Introduction

As the electric vehicle (EV) market expands, Li metal anode batteries are gaining attention due to their high energy density. Solid electrolytes, such as polymer, sulfide, and ceramic, have gained popularity in the last decade due to their non-flammable characteristics and good mechanical properties that can prevent Li dendrite formation [1–3]. Among these solid electrolytes, ceramic electrolytes exhibit high ionic conductivity, high transference number, wide electrochemical window, and thermal stability [4–6]. However, ceramic electrolytes, such as garnet type Li7La3Zr2O12 (LLZO), Li0.33La0.557TiO3 (LLTO), and NASICON type Li1.5Al0.5Ge1.5(PO4)3 (LAGP), Li1.3Al0.3Ti1.7(PO4)3 (LATP), are difficult to fabricate because of the high temperature and pressure requirement during the manufacturing process [7, 8]. In addition, they are brittle and have difficulty in achieving good interfacial contact with the Li metal anode and composite cathodes [9]. On the other hand, polymer electrolytes, such as PEO (poly-ethylene-oxide) electrolytes, have lower manufacturing cost compared to ceramic electrolytes and have good flexibility so that they can be easily contacted with both electrodes. Nevertheless, the poor ionic conductivity at room temperature combined with low transference number, which leads to large potential drops across the electrolyte remains a major bottleneck to their commercialization [10–12]. Therefore, many research efforts have been devoted to improving ionic conductivity and the transference number of polymer electrolytes. Although some gel polymer electrolytes overcome these shortcomings, their mechanical strength is still insufficient to prevent Li dendrite growth [13, 14].

Another approach to solid electrolytes is CEs, synthesized by mixing polymer electrolytes and ceramic powders with the aim of achieving high ionic conductivity, high transference number, and reasonable mechanical strength compared to polymer electrolytes [9, 15–18]. The most commonly used CEs consist of PEO polymer and LLZO powders with the expectation of enhanced Li-ion transport through LLZO will compensate for the lower transport properties of PEO. Therefore, understanding the Li-ion transport mechanisms in the CEs is essential to optimize the total ionic conductivity and design a solid electrolyte. However, there have been controversies on the pathway Li-ions take to transport through CEs [19–21]. The possible Li-ion pathways in CEs are as follows: (1) through only the polymer phase, (2) through the polymer phase and ceramic fillers, and (3) through the interphase layer between polymer and ceramic fillers. First, Li-ion may move only through the polymer phase if the interfacial resistance between the polymer and ceramic is very high, which can be caused by various factors, e.g., surface impurities on the ceramic fillers. There are several shreds of evidence from experimental studies showing that the total ionic conductivity of CE decreases even when the number of inorganic fillers increase [22–25]. Second, if the interfacial resistance is low, Li-ions can pass through ceramics fillers instead of polymer. Gupta and Sakamoto [17] confirmed that in CE consisting of PEO-LiTFSI/LLZTO, by removing surface impurities from the LLZTO (Li6.4La3Zr1.4Ta0.6O12) fillers through heat treatment, the interfacial resistance can possibly be decreased to 100 \(\Omega \cdot {\mathrm{cm}}^{2}\) and the overall ionic conductivity of the CE improves significantly.

In contrast, another hypothesis suggests that Li-ions pass through the interfacial layer on the ceramic surface rather than conducting through the ceramic. According to these studies, the added ceramic fillers can improve the polymer chain mobility, or possibly form a space charge region due to the positively charged oxygen vacancies on the filler’s surfaces, leading to a fast conduction pathway of Li-ions. Li et al. [26] developed the random resistor model consisting of a space charge region between PEO matrix and Ga-doped LLZO filler particles within the composite electrolyte, and compared the computationally predicted conductivity with experimentally observed values, which clearly indicate enhancement in ionic conductivity due to the addition of the LLZO fillers. Using the percolation model, Zaman et al. [21] also explained that enhanced transport in CE is due to the interactions between polymer and Al-LLZO, which could increase ionic conductivity at the surface area of the inorganic particle. Note that even if Li-ions pass through the ceramic phase, the total ionic conductivity may not improve depending on the amount of interfacial resistance, particle size, and particle shape. Finally, if the volume fraction of ceramic fillers exceeds the percolation threshold, the resulting percolating network of ceramic fillers could lead to an even higher total ionic conductivity. Lee et al. [27] showed that the enhanced total ionic conductivity of LTAP (Li1.3Ti1.7Al0.3(PO4)3)-PTFE (polytetrafluoroethylene) composite electrolyte is 2.94 × 10−4 S/cm, and suggested that Li-ions migrate through the conducting ceramic powder.

Several experimental studies have demonstrated that Li-ions pass through ceramic particles, and not the interphase layer between ceramic and polymer. Zheng et al. [20] employed solid-state NMR to reveal that Li-ion migrates mainly through LLZO particles. Sun et al. [28] showed that Li-ion hopping in highly dense LLZO particles leads to enhanced ionic conductivity and transference number of CE, using X-ray photoelectron spectroscopy and Raman analysis. In addition, Dong et al. [29] revealed the low ionic conductivity at the interface between PEO polymer and Si/SiO2 substrates because of the formation of immobile interfacial zone. This indicates the possibility of restricted PEO polymer chain motion at the polymer/ceramic interface.

Unlike the different hypothesis related to Li-ion transport, it is universally accepted that incorporation of ceramic fillers leads to enhancement in the mechanical properties of the composite electrolytes [4, 30, 31]. The addition of only 20 to 50% volume fractions of oxide-based ceramic fillers (such as LAGP) leads to a ten-fold increase in the elastic properties for the composite material as compared to bare PEO [32]. However, this ten-fold increase still results in the mechanical stiffness two to three orders of magnitude smaller than that of lithium metal anodes. To stabilize the lithium deposition, solid electrolytes with elastic modulus close to, or larger than, that of lithium metal needs to be adopted [13, 33]. As a result, the adoption of composite electrolytes with small amount of ceramics appear to be insufficient for preventing dendritic growth [34]. Therefore, there remains a pressing need to understand how the change in volume fraction of the ceramic impacts the relevant transport and mechanical properties of the CE, and the optimum design of CEs to prevent dendrite in Li metal batteries needs to be determined.

In this work, we provide insights into these aspects using a two-dimensional (2D) mathematical model for a PEO-LLZO composite electrolyte. A Li|CE|Li symmetric cell is modeled, and different volume fractions of LLZO is examined. The potential drop at the interface is estimated using experimentally observed interfacial resistance values. Finally, we compare the total ionic conductivity and effective transference number with existing literature values. Furthermore, the mechanical strength of CEs is also calculated using the theory of linear elasticity to show how much LLZO particles within the CEs need to be added to inhibit Li dendrite growth.

Methodology

The focus of the study is to understand the effect of varying LLZO ceramic content in PEO polymer-based CEs on the total ionic conductivity, effective transference number, and mechanical strength. Based on the concentrated solution theory [35], a two-dimensional (2D) Li|CE|Li symmetric cell model is developed to describe the internal behavior of CEs. Figure 1 shows the schematic diagram of a symmetric cell having CE thickness, L. In Fig. 1a, Li-ion is produced at the Li metal electrode on the left (x = 0) and is consumed at the opposite electrode (x = L). In the PEO polymer phase, mass conservation and charge conservation are considered to obtain Li-ion transport and time-dependent potential distribution. In the solid phase of the LLZO particle, we only considered the charge conservation because there was no internal concentration polarization due to the transference number being unity. When current passes through CE, a potential gradient is generated in each of the different phases depending on the ionic conductivity of the electrolyte. Due to the lower ionic conductivity of PEO polymer compared to LLZO, a steeper potential gradient can be observed in the PEO polymer phase.

Following the junction potential theory [35], the PEO-LLZO interface is assumed to be a selective interface that only allows the lithium cations to be transferred to the LLZO phase among all the various ions available in the PEO polymer electrolyte. Therefore, a potential drop, called junction potential, occurs depending on the charge transfer resistance at the PEO-LLZO interface. To simplify the ion transport mechanisms of CE as mentioned above, the model adopted in this work follows several assumptions. All LLZO particles are single crystals of spherical shape, and their sizes are constant. In addition, there are no voids/pores and impurities in CEs. All transport properties, such as diffusion coefficient, ionic conductivity, transference number, and thermodynamic factor, are constant values depending on salt concentration. At the PEO-LLZO interface, only the interface resistance is considered as the charge transfer resistance, which is assumed to have a linear relationship with the junction potential. Other effects such as the self-conductivity of the interfacial layer, or the influence of the internal ion pairs near the PEO-LLZO interface, are not considered in this study. A more detailed explanation of the relationship between potential and applied current at the PEO-LLZO interface is covered in the “PEO-LLZO interfaces” section. We used the COMSOL Multiphysics V5.6 for this work.

The mechanical strength of the composite electrolytes is estimated via their elastic stiffness, particularly Young’s modulus. For calculating the elastic properties of the two-dimensional electrolyte domain, finite element-based techniques are adopted, where both the PEO polymer and the LLZO ceramics are assumed to behave as linearly elastic solids [34, 36]. Appropriate magnitudes of Young’s modulus for the individual polymer and ceramic phases are provided as input parameters [37–39]. Due to its inherent softness, the elastic stiffness of PEO is several orders of magnitude smaller than LLZO. The entire computational methodology is built on small-strain and small-displacement approximations [40, 41]. The schematic of the computational approach adopted to estimate the effective Young’s modulus is shown in Fig. 10 (see Appendix). The bottom and left sides of the electrolyte domains are constrained such that they can accommodate unrestricted movement along the horizontal (x-axis) and vertical (y-axis) directions, respectively. Tensile load is applied on the top boundary, and the corresponding stresses and strains at the top of the electrolyte are estimated. Young’s modulus of the composite electrolyte is the ratio between the stresses and strains at the top of the electrolyte domain, derived according to Hooke’s law. No interfacial imperfections between the polymer and ceramic electrolytes are assumed from a mechanical standpoint, which renders continuity in force and displacement across the PEO/LLZO interface. Due to the isotropy in the mechanical properties of the individual polymer and ceramic phases, and the uniform distribution of the LLZO fillers, the entire composite is assumed to behave as an isotropic solid, and the possibility of any anisotropy in their mechanical properties are completely neglected [30, 42, 43]. Detailed description of the governing equations used to estimate the stress distribution within the composite electrolyte domain will be discussed later.

PEO-LLZO interfaces

LLZO phase has a cation transference number of one. Therefore, the interface between PEO and LLZO allows only cations to penetrate. Figure 1a represents the potential difference between two different phases (called the junction potential, \({\phi }_{\mathrm{j}}\)) when Li-ions pass through the interface between PEO and LLZO [35].

$${\phi }_{\mathrm{j}}={\phi }_{2}^{P}-{\phi }_{2}^{L}$$
(1)

Note that the superscripts P and L indicate PEO phase and LLZO phase, respectively. Since the anions cannot pass through the interface, if we ignore the solvent, the chemical potential of the cations between two phases should be equal at equilibrium.

$${\mu }_{+}^{P}={\mu }_{+}^{L}$$
(2)

The chemical potential of the cations at each phase can be defined as follows:

$${\mu }_{+}^{P}=RT\mathrm{ln}{c}_{+}^{P}+{z}_{+}F{\phi }^{P}$$
(3)
$${\mu }_{+}^{L}=RT\mathrm{ln}{c}_{+}^{L}+{z}_{+}F{\phi }^{L}$$
(4)

where \({c}_{+}^{P}\), \({c}_{+}^{L}\) are salt concentrations in PEO and LLZO, respectively, \({z}_{+}\) is a charge number of cations, (\({z}_{+}\)=1), R is the gas constant, T is the temperature, and F is the Faraday constant. From the Eqs. (2)–(4), we can get the junction potential at the equilibrium condition.

$${\phi }_{2}^{P}-{\phi }_{2}^{L}=\frac{RT}{{Fz}_{+}}\mathrm{ln}\left(\frac{{c}_{+}^{L}}{{c}_{+}^{P}}\right)$$
(5)

If there is applied current at the interface, there can be additional potential differences due to interfacial resistance, \({R}_{\mathrm{if}}\). Here, we assumed a linear relationship between penetrated current density, \({i}_{\mathrm{if}}\), and the potential difference due to interfacial resistance is as follows.

$${\phi }_{2}^{P}-{\phi }_{2}^{L}={R}_{\mathrm{if}}{ i}_{\mathrm{if}}+\frac{RT}{{Fz}_{+}}\mathrm{ln}\left(\frac{{c}_{+}^{L}}{{c}_{+}^{P}}\right)$$
(6)

Then, we applied the Butler-Volmer equation for the current–potential relationship at the PEO and LLZO interface that can be written as:

$${i}_{\mathrm{if}}={i}_{0,\mathrm{if}}\left[\mathrm{exp}\left(\frac{F\overline{\eta }}{2RT}\right)-\mathrm{exp}\left(-\frac{F\overline{\eta }}{2RT}\right)\right]$$
(7)
$$\overline{\eta }= {\phi }^{C}-{\phi }^{P}-\Delta {\mu }_{eq}$$
(8)
$${i}_{0,\mathrm{if}}=\frac{RT}{{FR}_{\mathrm{if}}}$$
(9)
$$\Delta {\mu }_{eq}= \frac{RT}{{z}_{+}F}\mathrm{ln}\left(\frac{{c}_{+}^{P}}{{c}_{+}^{L}}\right)$$
(10)

where \({i}_{0,\mathrm{if}}\) is exchange current density, which is defined from the Tafel Eq. (9).

Governing equations for transport property estimation

The concentrated solution theory is used to describe the characteristics of a Li|CE|Li symmetric cell. The 2-D mass transport equation is used to demonstrate time-dependent concentration distribution across the PEO phase.

$$\frac{dc}{dt}=\nabla \cdot \left[D\left(1-\frac{d\mathrm{ln}c}{d\mathrm{ln}{c}_{0}}\right)\;\nabla c\right]-\frac{{i}_{2}\cdot \nabla {t}_{+,\mathrm{ PEO}}^{0}}{F}$$
(11)

where \(c\) is the salt concentration, \({c}_{0}\) is the solvent concentration, \(D\) is the diffusion coefficient of PEO, and \({t}_{+,\mathrm{ PEO}}^{0}\) is the transference number of PEO. We note that \(\frac{d\mathrm{ln}c}{d\mathrm{ln}{c}_{0}}\) is assumed to be zero in this study. Equation (11) can be solved with two boundary conditions at electrodes and one boundary condition at PEO-LLZO interfaces given below.

$${\left.-D\nabla c\right\vert}_{x=0}=\frac{(1-{t}_{+,\mathrm{ PEO}}^{0})}{F}{i}_{\mathrm{app}}$$
(12)
$${\left.-D\nabla c\right\vert}_{x=L}=-\frac{(1-{t}_{+,\mathrm{ PEO}}^{0})}{F}{i}_{\mathrm{app}}$$
(13)
$${\left.-D\nabla c\right\vert}_{ PEO-LLZO\; interfaces}=-\frac{(1-{t}_{+,\mathrm{ PEO}}^{0})}{F}{i}_{\mathrm{if}}$$
(14)

Charge conservation in CE under constant current conditions is expressed using modified Ohm’s law as

$$\nabla \cdot ({i}_{2,\mathrm{PEO}})= 0$$
(15)
$$\begin{aligned}{i}_{2,\mathrm{PEO}}=&\; -{\kappa }_{PEO}\nabla {\phi }_{2}^{P}+\frac{2RT{\kappa }_{PEO}}{F}\\&\left(1+\frac{\partial \;\mathrm{ln}\;{f}_{\pm }}{\partial\; \mathrm{ln}\;c}\right)(1-{t}_{+,\mathrm{PEO}}^{0})\nabla (\mathrm{ln}\;c)\end{aligned}$$
(16)
$$\nabla \cdot ({i}_{2,\mathrm{LLZO}})= 0$$
(17)
$${i}_{2,\mathrm{LLZO}}= -{\kappa }_{LLZO}\,\nabla {\phi }_{2}^{L}$$
(18)

where \({i}_{2,\mathrm{PEO}}\) and \({i}_{2,\mathrm{LLZO}}\) are the current densities by moving Li-ions in the PEO phase and LLZO phase, respectively. \({\kappa }_{PEO}\) and \({\kappa }_{LLZO}\) are the ionic conductivities of PEO and LLZO, respectively, and \(1+\frac{\partial \;\mathrm{ln}\;{f}_{\pm }}{\partial \;\mathrm{ln}\;c}\) is a thermodynamic factor, where \({f}_{\pm }\) is the mean activity coefficient and \(c\) is the concentration value. Since the transference number of LLZO was set to 1, the second term from the right of Eq. (16) will be zero. Thus, Ohm’s Law as shown in Eq. (18) is solved for the LLZO phase. Here, the constant current boundary conditions are applied at both electrodes and CE. One assumption here is that the Li metal electrodes and CE are in contact with only the PEO polymer, not the LLZO particles.

$${\left.{i}_{2,\mathrm{PEO}}\right\vert}_{x=0}= {i}_{\mathrm{app}}$$
(19)
$${\left.{i}_{2,\mathrm{PEO}}\right|}_{x=L}= -{i}_{\mathrm{app}}$$
(20)

To solve the above equations, \({\left.{\phi }_{2}^{P}\right|}_{x=L}\) is set to 0 V as a ground condition, so \({\left.{\phi }_{2}^{P}\right|}_{x=0}\) is considered as the working voltage of the Li–Li symmetric cell.

Governing equations for mechanical property estimation

The quasi-static equilibrium equations are solved for estimating the stress distribution within the polymer and ceramic phase of the composite electrolytes [44]:

$$\overset\rightharpoonup\nabla\cdot\boldsymbol\sigma+\overset\rightharpoonup b=0$$
(21)

Here, \({\varvec{\sigma}}\) indicates the stress tensor, \(\overset\rightharpoonup{\triangledown}\) is the gradient operator, and \(\overset\rightharpoonup{b}\) denotes the body force, which is assumed to be zero in the present context \((\overset\rightharpoonup{b}=0)\). The stresses are expressed in terms of strains \(\left({\varvec{\epsilon}}\right)\) through the constitutive relations [40]:

$$\mathbf{\sigma} =\mathbf{C}^{el}:\;\mathbf\epsilon$$
(22)

where \({{\mathbf{C}}}^{{\mathbf{el}}}\) indicates the elastic stiffness tensor. The elastic stiffness tensor is written as a function of Young’s modulus \(\left(YM\right)\) and Poisson’s ratio \(\left(\nu \right)\) of the individual phases [44]: 

$$C^{el}=\frac{YM}{\left(1+\nu\right)}\mathbb{I}+\frac{YM\cdot\nu}{\left(1-\nu^2\right)}\boldsymbol I\otimes\boldsymbol I$$
(23)

where \(\mathbb{I}\) is the symmetric part of the 4th rank identity tensor, and I denotes the 2nd rank identity tensor. Different values of \(YM\) and \(\nu\) are adopted for the polymer and the ceramic domains, which are provided in Table 1. The computational domain is constrained with appropriate boundary conditions to avoid any rigid body motion. Displacement along the x \(\left({u}_{x}\right)\) and y \(\left({u}_{y}\right)\) directions at the left and bottom boundaries, respectively, are constrained:

$${u}_{x}\left(x=0\right)=0$$
(24)
$${u}_{y}\left(y=0\right)=0$$
(25)

Force, \({P}_{y}\), is applied at the top boundary along the y-direction, which results in a displacement of \({\delta }_{y}\), in the y-direction. The stress \(\left({\sigma }_{yy}\right)\) and strain \(\left({\epsilon }_{yy}\right)\) along the y-direction are defined as:

Table 1 Summary of model parameters
Full size table
$${\sigma }_{yy}={P}_{y}/{L}_{x}$$
(26)
$${\epsilon }_{yy}={\delta }_{y}/{L}_{y}$$
(27)

Here, \({L}_{x}\) and \({L}_{y}\) are the lengths of the computational domain along the x-direction and y-direction, respectively. According to Hooke’s law, Young’s modulus of the composite electrolyte is estimated as the ratio between the stress and the strain:

$$Y{M}_{Electrolyte}={\sigma }_{yy}/{\epsilon }_{yy}$$
(28)

Due to the inherent isotropy in the mechanical properties of the electrolyte, elastic modulus along the x-direction is assumed to be the same as the y-direction, and not calculated separately. The detailed boundary conditions externally applied loads, and procedure to estimate the effective Young’s modulus for the composite electrolyte are also outlined in Fig. 10 (see Appendix).

Assumptions

In order to keep the modeling framework computationally tractable several assumptions are made, which limits the generalized application of the quantitative conclusions obtained from this study. However, the qualitative trends are general and should be applicable in 3D realistic polymer/ceramic composite electrolyte materials. Some of the assumptions are listed below:

  1. 1.

    The entire composite electrolyte is analyzed using a 2D (two-dimensional) modeling framework, instead of a more detailed 3D (three-dimensional) one. This assumption fails to capture several subtleties of the realistic microstructures, such as, smaller percolation thresholds for spheres in 3D as compared to circles in 2D [48].

  2. 2.

    The polymer/ceramic microstructures within the composite electrolyte are generated by randomly adding ceramic particles within the polymer matrix. Realistically, during the synthesis of composite electrolytes, distribution of the LLZO and/or LAGP particles should be determined by the ceramic-ceramic and ceramic-polymer surface energies [49]. If the ceramic-polymer surface energies are lower, a uniform distribution of ceramic particles, without any overlap, is desirable. If the ceramic-ceramic interaction energies are smaller than, or equivalent to, that of ceramic-polymer interaction energies, aggregation of ceramic particles is desirable.

  3. 3.

    Even though any experimentally manufactured particles demonstrate a log-normal size distribution [50], in the present context, either a uniform size distribution, or a constant size, of ceramic particles is assumed in order to simplify the microstructure generation process.

  4. 4.

    Inner overlap between the ceramic particles is not taken into consideration because the composite electrolytes are assumed to be fabricated through a simple mixing and casting procedure. Overlapping ceramic particles are only observed if the ceramics are sintered together prior to mixing with the polymer salt [51], which is not the case being analyzed here.

All these assumptions help to simplify the computational analysis, and leads to results that qualitatively correlate well with the realistic systems.

Results and discussion

Using the developed transport model, we performed the simulation study on the Li|CE|Li symmetric cells containing the 10 to 50% volume fraction of LLZO as shown in Fig. 1b. The cell thickness was set to 25 \(\upmu\)m, similar to the porous separator used in the conventional LIBs. The diameter size of all LLZO particles was set to be the same as 0.9 μm. It is also assumed that there is no inner overlap between LLZO particles, and no particle come in direct contact with Li metal. The operating temperature was set to 50 °C, and 0.2 mA/cm2 was applied for 6 min until reaching the steady-state condition. All model parameters used in this study are summarized in Table 1. A detailed description of the ceramic particle type and corresponding microstructure is also provided in Table 2.

Table 2 Description of LLZO particle microstructure adopted during the estimation of transport properties (conductivity and transference number) and mechanical properties (elastic modulus)
Full size table

Figure 2a shows the effective ionic conductivity of Li|CE|Li cells at LLZO volume fraction 0%, 10%, 20%, 30%, 40%, and 50%. To determine the effective ionic conductivity of CE, we used the line averaged value of the potential at (x = 0) at the initial time (t = 0) as the below equation.

Fig. 1
figure 1

a Schematic model description of a Li|CE|Li symmetric cell of a composite electrolyte considering the potential drop at the interface of PEO and LLZO, and b computational model domain with different volume fractions of LLZO

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Fig. 2
figure 2

The total ionic conductivity of CEs depends on the magnitude of interfacial resistance between LLZO particles and polymer and the volume fraction of LLZO

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$${\kappa }_{eff}=\frac{{i}_{\mathrm{app}}\cdot L}{{(\left.{\phi }_{2}^{P}\right\vert}_{x=0})}$$
(29)

Interestingly, the effective ionic conductivity decrease with increasing volume fraction of LLZO, for \({R}_{\mathrm{if}}\) other than 0.1 \(\Omega \cdot {\mathrm{cm}}^{2}\). In addition, it was confirmed that the model results with the \({R}_{\mathrm{if}}\) value of about 4 \(\Omega \cdot {\mathrm{cm}}^{2}\), obtained as a fitting parameter, showed similar results to the experimental study for PEO-LiTFSI/LLZO composite electrolyte [22]. So far, on a practical level, \({R}_{\mathrm{if}}\) can be reduced to about 2.4 \(\Omega \cdot {\mathrm{cm}}^{2}\) by removing impurities on the surface of ceramic powder [52]. The effective ionic conductivity was reduced by half compared to pure PEO polymer when the volume fraction was 40%.

To investigate the other characteristics of CEs according to \({R}_{\mathrm{if}}\) and LLZO volume fractions, we focus on the time-dependent internal behavior while applying a constant current to the Li|CE|Li cell. Figure 3 shows the results of the potential drop across the CE with time in symmetric cells with 10% and 40% volume fractions of LLZO and \({R}_{\mathrm{if}}\) of 1.0 \(\Omega \cdot {\mathrm{cm}}^{2}\) and 104 \(\Omega \cdot {\mathrm{cm}}^{2}\). The applied current density was set to 10.0 A/m2. In the modified Ohm’s law of Eq. (15), the cell potential, or potential drop, across the symmetric cell can be assumed to consist of two distinct terms: ion migration and ion diffusion. At time zero, the concentration gradient across the cell is negligible, so the cell potential depends only on the migration term. As the concentration gradient develops, the cell potential increases with time due to the effect of the diffusion term, and it reaches a steady-state condition after sufficient time. When the \({R}_{\mathrm{if}}\) is set to a small value of 1.0 \(\Omega \cdot {\mathrm{cm}}^{2}\), it is observed that the steady-state potential decreases slightly as the volume fraction of LLZO filler increases. Note that, for pure PEO cell, the cell potential at a steady state is about 76 mV, and so the potential drop of CE shows a smaller value than that of the pure PEO cell regardless of the volume fraction when the \({R}_{\mathrm{if}}\) is less than 1.0 \(\Omega \cdot{\mathrm{cm}}^{2}\). However, at a high value of \({R}_{\mathrm{if}}\) with 104 \(\Omega \cdot {\mathrm{cm}}^{2}\), the potential drop at steady-state increases significantly as the LLZO volume fraction is increased from 10% (64 mV) to 40% (240 mV). In addition, since the development of cell potential by diffusion is slower than that by migration, the time for cell potential to reach a steady state is delayed by about 180 s.

Fig. 3
figure 3

The time-dependent potential drop across the symmetric cell at different volume fractions of LLZO and interfacial resistance. The current density with 10.0 A/m2 was applied for 6 min

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Fig. 4
figure 4

a Contour plot for polarization of Li concentration in polymer phase and b plot of Δc depends on the magnitude of interfacial resistance between LLZO particle and polymer

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Fig. 5
figure 5

Potential gradient and current flow depend on the magnitude of interfacial resistance between LLZO particle and polymer at 40% of LLZO volume fraction at the steady-state condition: a Rif = 1 Ω ∙ cm2 and b the Rif = 104 Ω ∙ cm2

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Fig. 6
figure 6

Effective transference number of composite electrolytes with dependence on interfacial resistance and volume fraction

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At the steady state, the \({R}_{\mathrm{if}}\) affects the concentration gradient across the cell for the ion diffusion process. As the vol ume fraction of LLZO increases in the CEs, the tortuosity of the diffusion path increases. When the \({R}_{\mathrm{if}}\) value is large, Li-ions tend to transport through PEO instead of LLZO particles. In this case, Li-ion diffuses through the long pathway at a high-volume fraction of LLZO and that leads to lower effective diffusivity of CE. Figure 4a shows the contour plot of the concentration gradient of Li-ions at the steady state inside the two different Li|CE|Li cells as shown in Fig. 3. At the small \({R}_{\mathrm{if}}\) with 1 \(\Omega \cdot {\mathrm{cm}}^{2}\), the concentration gradient of both the cells is the same, around 600 mol/m3. (Fig. 4b) However, at the large value of \({R}_{\mathrm{if}}\) with 104 \(\Omega \bullet {\mathrm{cm}}^{2}\), a significant difference in the concentration gradient was observed at 10% and 40% volume fraction of LLZO (Fig. 4b). In this case, if a high current is applied to the cell, the Li-ion concentration can reach zero value, leading to cell failure. The current at which Li-ion concentration reaches zero is also called the limiting current, which is one of the important factors in electrolyte development. Overall, when the LLZO volume fraction is 10%, the concentration gradient, ∆c increases gradually with increasing interfacial resistance, but for 40% LLZO volume fraction, the concentration gradient increases drastically with increasing interfacial resistance.

To see whether the Li-ions pass through PEO or LLZO phases depending on the magnitude of \({R}_{\mathrm{if}}\) values, the potential distribution and current flow path are simultaneously represented in Fig. 5. The arrow shown in the enlarged part is the current flow vector, indicating the direction and magnitude of the flow of Li-ions. Figure 5a shows that when the \({R}_{\mathrm{if}}\) is low, Li-ions mainly pass through LLZO particles rather than the PEO. On the other hand, when the \({R}_{\mathrm{if}}\) is high, most Li-ions migrate mainly through the PEO rather than LLZO particles. In this case, the potential drop becomes as severe as about 244 mV at the steady state, as seen in Fig. 3.

Fig. 7
figure 7

Estimation of normalized effective Young’s modulus with increasing ceramic volume fraction within the PEO/LAGP composite electrolyte. a Predictions with the pristine elastic modulus of the polymer phase. Black solid line indicates the effective elastic modulus of the composite as predicted by the developed computational model. The black dash-dot and dash-dash lines indicates the effective modulus as estimated from the assumptions of uniformly distributed strain and stress, respectively, within the polymer and ceramic phase of the computational domain. The red cross symbols indicate experimentally observed numbers as obtained from existing literature [37]. b Blue solid line indicates effective Young’s modulus of the PEO/LAGP composite with the incorporation of a modified polymer with stiffness higher than the pristine one, which provides a good correlation with the experiments. The black line denotes the model predictions with pristine polymer modulus, which substantially underestimates the experimental observations. It is argued that incorporation of the ceramic particles hinders the polymer chain motion and leads to a mechanically stiffer polymer. The LAGP particle size is maintained between 300 and 500 nm (consistent with the experiments) for all the volume fractions being investigated here

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Fig. 8
figure 8

Effective Young’s modulus of the PEO/LLZO electrolyte with increasing volume fraction of LLZO. Decreasing size of the LLZO particles leads to an increase in effective modulus due to the formation of percolated networks with smaller LLZO particles at larger LLZO volume fractions (see Fig. 14 in the Appendix section)

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The cation transference number of the electrolyte is the indicator of how effectively cations are passing through the electrolyte compared to anions. Although the cation transference number of pure PEO is as low as about 0.11, it is expected that the effective transference number of CE can be improved when Li-ions mainly pass through the LLZO particles, which has a high transference number (~ 1). As shown in Fig. 11 (see Appendix), the effective transference number of CE is calculated from the ratio of the amount of ionic current flow of PEO and LLZO for cross-section lines at the CE center region using the equation shown below:

$${t}_{+}^{eff}=\frac{\frac{\sum \left({t}_{+,\mathrm{ PEO}}^{0}{\times i}_{P,i}+{{t}_{+,\mathrm{LLZO}}^{0}\times i}_{L,i}\right)}{\left({i}_{P,i}+{i}_{L,i}\right)}}{\#\; of \;cross \;section}$$
(30)

where the \(i\) is the index of the cross section. Figure 6 shows the total ionic conductivity and effective transference number of CEs at varying volume fractions of LLZO. Since most Li-ions tend to transport through the PEO at the high \({R}_{\mathrm{if}}\) with 10 [4] \(\Omega \cdot {\mathrm{cm}}^{2}\), the effective transference number as a function of the LLZO volume fraction is almost the same as that of PEO (\({t}_{+,PEO}^{0}\) = 0.11). On the other hand, at low \({R}_{\mathrm{if}}\) with 1 \(\Omega \cdot {\mathrm{cm}}^{2}\), the effective transference number is improved by ~ 4 times as the LLZO volume fraction is increased to 50%. This is because Li-ions mainly pass through the LLZO rather than the PEO phase. Similarly, Fig. 12 (see Appendix) shows the results when the diameter of the LLZO particle is about 4 μm. As the particle size of the LLZO increases, the total interface area between PEO and LLZO decreases at the same volume fraction of LLZO. Therefore, at the low interfacial resistance, the frequency of Li-ion crossing the PEO-LLZO interface decreases, resulting in the improved total ionic conductivity compared to the small particle size of LLZO. It is also observed that the effective transference number increases to 0.6 at the LLZO volume fraction of 50%. When the interfacial resistance is high, the effective ionic conductivity decreases more than in the case of small LLZO particles due to an increase in the tortuosity in CE with large LLZO particles. According to the above results, it is not discernible whether the effective ionic conductivity can be improved through the formation of percolated network of LLZO particles.

Fig. 9
figure 9

Phase map between normalized electrolyte modulus and transference number indicating the domains where lithium dendrites should grow (yellow region) and where stable deposition (green/turquoise region) of Li is possible. The computational model used to generate this phase map have already been described in earlier publications (Barai et al. PCCP 2017 20,493 and Barai et al. JES 2018 A2654). The Young’s modulus and transference numbers of presently available PEO/LLZO electrolytes are denoted by the blue, red, and black lines, which corresponds to LLZO particles of diameter 4 μm, 900 nm, and 150 nm, respectively. One obvious way to achieve stable deposition of Li with PEO/LLZO composite electrolytes is to decrease the interfacial resistance between the PEO polymers and LLZO ceramics, and increase the volume fraction of LLZO within the composite electrolyte above 40%. Location of a PEO/75vol%LLZO composite electrolyte is pointed out by the cyan square

Full size image

A computational framework is developed to estimate the effective elastic properties of polymer/ceramic composite electrolytes by solving the set of governing equations provided in the Methodology section (see Eqs. (21)–(28)). Even though the main focus is on the estimation of the effective elastic modulus of PEO/LLZO composites, model validation is conducted using a PEO/LAGP composite due to the availability of systematic experimental data in the literature [32]. The comparison between the experimentally observed and computationally predicted effective elastic modulus of PEO/LAGP composite electrolytes are demonstrated in Fig. 7a by the red cross symbols and black lines, respectively. The experimental results are available for LAGP volume fractions from 0.0 to 0.38, where approximately fivefold increase in the elastic properties of the composite is observed as compared to bare PEO. The solid black line in Fig. 7a indicates effective Young’s modulus as predicted by the developed finite element-based computational framework, which substantially underestimates the experimentally observed numbers, especially at higher volume fractions of LAGP. Note that no interfacial imperfections between the polymer and ceramics are assumed in the developed model, which means the forces and displacements at the polymer/ceramic interfaces are continuous. To be consistent with the experiments, the size of the ceramic particles is maintained between 300 and 500 nm (detailed description of the size and size distribution of the LAGP particles is provided in Table 2) [32]. All the electrolyte modulus values reported in this study are normalized by the elastic properties of metallic lithium. The black dash-dot line in Fig. 7a indicates the effective elastic properties as predicted by the rule of mixture while assuming uniform distribution of strain within the polymer and ceramic domains, which provides an upper limit for the effective properties [43, 53]. On the other hand, the black dash-dash line in Fig. 7a indicates the effective elastic properties while assuming uniform distribution of stress within the polymers and ceramics, which provides a lower limit to the effective electrolyte modulus under the approximation of zero interfacial imperfections [53]. Note that the developed computational methodology is capable of predicting the effective properties of polymer/ceramic composites that contains any amount of filler particles, and not limited by any dilute approximations. Hence, in Fig. 7a, effective elastic modulus of the composite electrolytes is reported for ceramic volume fractions ranging from 0.0 to 1.0, which also resides well within the Voigt (upper) and Reuss (lower) bounds.

As shown in Fig. 7a and the solid black line in Fig. 7b, under the assumption of mechanically perfect polymer/ceramic interfaces, the magnitude of computationally predicted Young’s modulus is substantially lower than the experimental observations. Hence, incorporation of imperfections at the polymer/ceramic interface in terms of discontinuity in force and/or displacements will not help to obtain better comparison with the experiments, because these imperfections tend to further lower the effective properties [54]. Certain mechanisms need to be adopted in the computational framework that helps to enhance the elastic properties of the composite electrolytes. Earlier studies have adopted the existence of interphase layers with finite thickness between the polymer and the ceramic particles which is expected to demonstrate higher elastic properties than bare PEO [34, 55, 56]. In the present context, existence of no such interphase layers is considered while solving the transport equations, which should be reflected in the mechanical analysis as well. Accordingly, it is assumed that the presence of the ceramic particles hinders the movement of the polymer chains, and effectively enhances the mechanical properties of the entire polymer domain [29, 57]. This minor change in polymer chain motion is expected to not influence the ionic transport properties within the polymer. Increment in polymer modulus depends on the amount of ceramic present within the composite because the presence of more ceramic particles should exert enhanced hindrance to the polymer chain motion. To get a good correlation with the experimentally observed elastic modulus of the PEO/LAGP composites, the necessary increment in polymer modulus is demonstrated in Fig. 13 (see Appendix) by the cyan squares as a function of the ceramic volume fraction. It is evident that increasing ceramic volume fraction results in larger increase in polymer modulus. The solid black line if Fig. 13 (see Appendix) provides a model fit that can capture the increase in polymer modulus with increasing ceramic fraction:

$$\frac{Y{M}_{Polymer,Modified}}{Y{M}_{Polymer,0}}=1+\left(A\cdot \mathrm{exp}\left(-m\cdot v{f}_{Ceramics}\right)\right)$$
(31)

Here, \(Y{M}_{Polymer,Modified}\) and \(Y{M}_{Polymer,0}\) indicate the modified and pristine Young’s modulus of the polymer phase, \(v{f}_{Ceramics}\) is the volume fraction of the ceramic particles, and \(A\) and \(m\) are the fitting parameters. The best fit with experimental results is obtained with \(A=0.65\) and \(m=7\). Incorporation of the ceramic volume fraction-dependent modified elastic modulus of the polymer phase provides a better correlation with the experimental results, which is shown in Fig. 7b by the blue solid line. The ceramic volume fraction in Fig. 7b is varied from 0.0 to 0.4 (instead of 0.0 to 1.0) in order to highlight the initial differences between the experimental and computational results, and improvement in the model predictions through the adoption of stiffer polymers. In the following study, a similar ceramic volume fraction-dependent increase in Young’s modulus of the polymer phase will be assumed. Note that the increment in polymer stiffness can very well be a function of the type of ceramic particles, for example, the correlation obtained for LAGP fillers may not be strictly applicable to PEO/LLZO composites [58]. Also, the size and size distribution of the ceramic particles can substantially impact the enhancement in polymer modulus [18]. However, due to the lack of experimental data, the correlation reported in Eq. (31) will be used in the subsequent analysis for LLZO fillers with different particle sizes.

Once the developed computational model for estimating mechanical properties is validated against realistic experimental data (with PEO/LAGP composites), all the subsequent analysis will be conducted on PEO/LLZO composite electrolytes, which is the main focus of the present study. Various size and size distribution of LLZO particles have been used in the existing literature for constructing composite electrolytes, which ranges from tens of nanometers to tens of microns, demonstrating significant influence on the overall conductivity [18, 26, 30, 31]. Such a wide variation in the filler particle size is expected to show some influence on the mechanical properties of the PEO/LLZO composites as well. Accordingly, the effective electrolyte modulus is plotted in Fig. 8 as a function of LLZO volume fractions for different sizes of the filler particles ranging between 150 nm to 4 μm. Unlike the PEO/LAGP composites, where the size of the filler particles ranged between 300 and 500 nm, for PEO/LLZO composite electrolytes, a constant magnitude of the LLZO particle size is considered. It is evident that decreasing the LLZO particle size helps to achieve faster increase in electrolyte modulus. For example, incorporation of 40% volume fraction of LLZO would result in fourfold increase in electrolyte modulus as compared to bare PEO if 4-μm sized LLZO particles are used. On the other hand, addition of the same 40% volume fraction of LLZO particles of size around 150 nm results in a tenfold increment in electrolyte elastic properties. To understand the reason behind this extra enhancement in mechanical properties with smaller LLZO particles, the PEO/LLZO composite microstructures are shown in Fig. 14 (see Appendix) for three different filler particle sizes of 900 nm, 400 nm, and 150 nm. The volume fraction of LLZO is maintained constant at 0.4 for all the three microstructures. It can be concluded from Fig. 14 (in the Appendix) that formation of mechanically percolated networks becomes evident with decreasing size of the LLZO particles. Percolated network of LLZO particles can efficiently carry the externally applied load, which results in higher magnitudes of effective Young’s modulus [59]. Hence, adoption of smaller LLZO particles with the propensity of forming mechanically percolated networks can be beneficial from the elastic stiffness standpoint. Please note that formation of a mechanically percolated network of LLZO does not necessarily indicate that the LLZO particles will also demonstrate electrochemical percolation. Also, note that in the present study, the LLZO pa rticles are assumed to be uniformly distributed within the PEO matrix irrespective of the size of the filler particles. This is not strictly true in realistic materials because the smaller LLZO particle may prefer formation of aggregates to minimize their surface energy. Formation of these agglomerates tend to destroy the percolated LLZO microstructures, and the subsequent enhancement in transport and mechanical properties are lost [31]. However, due to the lack of any such detailed experimental data, the enhancement in mechanical properties obtained by lowering the LLZO particle size, as predicted by the present calculations, will be used in the subsequent analysis.

The main focus of this study is to investigate whether the adoption of PEO/LLZO composite solid electrolytes will really help to stabilize the lithium deposition process under high current densities by mitigating the dendrite formation and growth. To elucidate the impact of the mechanical and transport properties of solid electrolytes in stabilizing the lithium deposition process, the authors previously published detailed computational analysis of lithium deposition mechanism on dendritic protrusions and developed phase maps indicating zones of dendrite growth and stable deposition [13, 60]. Along with charge and mass transport within the solid electrolytes, mechanical force balance was also taken into consideration while generating the stability maps [13]. Lithium deposition at the electrode electrolyte interface were assumed to follow the Butler-Volmer reaction kinetics, which depends not only on the potential and salt concentration, but also on the mechanical stress state of the electrode and the electrolyte domains [33]. A similar phase map between the electrolyte modulus and transference number is shown in Fig. 9, where the yellow domain indicates possibility of dendrite growth, and the green/turquoise region indicates stable deposition of lithium. It is evident from the phase map that increasing either Young’s modulus of the electrolyte, or the transference number, or both, can help to stabilize the lithium deposition process. The stability map is divided into three domains which are associated with the elastic and plastic deformations of lithium and the composite electrolyte [13, 60]. The intermediate domain, where plastic flow of lithium occurs along with elastic deformation of the electrolyte, substantially helps to suppress the dendritic protrusions, and eventually stabilizes the lithium deposition process. Please note that the entire phase map is constructed assuming no microstructural heterogeneity within the solid electrolytes. Realistically, at a microscopic length scale, the composite electrolyte contains PEO and LLZO domains, which shows very different mechanical and transport properties capable of creating inhomogeneous distribution of reaction current at the electrode/electrolyte interface [61, 62].

The experimentally investigated PEO/LLZO composite electrolytes usually contain very small (~ 2% volume fraction) to approximately 50% volume fraction of ceramic fillers [18, 32]. The size of the filler particles can vary significantly from tens of nanometers to several microns [18, 26, 31, 58, 63]. The computationally predicted Young’s modulus and transference number of these polymer/ceramic composite electrolytes are shown in Fig. 9 by the blue, red, and black lines for LLZO particle sizes of 4 μm, 900 nm, and 150 nm, respectively. The filler particle fraction increases from 0.0 to 0.4 along each of the lines. Negligible interfacial resistance is assumed between PEO and LLZO while estimating the effective transference numbers for the composite electrolytes \(\left({R}_{\mathrm{if}} \sim 1\;\Omega \cdot {\mathrm{cm}}^{2}\right)\). It is evident that bare PEO resides within the zone of dendrite growth, which is supported by multiple literature showing lithium dendrites within PEO solid electrolytes [64, 65]. However, even with 40% volume fraction of ceramic particles, it is difficult to reach the zone of stable deposition, which is true irrespective of the size of the ceramic particles. Interestingly, increasing the volume fraction of LLZO to 75% can help to achieve stable lithium deposition, which is highlighted by the cyan square symbol. Due to the difficulties associated with packing such large volume fractions of LLZO within the PEO matrix with uniform particle size, a size distribution of LLZO is approximated in the present study, which is consistent with the experimentally observed log-normal distributions [50]. The average particle is around 2 μm in this particular sample with 75% volume of LLZO. Enhancement in the electrolyte modulus with 50%, or more, volume fraction of LLZO is depicted in Fig. 15 (see Appendix), where incorporation of 80% LLZO volume fraction leads to a 32-fold increase in elastic modulus as compared to that of bare PEO. Please note that the enhancement in transference number reported in Fig. 9 is subjected to the assumption that the interfacial resistance between PEO and LLZO is negligibly small, which has not yet been achieved using experimental means [22]. Experimentally observed interfacial resistance between PEO and LLZO \(\left(\sim 100\;\Omega \cdot {\mathrm{cm}}^{2}\right)\) leads to minimal ion transfer between the polymer and ceramic domains and render negligible increase in transference numbers [17]. Hence, to enable stable deposition of lithium, the following two aspects needs to be improved for every PEO/LLZO composite electrolytes:

  1. (a)

    Decrease interfacial resistance between the PEO and LLZO.

  2. (b)

    Increasing the LLZO volume fraction within the PEO matrix.

It is evident from the present study that lowering the charge transfer resistance between PEO and LLZO to something around \(1\;\Omega \cdot {\mathrm{cm}}^{2}\) (see Figs. 6 and 12 (in Appendix)), and increasing the LLZO volume fraction to 60–70% through the adoption of multimodal particle sizes, can possibly help to stabilize the lithium deposition with composite electrolytes.

As alluded to in the earlier paragraph, the phase map between elastic modulus and transference number shown in Fig. 9 assumes a homogeneous interface between the Li metal electrode and the solid electrolyte. However, in a real polymer/ceramic CE, there exist two distinct domains, the polymer domain and the ceramic region, which demonstrates very different mechanical and transport properties. The presence of such a heterogeneous electrolyte right on top of the Li metal anode can substantially alter the current distribution at the electrode/electrolyte interface, which can eventually lead to nucleation of dendrites. To understand the inhomogeneity in reaction current density due to the presence of the polymer and ceramic domains adjacent to the Li electrode, and their influence on the overall dendrite growth, more detailed study needs to be conducted while focusing specifically on the distribution of reaction current at the interface between Li metal and the heterogeneous polymer/ceramic CE.

Conclusion

In this work, the transport and mechanical properties of PEO-LLZO CEs were estimated via continuum modeling at different LLZO volume fractions ranging from 10 to 50% and interfacial resistances from 0.1 to 104 \(\Omega \cdot {\mathrm{cm}}^{2}\). Interfacial resistance plays a critical role in enabling Li-ion transport through highly conductive ceramic particles. Existing experimentally observed interfacial resistance values \((\sim 1\;\Omega \cdot {\mathrm{cm}}^{2}\)) negatively impact the total ionic conductivity as LLZO content increases in the CEs. Besides the interfacial resistance, in the presence of LLZO particles transport of Li-ions through the tortuous path in the PEO phase decreases overall ionic conductivity. At low interfacial resistance values \((\sim 1\;\Omega \cdot {\mathrm{cm}}^{2}\)), Li-ions prefer to travel through LLZO particles, and the overall transference number of CE increases to 0.6 at 50% LLZO volume fraction. Some strategies to improve the ionic conductivity of CEs are to minimize the interfacial resistance between PEO and LLZO and/or decrease the overall interfacial area by increasing the LLZO particle size.

Interestingly, the mechanical properties (Young’s modulus) of the CE demonstrates a monotonic improvement with incorporation of LLZO based ceramic fillers. Enhancement in the elastic stiffness of the polymer phase needs to be adopted to obtain good correlation with the existing experimental results [32]. Ceramic particle induced hindrance to the polymer chain motion is attributed as the main reason behind the increase in the elastic properties of PEO [29, 56, 59]. Also, smaller LLZO particles are assumed to form mechanically percolated networks capable of carrying enhanced amount of mechanical load that can eventually lead to superior elastic properties for the PEO/LLZO composites [59]. However, this conclusion does not directly correlate with the trend observed for the transport properties, where adoption of larger ceramic particles are beneficial because it helps to minimize the transport through the highly resistive PEO/LLZO interface. Hence, adoption of multimodal particle size may be beneficial to improve both the transport and mechanical properties. Also, multimodal particle sizes, or very wide distribution of LLZO particles, can help to fit large volume of ceramics within the CE, which can eventually be able to reach the domain where stable deposition of lithium is possible.