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Space-charge effects at the LLZO/PEO interface Doriano Brogioli, Frederieke Langer, Robert Kun, and Fabio La Mantia ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b19237 • Publication Date (Web): 01 Mar 2019 Downloaded from http://pubs.acs.org on March 2, 2019

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ACS Applied Materials & Interfaces

Space-Charge Effects at the LLZO/PEO Interface Doriano Brogioli,∗,† Frederieke Langer,‡ Robert Kun,‡,¶,§ and Fabio La Mantia† †Universit¨ at Bremen, Energiespeicher– und Energiewandlersysteme, Bibliothekstraße 1, 28359 Bremen, Germany ‡Fraunhofer Institute for Manufacturing Technology and Advanced Materials IFAM, Wiener Str. 12, 28329 Bremen, Germay ¶University of Bremen, Innovative sensor and functional materials, Badgasteinerstraße 1, 28395 Bremen, Germany §University of Bremen, MAPEX Center for Materials and Processes, Bibliothekstraße 1, 28359 Bremen, Germany E-mail: [email protected] Abstract Composites consisting of garnet type Li7 La3 Zr2 O12 (LLZO) ceramic particles dispersed in a solid polymer electrolyte, based on poly(ethylene oxide) (PEO), have recently been investigated as a possible electrolyte material in all solid state Li-ion batteries. The interface between the two materials, i.e. LLZO/PEO, is of special interest for the transport of lithium ions in the composite. For obtaining the desired high ionic conductivity, Li+ ions have to pass easily across this interface. However, previous research found that the interface is highly resistive. Here, we further investigate the interface between Al-substituted LLZO and PEO-LiClO4 electrolytes in the frame of a theoretical description, which is based on space charge layers. By theoretical calculations supported by experiments we find that the interface is highly resistive. From the

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results, we have deduced that the highest contribution to this resistance comes from a high activation energy and not from electrostatic repulsion of lithium.

Keywords solid state electrolytes, electric double layer, stern layer, electrochemical impedance spectroscopy, simulation of interface

Introduction Bulk-type solid state batteries are considered a promising alternative for future energy storage devices. 1 The proposed battery consists of a Li metal anode and a high voltage cathode separated by a thin layer of the solid Li+ ion conducting electrolyte that also acts as a separator. While commercially available cathode materials can be used also in solid state batteries, a completely new electrolyte has to be developed. A multitude of materials have been proposed as electrolytes ranging from oxide ceramics, such as NaSiCON type 2,3 Li1+x Alx Ti2-x (PO4 )3 and garnet type 4–6 Li7 La3 Zr2 O12 (LLZO) to glasses (e.g. LiPON 7,8 ), sulphides 9,10 ) and polymers. 11,12 Among these, LLZO has emerged as a promising material due to its chemical stability against Li metal, its broad electrochemical window and high Li+ ion conductivity. 4 However, its ceramic nature makes the material brittle and difficult to process into thin membranes. Extensive effort is necessary to achieve thin layers. 13 On the other hand, solid polymer electrolytes (SPEs), 11,14 most commonly based on poly(ethylene oxide) (PEO), are easily processed into free standing thin membranes using solvent based tape casting methods. 11,14 However, SPEs consisting of PEO with Li+ ion containing salts (e.g. LiClO4 , 15 LiCF3 SO3 (triflate) 16 and LiN(CF3 SO2 )2 (LiTFSI) 17 ), dissolved by the PEO matrix, often fall short in terms of room temperature ionic conductivity and mechanical stability. Recently, attempts have been made to combine both materials, PEO-based SPE and 2


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LLZO, in a composite with the aim to profit from possible synergistic effects: 18–20 the SPE allows to produce thin membranes, while the LLZO should enhance the conductivity. Computational calculation for optimised composite composition have been done by Kalnaus et al. 21 Experimentally, the composite is usually manufactured by homogeneously dispersing LLZO particles in an SPE matrix; 22 also dispersion of LLZO nanowires have been made 23 and infiltration of an LLZO scaffold with an SPE has been attempted. 24,25 The aim is to facilitate the processing of LLZO by mixing it with SPE, thereby making tape casting processes for the composites possible. Furthermore, the flexibility of the received composite electrolyte is increased compared to pure LLZO electrolytes, while the mechanical stability is increased compared to SPEs. Most importantly, the composite should have high ionic conductivity, in the best case close to the conductivity of LLZO. In an attempt to further improve mechanical and electrochemical properties of LLZO/polymer composites, liquid components, such as TEGDME, 26 PEG 27,28 or ionic liquids, 29 have been introduced into the system. Flexible, about 100 µm freestanding membranes with LLZO content of up to 40 vol% (75 wt%) can be prepared, 19 with advantages in the mechanical processing; however, the results on ionic conductivity are unclear. While some publications report an enhanced ionic conductivity of the composite compared to the pure SPE, 22,23,30 others do not notice a significant enhancement, but even a negative effect on the Li+ ion conductivity. 19,31 Therefore, researchers have speculated on the presence of an unsuitable interface between LLZO and PEO, which is resistive to Li+ ion conduction. 32 The addition of nanosized ceramic filler particles is known to enhance the ionic conductivity in SPEs by reducing the crystallinity of the polymer host matrix and favourable interaction between the ceramic surface groups, the Li+ ions and the polymer chains. 33–35 Nevertheless, Li+ ion conductivity in SPEs remains lower than ionic conductivity in ceramic electrolytes such as LLZO. Therefore, in a composite electrolyte with high ionic conductivity, the majority of Li+ ion conduction needs to take place in the LLZO phase. In consequence, in a composite with homogeneously

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dispersed particles, the Li+ ion needs to cross the SPE/LLZO interface. Zheng et al. 36 investigated preferred Li ion pathways and found a preferential conduction though the LLZO phase. Hence, a low ionic conductivity indicates a resistive SPE/LLZO interface. In order to verify this hypothesis, controlled systems consisting of layers of SPE (PEO and LiClO4 ) contacted to a sintered LLZO pellet (i.e. Cu/SPE/LLZO/SPE/Cu) have been tested; 32 thanks to the geometry, the current lines are almost known a priori and they must cross the interfaces, thus enabling a separate study of the interface LLZO/SPE. Indeed, by means of electrochemical impedance spectroscopy, an additional limiting process was found, which was attributed to the LLZO/SPE interface. 32 The aim of the study presented here is to investigate the origin of the high LLZO/SPE interface resistance. It has been proposed that space charge phenomena, i.e. the formation of an electric double layer, play a role in the determining the interface resistance. 37–39 In this study, we evaluate the possible relevance of such phenomena to the interface resistance. After developing a theoretical framework, which describes the ion transfer at the interface through a Butler-Volmer like equation, we compare the theoretical results with the experimental ones, obtained at different concentration of salt in the SPE. According to Nernst law, the wide concentration range used in the experiments allows us to explore a range of the order of 180 mV in voltage across the interface. Finally, we discuss the obtained parameters in terms of influence on the resistance of the interface. It is important to notice that our analysis is applied to a specific procedure of preparation of the materials, which gives rise to quite high interface resistance, but it is meant as general analysis, which enables a comparison between the materials produced by different procedures, while it does not support the idea that the discussed preparation method is more effective than others. Even more generally, the present work can be seen as an example of application of an analytical method, aimed at the experimental detection of the presence of space-charge effects.

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Experimental Copper foil

PEO Spacer LLZO

Copper sputtering

Figure 1: Setup of model system (schematic) before assembling (upper) and after temperature treatment (lower). The copper foil has a diameter of 10 mm and a thickness of 14 µm; PEO has a diameter of 8 mm and a thickness of 0.5 mm; the spacer has a hole with diameter of 8 mm and an outer diameter of 11 mm; the LLZO has a diameter of 9 mm and a thickness of 1 mm, with the lower side sputtered with copper. In order to reduce the number of components and thereby, increase the accuracy of electrode positioning, asymmetric model systems (see Fig. 1) have been developed. Asymmetric model systems consisted of an LLZO pellet and a layer of SPE. The concentration of Li+ ions in the SPE was varied by applying an electrolyte of well-defined concentration. After measurement, the SPE was removed and a new electrolyte layer was applied to the same LLZO pellet. The concentration of Li+ ions in the SPE was thereby, subsequently increased. Long equilibrium times at high temperature before measurement ensured that the concentration gradient was allowed to settle. LLZO pellets with nominal composition Li6.25 Al0.25 La3 Zr2 O12 were prepared as described before. 6,32 In short, the constituting metal ions were co-precipitated from aqueous solution, dried and mixed with a Li salt. A 10 mol% excess of Li salt was used to compensate the loss of Li+ ions during heat treatments. The mixture was calcined at 850◦ C for 6 hours, ground and pressed into pellets. Then, they were sintered in a Y-stabilized ZrO2 crucible in air covered with calcined powder to minimize the loss of Li+ ions. After sintering, the pellets were 5



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polished and placed in a tube furnace. The LLZO pellets were calcined at 850◦ C for 2 hours in Ar atmosphere to remove any carbonate contamination and prevent further contamination with water or carbon dioxide from ambient air. Then, the pellets were transferred into an Ar filled glovebox (O2 and H2 O below 0.1 ppm, MBraun) without exposure to air. All further steps were carried out in the glovebox. One surface of the pellets was polished inside the glovebox and, afterwards, a round shape Cu blocking electrode with a diameter of 8 mm was applied by magnetron sputtering. In total, three pellets were prepared. PEO-based SPEs were prepared by conventional tape casting method. All chemicals were dried in vacuum before use. PEO (Mv = 105 g/mol, Sigma Aldrich) and LiClO4 (99.9%, Sigma Aldrich) were dissolved in acetonitrile (anhydrous, Sigma Aldrich). In this paper, we follow the usual notation for expressing the Li+ ion concentration in the SPE as the molar ratio between ethylene oxide (EO) monomer units and Li+ ions; e.g. a ratio EO:Li = 20:1 is denoted P(EO)20 Li. We studied a wide concentration range of Li+ ions, from P(EO)10,000 Li to P(EO)10 Li, thus covering three decades. A P(EO)10 Li stock solution was prepared and diluted by addition of PEO to obtain the electrolytes P(EO)20 Li, P(EO)50 Li and P(EO)100 Li. These solutions were further diluted in turn. Dilution steps were carried out until concentration of P(EO)10,000 Li was prepared. The solutions were cast into PTFE dishes and first slowly dried in glovebox atmosphere. The membranes were further dried in vacuum to remove all traces of solvent. The dried membranes were heated and pressed into sheets of 500 µm thickness. The model systems were assembled as follows (see also Fig. 1). An LLZO pellet was put into the custom made sample holder (glass cell) with the untreated surface pointing upwards. Circular pieces of SPE (8 mm diameter) were taken from the prepared sheets (thickness 500 µm) and placed inside a polyether ether ketone (PEEK) spacer with an inner diameter of 8 mm. The prepared polymer electrolyte was then placed on top of the LLZO pellet (on the untreated LLZO surface). A circular piece of smooth Cu foil was placed on top of the SPE, as a blocking electrode and the sample holder was sealed gastight.

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The samples were placed inside a climate chamber with the SPE facing downwards to prevent the melted SPE from flowing onto the LLZO pellet beneath the spacer. The climate chamber was set to 80◦ C and the system was allowed to equilibrate for 16 hours. Then the temperature was set to 70◦ C and impedance measurement was performed after two hours dwell time. Impedance spectroscopy measurement was carried out using a Novocontrol Alpha-A impedance analyser in a frequency range of 10 MHz to 1 Hz with an amplitude of 10 mV. Measurements at various temperatures and concentrations were performed. In order to eliminate the effect of the pellet’s microstructure (i.e., interface quality) on the deviations in the EIS analysis, same pellet was used in the subsequent measurements using increasing Li-salt concentration in the SPE matrix. Starting with the lowest Li-salt concentration in the SPE matrix, temperature dependent EIS analyses were carried out. After each set of measurements, in order to change the concentration in the SPE, the SPE/LLZO specimen was cooled down to room temperature and the SPE was physically removed: at room temperature, the SPE forms a relatively hard film that is easily peeled off from the pellet’s surface. Despite this, we assume that the full removal of the SPE residues from the pellet’s surface microporosity is not achieved in this way. However, in our procedure the Li-salt concentration is steadily increased between the consecutive analyses and the effect of concentration change is clearly seen. Thus we conclude that the rather long equilibration time prior to the EIS at elevated temperatures ensures the homogeneous diffusion of Li-salt across the bulk of the SPE membrane, including the SPE residues trapped in the pellet’s surface porosity. To ensure that the SPE uniformly coats the porous surface of the LLZO pellet, the assembled structure system was placed in a vacuum oven inside the glove box. The oven was heated to 90◦ C and when the temperature was reached, vacuum was applied to the cell to extract all gas from the sample. For reasons of comparison, samples of the prepared SPE were also measured.

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Model of the electric double layer at the interface LLZO

SPE

+

Li

Li Empty site

+ -

ClO 4

Stern layer

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σ cr = σ

ϕ

σ sol =- σ ∆ϕ sol

∆ϕ

∆ϕ Stern ∆ϕ cr x

Figure 2: Scheme of the interface between LLZO and SPE. Green disks represent Li+ ions. Vacancies (empty sites in LLZO) and anions (ClO4 – in PEO) are represented by red disks. Accumulation of charges near the interface, constituting the electrical double layer, is shown. The drawing is not in scale. A scheme of the interface between LLZO and SPE is shown in Fig. 2. Both phases contain Li+ ions, together with other charges, so that electroneutrality holds in the bulk of the materials. At the interface, an enrichment/depletion of Li+ ions takes place, due to differences in chemical or electrical potential, leading to the formation of diffuse layers with opposite net charge in the two phases. The process reaches the equilibrium when the electric field generated between the two charge distributions counterbalances the driving force of the migration. We derive the mathematical model of the diffuse layers following the traditional derivation of Gouy-Chapmann-Stern equation. 40 However, in our model we additionally consider the excluded-volume effects by assuming that the two phases are composed by a finite set of discrete sites, which can be empty or occupied by at most one of the ions, in analogy with the Bikerman lattice model. 41–44 We consider separately the two phases, LLZO and SPE; then we assume that they are 8


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separated by the Stern layer. The calculation allows us to evaluate the relation between the voltage across the whole interface and the surface charge density. We model the passage of the ions by means of the Butler-Volmer equation, using the concentrations of ions calculated according to the above-described procedure.

General formulation of the electric double layer We consider a flat interface between two phases; the x axis is perpendicular to the surface; the phase starts at x = 0 and the bulk is at x → +∞. The Poisson equation is: 1 ∂2 ϕ = − ρ (ϕ) . ∂x2 ǫ0 ǫr

(1)

where ϕ is the electric potential ρ is the space charge density, ǫ0 is the dielectric constant of vacuum and ǫr is the relative dielectric constant. We remark that ρ is written as a function of the potential rather than of the position x. We rewrite this equation as: ∂2 kB T ∂ ϕ= R (ϕ) , 2 ∂x ǫ0 ǫr e ∂ϕ

(2)

where kB is the Boltzmann constant, T is the temperature, e is the electron charge, and R (ϕ) is defined by: e ∂ R (ϕ) = − ρ (ϕ) . ∂ϕ kB T

(3)

In order to univocally identify R (ϕ), we impose:

R (ϕ = ϕ+∞ ) = 0,

where ϕ+∞ is the potential in the bulk of the phase.

9


(4)


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We further modify Eq. 2 by multiplying by ∂ϕ/∂x and integrating: Z

∂2 ∂ ϕ ϕdx = ∂x2 ∂x

Z

∂ kB T ∂ R (ϕ) ϕdx ǫ0 ǫr e ∂ϕ ∂x

(5)

By changing the integration variable on the right-hand side and integrating by parts on the left-hand side, we get:

∂ ϕ ∂x

2

2kB T R (ϕ) + C. ǫ0 ǫr e

=

(6)

For obtaining the value of the integration constant C, we evaluate the equation in the bulk, for ϕ = ϕ+∞ ; using Eq. 4 and assuming ∂ϕ/∂x = 0 in the bulk, we get C = 0. We can thus calculate: r 2kB T ∂ ϕ=± R (ϕ); ∂x ǫ0 ǫr e

(7)

where the sign depends on the physical description, and it will be decided later. Now we calculate the surface charge density σ, by integrating the space charge density ρ across the diffuse layer, from the surface x = 0 to the bulk x → +∞: Z

σ=

+∞

ρ (x) dx.

(8)

0

We change the integration variable to ϕ:

σ=

Z

ϕ+∞

ρ (x) ∂ϕ ∂x

ϕ0

dϕ,

(9)

where ϕ0 is the potential at x = 0, and we rewrite the integral in terms of R: r

σ=±

ǫ0 ǫr kB T 2e

Z

ϕ+∞

ϕ0

10

∂R(ϕ) ∂ϕ

p

R (ϕ)


dϕ.

(10)

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By changing the integration variable to R: r

σ=±

ǫ0 ǫr kB T 2e

Z

0

R(ϕ0 )

dR √ . R

(11)

We finally get: σ = −sgn (∆ϕ)

r

2ǫ0 ǫr kB T p R (ϕ0 ), e

(12)

where sgn is the sign function and ∆ϕ = ϕ0 − ϕ+∞ is the potential difference between the plane x = 0 and the bulk (see Fig. 2, arrows marked with ∆ϕsol and ∆ϕcr ). Here we assume that the material contains a finite set of discrete sites, which can be empty or occupied by at most one of the ions, in analogy with the Bikerman lattice model. 41–44 In order to calculate R, we express the space charge density ρ of Eq. 3 in terms of the probability Pj (ϕ) that a site, at potential ϕ, is occupied by the j-th ion: e2 X ∂ zj Pj (ϕ) , R (ϕ) = − ∂ϕ vkB T j

(13)

where zj is the charge if the j-th ion expressed with respect to the electron charge e and v is the ratio between the volume of the material and the number of sites that it contains. Together with Eq. 4, we thus get:

R (ϕ0 ) =

Z

ϕ+∞ ϕ0

e2 X zj Pj (ϕ) dϕ. vkB T j

(14)

In the following sections we show the calculation of R separately for the SPE and for the garnet LLZO.

Polymer PEO The polymer, PEO, is highly polar, with a glass transition temperature below room temperature; it is thus able to dissolve various salts, including LiClO4 used for the present work.

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We assume that it contains finite and discrete sites that can be empty or contain an ion, either positive or negative; their distribution is random, as schematically shown in Fig. 2 (the sites are the circles; the ions occupying them are the red and green disks). We evaluate the spatial density of the sites from the molarity of LiClO4 at saturation, csat =5 M; we assume that, at saturation, all the sites are occupied, hence the molarity of the sites is 2csat . From the same information, we evaluate the volume available per site, vsol , i.e. the ratio between the volume of a given amount of polymer and the number of sites it contains:

vsol =

1 , 2NA csat

(15)

where NA is the Avogadro’s number. Analogously, the molar fraction xsol of Li+ ions with respect to all the sites (empty or occupied) is expressed as the ratio between the molarity c of Li+ ions (equal to the molarity of LiClO4 ) and the molarity of the sites (2csat ): xsol =

c . 2csat

(16)

We assume that both ions have the same absolute charge zsol ; it is 1 for the ions generated by the dissociation of LiClO4 . The value of the relative dielectric constant is in the range 10-30 (see Ref. 45 ); we assume ǫsol = 20. The probabilities P0 , P+ and P− represent the probability of a site to be empty, occupied by the cation or the anion respectively. Following the grand-canonical approach: 0 µ 1 exp P0 (ϕ) = Z k T B+ 1 µ − eϕ P+ (ϕ) = exp Z k T −B µ + eϕ 1 , exp P− (ϕ) = Z kB T

12


(17) (18) (19)

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where Z is a normalization constant, obtained by the condition:

P+ (ϕ) + P− (ϕ) + P0 (ϕ) = 1;

(20)

µ0 , µ+ and µ− are the chemical potentials of the empty sites and of the cations and anions respectively. Without losing generality, we assume µ0 = 0. We further impose that the probabilities P+ and P− equal xsol in the bulk, i.e. for ϕ = ϕ+∞ ; this condition defines the values of µ+ and µ− . We find:

P+ (ϕ) =

P− (ϕ) =

+∞ exp −e ϕ−ϕ kB T ϕ−ϕ+∞ 1−2xsol + 2 cosh e xsol kB T +∞ exp e ϕ−ϕ kB T ϕ−ϕ+∞ 1−2xsol + 2 cosh e kB T xsol

(21)

(22)

As explained above, from the probabilities it is possible to calculate the relation between σ and ∆ϕ; σsol is the surface charge density in the diffuse layer in the polymer, and ∆ϕsol is the difference between the potential at interface and the potential in the bulk (see Fig. 2).

σsol (∆ϕsol ) = −sgn (∆ϕsol )

r

2kB T ǫ0 ǫsol vsol

s

log 1 + 4xsol sinh

2

zsol e∆ϕsol 2kB T

(23)

The molar fraction of Li+ ions at interface is finally obtained from the probability at ϕ = ϕ+∞ : exp x˜sol (∆ϕsol ) =

1−2xsol xsol

− ∆ϕksolB Tzsol e

+ 2 cosh

∆ϕsol zsol e kB T

,

(24)

and an analogous calculation gives the molar fraction of the anion:

xÃ sol (∆ϕsol ) =

exp 1−2xsol xsol


+ 2 cosh

13



(25)


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Garnet, LLZO The garnet, LLZO, that we prepared and used in the experiments has composition Li6.25 Al0.25 La3 Zr2 O12 . The crystal has space group Ia¯3d, and each cell contains 8 formula units. The Li+ ions are hosted in two sites, the 24d and 96h; only the latter enables the movement of Li+ ions, corresponding to 12 sites per formula unit. 46 From neutron and X-ray diffraction measurements, 47 the occupancy of such sites is 0.37. We call xcr the molar fraction of the empty 96h sites, i.e. xcr =0.63. In the scheme of Fig. 2, the empty sites are represented as negatively charged holes, while the sites occupied by lithium are electrically neutral; the net charge of the holes, with a concentration of 63%, is counterbalanced by background positive charges (schematically represented by the background pale green color). The site volume vcr = 2.25×10−29 m3 is the ratio between the volume of a crystallographic cell 47 and the number of the mobile sites it contains. 46 zcr = 1 is the absolute charge of the hole. The dielectric constants (relative permittivities) of doped c-LLZO materials have been reported 23 to be in the range of 40-60; we assume ǫcr = 55. The probabilities PH and P0 represent the probability of a site to be empty (it is a hole) or occupied by Li+ ions respectively. Following the grand-canonical approach: + µ + eϕ 1 exp PH (ϕ) = Z kB T 0 µ 1 , exp P0 (ϕ) = Z kB T

(26) (27)

where Z is a normalization constant, obtained by the condition:

P+ (ϕ) + P0 (ϕ) = 1;

(28)

µ0 and µH are the chemical potentials of the empty sites and of the Li+ ions respectively. Without losing generality, we assume µ0 = 0. We further impose that the probabilitiy PH

14


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equals xcr in the bulk, i.e. for ϕ = ϕ+∞ ; this condition defines the value of µH . We find:

PH (ϕ) =

xcr +∞ xcr + (1 − xcr ) exp −e ϕ−ϕ kB T

(29)

Following the procedure outlined above, we calculate the relation between σ and ∆ϕ; σcr is the surface charge density in the diffuse layer in the garnet, and ∆ϕcr is the difference between the potential at interface and the potential in the bulk (see Fig. 2). The relation is: r

2kB T ǫ0 ǫcr σcr (∆ϕcr ) = −sgn (∆ϕcr ) vcr s ∆ϕcr zcr e ∆ϕcr zcr e + 1 − xcr + log xcr exp −xcr kB T kB T

(30)

Molar fraction of the empty sites (holes) at interface is:

x˜cr (∆ϕsol ) =

xcr xcr + (1 − xcr ) exp − ∆ϕkcrB Tzcr e

(31)

It is worth noting that the molar fraction of Li+ ions is 1 − x˜cr .

Evaluation of the voltage vs. charge relation across the interface In order to preserve the electroneutrality of the whole interface, the surface charge densities in the two phases, i.e. in the garnet LLZO and in the polymer SPE, are equal in absolute value and opposite; we arbitrarily call σ (the surface charge density of the whole interface) the charge density in the garnet: σ = σcr = −σsol .

(32)

We assume that the two phases are separated by a layer without charges, the so-called Stern layer (see Fig. 2), with thickness hStern . Its value is not known and will be discussed in relation with the experimental results; it should be of the order of a fraction of the garnet

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cell size or of the width of the polymer chain, i.e. in the 0.1-1 nm scale. We define ∆ϕStern , the potential difference across the Stern layer, as the difference between the potential at interface with the garnet and the interface with the polymer (see Fig. 2). We assume that the Stern layer behaves as an electrostatic capacitor:

σ = ∆ϕStern

ǫ0 ǫStern . hStern

(33)

The relative permittivity of the Stern layer, ǫStern , is usually smaller at the interface than in the surrounding medium, so we assume ǫStern =5. We define the voltage ∆ϕ across the whole interface (SPE, Stern layer, LLZO) as the difference between the potential of the bulk of the polymer and the garnet:

∆ϕ = −∆ϕsol + ∆ϕcr − ∆ϕStern .

(34)

The σ vs. ∆ϕ relations in the three regions, i.e. Eqs. 23, 31 and 33, together with the consistency conditions, i.e. Eqs. 34 and 32, can be easily solved numerically. Alternatively, it is possible to invert analitically Eqs. 23 and 33, and calculate σ and ∆ϕ parametrically in ∆ϕcr :

σ = σcr (∆ϕcr )

(35)

∆ϕ = −∆ϕsol (−σ) + ∆ϕcr − ∆ϕStern (−σ) .

(36)

We can thus calculate the differential capacitance of the interface:

Cint (∆ϕ) = −

∂ σ (∆ϕ) . ∂∆ϕ

(37)

In order to evaluate the resistance of the interface, we assume that the crossing of the interface is a thermally-excited process, so that the current vs. overvoltage relation can be

16


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modelled with the Butler-Volmer equation. We calculate the concentrations at the interface by means of Eqs. 24, 25 and 31. In the limit of vanishing overvoltage, and with α = 1/2, we find the relation: 1 0 p . Rint (∆ϕ) = Rint ˜cr ) x˜sol x˜cr (1 − x˜sol − xÃ sol ) (1 − x

(38)

0 where Rint is an arbitrary constant.

It is worth noting that the experimental method does not allow us to apply a direct current (the electrodes are “blocking”) and the impedance measurements introduce only small overvoltages; hence, ∆ϕ will be determined by the difference of chemical potential of Li+ ions between the two phases, in turn depending on the concentration of LiClO4 : ∆ϕ = ∆ϕ0 −

xsol kB T , log e 1 − xA sol − xsol

(39)

where ∆ϕ0 is arbitrarily set equal to the potential at xsol = 1/3 (the molar fraction at which 1/3 of the sites are free, 1/3 are occupied by cations and 1/3 are occupied by anions).

Examples of results of the model Figure 3 reports results of the models for various cases. We see that the resistance increases dramatically at large positive and negative cell voltages. By inspecting the molar fraction distribution at the interface, we see that, in this case, resistance increase is mainly due to depletion of Li+ ions, for negative and positive cell voltage respectively. We remind that such depletion is the consequence of the migration of Li+ ions, which takes place as a consequence of a different chemical potential of Li+ ions in the two phases. At large negative voltages, Li+ ions tends to migrate from SPE to LLZO, thus leaving no mobile Li+ ion in the SPE. At large positive voltages, Li+ ions migrates from LLZO to SPE, filling the SPE and depleting the empty sites, which are necessary for enabling further jumps across the Stern layer. In the following we compare our experimental results to the theoretical framework, in order to 17



(a)

Resistance R (a.u.)

102

hStern=0.5 nm hStern=1 nm hStern=2 nm

101

100

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Cell voltage ∆ϕ (V)

1

(b) 100

Molar fraction x

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 33

10-1

10-2 xPEO Li+ 10

xLLZO Li+

-3

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Cell voltage ∆ϕ (V)

1

Figure 3: Example of results of the model. (a) resistance of the interface, as a function of the cell voltage, for various Stern layer thicknesses hStern , with concentration of LiClO4 in the polymer of 1 M. (b) molar fraction at interface of Li+ ions in PEO and LLZO, with concentration 1 M of salt in the polymer and hStern =0.5 nm.

18


Page 19 of 33

have further hint on the origin of the high resistance at LLZO/SPE interface.

Analysis of the impedance spectroscopy measurements 80

PEO-10 PEO-50 PEO-200 PEO-1k PEO-5k

70 60 -Im(Z) (kΩ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


50 40 30 20 10 0 0

10

20

30 40 50 Re(Z) (kΩ)

60

70

80

Figure 4: Nyquist plots of impedances of the model system (SPE layer on LLZO pellet, with blocking Cu electrodes on both sides), measured at various concentrations of LiClO4 in PEO, from P(EO)10 Li to P(EO)5,000 Li at 70◦ C. The results refer to a single sample of LLZO pellet. The Nyquist plots of impedances measured for a representative model system consisting of one LLZO pellet with increasing concentration of Li+ ions in the SPE are depicted in Fig. 4. The Nyquist plots display one well resolved high frequency semicircle and a second, less clearly resolved semicircle at intermediate frequencies and a straight line at low frequencies. It can be seen that both the high frequency semicircle increases with decreasing concentration of Li+ ions in the SPE. In contrast, the semicircle at intermediate frequencies does not change that much. In Fig. 5a, we show the equivalent circuit used to fit the impedance spectra. The Cu electrodes appear to be not completely blocking as expected, giving rise to a slight bending of the low-frequency branch that we interpret as the presence of a parasitic reaction; thus we model them with an RC circuit, composed by a capacitor Ch and Rct representing the 19



(a) Ch

Cint

Rct

Rint

Rs

Cs

β; Rp

(b) Measured Model

14 12 10 -Im(Z) (kΩ)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 33

Bulk

8 6 4 Interf.

Electr.

2 0 0

2

4

6 8 10 Re(Z) (kΩ)

12

14

Figure 5: (a) equivalent circuit used for fitting the data. The parameters β and Rp are needed to take into account the porosity of the material; we set β = 0. (b) Nyquist plot of the impedance measured for a single LLZO pellet and a concentrations of LiClO4 in PEO P(EO)500 Li, with the result of the fit with the equivalent circuit.

20


Page 21 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


parasitic reactions. The series resistance of the SPE and LLZO bulk is represented by Rs . The parasitic capacitance in series to the instrument input is modelled by Cs . The interface is modelled by the parallel of the interface resistance Rint and capacitance Cint . The interface is not flat, hence we also consider the effect of porosity. Following Ref., 48 we write the ideal 0 admittance Yint of the interface as:

0 Yint = iωCint +

1 ; Rint

(40)

then we define the characteristic parameter of the porous material gamma: q 0 , γ = Rp Yint

(41)

where Rp is a resistance describing the porosity; we finally calculate the impedance Zint of the porous interface: Zint = Rp

cosh (γ) + β β + γ sinh (γ) 2

,

(42)

where β is another parameter describing the porosity; we will assume β = 0. In Fig. 5b, we show an example of fit of an impedance spectrum with the model. The semicircle at the high frequency (left) is attributed to the parallel of the resistance Rs and parasitic capacity of the instrument Cs , and thus it should be regarded as an instrumental artefact. The tail at low frequency is due to the almost blocking electrodes, modelled by the constant phase element with parameters Ch and n, and a high value of Rct . We performed a fitting procedure on all the samples with various concentrations of salt in PEO, following the smoothing algorithm described in reference. 49 Figure 6 shows the fitted values of the resistance Rs . The resistance of the SPE alone (without LLZO) is also reported. The resistance of SPE appears to be inversely dependent on the concentration of Li+ ions in the PEO matrix at the low concentrations. The resistance decreases with increasing Li+ ions concentration, reaches a maximum for about 1.28 M,

21



100000

Complete cell SPE alone

10000 Rs (Ω)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 33

1000

100 1

10 100 1000 LiClO4 concentration (mM)

10000

Figure 6: Values of Rs obtained by fitting the experimental data; average and standard deviation of three LLZO samples shown. Results for SPE alone (without LLZO) are also shown. corresponding to P(EO)20 Li, and increases again. This is in agreement with previous studies concerning the influence of Li+ ions concentration on Li+ ions conductivity in SPEs. 15,50 A similar phenomenon is commonly observed in solutions and is attributed to the increase of viscosity. The values of Rs measured for the complete cell (including the LLZO) are similar to the SPE alone at low concentrations, where the major role is played by the SPE; at higher concentrations, the curves are displaced by an almost constant resistance of 636 Ω on average, which we attributed to the LLZO material. The parameters connected to the copper electrodes show a quite large variability, compatible with the difficulty of forming a good contact: Rct is between 100 kΩ and 1 MΩ, while Ch is of the order of a few µF (data not shown). The value of Cs is kept constant and depends on the instrument. In our case, it was equal to 9.5 pF. The most relevant feature, for the scope of the present work, is the semicircle at intermediate frequencies, which has previously been attributed to the Li+ ions transfer across the LLZO/SPE interface. 32 In this region, the Nyquist plot shows a semicircle, given by the

22


Page 23 of 33

parallel of the resistance Rint and Cint of the interface, distorted by the effect of porosity.

Discussion of the measured interface resistance 100000

Space-charge model Model without space-charge

10000 Rint (Ω)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1000

100 1

10 100 1000 LiClO4 concentration (mM)

10000

Figure 7: Values of interface resistance Rint as a function of LiClO4 concentration in PEO, evaluated from impedance spectroscopy measurements; averages and standard deviation for three samples are shown. The red line shows the results of the model, with Stern layer thickness hStern =0.1 nm and ∆ϕ=-600 mV. The green line is the result of a model without space-charge effects.

Figure 7 shows the values of interface resistance Rint as a function of the concentration of Li+ ions in PEO. The experimental values follow the model for low concentrations, however at high concentrations the expected minimum is not clearly observed. This is mostly caused by the quite large statistical error of the manually assembled samples and by the relatively shallow minimum in the model. So, we can conclude that, within the statistical error, the model correctly describes the observed results. Discrepancies could also arise at high concentrations due to a non-ideal behaviour of the system. Values of the parameter hStern between 0.06 and 0.12 nm and ∆ϕ0 between -700 and -300 mV give good fits of the curve in Fig. 7. The Root mean square relative fit error as a function of the fit parameters hStern and ∆ϕ0 is shown in Fig. 8. We can notice that hStern 23



0

-0.2

∆ϕ0 (V)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 33

-0.4

-0.6 3 2.5 2 1.5 1 0.5

-0.8

-1 0

0.05

0.1 hStern (nm)

0.15

0.2

Figure 8: Root mean square relative fit error as a function of the fit parameters hStern and ∆ϕ0 . are ∆ϕ0 are correlated. It is useful to compare the results of experiments and calculations with a model in which the space-charge effect is neglected. This model is obtained by assuming that at the interface the concentration of lithium is the same as in the phases’ bulk. Under this condition, the resistance (Eq. 38) is rewritten as follows: 0 p Rint (∆ϕ) ≈ Rint

1 . xsol xcr (1 − 2xsol ) (1 − xcr )

(43)

This function is shown in Fig. 7.

We observe a decrease of the resistance with increasing concentration, up to a concentration of 100 mM, with power-law dependence. Indeed, an exponent 1/2 for xsol can be seen in Eq. 43, for small values of xsol . The discrepancy observed between experimental data and the green line, for concentrations more than 100 mM, can be accounted for by the space-charge effect. We can thus conclude that the space-charge effect is clearly measured by our experimental method. We can notice that the space-charge effect increases the resistance, up to almost one

24


Page 25 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


decade for the highest concentrations. This effect is detrimental for the conductivity of the LLZO/PEO composites. Conversely, in principle, the resistance could be decreased by almost one decade by decreasing the space-charge effect. However, the observed interface resistance is several of magnitudes more than the resistances due to the bulk properties of the materials in the composites, 32 hence removing the space-charge effects would not make the composites work better. 0 By means of the transition state theory, 51,52 we can relate Rint to an activation energy

barrier: ∆E

0 Rint = Ae kB T

(44)

where ∆E is the height of the energy barrier and A is a proportionality constant depending on the width and shape of the barrier. Typically, a large activation barrier is also associated to a large pre-factor A; we thus conclude that the reason for the high resistance is the presence of a high activation energy barrier ∆E.

Conclusion We presented a model of the LLZO/SPE interface, which takes into account the space-charge effects and is able to give an evaluation of the interface resistance. We experimentally studied the LLZO/SPE interface by means of electrochemical impedance spectroscopy measurements. By fitting the spectra with an equivalent circuit, we were able to evaluate the interface resistance. We performed the experiments at various concentrations of Li+ in the SPE, thus changing the chemical potential of the ion. The graph of interface resistance versus concentration can be satisfactorily fitted by the results of the model, thus suggesting that the observed dependence is due to the development of space-charge effects. The observed resistance varies by approximately a factor 3 in the considered concentration range; the minimum corresponds to the absence of an adverse effect of the space-charge 25



Page 26 of 33

distribution on the conduction. We can thus conclude that the space-charge effect does not account for the large value of the interface resistance that we obtain, which therefore can be attributed to a high activation energy barrier of the lithium-ion transfer between the two phases.

Acknowledgments This research has been supported by the Institutional Strategy of the University of Bremen “Ambitious and Agile”, funded by the German Excellence Initiative (DFG ZUK66/1).

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Solution inlet

33


Polymer

Stern layer

Graphical TOC Entry

Garnet

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PEO interface - ACS Applied

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