Ion-chain local interactions in polyether–salt complexes

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Electrochimica Acta, Vol. 43, Nos 10±11, pp. 1371±1374, 1998 # 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0013±4686/98 $19.00 + 0.00 S0013-4686(97)10046-9

Ion-chain local interactions in polyether±salt complexes L. D. Carlosa* and A. L. L. Videirab b

a Departamento de FõÂ sica, Universidade de Aveiro, 3800 Aveiro, Portugal Departamento de FõÂ sica, Universidade de EÂvora, 7001 EÂvora Codex, Portugal; Centro de MatemaÂtica e Aplicac° oÄes Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1699 Lisboa Codex, Portugal; and Academia Militar, Pac° o da Rainha, 29, 1198 Lisboa Codex, Portugal

(Received 16 September 1996; accepted 16 January 1997) AbstractÐRepresenting polyether±salt systems by chains of interacting coordination shells, each of which de®ned by a cation and by its nearest ligands (ether±oxygens and eventually anions), the interaction potential between a given coordination shell and all the others closest to itÐthe inter-shells potentialÐis derived in terms of two-electron polarization e€ects. Values are presented for crystalline monovalent-based poly(ethylene oxide), PEO , complexes, while for non-crystalline divalent electrolytes only an upper limit is estimated. For the eutectic concentration of PEO/Eu3+ electrolytes, the inter-shells energy e is also evaluated by relating the empirical value of the nearest-ligands local-®eld potential with the variation of Eu3+ concentration. The value obtained by this procedure was e = 554.2 cmÿ1, while by the two-electron polarization potential they were e = 520.3 cmÿ1, for 10 nearest oxygens and e = 572.4 cmÿ1, for 11 oxygens, for a mean radius of the ®rst coordination shell R=2.4 AÊ and a minimum distance between nearest shells R0=5.3 AÊ. This last method also permits the determination of the number of Eu3+-nearest oxygens. # 1998 Published by Elsevier Science Ltd. All rights reserved Key words: polymer electrolytes, ion-chain e€ective ®eld, local coordination.

INTRODUCTION In spite of extensive work performed over the last twenty years on polymer electrolytes containing mono and multivalent cations, their ion-chain local structures are practically unknown. In fact, only for a few crystalline monovalent-based poly(ethylene oxide), PEO , electrolytes the depicting of the ionchain local con®guration as forming oxygen-lined helical turns, with the cation located within the polymer cavity, has been con®rmed by single crystal X-ray di€raction [1, 2]. Following this local description, polyether±salt systems are here represented by chains of interacting coordination shells, each of which de®ned by a cation at its center and by its nearest ligands (ether ±oxygens and, eventually, anions) at its boundary. In this model, the ion-chain local dynamicsÐwhich *Author to whom correspondence should be addressed.

changes with the number of interacting shells, that is, with the number n of ether oxygens in the polymer chain per cationÐresults jointly from each shell's cation±nearest ligands interactionÐthe intrashell interactionÐand from the interaction between a given coordination shell and all the other shells in its immediate vicinityÐthe inter-shells interaction.

THE INTER-SHELLS POTENTIAL For a given shell, the polarization of one of its ligands located at R~ from the cation (taken as the origin) by one of the valence electrons of the neighbouring shell, at r~1, is taken into account; one of the remaining electrons at r~2 interacts with the multipole moments induced in the polarized ligand [3]. In each shell the polarized ligands are attractively coordinated to the cation, and therefore, the interaction between a given shell and all the others closest to it, separated by an average inter-shells

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L. D. Carlos and A. L. L. Videira

distance Rss, is repulsive and is then written as Vssn ˆ

XX

Fn …abk†

  …^r2 † Ra Ca‡n;a‡m …^r1 † Rb Cb‡n;b‡m

aba kbm



b‡n ÿb ÿ m

a‡n a‡m

…r1 †a‡n‡1

…r2 †b‡n‡1

 k ^ c …R†; ÿa ‡ b k;bÿa

…1†

where C`,q=(4p/2` + 1)1/2 Y`,q are spherical tensors, Fn(abk)Ðexpressed in terms of 3-j symbols ( ) and of 6-j symbols { }Ðis   a b k Fn …abk† ˆ…ÿan e2 †…ÿ1†nÿm‡b …2k ‡ 1† 0 0 0   a a‡n n ‰…a ‡ n†; …b ‡ n†Š1=2 b‡n b k    2a ‡ 2n 2b ‡ 2n 1=2 ; 2a 2b and an is the nth multipole polarizability of the ligands. Taking only the dipolar contribution, assuming spherical symmetry in the space of the ligands valence electrons (k = 0), r1=r2=Rss+R, and summing over these electrons and over the N ~ equation (1) becomes ligands of valence Zi at R, Vss ˆ

N 1 X 1X R2a i ai e2 Zi …a ‡ 1† : 2 i …Ri ‡ Rss †2a‡4 aˆ0

i

…2†

Considering only the ®rst two terms of the expansion (RsswRi), and considering that all the ligands are at the same distance R from the cation, the inter-shells interaction is then   N 1X 1 2R 2 : …3† Vss ˆ ai e2 Zi ‡ 2 i …R ‡ Rss †4 …R ‡ Rss †6 Monovalent-based PEO electrolytes The corresponding inter-shells potentials for the only crystalline monovalent-based PEO electrolytes Table 1. Coordination number N, nearest cation±ligands average distance R (AÊ), inter-shells distance Rss (AÊ), and inter-shell potential Vss (cmÿ1) for a few monovalent-based PEO electrolytes

PEO3NaI PEO3NaSCN PEO3NaClO4 PEO3LiCF3SO3 PEO4KSCN PEO4NH4SCN a

N

R

Rss

a

2.71 2.47 2.50 2.10 2.87 2.96

2.99 5.69 3.35 4.41 2.34 2.31

5 6b 5c 5c 7d 7d

for which the local structure has been experimentally determined are given in Table 1. For PEO3NaI, Wright [4] estimates the total interaction energy between the Na+ and Iÿ ions and the 3OE polymer segments of neighboring helices as being e(3EO-Nai-inter) 120±30 kJ (mol of 3EO units)ÿ1 (11700±2500 cmÿ1), a value which is of the same order of magnitude as the one calculated in Table 1. Divalent-based PEO electrolytes. The local structural information (EXAFS) is able for divalentbased PEO electrolytes to furnish only an upper limit for Vss. For example, in PEOnZnI2 (30 rn r4): Vss<600 cmÿ1, and PEOnZnBr2 (20 rn r6): Vss<900 cmÿ1. Trivalent-based PEO electrolytes: PEOnEuBr3. Very little is known about the ion-chain structure of trivalent-based polymer electrolytes. In particular, there are very few results for R [5, 6], and none for Rss. Here, the minimum inter-shell distance, Rss=R0, for which Vss does not yet alter R is determined. For this, following Morrison [7], the intershell potential Vs, also due to two-electron polarization e€ects, is written as   1X 2 1 22 Vs ˆ ÿ i e Zi 4 ‡ 6 ; …4† 2 R R

Vss 1453.23 415.46 915.85 526.65 3742.49 3638.94

3 oxygen and 2 iodine ligands, a(O2ÿ) = 1.349, a(Iÿ) = 5.013 AÊ3. b 4 oxygen and2 nitrogen ligands, a(N3ÿ) = 2.684 AÊ3. c oxygen ligands. d 5 oxygen and 2 nitrogen ligands

where r is an ion-dependent, host-independent quantity, which corrects the Hartree±Fock expectation values of the even powers of the 4¦ electrons radial distances. Then, imposing a balance between the repulsive Vss (equation (3)) and the attractive Vs (equation (4)) dVs dVss ˆ …5† dR  dR RˆR0

RˆR

one obtains the equation …R ‡ R0 †7 …R 2 ‡ 32 † ‡ R 7 R20 ‡ 2R 8 R0 ‡ 4R 9 ˆ 0 …6† from which the minimum inter-shells distance R0 is determined. Taking the value R=2.4 AÊ for 80r n r5 [5, 6], and r2=0.1666 AÊ2 into equation (6), one gets R0=5.3 AÊ. Table 2 presents the results for the inter-shells potential for a number of europium-nearest oxygens between 8 and 12, a value in accordance with both the determined C2v local symmetry group [6, 8], generally associated with a coordination number around 8, and with a recent result on La3+-based PEO electrolytes [9], which indicates between 9 and 10 oxygens surrounding the lanthanum cation. Local-®eld intra-shell potential Next, for PEOnEuBr3, the e€ect of the increase of Eu3+ concentration from n = 80 to n = 5 on the interaction potential between the europium and its surrounding oxygensÐthe intra-shell potentialÐ

Interactions in polyether±salt complexes Table 2. Inter-shell potential, Vss(cmÿ1), for a number of nearestoxygen ligands N between 8 and 12 (R=2.4 AÊ, and Rss=R0=5.3 AÊ). N

8

9

10

11

12

Vss

416.3

468.4

520.3

572.4

624.4

is represented. As usual, this potential is expressed in terms of the empirically determined Bk,q (ÿk R q Rk). Relating these parameters with the interaction energy between nearest shellsÐthe intershell potentialÐthe inter-shells interaction energy e is obtained, by an independent method, for the PEO-EuBr3 stablest concentration n 112. The emission spectra of Eu3+-based electrolytes present a series of sharp lines assigned to transitions between the Stark components of the 5D0,1 and 7F04 levels [6, 8, 10], of which, the only transitions considered here are the dominant electric-dipole 5 D0 47F2 and the magnetic-dipole 5D0 47F1 lines. The eigenvalues of the matrix elements of the local®eld perturbation for the 7F1,2 Stark sublevels are expressed in terms of the non-zero B2,q and B4,q (the k = 0 term not being included, as it merely shifts the energetic con®guration as a whole), and are ®xed by the best ®tting between the observed Stark energies and the calculated eigenvalues [6, 8]. Considering that only the nearest ligands contribute signi®cantly to the phenomenological intra-shell potential, the most relevant parameters are the B4,q. Since for lanthanide complexes, the B4,q predicted by the point-charge model are in good agreement with the corresponding empirical values, in the case of PEOnEuBr3, the e€ect of the increase of n on the intra-shell potential is represented expressing these B4,q in terms of that model. These parameters change as the Eu3+ concentration increases from n = 80 to n = 5 [6, 8]. The connection between the electrolyte global behaviorÐexpressed by the ion's concentration increaseÐand the local interaction within each coordination shellÐexpressed by the sum of all the B4,qÐis given by the changes on the ligands valence. In the model presented here of PEO-EuBr3 electrolytes as a set of interacting luminescent shellsÐthe number of which increases with the amount of complexed-Eu3+Ðthe phenomenological intra-shell potential (and, therefore, the ligands valence) must re¯ect the increase of the interaction between a given shell and all the others in its immediate vicinity with the decrease of the oxygen±cation ratio. For a europium concentration increase from n = 80 to n = 32, the coordination shells are spaced so far apart that they do not interact with each other. In the range 32 rn r8, the number of luminescent shells present in the electrolytes is such that the distance between nearest shells decreases in such a way that the induced ligands valence charge of each shell also becomes smaller.

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This, in turn, is associated with a corresponding diminution of the intra-shell potential. Hence, the inter-shells interaction begins to become relevant without yet altering the average Eu3+-ligand oxygens distance. For n < 8, the electrostatic repulsion between nearest shells increases, their mutual interaction decreases, and the electrolytes are not found any more in their stablest phaseÐthe eutectic concentration n 1 12. This behavior of the intra-shell dynamics with the increase of the Eu3+ concentration shows precisely the same trend as the observed dependence with n of the ratio between the 5D0 47F2 and 5D0 47F1 average intensities, I0-2/ I0-1, which slowly diminishes between 80 r n r 12, attains a minimum for n 112, and sharply increases for n < 12 [10]. As it is well known that the relative intensities of these two transitions (the only ones being considered here) are hypersensitive to the nature of the Eu3+-ligand surroundings, the intrashell potential must, somehow, incorporate the ratio I0 ÿ 2/I0 ÿ 1. The strength of this potential is 0(2 K + 1)s2k = 4, where de®ned as s2LF s2k = 4=(2k + 1)ÿ1Sq(B4,q)2 is the corresponding Leavit's quadratic rotational invariant [11]. Then the proposition is made that, in terms of this strength, the phenomenological intra-shell potential is related to the Eu3+ concentration by ! 2 1=2 X ˆ e…1 ÿ e1ÿn=n0 †2 ; …7† sLF  B4;q q

where e is the interaction energy between a given coordination shell and all the other shells closest to it, corresponding to the Eu3+ stablest concentration n0. Inserting into equation (7) the value for sLF obtained from the phenomenological parameters [6, 8], e and n0 are evaluated by the least squares ®t (Fig. 1). The results obtained are e = 554.2 cmÿ1 and n0=12.9. This value for the Eu3+ stablest concentration is precisely the one given by the morphological results (DSC, XRD, etc.) [12], thus bearing out the relation proposed in equation (7). CONCLUSIONS Representing polyether±salt systems by a set of interacting coordination shells, de®ned by the cation and its nearest ligands, the interaction potential between a given coordination shell and all the others in its immediate surroundings is derived. The values obtained for monovalent-based crystalline PEO electrolytes are given in Table 1. For non-crystalline divalent electrolytes, it is only possible to estimate an upper limit for the inter-shell potential. For PEO-EuBr3 electrolytes, this inter-shells potential is dealt with in two separate ways, one involving the same two-electron polarization e€ects, and the other relating the empirical value of the nearestligands local-®eld potential with the variation of the Eu3+ concentration.

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L. D. Carlos and A. L. L. Videira

Fig. 1. Dependence of the strength of the nearest-ligands local-®eld interaction SLF (cmÿ1) on the O/Eu ratio (n) for PEOnEuBr3 (n = 5,8,12,16,20,24,28,32,80). Fitting parameter D = 40 cmÿ1.

The inter-shells interaction energy, e, calculated by the two-electron polarization potential, is e = 520.3 cmÿ1, for 10 nearest oxygens, and is e = 572.4 cmÿ1, for 11 oxygens/ The value of e evaluated by the phenomenological local-®eld potential is e = 554.2 cmÿ1. The agreement between these two independent methods indicates that the concentration for which the inter-shells interaction does not yet alter the intra-shell e€ective ®eld is the stablest eutectic concentration. In addition, this agreement suggests that the screening e€ects of the noncomplexed oxygens, included in the inter-shell potential, equation (4), are well accounted for by the dipole polarization, by the number of ligands, and their valence. On the other hand, although, as previously referred to [5], the dependence of the intrashell potential with the Eu3+-ligands distance, equation (4), is essentially correct, the screening e€ects of the 5s and 5p shells is eventually over estimated. This may be due to the distance between nearest shells being more than twice the europiumoxygens distance. ACKNOWLEDGEMENTS L.D. Carlos wishes to thank the Fundac° aÄo Calouste Gulbenkian for co-®nancing traveling expenses.

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