as “post-mortem” after the electrochemical characterizations. Finally, the cells were thermosealed into “coffee bags” (from Protective Packaging Ltd.) in order to carry out the electrochemical experiments outside of the glovebox. The conductivity of the samples was measured by ac impedance spectroscopy using a Solartron 1260 frequency analyzer. The frequency was varied from 107 to 10−2 Hz, and the ac voltage amplitude was fixed at 10 mV. The spectra were modeled with an equivalent electrical circuit (see inset in Figure 2) and fitted using ZView software (Scribner Inc.). The sample temperature was varied

a lithium salt.1,2,16 They showed that the conductivity of the PS−PEO/Li electrolytes with [OE]/[Li] = 10−50 increases with molecular weight of the PEO block, in contrast with the behavior of the PEO homopolymers. Such a surprising behavior was nevertheless theoretically predicted recently.17 In this work, we investigate the ionic transport within linear PS−PEO−PS triblock copolymers based on PEO molecular weights of 9 and 10 kg mol−1 as middle block. Block copolymers with various PEO weight fractions have been prepared using either atom transfer radical polymerization (ATRP) or nitroxide mediated polymerization (NMP). A copolymer with a PEO molecular weight of 35 kg mol−1 was also investigated for comparison. Although the self-assembly properties of the BAB triblocks are very similar to those of the BA diblocks,18,19 the former materials are known to show better mechanical properties.20−22 This may essentially arise from “bridging” configurations of BAB macromolecules, in which the central A block is anchored to two distinct B domains. We obtained BCE by doping the copolymers with the LiTFSI salt. The periodicity of the microstructure and the morphology of the block copolymer electrolytes were examined by small-angle X-ray scattering (SAXS). The conductivity of the BCE was investigated by impedance spectroscopy and compared with the conductivity of a 35 kg mol−1 PEO homopolymer. We modeled the transport properties above the percolation threshold by taking into account three factors: (i) the conductivity of the PEO homopolymer, (ii) the topology of the conducting network described by the tortuosity parameter,23,24 and (iii) the influence on the tortuosity of the effective volume fraction of the PEO phase useful for the conduction, taking into account a “dead zone” excluded from conduction at the PS/PEO interface.

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EXPERIMENTAL SECTION

Materials. Triethylamine (TEA) (99%), 2-bromoisobutyryl bromide (98%), styrene (Sty) (99%), N,N,N′,N″,N″-pentamethyldiethylenetriamine (99%) (PMDETA), and copper bromide (98%) were all obtained from Aldrich and used as received. BlocBuilder (>99%), an alkoxyamine based on the nitroxide SG1 (N-tert-butyl-N-[1diethylphosphono-(2,2-dimethylpropyl)] nitroxide) and the 1-carboxy-1-methylethylalkyl moiety, was kindly provided by Arkema (France). Poly(ethylene oxide) (PEO) of molar masses 9 kg mol−1 (provided by batScap Company), 10 kg mol−1, and 35 kg mol−1 (Aldrich) were used as received. These molar masses were given by the suppliers. All solvents and other reagents were synthesis grade and used without further purification. Synthesis of the Triblock Copolymers. The PS-b-PEO-b-PS triblock copolymers based on PEO of molar masses of 9 and 35 kg mol−1 were prepared by the ATRP method. The synthesis procedure of these triblock copolymers was described elsewhere.25 The series of PS-b-PEO-b-PS triblock copolymers based on the PEO of molar mass 2660

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from 10 to 70 °C, by steps of 10 °C, thanks to a climatic chamber Voetch 4002. A stabilization time of 1 h was used at each step. The reproducibility of the data during heating and cooling was systematically checked.

(2qmax) and third (3qmax) scattering peaks prove the existence of a lamellar structure, which is not surprising for this almost symmetric copolymer. The second example (Figure 1b), SEO9S_37, is a copolymer with high PS content. Here again, a well-defined structure is observed. The first scattering peak appears at qmax = 0.35 nm−1 (D = 18.0 nm). The average domain size also spans about 17 periods. The positions of the second and third scattering peaks are 2qmax and √7qmax, respectively, which is characteristic of a hexagonal structure where the √3qmax peak is missing. The extinction of this peak is simply due to the very small value, at this specific scattering vector modulus, of the form factor of the PEO cylindrical core in the PS matrix for this particular composition of 35 vol % of PEO.32 Nevertheless, the observation of the (21) scattering peak, at √7q max , proves the 2-dimensional hexagonal morphology of this compound. Samples with high PEO content do not show well-defined structures by SAXS. For example, the diffractogram of SEO9S_74 (Figure 1c) is typical of that of samples with high PEO content. Only a very broad peak can be observed at qmax = 0.58 nm−1 (D = 10.8 nm). Although the period of the microstructure can be extracted with reasonable accuracy, due to the absence of higher order scattering peaks, on the basis of SAXS patterns alone, the morphology remains unclear. As previously discussed,25 our synthesis route leads to samples with high polydispersities, typically about 1.2−1.4 (see Table 1), but our previous TEM observations25 confirmed that PEOrich block copolymers have lamellar short-range order because of the crystalline character of PEO. For lamellar samples, the thickness of the PEO domains can be derived from the period D. We define RPEO (Table 2) as half

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RESULTS Microsegregation of the BCE. Figure 1 shows several examples of SAXS patterns of the as-produced BCE films. For

Table 2. Structural Data for the Block Copolymer Electrolytes composition

EO/Lia

D = 2π/qmaxb (nm)

morphologyc

RPEOd (nm)

ε(λ)e (%)

SEO9S_74 SEO9S_56 SEO9S_37 SEO10S_68 SEO10S_62 SEO10S_42 SEO35S_75

20 20 30 30 30 30 20

10.8 13.6 18.0 13.7 14.6 16.5 20.9

L L HC L L L L

8.4 8.2 7.1 9.8 9.5 7.6 16.5

46 35 20 46 42 25 64

a

Number of ethylene oxide units per Li+ ion. bDomain spacing from SAXS. cMorphology (L: Lamellae; HC: hexagonal compact). d Dimension of the PEO domains. eVolume fraction of the conducting phase taking into account the excluded region.

Figure 1. SAXS diffractograms of block copolymer electrolytes PS-bPEO-b-PS doped with LiTFSI salt. SEO9S series with PEO central block of 9 kg mol−1 and PS blocks: (a) 56, (b) 37, and (c) 74 wt % PEO. The arrows point to the higher order diffraction peaks.

each pattern, the position of the first scattering peak provides the characteristic period of the microstructure while weaker scattering peaks at larger q values, when present, help identifying the symmetry of the mesophase. Samples with PEO/PS proportion (volume fraction ratio) lower than 0.6 show well-defined microstructures. In contrast, samples with higher PEO/PS proportion do not show a well-defined structure, but their local order is known to be lamellar.25,29,30 For example, the SAXS pattern of the SEO9S_56 (Figure 1a) compound displays a first scattering peak at qmax = 0.46 nm−1, which corresponds to a period D = 13.6 nm. A domain size of ∼230 nm (around 17 periods) can be inferred from the peak width (corrected for instrumental resolution) by using the Scherrer formula.31 Moreover, the positions of the second

the thickness of the conducting domains made of PEO loaded with LiTSFI salt. From the molar ratio EO/Li and the densities of PEO and LiTFSI, the volume fractions of LiTFSI and PEO in these conducting domains can be calculated. Then, from the molar masses and densities of the PS and PEO blocks, the volume fraction of the whole conducting phase (PEO + LiTFSI) in the BCE can be determined. Finally, RPEO is obtained by multiplying this volume fraction with the lamellar period determined by SAXS. At this point, a comparison of RPEO in the absence and in the presence of salt could be interesting, but it cannot be made quantitative because of the PEO chain folding in the absence of salt.25 For example, for the SEO9S series, we observed a decrease of RPEO from typically 5 to 2.5 nm as the PS 2661

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proportion was increased. In contrast, in the presence of salt, RPEO remains fairly constant and significantly higher: RPEO ∼ 8 nm for the whole SEO9S series. This higher value actually agrees with the higher interaction parameter χeff in the presence of salt.33 Moreover, the constant value of RPEO means that there is little chain folding in the presence of salt. Table 2 summarizes the structural data obtained from the SAXS experiments. Ionic Conductivity. As an example of raw conductivity data, Figure 2 shows a typical impedance spectrum obtained at

Figure 3. Conductivities versus temperature for the triblock BCE with PEO 9 kg mol−1 central block: SEO9S_36 (■), SEO9S_54 (∗), and SEO9S_74 (□). The salt doping is OE/Li = 20. The data obtained for the homo PEO9 are also shown for comparison (▲). The error bars correspond to the dispersion of the data obtained for three samples.

which means that the PEO lamellae most probably percolate throughout the whole sample. Furthermore, the smooth evolution of the conductivity in the explored temperature range shows that the crystallization of PEO is strongly hampered in BCE, in agreement with the effect of PEO confinement within nanometric domains.25 Consequently, these two BCE present a higher conductivity than that of the PEO homopolymer below its melting temperature, i.e., below 40 °C. Figure 4 shows, in Arrhenius coordinates, the variation of the conductivity of BCE with similar proportions of PEO (∼75 wt %) but differing by the molecular weight of the central PEO block, i.e., 9 and 35 kg mol−1. The conductivities of the PEO homopolymers of 9 and 35 kg mol−1 are added for comparison.

Figure 2. Impedance spectrum (empty triangles) recorded at 25 °C on the SEO9S_74 with EO/Li = 20. The line corresponds to the best fit using the equivalent circuit given in inset.

25 °C for the SEO9S-74 BCE with EO/Li = 20. The spectrum is composed at high frequencies of a flattened loop corresponding to the electrical response of the electrolyte (modeled by the BCE resistance in parallel with a constant phase element), followed at low frequencies by an almost straight line characteristic of ion-blocking electrodes (modeled by a constant phase element). The electrical equivalent circuit is inserted in Figure 2. The inductance in series with a resistance in this circuit takes into account the cables and electrical contacts.28 Finally, the conductivity was deduced from the BCE resistance and the geometry of the cell. The temperature dependences of the ionic conductivities of the three BCEs based on the 9 kg mol−1 PEO are plotted in Arrhenius coordinates in Figure 3. The errors bars are estimated from the dispersion of the results obtained on three different cells. For comparison, the data obtained for the homopolymer PEO 9 kg mol−1 are also shown. As expected, the conductivity of the homopolymer dropped by 1 order of magnitude below 40 °C due to PEO crystallization. From this figure, the effect of BCE composition, at constant PEO molecular weight, can be assessed. As expected, the conductivity increases with PEO content. However, the conductivity of the 37 wt % PEO BCE is lower than those of the other compounds by 2 orders of magnitude. Such a low conductivity suggests that the PEO cylinders, in hexagonal morphology, do not percolate within the continuous PS matrix through the whole sample thickness (150 μm). Moreover, the large dispersion of the data obtained on three different samples with the same composition also suggests that the percolation threshold may lie around this composition. In contrast, the two lamellar samples with 56 and 74 wt % PEO, display not only a much higher conductivity but also very reproducible data,

Figure 4. Conductivities versus temperature for the BCE with two PEO of different molecular weights and approximately 75 wt % of PEO: SEO9S_74 (□) and SEO35S_77 (△). The salt doping is OE/Li = 20. The data obtained for the homo-PEO9 (■) and PEO35 (▲) are also shown for comparison. 2662

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show that the BCE samples are not aligned at all and that the lamellae are therefore randomly oriented. In this case, Sax and Ottino, based on geometrical considerations, proposed the value 3/2 for the tortuosity.23 Balsara et al. recently used the same approach to analyze their data on the conductivity of diblock BCE.2,16 In a further step, the dependence of the tortuosity on the volume fraction of the conducting phase, ε, should be introduced. Various phenomenological expressions for τ(ε) have been so far proposed to describe this dependence.44−49 Among these, Weissberg suggested48 the following equation for a random distribution of particles dispersed in a conducting solution:

As expected, above the melting temperature, the PEO homopolymer of 9 kg mol−1 presents a slightly higher conductivity than the 35 kg mol−1 homopolymer,12,13 whereas the conductivity of the BCE containing the highest molecular weight PEO is the highest of the two BCE. This counterintuitive result actually agrees well with the previous observations of Balsara et al. for diblock PS−PEO-based BCE.1,2,16

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DISCUSSION

Ion transport in complex heterogeneous media is an issue that arises in various fields such as cement chemistry, ion-exchange chromatography, electrochemistry, or geophysics.34−37 Ion transport in a composite structure depends on the bulk conductivity of the conducting phase, on the topological parameters of the conducting network, and on the properties of the region at the interface between the conducting and insulating phases. The structural parameters generally considered are the interface area, the size distribution of the conducting domains, and the conducting network topology that may be quantified by the tortuosity, τ.38 In this paragraph, we discuss this concept of tortuosity that was originally defined by Carman to explain the difference between modeled permeability values and experimental observations.39 τ is often used, as an adjustable parameter, in models for chromatography, membrane separation applications, or charge transport through composite electrodes.36 Carman proposed the determination of the tortuosity from conductivity measurements on porous media filled with an electrolyte of conductivity σ0. In an isotropic random composite medium, the conducting network available for current flow has a volume fraction ε that is exactly the porosity of the medium. However, it appears experimentally that all the conducting pathways are not used with the same efficiency for transport. The effective conductivity, σ, of a composite is therefore smaller than σ°ε by a factor 1/τ, with τ larger than 1:

σ=

σ 0ε τ

τ = 1 − p ln ε

(2)

The value of p is specific of the pore topology with values of p = 0.49 for spheres,47 p = 0.53 for cubes, and p = 0.86−3.2 for plates.49 This equation has been supported by both experiments24,50,51 and simulations.52 Because the ordered microstructure in our BCE only spreads over a small domain size (200−300 nm), the BCE can be considered as a random packing of small (spherical or cubic) grains with lamellar order. Then, by analogy to Weissberg’s work, the BCE tortuosity could be theoretically described by using eq 2 with p = 0.5, in first approximation. This theoretical value of the tortuosity can then be compared to that directly obtained from eq 1, using the measured conductivity and, for σ0, the conductivity of the PEO−LiTFSI complex of 35K PEO molecular weight (i.e., of large enough molecular weight). In Figure 5, the experimental tortuosity obtained from eq 1, for the BCE that percolate, is compared to the constant value of

(1)

Hence, 1/τ can be defined as the fraction of conducting volumes that have the same transport efficiency as the bulk electrolyte, the complementary fraction being perfectly inefficient. The tortuosity is a very general concept used in many different fields.38,40−43 By extension, in the present case of BCE composed of conducting PEO−LiTFSI domains dispersed within an insulating PS phase, eq 1 can be used to relate the effective conductivity σ of the BCE to the conductivity σ0 of the pure PEO−LiTFSI complex (with the same EO/Li ratio as the BCE) to ε the volume fraction of the PEO−LiTFSI conducting phase and to τ the tortuosity of the PEO network. Our experiments show that all the BCE under study whose conductivity can be reliably measured present exclusively a lamellar microsegregation. Therefore, the only structure that we discuss in the following is the lamellar one. For this geometry, two extreme cases can, a priori, be considered: the lamellae may either be parallel to the electrodes, leading to a nonpercolating conducting medium and an infinite tortuosity. On the contrary, the lamellae may be perpendicular to the electrodes; then the whole PEO−LiTFSI volume would conduct as the bulk phase and the tortuosity would be equal to 1. Actually, our samples lie in between these two extreme cases. Indeed, our SAXS results

Figure 5. Evolution of the tortuosity with the composition of the BCE, using eq 1 without exclusion (□) and eq 3 with an exclusion layer of thickness λ = 1.6 nm (■). In this representation, the best fit of the data by the model with constant tortuosity is shown as the horizontal dashed blue line whereas that by the model with variable tortuosity (Weissberg’s equation) is shown by the solid line.

3/2 and to the theoretical expression (eq 2 with p = 0.5). It is clear that neither the 3/2 value (dashed line) nor the theoretical law (solid line) does properly describe the experimental data (open symbols). However, the PEO−LiTFSI volume fraction useful for conduction may actually be smaller than expected due to the effect of the PS/PEO interface on the properties of PEO. Such argument, already suggested by our previous study on the melting of confined PEO,25 is also supported by very recent 2663

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1.5 ± 0.1 nm. This value is indeed very close to the one obtained here for the layer in which PEO is not efficient for conductivity. A slow relaxation process is known to occur near rigid PS domains, for PEO54,55 as for other flexible chains like PI.56 We suggest that the reduced mobility of the PEO chains close to the PS interface could account for both the absence of crystallization previously observed and the absence of conductivity observed here, in an excluded zone with a thickness of 4−5 EO units. The thickness of the excluded zone does not seem to depend on the EO/Li ratio either, at least in the EO/Li range (20−30) explored in this work. This means that the values of conductivity of the BCE and of the parent PEO homopolymer, respectively σ and σ0, depend on the EO/Li ratio in a similar manner. This is actually not so surprising since the ion transport in the BCE occurs within the PEO moieties. Finally, as already noted by Balsara et al. for diblock BCE,16 the existence of an excluded zone can explain the unexpected behavior of the BCE compared to the homopolymers, i.e., the increase of BCE conductivity upon increasing PEO molecular weight. Indeed, for a similar BCE composition (a given PEO/ PS ratio) the proportion of the excluded zone in a high-Mw PEO-based BCE will be smaller compared to a low-Mw PEObased BCE, leading to a higher proportion of conducting PEO together with a lower tortuosity. The impact of these two parameters (excluded volume and tortuosity) completely overcomes the small dependence of the conductivity on molecular weight in homo-PEO above the entanglement weight.

theoretical predictions17 and by the results of Balsara et al. that have shown that the electrical charges are segregated to the center of the lamellae in diblock BCE.53 Assuming that the covalent anchoring of PEO moieties at each end to a hard PS domain (bridge or loop conformations) hampers segmental mobility near the PEO/PS interface, we propose that a part of the PEO chains near the PS wall does not contribute to the overall charge transport process. From the thickness λ of the excluded region at the interface, one can calculate ε(λ) = (RPEO − 2λ)/D, the volume fraction of PEO−LiTFSI useful for conduction, and finally the tortuosity from σ=

σ 0ε(λ) τ (ε )

(3)

The tortuosities obtained with eq 3, taking into account the excluded PEO interfacial volume, are given in Figure 5. A good description by the theoretical curve is reached for λ = 1.6 ± 0.1 nm, which corresponds to 4−5 EO segments. Even when an excluded region is introduced, the constant τ = 3/2 value (dotted straight line in Figure 5) cannot account for the experimental dependence of the tortuosity on the volume fraction of the conducting phase, ε. With our model, any value of p from 0.5 to 3 could be used to fit the data by altering the value of λ. For example, the rather extreme value p = 3 can fit the experimental data without the need of any excluded zone, but this would lead to tortuosity values reaching up to τ ∼ 4. This would be the sign that the lamellae are essentially aligned parallel to the electrodes. However, SAXS data prove that the lamellar domains are completely isotropically distributed. In this case, previous work has shown that the value of the tortuosity is τ = 3/2.23 Then, the model keeps physical meaning only for values of tortuosity close to 3/2, which is the case only if p remains close to 0.5. The thickness of the excluded zone seems independent of the molecular weight in the range of the PEO molecular weight explored here (from 9 to 35 kg mol−1). The values of ε are reported in Table 2 for λ = 1.6 nm. Figure 6 displays a schematic representation of the local organization that shows the exclusion region at the interface between the PEO and PS domains. Recently, we proved the existence, at the PEO/PS interface, of a PEO layer that does not crystallize.25 We found that the thickness of this layer corresponds to 4−5 monomer length, i.e.,

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CONCLUSION We propose here an approach of the ion transport through PEO-based BCE that takes into account not only the tortuosity of the PEO network but also its dependence on the effective volume fraction of conducting PEO, due to a “dead zone” at the interface between domains. This dead layer has the same thickness as the layer in which PEO crystallization is hampered, i.e., 4−5 EO units. It could arise from a reduced PEO chain mobility at the PEO/PS interface. Moreover, we also show that the tortuosity law proposed by Weissberg, which explicitly depends on the proportion of conducting phase, gives a fairly precise description of the ion transport in BCE. Furthermore, this model is independent of the EO/Li ratio in the domain of concentration explored (which is the most widely used in practice). We believe that our approach provides a useful tool to analyze the properties of BCE in terms of their ionic conductivity and will help to design more efficient BCE for battery applications.

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ASSOCIATED CONTENT

S Supporting Information *

Synthesis by NMP of PS-b-PEO-b-PS triblock copolymers. This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

Figure 6. Schematic representation of the microstructure of the BCE. Solid wavy lines represent the PS chains whereas the dashed wavy lines show the PEO chains. The vertical dashed lines show the PS−PEO interface. The hatched red areas show the zone devoid of ionic transport, and the blue areas are PEO crystallites.

*E-mail [email protected] (R.B.). Notes

The authors declare no competing financial interest. 2664

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ACKNOWLEDGMENTS The authors gratefully acknowledge the ANR (Agence Nationale de la Recherche, “STOCK-E-09_03” Program, COPOLIBAT Project) for financial support.

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dx.doi.org/10.1021/ma500420w | Macromolecules 2014, 47, 2659−2665