On the extraction of ion association data and transference numbers from ionic diffusivity and conductivity data in polymer electrolytes

Electrochimica Acta 102 (2013) 451–458

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On the extraction of ion association data and transference numbers from ionic diffusivity and conductivity data in polymer electrolytes Nicolaas A. Stolwijk a,∗ , Johannes Kösters a , Manfred Wiencierz a , Monika Schönhoff b a b

Institut für Materialphysik, University of Münster, Wilhelm-Klemm-Str. 10, 48149 Münster, Germany Institut für Physikalische Chemie, University of Münster, Corrensstr. 28/30, 48149 Münster, Germany

a r t i c l e

i n f o

Article history: Received 30 January 2013 Received in revised form 2 April 2013 Accepted 3 April 2013 Available online 17 April 2013 Keywords: Poly(ethylene oxide) Nernst–Einstein equation Charge diffusivity Ion pairing PFG-NMR Radiotracer

a b s t r a c t The degree of ion association in polymer electrolytes is often characterized by the Nernst–Einstein deviation parameter , which quantifies the relative difference between the true ionic conductivity directly measured by electrical methods and the hypothetical maximum conductivity calculated from the individual ionic self-diffusion coefficients. Despite its unambiguous definition, the parameter  is a global ∗ quantity with limited explanatory power. Similar is true for the cation transport number tcat , which relies on the same ionic diffusion coefficients usually measured by nuclear magnetic resonance or radiotracer methods. Particularly in cases when neutral ion pairs dominate over higher-order aggregates, more specific information can be extracted from the same body of experimental data that is used for ∗ the calculation of  and tcat . This information concerns the pair contributions to the diffusion coefficient of cations and anions. Also the true cation transference number based on charged species only can be deduced. We present the basic theoretical framework and some pertinent examples dealing with ion pairing in polymer electrolytes. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Over the last decades solid polymer electrolytes (SPEs) and room temperature ionic liquids (ILs) have been extensively investigated for applications in batteries, electrochemical solar cells, smart windows, etc. [1,2]. In this context, optimal ion transport properties require not only high ionic mobilities but also high concentrations of charged or ‘free’ ions. Therefore, ion association has been a crucial issue in most studies of mass and charge transport in SPEs and ILs. The most elementary form of ion association is pair formation. In particular, neutral ion pairs contribute to the diffusivity of the cation and the anion but they do not contribute to the electrical conductivity . Also the occurrence of higher-order clusters, such as triplets, diminishes the number of charged species, and moreover, tends to reduce the average mobility of the charge carriers. To quantify the impact of ion association, or more specifically ion pairing, one usually calculates the Nernst–Einstein deviation parameter  [3]. This so-called delta parameter is defined by the relation  = diff (1 − ) ,

∗ Corresponding author. Tel.: +49 2518339013. E-mail address: [email protected] (N.A. Stolwijk). 0013-4686/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.electacta.2013.04.028

(1)

where  diff is the hypothetical overall conductivity derived from the joint individual contributions of all ionic diffusion coefficients ∗ and D∗ ) obtained either by pulsed field gradient nuclear mag(Dcat an netic resonance (PFG-NMR) or radiotracer diffusion (RTD) [3]. For monovalent electrolytes  diff is written as diff =

e2 Cs ∗ ∗ ), (Dcat + Dan kB T

(2)

where kB denotes the Boltzmann constant, T is temperature, e is elementary charge, and Cs is the known total salt concentration (number density of molecules). It should be emphasized that these diffusion coefficients, marked by an asterisk, are concentrationweighted averages over all cation-containing and anion-containing mobile species, respectively. For full dissociation of the salt, all ions contribute to charge transport, so that  = 0 holds under the provision that ionic motion occurs in an uncorrelated manner. Conversely, in the case of strong cation–anion pairing the concentration of charge carriers becomes very low, which is reflected by  approaching the value of 1. In spite of its usefulness as a global indicator of the degree of ion association, the true meaning of the empirically defined parameter  remains obscure. Nevertheless, in many studies on electrolytes, the value of  serves as a major characteristic. However, in cases where neutral ion pairs dominate over higher-order aggregates, the meaning of  can be specified. For instance, this situation occurs for polymer electrolyte with a sufficiently low salt concentration.

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Another global characteristic of electrolytes is the cation transport number given by ∗ tcat =

∗ Dcat ∗ + D∗ , Dcat an

(3)

which relates the cation diffusivity to the sum of the ion diffu∗ and D∗ will generally contain contributions sion coefficients. Dcat an ∗ must be distinof ion pairs and larger aggregates. Therefore, tcat guished from the cation transference number t+ , which only refers to charged species. It is generally believed, however, that transference numbers cannot be directly determined from conductivity and diffusion data. In this work, we will derive an expression for the Nernst–Einstein deviation parameter  under conditions where ion association is governed by the formation of neutral pairs. In addition, the pair contributions to the diffusion coefficient of ∗ . Within the same cations and anions are deduced from  and tcat simple theoretical framework, the effective diffusivities of the free ionic species and ion pairs are calculated either directly from the measured conductivity and diffusion coefficients or indirectly ∗ . It will be also shown, how t can be obtained from using  and tcat + the same experimental data. Results will be given for polymer electrolytes based on poly(ethylene oxide) (PEO) with salts of different type and concentration including the alkali metal iodide NaI and the ionic liquids 1-ethyl-3-methyl-imidazolium iodide (EMImI) or 1-ethyl-3-methyl-imizadolium bis(trifluoromethylsulfonyl)imide (EMImTFSI). 2. Charge diffusivity and Haven ratio The definition of  diff in Eq. (2) involves the conversion of diffusivity into a quantity with the dimensions of a conductivity. This is based on the Nernst–Einstein equation, which in its reversed form, transforms the measured conductivity to the charge diffusivity D , i.e., kB T D = . Cs e2

(4)

It should be emphasized that the denominator contains the known total salt concentration Cs instead of the a priori unknown concentration of free (dissociated) ions. With this definition, D is closely related to the molar conductivity given by  = /cs , where cs is the same salt concentration in molar units. Eq. (1) compares mass and charge transport in units of conductivity. Alternatively, such a comparison can be made on the diffusivity scale by means of the Haven ratio, HR given by [4,5] HR =

∗ + D∗ Dcat an . D

(5)

Combining this expression with Eqs. (2) and (4) it readily follows that the identity relationship  1 = diff HR

(6)

must hold. The ratio / diff = imp /NMR has been termed the ‘ionicity’ [6] or the ‘dissociation degree’ [7] of the electrolyte under investigation. Here, the indices of the molar conductivity  refer to ‘impedance measurements’ and ‘PFG-NMR analysis’, respectively. Using Eqs. (1), (5) and (6) we obtain for the delta parameter the expression 1 D =1− =1− ∗ ∗ , HR Dcat + Dan

salt. Higher values of HR (>1) reflect an increased degree of association. Approaching the limit of complete association, HR diverges to infinitely high values.

(7)

which only contains diffusion coefficients as parameters. Also the Haven ratio can be used as a measure of ion association. For HR = 1 we have  = 0, which implies full dissociation of the

3. Predominance of ion pairing 3.1. Basic equations In SPE-complexes of moderate or low salt concentration, the formation of cation–anion pairs according to the reversible reaction cat+ + an−  pair

(8)

appears to be the predominant ion-aggregation process [8–10]. Thus, neglecting higher-order clusters, the ion-specific diffusion ∗ and D∗ ) may be coefficients measured by PFG-NMR or RTD (Dcat an written as ∗ eff eff Dcat = Dcat + + Dpair ≡ (1 − rpair )Dcat+ + rpair Dpair

(9)

∗ eff eff = Dan Dan − + Dpair ≡ (1 − rpair )Dan− + rpair Dpair ,

(10)

where rpair = Cpair /Cs denotes the relative concentration of neutral ∗ and D∗ represent the total pairs (or pair fraction). Consistently, Dcat an diffusion coefficient of cations and anions, respectively, averaged over all relevant mobile species. On the contrary, the plain D symbols (Dcat+ , Dan− , and Dpair ) designate the true (or inherent [11]) diffusivities of the individual species (cat+ , an− , and pair), which are related to the respective mobilities. Furthermore, D’s labelled with eff , Deff , and Deff ) are so-called effective the superindex ‘eff’ (Dcat + an− pair diffusion coefficients which also involve the relative concentration of the respective species (rpair or 1 − rpair ). Thus, in this notation D is given by the sum of the contributions of charged species, i.e., eff eff D = Dcat + + Dan− ≡ (1 − rpair )Dcat+ + (1 − rpair )Dan− .

(11)

Eqs. (9)–(11) apply to spatially homogeneous SPEs with uniform charge distributions in thermal equilibrium. The various selfdiffusivities entering the model characterize the purely statistical (entropy-driven) diffusion and thus do not contain contributions due to thermodynamical (enthalpic) or electrical driving forces. Although the measurement of D involves electrical fields, the small amplitudes and the high ac frequencies employed warrant negligible deviations from the ideal reference state. It should be mentioned that the evaluations in this paper will be eff restricted to the level of ‘effective diffusivities’. To decompose Dpair eff in 1 − r in rpair and Dpair or Dcat + pair and Dcat+ further assumptions about the temperature dependence of the pair fraction and the true diffusivities are necessary [12,13]. Thus, ignoring the right-hand sides of Eqs. (9)–(11), we have 3 equations with 3 a priori unknowns, eff , Deff , and Deff . Solving this equation system yields the viz., Dcat + an− pair effective diffusivities as a function of the measured data, i.e., eff Dcat + =

1 ∗ ∗ + D ) (D − Dan 2 cat

(12)

eff Dan − =

1 ∗ ∗ (D − Dcat + D ) 2 an

(13)

eff Dpair =

1 ∗ ∗ − D ). (D + Dcat 2 an

(14)

Altogether, the effective diffusivity of each ionic species can be obtained from the experimental data in a straightforward manner. Conceiving a potentially strong role for ion pairs, it is useful to specify the relative pair contributions to cation and anion transport

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individually. To this aim, we introduce by using Eqs. (9) and (10) cat and f an , i.e., the fractional pair components fpair pair cat fpair

=

an fpair =

eff Dpair

=

eff + Deff Dcat + pair eff Dpair eff + Deff Dan − pair

=

eff Dpair ∗ Dcat

eff Dpair

.

∗ Dan

eff Dcat + eff + Deff Dcat + an−

eff Dcat +

=

D

.

(16)

an = 1 − f an . Then, instead of using Eqs. (12)–(14) and similarly fan − pair the effective diffusion coefficients of unpaired and paired ionic species may be obtained from

(17)

note, however, that Eq. (17) may be reduced to t+ = Dcat+ /(Dcat+ + eff and Deff (cf. Eq. (11)) Dan− ), since the joint factor 1 − rpair of Dcat + an− cancels in numerator and denominator.

3.2. Meaning of the Nernst–Einstein deviation parameter Within the present scenario, the significance of the parameter  ∗ + D∗ can be evaluated. As a first step, the division of Eq. (14) by Dan cat yields with the aid of Eq. (7) the expression eff Dpair ∗ (Dan

∗ )/2 + Dcat

,

(18)

eff . As a second which relates  to the effective pair diffusivity Dpair step, we introduce the mean fractional pair component fpair  as the cat and f an , i.e., weighted average of fpair pair ∗ cat ∗ an fpair  ≡ tcat fpair + tan fpair =

eff Dpair ∗ + D∗ )/2 (Dcat an

.

(19)

∗ given by Eq. (3) This equation contains the transport numbers tcat ∗ = 1 − t ∗ . Comparing Eq. (18) with Eq. (19) reveals the idenand tan cat tity relationship

 = fpair  .

(20)

Thus,  represents the mean pair contribution to the mean ionic diffusivity, which involves averaging over cations and anions. Simple algebra shows that the weighted arithmetic mean expressed in Eq. (19) equals the harmonic mean given by 1 fpair 

=

1 2



1 cat fpair

+

1 an fpair



,

(21)

which involves the reciprocal values of the fractional pair components. The (seeming) absence of weight factors in Eq. (21) relates to the fact the number of cations and anions are equal for the considered electrolytes. Eq. (21) implies that  = fpair  takes low values if one of the fractional pair components is much smaller than unity. Several studies on SPEs and ILs report data for transport numcat and f an . bers and delta parameters but do not evaluate fpair pair However, these ion-specific pair contributions can be readily calculated from Eqs. (3), (15), (16) and (18) as cat fpair =

 ∗ , 2tcat

an fpair =

 ∗ ) . 2(1 − tcat

eff Dcat +

(15)

Thus, in principle, t+ is a fractional component on the level of the cat in Eq. (15). We effective diffusivities and hence related to, e.g., fpair

=

For completeness we note that the contribution of free cations (cat+ ) to the overall cation diffusivity is given by cat fcat + ≡

cat and f an , which may be also taken as Hence, the calculation of fpair pair pair-related transport numbers of cations and anions, respectively, can be done exclusively on the basis of experimental data. The cation transference number t+ characterizes the share of cations in overall charge transport. In the present terminology, t+ can be written as

t+ =

453

(22)

∗ Dcat

cat = 1 − fpair ,

eff cat ∗ Dcat + = (1 − fpair )Dcat ,

(23)

eff an ∗ Dan − = (1 − fpair )Dan

(24)

and eff cat ∗ an ∗ Dpair = fpair Dcat = fpair Dan ,

(25)

respectively. In the following, the equations derived or recapitulated in this section will be used for the characterization of three pertinent saltin-polymer complexes. 4. Examples of ion association in polymer electrolytes 4.1. Introductory remarks The present article focusses on theoretical concepts of ion transport and it elucidates the meaning of the chemo-physical parameters involved. The usefulness of our approach is demonstrated with the aid of three different PEO–salt complexes that were recently investigated at our laboratories, i.e., PEO500 NaI [14,15], PEO60 EMImI (cf. a recent report on a closely similar system [12]), and PEO20 EMImTFSI [16]. These complexes differ not only in the type of salt, either inorganic salt or ionic liquid, but also by the salt concentration, as indicated by the given EO-to-salt molar ratios. The choice of the prototype electrolytes was guided by the idea to present a great diversity in data constellations leading to mutual disparities in ion association and transference numbers. Potential application in batteries or other devices is not an important viewpoint in this basic study. The experimental methods comprise electrochemical impedance spectroscopy (EIS), PFG-NMR, and radiotracer diffusion (RTD). For experimental details, the reader is referred to a very recent paper [16] and to other publications by our group [12,14,15,17]. 4.2. PEO500 NaI The PEO–NaI system has been extensively studied in earlier work of our group [14,15]. Fig. 1a shows the experimental data of the PEO500 NaI complex, which is characterized by a very low salt concentration given by the molar ratio EO:NaI = 500:1. Specifi∗ , D∗ , and D are displayed over a cally, the diffusion coefficients DNa  I wide temperature range (70–200 ◦ C) within the amorphous phase ∗ and D∗ result from RTD experiments region of the electrolyte. DNa I 22 using the isotopes Na and 125 I [15], while D was deduced from EIS measurements [14] using the Nernst–Einstein equation (4). ∗ (triangles) at all It is seen in Fig. 1a that DI∗ (squares) exceeds DNa temperatures. This agrees with the common observation that the anion is faster than the cation in conventional SPEs containing alkali metal salts. Furthermore, it is striking that the D data (circles) exhibit a much weaker T-dependence. At lower temperatures D ∗ and even approaches the virtual extrapolation of is larger than DNa ∗ the DI data. By contrast, at higher temperatures D increasingly falls below both tracer diffusivities leading to a difference in magnitude ∗ . This behavior involves by down to a factor of 1/6 with respect to DNa ∗ near 100 ◦ C. a cross-over of D and DNa Assuming the occurrence of unpaired and paired ions but no formation of higher-order clusters, the effective diffusivity of each species can be obtained by Eqs. (12)–(14). To this aim, spline curves

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Most diffusion studies on polymer electrolytes do not utilize eff , Deff , and Deff from the experthe possibility to extract Dcat + an− pair imental data. Instead, the deviation parameter  is commonly evaluated either by Eqs. (1) and (2) or, equivalently, by Eq. (7). Using the latter equation we obtain for PEO500 NaI the solid curve annotated by  in Fig. 1b. It is seen that  varies from about 0.43 at low T to 0.92 at high T. These values signify a moderate to high degree of ion association with a strong temperature dependence but they do not allow for a discrimination between cations and anions. ∗ is In published work, often the cation transport number tcat ∗ was calculated with reported in addition to . For PEO500 NaI, tcat the aid of Eq. (3) leading to the lower solid curve in Fig. 1b. We find a moderate temperature dependence with values between 0.24 (low ∗ < 0.5, these data are consistent with T) and 0.46 (high T). Since tcat ∗ the observation that DNa < DI∗ over the entire range. ∗ , f cat and f an are obtained from Having determined  and tcat pair pair Eq. (22) and displayed in Fig. 1b as dashed curves. The difference between both data is significant. The fractional pair component to cat ) amounts to more than 90% for all temperatures, Na transport (fpair

T/°C 190 -6

70 PEO500NaI

eff Dpair

2

Diffusivity/cm s

–1

10

130

eff

-7

10

Dan–

*

DI

*

DNa Dσ (a)

-8

10

eff

Dcat+

1.0

Numbers, Fractions

cat

0.8

Δ an fpair

0.6 0.4

fpair

*

tcat

0.2

(b)

t+ 0

2.2

2.4

2.6

(1000/T)/K

2.8

3.0

–1

Fig. 1. (a) Measured (symbols) and deduced (lines) diffusion coefficients characterizing ion transport in PEO500 NaI as a function of reciprocal temperature, as indicated. The experimental data were taken from earlier work [14,15]. The effeceff eff eff − 0 tive diffusivities of Na+ (Dcat (Dan + ), I − ), and NaI pairs (Dpair ) were deduced by using Eqs. (12)–(14). (b) Additional parameters derived from the measured diffusion ∗ coefficients comprising the cation transport number tcat , the transference number cat an and fpair . t+ , the deviation parameter , and the fractional pair components fpair

eff almost matches which complies with the above finding that Dpair an ∗ DNa (cf. Fig. 1a). In contrast, fpair adopts values between 0.3 and 0.86, thus showing a distinct dependence on temperature. Consequently, DI∗ is made up of substantial contributions from both free I− ions and bound NaI0 pairs. We note that the three effective diffusivities ∗ , and one of the experimental diffusion can be retrieved from , tcat coefficients. Our approach also allows for the extraction of the cation transference number t+ , i.e., by using Eq. (17). The result is given by the low-lying dash-dotted line in Fig. 1b, which extends from 0.04 at low T to 0.03 at high T. The great disparity with the cation transport number is easily recognized. Hence, for most practical applications PEO500 NaI suffers from at least three disadvantageous characteriscat , and a low value tics: a low salt concentration, a high value of fpair of t+ . Nevertheless, this complex represents an interesting model system.

4.3. PEO60 EMImI were calculated to represent each set of experimental data [15] (see Section 5) but omitted in Fig. 1a for clarity. Then, the solid and dashed curves in Fig. 1a are produced by mathematical summation and subtraction. Some salient features can be summarized eff as follows: firstly, Dcat + is much lower than all other diffusion coefficients. Secondly, this implies based on Eq. (9) that the over∗ is almost fully carried by the NaI0 pairs. This all Na diffusivity DNa situation is demonstrated by the dashed line virtually matching ∗ . Thirdly, according to Eq. (11) D the triangles that represent DNa  eff  Deff holds (cf. essentially relies on the free anions, since Dcat + − an eff curve intersects solid lines in Fig. 1a). As a consequence, the Dan − the D data (circles). According to the right-hand sides of Eqs. (9) and (10), the eff = (1 − r effective cation diffusivity is given by Dcat + pair )Dcat+ and

eff = (1 − r similarly Dan − pair )Dan− . Since both effective diffusivities share the same fraction of free ions (1 − rpair ), the great disparity eff and Deff revealed by Fig. 1a reflects corresponding between Dcat + an− differences in the values of Dcat+ and Dan− . This means that the inherent or true diffusivity of Na+ (Dcat+ ) is much smaller than that of I− (Dan− ), which is due to the fact that only the (small) cations are coordinated to the oxygen atoms in the PEO chains and thus tightly eff and Deff curves in Fig. 1a run nearly bound. Nevertheless, the Dcat + an− parallel. This bears out the established notion that the mobility of both Na+ and I− is strongly connected with the segmential motion of the polymer matrix [18].

In the PEO60 EMImI electrolyte, the inorganic salt is replaced by an IL iodide containing the organic cation 1-ethyl-3-methylimidazolium (EMIm). With the molar ratio EO:IL = 60:1, the ion density is much higher than for PEO500 NaI but still rather low for practical applications. Iodide-based electrolytes are crucial to dye sensitized solar cells, which usually rely on charge transfer /I− redox couple [12,19]. The amorphous range mediated by the I− 1 3 investigated for PEO60 EMImI extends from 70 ◦ C, just above the PEO melting temperature, to 120 ◦ C, just below the onset of IL precipitation, as independently monitored by differential scanning calorimetry. The diffusion experiments involved three different methods: conductivity measurements by EIS for D , 125 I radiotracer ∗ . The data have not been diffusion for DI∗ , and 1 H PFG-NMR for DEMIm published before but the results on a similar system are described in recent work [12]. Fig. 2a shows that cation and anion migrate almost equally fast, despite the great differences in size and structure of these ions. This ∗ ≈ 0.5 in Fig. 2b, which is represented by the is also reflected by tcat lower, nearly horizontal line. Similar observations have been made for related polymer electrolytes and pure ILs based on the EMIm ∗ cation [12,20]. Compared to DEMIm and DI∗ , the charge diffusivity is clearly smaller, while the difference monotonically increases from low to high temperature. These features point to a high degree of ion association, as expressed by  close to 0.8 in Fig. 2b. Apparently, ion pairing increases with increasing temperature, as generally cat and f an do observed in SPEs [3,15]. Fig. 2b also displays that fpair pair

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455

T/°C 120

T/°C

90

70

-5

PEO60EMImI

2

*

DEMIm

eff

Dan–

*

DI

eff

Dcat+

-6

10

eff

Dan– Dσ

-7

10

(a)

*

DEMIm

(a)

Dσ

70

–1

-7

10

2

–1

eff

Dpair

eff

Dcat+

120

PEO20EMImTFSI Diffusivity/cm s

-6

10 Diffusivity/cm s

10

170

eff Dpair

* DTFSI

-8

10

1.0

1.0

0.8 Δ

0.6

Numbers, Fractions

Numbers, Fractions

an

fpair cat

fpair

t+ *

tcat

0.4 0.2 0 2.5

(b) 2.6

2.7

2.8

(1000/T)/K

2.9

3.0

–1

Fig. 2. (a) Measured (symbols) and deduced (lines) diffusion coefficients characterizing ion transport in PEO60 EMImI as a function of reciprocal temperature, as indicated. The experimental data result from hitherto unpublished work. The effeceff eff eff − 0 tive diffusivities of EMIm+ (Dcat + ), I (Dan− ), and EMImI pairs (Dpair ) were deduced by using Eqs. (12)–(14). (b) Additional parameters derived from the measured dif∗ , the transference fusion coefficients comprising the cation transport number tcat cat and number t+ , the deviation parameter , and the fractional pair components fpair an . fpair

not differ much from . This is a consequence of the finding that ∗ DI∗ ≈ DEMIm applies to this electrolyte. Nevertheless, t+ distinctly ∗ . exceeds tcat We note that the calculation of the ion pairing parameters was done with the aid of Vogel–Tammann–Fulcher (VTF) equations separately fitted to each experimental diffusivity. This procedure should not be taken as representative of a particular ion transport model. In the present context, it only serves for the purpose of smoothing and interpolation. Additional insight is given by the effective diffusivities and eff curve runs closely their T-dependence. Fig. 2a shows that the Dpair eff and Deff ∗ below the experimental DI∗ and DEMIm data, whereas Dcat + an− are distinctly lower. Thus, the contribution of EMIm+ and I− to the overall transport of cations and anions, respectively, is rather minor.

4.4. PEO20 EMImTFSI The PEO20 EMImTFSI complex is characterized by a fairly high concentration of ionic liquid, which is composed of two spatially extended ions, the EMIm cation and the TFSI (bis(trifluoromethylsulfonyl)imide) anion. The PEO–EMImTFSI system may be relevant to battery technology, particularly in mixtures with PEO–LiTFSI, where it may lead to enhanced Li transfer between the electrodes [21]. Similar effects have recently been

(b)

0.8 t+

0.6

*

tcat 0.4 0.2 0 2.2

an

fpair

Δ

cat fpair

2.4

2.6 (1000/T)/K

2.8

3.0

–1

Fig. 3. (a) Measured (symbols) and deduced (lines) diffusion coefficients characterizing ion transport in PEO20 EMImTFSI as a function of reciprocal temperature, as indicated. The experimental data were taken from earlier work [16]. The effeceff eff eff − 0 tive diffusivities of EMIm+ (Dcat (Dan + ), TFSI − ), and EMImTFSI pairs (Dpair ) were deduced by using Eqs. (12)–(14). (b) Additional parameters derived from the ∗ measured diffusion coefficients comprising the cation transport number tcat , the transference number t+ , the deviation parameter , and the fractional pair compocat an and fpair . nents fpair

examined for Na transport in ternary PEO20 NaI0.5 EMImTFSI0.5 and PEO20 NaI1 EMImTFSI1 complexes [16]. The present analysis concerns diffusion and conductivity data covering the temperature interval between 70 ◦ C and 170 ◦ C, where the electrolyte is in a stable amorphous state. The data were collected by means of EIS and PFG-NMR. Specifically, the 1 H resonances of EMIm and the 19 F signal of TFSI were monitored [16]. The experimental data are plotted in Fig. 3a. Also in this case the individually fitted VTF curves were omitted to avoid confusion. It is seen that the cation is slightly faster than the anion. This translates ∗ passin Fig. 3b to a spline curve for the cation transport number tcat ing through the middle region from 0.55 to 0.59. Most remarkable, ∗ however, is the observation in Fig. 3a that D exceeds both DEMIm ∗ . As an implication, the degree of ion association is rather and DTFSI low, as shown in Fig. 3b by the  curve running within the range cat and f an follow from 0.1 to 0.3. Like in Fig. 2b, also in this case fpair pair  closely at opposite sides of the corresponding line. Furthermore, the low degree of pairing leads to only small differences between ∗ and t . tcat + Inspecting the effective diffusivities in Fig. 3a, one notices the eff line. On the contrary, Deff and extremely low position of the Dpair cat+ eff approach the respective experimental diffusion coefficients Dan − ∗ ∗ and DTFSI are dominated at small distances. Thus, both DEMIm by their individual free-ion contributions, in particular at low temperatures.

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5. Discussion

Second, based on Eqs. (12)–(14) this implies that the following requirements are fulfilled:

5.1. General remarks on ion pairs and larger aggregates

∗ ∗ ∗ ∗ |Dcat − Dan | ≤ D ≤ Dcat + Dan .

In the preceding analysis, ion pairing was assumed to be the predominant process of ion association. This seems plausible for systems sufficiently dilute in salt, such as the examined polymer electrolytes. In such cases, unlike concentrated ILs, the pair concept is clear and relies on the spatial proximity of cations and anions. From the viewpoint of enthalpy, the first step of ion association will be the most favorable one due to the strong Coulombic interaction between the smallest charge carriers of opposite sign present in the electrolyte. Compared to this, the formation of triplets and larger aggregates yields at most a modest additional binding enthalpy. Furthermore, ion association in (classical) polymer electrolytes is essentially an entropic effect [22], which relates to (partial) release of cations from the coordinating ether oxygens and the corresponding gain in conformational freedom of the polymer chains. This gain will be substantial for pair formation but probably much less or even negligible for consecutive aggregation steps. Quantitative analysis shows that ion pairing in (dilute) SPEs with alkali metal salts is characterized by a delicate balance between a positive pair formation enthalpy Hpair and a corresponding positive entropy Spair , so that Gpair = Hpair − TSpair < 0 for the temperature range of interest [9,15]. This implies that ion pairing increases with increasing temperature, as is also manifested by the data in Figs. 1b, 2b and 3b. In addition, it has been found that the tendency to pair formation increases with decreasing salt concentration [3,15]. In contrast, the formation of ionic triplets and higher-order clusters is unlikely in dilute electrolytes, i.e., according to the mass action law. Ion pairs also appear in molecular dynamics simulations of PEO–IL systems but triplets do not play a prominent role therein [23,24]. Thus, the predominance of ion pairing over larger aggregates is well-founded for PEO500 NaI and it seems a reasonable assumption for PEO60 EMImI and PEO20 EMImTFSI. Information on ion association can be also obtained from infrared (IR) or Raman spectroscopy and related techniques [9,25,26]. These methods provide structural data on the abundance of ion pairs and isolated ions, which relate to the quantity rpair in this work. The pair fraction rpair does appear in Eqs. (9)–(11) but it is not further evaluated in the present paper. By contrast, we focus on the pair contribution to diffusion processes, which cannot be determined by IR/Raman spectroscopy, as it also involves the mobility of the ionic species. A further discussion of this issue is beyond the scope of this paper.

Third, the experimental diffusivities are subjected to similar but differing constraints, if it is assumed that, e.g., (cat2 an)+ triplets in conjunction with an− are the predominant charge carriers [10,27]. Fourth, examining all possibilities including higher-order clusters it can be concluded that the data constellation for PEO20 EMImTFSI excludes significant diffusion contributions of triplets and larger aggregates (see the full argument earlier given for the EMImBF4 ionic liquid [27], which exhibits a similar data constellation). Thus, the PEO20 EMImTFSI data are only consistent with the occurrence of EMImTFSI0 pairs in addition to the single ions EMIm+ and TFSI− . Evidently, this conclusion crucially relies on the relatively high values of D in this case. Within the same scenario, the diffusion picture for PEO500 NaI at low T excludes a significant influence of negatively charged clusters (see the full argument earlier given for PEO30 NaI [10]). Then at higher temperatures their contribution to mass and charge transport should be even less. Thus, I− is the only relevant negative charge carrier. By contrast, Na2 I+ is formally possible at low T. However, its mobility is expected to be very low since both Na+ ions are coordinated by the polymer [10]. Altogether, this supports the view that aggregates larger than pairs may be neglected for PEO500 NaI. For PEO60 EMImI, the experimental data in Fig. 2a do not impose narrow limits on the size of the charge carriers on logical grounds. , can neither Presumably, the occurrence of triplets, such as EMImI− 2 be excluded for chemical reasons. Therefore, more information on the PEO–EMImI system is needed to justify the present analysis based on single ions as the only relevant charged species.

5.2. Information on larger aggregates contained in the diffusion data Information on ion aggregation beyond neutral pair formation can be also directly deduced from the temperature dependence of ∗ , D∗ , and D in Figs. 1a, 2a and 3a, respectively. In particuDcat  an lar, the differences in magnitude among these measured diffusion coefficients provide clues to answering the question, whether, e.g., positively or negatively charged triplets may significantly contribute to ion transport [10,27]. Since the arguments are mathematically involved, only the global results of our analysis are presented here. A relatively clear case is given by PEO20 EMImTFSI, for which ∗ > D∗ holds true (cf. Fig. 3a). First, we notice that within D > Dcat an the present ‘pairs-only hypothesis’ of ion association physically eff , Deff , and Deff are obtained, that meaningful solutions for Dcat + an− pair is, none of these effective diffusivities result with a negative sign.

(26)

5.3. Artifacts due to experimental error In extreme cases the present derivation of the effective diffusivities may lead to seemingly inconsistent results. This situation was eff originally encountered for PEO500 NaI data in Fig. 1a. Here, Dcat + is at least an order of magnitude smaller than all other diffusion eff by Eq. (12) is very sensicoefficients. Thus, the calculation of Dcat + tive to experimental error, which is generally estimated to be about ∗ , D∗ , and D . Indeed, in the initial evaluation using 10% for Dcat  an VTF fits for interpolation and smoothing slightly negative values for eff were obtained, which relates to a (formal but not significant) Dcat + violation of the left-hand side of Eq. (26). To circumvent this artefact, the actual effective diffusivities depicted in Fig. 1a are based on a comprehensive model fit, which includes all experimental data, and moreover, numerically prevents physically inconsistent results [15]. Yet this fit offers an excellent representation of the measured diffusivities with a standard deviation below 5%. It should be further emphasized that the VTF equation may not be always a good choice for smoothing the experimental data. In fact, for PEO500 NaI with its strong ion-pairing tendency, particularly the T-dependence of D is affected by the factor 1 − rpair in Eq. (11) and not well represented by a VTF fit [14]. 5.4. Comparison of the different SPE complexes A discussion about the differences among the analyzed SPE complexes involves the effect of salt concentration as well as the nature of the cation and the anion. It is generally found that ion pairing increases with decreasing Cs (or increasing EO-to-salt ratio n) [3,28,29]. The same observation is made for the PEOn NaI system, where the  values for PEO500 NaI are distinctly higher than for, e.g., PEO30 NaI [15]. Unpublished data on the PEO–EMImI system show a similar dependence of the deviation parameter on Cs , whereas for PEO–EMImTFSI no further data are available. Altogether, the

N.A. Stolwijk et al. / Electrochimica Acta 102 (2013) 451–458

relatively high and low values of  for PEO500 NaI in Fig. 1 and PEO20 EMImTFSI in Fig. 3, respectively, may be effected by the difference in the composition parameter, i.e., n = 500 or 20. However, there must be other impact factors, since PEO60 EMImI with its intermediate salt concentration exhibits the highest degree of ion pairing ( ≈ 0.8) at most temperatures. The type of anion also plays a paramount role in pair formation. It is known that the spatially extended structure of TFSI− makes pairing less favorable energetically due to steric effects and charge delocalization. This is certainly true in combination with the bulky EMIm+ cation, which explains the low  data for PEO20 EMImTFSI. However, also for PEO–LiTFSI and related systems it has been found experimentally and theoretically that ion association is a minor effect [14,25,30,31]. In contrast, the comparatively small I− anion with its spherical symmetry promotes the coulombic interaction with the cation. A third factor influencing the pair formation tendency is the type of cation. Apart from its spatial structure, it is the coordination to the oxygen atoms in the PEO chains that determines which cation state is preferred, i.e., either the ‘isolated state’ with only polymer binding or the ‘pair state’ involving the anion. For PEO–NaI, the strong coordinative bonds of Na+ with PEO substantially reduces the conformational freedom of the polymer, so that the polymer entropy plays a significant role in ion pairing. As already indicated in Section 5.1, large positive values of Hpair and Spair explain the strong temperature dependence of  in PEO500 NaI (Fig. 1). On the contrary, the binding of EMIm+ to PEO is expected to be much weaker, from which the narrow dynamic range of  in PEO60 EMImI (Fig. 2) and PEO20 EMImTFSI (Fig. 3) can be understood. However, comparing these two complexes it has to be concluded that in PEO60 EMImI the nature of the anion, I− , results in a preference for the pair state, whereas in PEO20 EMImTFSI the isolated state predominates due to the properties of TFSI− . 5.5. Evaluation levels ∗ , In studies of ion transport comprising measurements of Dcat ∗ , and D three levels of evaluation may be distinguished: Dan 

(i) On the simplest level, the deviation parameter  is determined and discussed. The theoretical background is given by ∗ the Nernst–Einstein equation. Also the transport number tcat and the Haven ratio HR can be directly calculated from the diffusion data. Most experimental reports are restricted to this type of analysis. (ii) The second level is the subject of this paper; it deduces the three a priori unknown effective diffusivities by solving a set of three coupled linear equations. Then related quantities, such cat , f an , and t can be readily calculated. The benefit of this as fpair + pair approach is that it does not necessitate model assumptions other than the predominance of ion pairs over larger aggregates. Furthermore, only simple mathematical operations on the experimental data are needed but no numerical fitting procedures. Such level-two analysis could be supplementary applied to already published data subject to a level-one discussion only. (iii) A third evaluation level makes it feasible to split up each effective diffusivity in a true (or inherent) diffusivity and the relative abundance of the corresponding species (cf. Eqs. (12)–(14)). Such an extended approach requires assumptions about the basic type of T-dependence for the true diffusivities and the reaction constant of ion pairing [15]. This involves the introduction of parameters characterizing ionic mobility and association, i.e., VTF parameters, such as B, T0 , and the associated pre-exponential factors as well as the enthalpy and

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entropy of pair formation [13,15]. Within such a specific model, containing reasonable simplifications to reduce the number of ∗ , D∗ , and D can be simultaneously fitted free parameters, Dcat  an using a least-squares routine. 6. Conclusions Exploring theoretically the different effects of ion pairing on the diffusivity of cations, anions and overall charge in dilute polymer electrolytes has led to the following results: eff ), free anions (1) The individual contributions of free cations (Dcat +

(2) (3)

(4) (5)

(6)

eff ), and pairs (Deff ) are obtained by simple summation and (Dan − pair subtraction operations on the experimentally determined dif∗ , D∗ , and D ). fusion coefficients (Dcat  an The meaning of the Nernst–Einstein deviation parameter  is elucidated. Expressions for the relative contributions of neutral pairs to the cat and f an , respecdiffusion coefficient of cations and anions, fpair pair tively, are derived from  and the cation transport number ∗ . tcat The present approach allows for the determination of the cation ∗ . transference number t+ , which may greatly differ from tcat The application of the theoretical framework to three different polymer electrolytes containing an alkali metal salt or an ionic liquid reveals mutual disparities in the degree of ion association, the impact of pair formation on ion transport, and the dependence of these properties on temperature. ∗ , D∗ , and D three levels of In simultaneously analyzing Dcat  an evaluation can be distinguished. The present work reveals and emphasizes the usefulness of level-two evaluation, involving only elementary algebra without model fitting.

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