A one-dimensional model of a Li/LiPON thin-film micro battery was developed by Fabre et al. [18] and is depicted schematically in Figure 1a. The x-axis, which points out the characteristic lengths of the model is directed from the negative towards the positive electrode; the interface between the negative electrode and the solid electrolyte hosts its origin.

Authors aimed at keeping their model as simple as possible, by introducing proper assumptions in order to come up with a reduced set of input parameters that can be measured from purposely designed experiments. The model is isothermal with no self-heating, neglects volume alterations during charge/discharge, and assumes that redox activities occur solely on the surface area, which remains unchanged throughout cycling. The negative electrode is a metallic film of lithium with negligible Ohmic losses. The ionic transfer in the solid electrolyte is described by a single ion conduction model: as such, the concentration of lithium ions across the solid electrolyte ${\mathrm{Li}}^{+}$ is uniform. This property straightforwardly comes out as long as electro-neutrality approximation holds, a property of the governing equations largely discussed in [22,24,25].

Accordingly, the model traces lithium ${\mathrm{Li}}^{\oplus }$ diffusion and electron $e^{-}$ migration in the positive electrode, single ions ${\mathrm{Li}}^{+}$ migration in the solid electrolyte, and charge-transfer kinetics at the electrode/electrolyte interfaces. Three unknown fields are required, the concentration of lithium in the positive electrode $c_{{\mathrm{Li}}^{\oplus }}(x,t)$, the electric potentials ${\varphi }_e(x,t)$ and ${\varphi }_c(x,t)$ in the solid electrolyte and in cathode, respectively.

The electric potentials in the positive electrode ${\varphi }_c$ and in the solid electrolyte ${\varphi }_e$ are related to the current densities ${\overrightarrow{i}}_c(x,t)$ and ${\overrightarrow{i}}_e(x,t)$ by means of Ohm’s law, through the electrical $k_c$ and ionic $k_e$ conductivities, respectively:

The third governing equation in the positive electrode is a planar solid-state diffusion equation, which describes the (neutral, in the sense detailed beforehand of shielded positive ion) transport of lithium ${\mathrm{Li}}^{\oplus }$ in the electrode: with diffusion coefficient ${\mathrm{D}|}_{{\mathrm{Li}}^{\oplus }}$. A concentration-dependent ionic diffusion coefficient was studied in [18], which resulted in more accurate outcomes.

Denoting with $\overrightarrow{n}$ the outward normal to the surface at boundary, the two boundary conditions required for Equation (4c) are the lithium flux density at the reaction surface and the zero-flux condition for lithium ${\mathrm{Li}}^{\oplus }$ at the electrode/collector interface:

Conditions for ${\varphi }_e$ and ${\varphi }_c$, either at the anode-electrolyte ($a/e$) or at the electrolyte-cathode ($e/c$) interfaces are modeled via Butler-Volmer equations in Fabre et al. [18], in the form with s either $a/e$ or $e/c$ and $\alpha$ the so-called anodic (cathodic) charge transfer coefficient, usually both taken to be equal to $0.5$. The overpotential is the difference between the jump $〚\varphi 〛$ of the electric potential at the electrolyte/electrode interface (always defined as the electrode potential minus the electrolyte potential) and the open circuit potential (OCP) measured experimentally or calculated theoretically as in [33]. The exchange current $i_{a/e}^0$ is given by: where $k_2$ and $k_{-2}$ are the forward and backward reaction rate constants for the reaction (2), $c_{{\mathrm{Li}}^{+}}^{sat}$ is the saturation concentration of lithium in the electrolyte. The exchange current $i_{e/c}^0$ is given by: where $k_1$ and $k_{-1}$ are the forward and backward reaction rate constants for the reaction (1) and $c_{{\mathrm{Li}}^{\oplus }}^{sat}$ represents the saturation lithium concentration within the cathode. Replacing $c_{{\mathrm{Li}}^{+}}(0,t)$ with the initial, uniform concentration of lithium in the electrolyte, the exchange current densities $i_{a/e}^0\left(t\right)$ and $i_{e/c}^0\left(t\right)$ simplify as where $k_{-}^{app}$ and $k_{+}^{app}$ are apparent rate constant for negative electrode and defined as the multiplication of the terms in Eqs. (9) and (10).

Galvanostatic boundary conditions are eventually imposed, where $i_{bat}$ is the given galvanostatic current flowing across the 1D battery. The potential is set arbitrarily to ${\varphi }_e(0,t)=0$.

Landstorfer and coworkers studied in [19] a non-porous electrode and a crystalline solid electrolyte. As in [18], they assumed a solid electrolyte with one mobile species (${\mathrm{Li}}^{+}$) and a uniform concentration of vacancies $c_{{\mathrm{n}}^{-}}$ that remains unaltered in time. The model entails a novel view of the electrode/solid electrolyte interface, which consists of an intermediate layer and a space charge region within the electrolyte. A visual representation of the model is given in Figure 1b.

Once ions randomly intercalate in the lattice structure of graphite or ${\mathrm{LiCoO}}_2$, a simple Fickian law diffusion analogous to Eq. (4c) accounts for ionic transport. The electric potential ${\varphi }_c$ is considered to be uniform within the electrodes, neglecting the ohmic loss (depicted in [18] via Eq. (4b)), while ${\varphi }_e$ influences the ionic transport in the electrolyte, ruled by non-equilibrium thermodynamics. The mass balance law relates the concentration of lithium $c_{{\mathrm{Li}}^{+}}(x,t)$ to the actual flux of cations ${\overrightarrow{h}}_{{\mathrm{Li}}^{+}}(x,t)$

Using the standard linear relationship of Onsager type2 between the flux ${\overrightarrow{h}}_{{\mathrm{Li}}^{+}}$ and the gradient of the electrochemical potential ${\overline{\mu }}_{{\mathrm{Li}}^{+}}$, the Clausis-Duhem inequality is satisfied a priori and thermodynamics consistency is granted, as largely discussed in [24,25]. The chemical potential is the functional derivative of the Gibbs free energy (or Helmholtz free energy according to [24,25]) with respect to the ionic concentration $c_{{\mathrm{Li}}^{+}}(x,t)$. The splitting of chemical and electrical potentials used in [19] is classical, as it is the free energy of mobile guest atoms interacting with a host medium, described by a regular solution model [36,37]. The electrochemical potential was eventually derived as the sum of the chemical and electrostatic potential as follows, with3 ${\theta }_{{\mathrm{Li}}^{+}}=c_{{\mathrm{Li}}^{+}}/c_{{\mathrm{Li}}^{+}}^{sat}$ . The mass flux to be inserted into the mass balance law (13) eventually holds:

The classical Nernst-Planck flux4 is attained if the dilute limit is assumed $1-{\theta }_{{\mathrm{Li}}^{+}}\sim 1$ and the energy of interaction vanishes $\chi |=0$.

The whole electrolyte is thought as consisting of a space charge region and a bulk region, as shown in Figure 1b. In the bulk region electroneutrality is imposed a priori, $c_{{\mathrm{Li}}^{+}}$ is defined by the (given and uniform) concentration of vacancies, and Ohm’s law (4a) allows recovering the electric potential. On the contrary, in proximity of the interfaces, where the concentrations of cations and anions differ, the electric potential is obtained by Gauss law, which provides, after constitutive prescriptions, the following Poisson equations: where ${\epsilon }_r$ denotes the relative permittivity of the electrolyte and ${\epsilon }_0$ the vacuum permittivity. Local electroneutrality is not enforced in the space charge region [19], rather a weak (i.e. global) electroneutrality is prescribed in the whole electrolyte thus allowing local deviations between cation and anion concentrations while keeping the overall amount of cations and anions equal.

The interface between the electrodes and the solid electrolyte has been modelled as consisting of two intermediate layers, treated as plate capacitors in terms of potential jumps and flux continuity. Authors borrowed from [42] the constitutive equation for potential jumps (defined as the potential at an electrode minus the one at the electrolyte) as with given surface capacitances $C_a$, $C_c$ and permittivities ${\epsilon }_a$, ${\epsilon }_c$.

Boundary conditions on fluxes at electrode - solid electrolyte interfaces resemble Butler-Volmer equations (7) in a form originally presented in [43] and named generalized Frumkin-Butler-Volmer equations. They read: where the reaction rate constants defined in reactions (1) and (2) are taken of Arrhenius type, i.e., $k_n={\tilde{k}}_n{e}^{E_n/RT}$, $n=1,2$. The Gibbs energies of activation $▵G_n$ are further parameters of the model and relate to the OCP in Butler-Volmer equations.

More complex one-dimensional mathematical models have been proposed in a series of publications from Notten’s group [26,44,45] for a micro-battery $Li/LiPON/LiCo{O}_2$. In these studies, ionic transport in the solid electrolyte involves the ionization reaction (3) of immobile, oxygen-bound lithium $\mathrm{LiO}$ to mobile ${\mathrm{Li}}^{+}$ ions and negatively charged vacancies. Charge-transfer kinetics at both electrode/electrolyte interfaces, diffusion and migration of mobile lithium ions in the electrolyte (${\mathrm{Li}}^{+}$) and positive electrode (${\mathrm{Li}}^{\oplus }$) were accounted for. In their most recent work, [20], additional features have been introduced, such as mixed ionic/electronic conductivity in the positive electrode, electrical double layers occurring at both electrode/electrolyte interfaces representing the space-charge separation phenomena that differ from [19], variable ionic and electronic diffusion coefficients that depend on the lithium concentration inside the positive electrode.

Figure 1c displays a discharge process. The anode consists of a lithium $\mathrm{Li}$ foil, the cathode of a ${\mathrm{LiCoO}}_2$ film, while LiPON is used as the electrolyte material. The current collector is tied from the top of the ${\mathrm{LiCoO}}_2$. As for [18] and [19], also this model is isothermal (no self-heating) and redox processes only occur at the interfaces between the electrolyte and the electrode layers; volume changes of the electrolyte during cycling are neglected and the active surface area does not change over cycling.

As a distinctive feature of this class of models, the ionic transfer in the solid electrolyte is not described by a single ion conduction model and the concentration of lithium ions through the solid electrolyte is generally not uniform even though electro-neutrality approximation holds. Denoting with $c_{{\mathrm{Li}}^{+}}$ the concentration of mobile ${\mathrm{Li}}^{+}$ ions, with $c_{\mathrm{LiO}}$ the concentration of immobile lithium, with $c_{{\mathrm{n}}^{-}}$ the concentration of uncompensated negative charges, and with $\delta$ the fraction of Li at equilibrium, the equilibrium concentration of the charge carriers is $c_{{\mathrm{Li}}^{+}}^{eq}=c_{{\mathrm{n}}^{-}}^{eq}=\delta c_0$ and the concentration of the remaining immobile lithium is $c_{\mathrm{LiO}}^{eq}=(1-\delta )c_0$. The overall rate of the charge carrier generation according to reaction (3) is

The ratio is the equilibrium constant of reaction (3) and is related to the the fraction of Li at equilibrium $\delta$, see [20]. In addition to earlier models, two electrical double layer capacitances $C_{a/e}^{dl}$ and $C_{e/c}^{dl}$ and a geometric capacitance $C_{geo}$ were introduced in [20]. As in [19], double layer capacitances attempt at capturing the response of the space-charge very narrow layers as for electric capacitors. Whereas the concept resembles [19], the implementation is different. Capacitors in [19] are described “in series”, whereas in [20] “in parallel” (compare Figures 1b and 1c). In view of this assumption, the current at electrode/electrolyte interfaces splits into two terms, the faradaic contribution that drives the reduction/oxidation charge-transfer reactions (1)-(2) ($i_s^{ct}$) and the non-faradaic contribution that feeds the double layer ($i_s^{dl}$), with s either $a/e$ or $e/c$.

During charging, the ${\mathrm{Li}}^{+}$ ions released from the positive surface must cross the solid electrolyte and are reduced into metallic Li at anode. Electrons, generated by the charge-transfer reactions (1) and (2), flow across the Li foils and the electronic collector, with potential drops that follow Ohm’s law similar to Eq. (4a). Transport of ionic concentrations5 $c_{{\mathrm{Li}}^{+}}$ and $c_{{\mathrm{n}}^{-}}$ in the solid electrolyte is ruled by the mass continuity equation (13), properly extended as in order to account for the reaction rate $w^{\left(3\right)}$ (see Eq. (23)). The generic mass flux ${\overrightarrow{h}}_{\alpha }$, with $\alpha ={\mathrm{Li}}^{+},{\mathrm{n}}^{-}$, is constitutively described by the Nernst-Planck law (18), which carries the electric potential as a further unknown field. Coupling with an additional relation is thus mandatory, to model the migration process. The most common selection for such an additional relation in battery modeling is the electro-neutrality condition

By substituting eq. (18) into eq. (25) and subtracting eq.(25a) from eq.(25b), two independent partial differential equations eventually arise:

To be solved they require the initial concentrations for $c_{{\mathrm{Li}}^{+}}$ and the Neumann conditions on fluxes at the left and right boundaries of the electrolyte

In the positive electrode a mixed ionic/electronic conductivity is considered. The mass balance equations that characterize the transport of ${\mathrm{Li}}^{\oplus }$ ions and electrons ${\mathrm{e}}^{-}$ are similar to Eqs. (25) for the solid electrolyte and write

The generic mass flux ${\overrightarrow{h}}_{\alpha }$, $\alpha =\{{\mathrm{Li}}^{\oplus },{\mathrm{e}}^{-}\}$ is constitutively described by the Nernst-Planck law (18). This choice of independent motion of electrons and ionic intercalated lithium makes the governing equations different from Fabre’s [18] equations ((4c) to be compared with (30a)).

Additionally, the model from [20] makes the positive electrode’s diffusion coefficients dependent on the concentration of ions. Experimental results show that the local electrochemical environment has a major impact on solid-state diffusion. The model further exploits the electro-neutrality approximation inside the cathode, implying $c_{{\mathrm{Li}}^{\oplus }}(x,t)=c_{{\mathrm{e}}^{-}}(x,t)$, and a space-time proportionality of the diffusion coefficients for ${\mathrm{Li}}^{\oplus }$ and ${\mathrm{e}}^{-}$. Classical mathematical passages allow to retrieve Eq. (4c) as the single PDE required to model the mass transport in the electrode, provided that the diffusivity ${\mathrm{D}|}_{{\mathrm{Li}}^{\oplus }}$ is replaced by a suitable combination of electron and ionic diffusivities.

The initial concentration of the charge carrier at $t=0$, when no concentration profile developed yet, is equal to its equilibrium concentration

Neglecting the impact of geometric capacitance, the Neumann boundary conditions related to fluxes at both the left and right boundaries of the electrolyte result in: with $i_{bat}\left(t\right)$ the given galvanostatic current flowing across the 1D battery, defined in Equation (12).

The non-faradaic current (dis)charging the electrical double layers $i_{\alpha }^{dl}\left(t\right)$, can be defined in derivative form as where the jump $〚\varphi 〛$ of the electric potential at the electrolyte/electrode interface is always defined as the electrode potential minus the electrolyte potential and $\alpha =a/e,e/c$. Equation (33) shall be compared with (21) in [19].

The faradaic current proposed in [20] emanates from charge transfer kinetics, in a form that extends Butler-Volmer equation (7) to the mass transfer-influenced conditions [46]. The expression of $i_{a/e}^{ct}$ at the metallic lithium electrode interface reads where ${\overline{c}}_{{\mathrm{Li}}^{+}}$ is the average bulk concentration of species ${\mathrm{Li}}^{+}$, ${\overline{c}}_{Li}$ is the bulk activity of the metallic Li, ${\alpha }_a$ is the charge transfer coefficient for reaction eq. (2), ${\eta }_a$ is the overpotential (8) of the charge transfer reaction at the negative electrode, and the exchange current $i_{a/e}^0$ is given by: with $k_2$ the standard rate constant for reaction eq. (2) . The expression of $i_{e/c}$ at the positive electrode interface is given by the Butler-Volmer equation (7), with the exchange current density with $k_1$ the standard rate constant for reaction eq.(1). The reader may refer to [20] for further details on these equations and on the geometric capacitance.

A model that accounts for two mechanisms of ionic conduction was recently published in [21]. Rooted in the thermo-mechanics of continua, the model builds upon the work of Raijmakers et al. [20] to enhance the description of vacancy replenishment in a LIPON solid electrolyte and applies to LLZO as well [47] according to Figure 3. The equations that depict ionic transfer in [21] are multi-scale compatible, too. This feature seems to be particularly pertinent for composite cathodes, as indicated in [16].

As eloquently detailed in [20], some of the Li ions in a solid electrolyte are thermally excited at standard conditions. The chemical ionization reaction (3) occurs, leaving behind uncompensated negative charges associated to a vacancy in the LIPON matrix at the place formerly occupied by lithium. Raijmakers et al. have depicted vacancies motion with the same conceptual formalism used for negative ions in liquid electrolytes, i.e., as able to move in the solid matter driven by an entropic Brownian motion together with migration within an electric field (see eq. (25b)). This picture is thermodynamically encapsulated in the constitutive law (18).

In [21,48] the dynamic filling of vacancies by neighboring positions has been explicitly modeled. To capture this process some ions, denoted henceforth with ${\mathrm{Li}}_{\mathrm{int}}^{+}$, move in a meta-stable interstitial state after the chemical ionization reaction (3) occurs, whereas the remaining ${\mathrm{Li}}^{+}$ ions hop and fill neighboring vacancies. This dynamic behavior is described by a further reaction where $k_f^{\mathrm{int}}$ and $k_b^{\mathrm{int}}$ are the rate constants for reaction (35). When $k_f^{\mathrm{int}}=0$, no interstitial mechanism is accounted for and the model restricts to a classical single ion conducting solid electrolyte [18].

In a nutshell, reaction (3) makes lithium ions capable to leave the host-sites and move within the complex amorphous LIPON structure, either by filling neighboring vacancies or to flow interstitially. The proportion of ions in these two mechanisms is governed by reaction (35). In this formalism, positive ions are the only moving species, whereby the concentration of negatively charged vacancies is the outcome of the motion process and do not possess any intrinsic motility. In this sense, there is no direct flow ${\overrightarrow{h}}_{{\mathrm{n}}^{-}}(x,t)$ of negative charges, contrary to eq. (25b), and the local concentration of vacancies is altered merely by the chemical ionization reaction eq. (35).

This conceptual picture frames into the following set of mass balance equations, which characterize the immobile lithium $\mathrm{LiO}$, the negative charges ${\mathrm{n}}^{-}$, the transport of the lithium ions ${\mathrm{Li}}_{\mathrm{int}}^{+}$ that go interstitial and the remaining ${\mathrm{Li}}^{+}$ that hop: having set according to reaction (35). The ratio is the equilibrium constant of reaction (35). The set of 4 mass balance equations (36a)-(36d) contains 5 unknowns, the concentrations $c_{\mathrm{LiO}}$, $c_{{\mathrm{n}}^{-}}$, $c_{{\mathrm{Li}}^{+}}$, and $c_{{\mathrm{Li}}_{\mathrm{int}}^{+}}$ plus the electric potential $\varphi$, which is constitutively related to the mass fluxes ${\overrightarrow{h}}_{\alpha }$. The additional required equation is Ampère’s law (with Maxwell’s correction) 6. To conclude the set of balance equations, the usual balance of forces in small strains was accounted for in [48]

The chemo-thermo-elastic strain $\epsilon$ is considered to be made up of two separate contributions: an elastic recoverable part after unloading ${\epsilon }^{el}$ and a swelling contribution due to the insertion of species in the host material ${\epsilon }^s$:

The swelling contribution ( $\alpha ={\mathrm{Li}}^{+},{\mathrm{Li}}_{\mathrm{int}}^{+}$ ) is assumed to be volumetric and proportional to the deviation $c_{\alpha }-c_{\alpha }^0$ from the reference concentration $c_{\alpha }^0$ by means of the chemical expansion coefficients ${\omega }_{\alpha }$ of species $\alpha$. They equal one third of the partial molar volumes at a given temperature.

To derive governing equations from the balance Eqs. (36), constitutive laws have been derived from a rigorous thermodynamic setting. The hopping mechanism is thermodynamically quite different from the interstitial motion. Therefore, making recourse to the same thermodynamic description for both mechanisms might be questionable. Aware of this limitation and inspired by [49,50,51] restricted to small strains, elaborating the electromagnetic contribution in the Helmholtz free energy $\psi$ from [24,25], the chemical potential (16b) of species $\alpha ={\mathrm{Li}}^{+},{\mathrm{Li}}_{\mathrm{int}}^{+}$ is extended to as detailed in [34,35]. A simple choice for the elastic part of the free energy density ${\psi }_{el}$ in the small strain range is the usual quadratic form where K, G are the bulk and shear modulus respectively and they are made dependent on species concentrations. The stress tensor ${\sigma }^e(\epsilon ,c_{\alpha })$ descends from thermodynamic restrictions (see [34] for details and extension to temperature dependency)

Note that the derivative $\partial {\psi }_{el}/\partial c_{\alpha }$, in eq. (39) is the sum of two contributions

The first emanates from the swelling part of the strain, and is present even if the material properties are independent on concentration of species. Nernst-Planck equation (18) is extended as follows with $\alpha ={\mathrm{Li}}^{+},{\mathrm{Li}}_{\mathrm{int}}^{+}$. The mobility tensor reads ( ${\theta }_{\alpha }=c_{\alpha }/c_{\alpha }^{sat}$ )

It accounts for saturation and in this differs from Eq. (14). The energetic and entropic contributions in the constitutive law (43) have been already described in eq. (17). The mechanical contribution to the mass flux is driven by the chemical expansion coefficient, and derives from thermodynamic consistency.

Mass balance equations, after inserting (43) into eqs. (36c)-(36d), do not form a complete set, because ionic transport entails movement of mass as well as of charge. In order to build a multiscale compatible theory, the generally accepted electroneutrality assumption cannot be taken, since it prevents to impose the conservation of energy across the scales: this concept has been illustrated with great detail in [24,29] and will not be elaborated here further. Multiscale compatibility is granted by using Ampere’s law (with Maxwell’s correction in the realm of small strains)

When multiscale is not invoked, the electroneutrality assumption can be called for.

A widespread choice for the initial conditions for concentrations and electric potential enforces equilibrium conditions. They hold

Three independet factors influence the equilibrium concentrations within the system: $c_0$ and the equilibrium constants for reactions (3) and (35). While $c_0$ can be determined with precision, estimating $K_{\mathrm{eq}}^{\mathrm{ion}}$ and $K_{\mathrm{eq}}^{\mathrm{int}}$ through experiments is fraught with substantial uncertainties. These three parameters are interrelated and play a role in defining the fraction of lithium existing in a mobile state at equilibrium, which is referred to as $\delta$, as detailed in reference [20]. $\delta$ holds,

Equation (49) can be readily transformed to express $K_{\mathrm{eq}}^{\mathrm{int}}$ in terms of both $\delta$ and $K_{\mathrm{eq}}^{\mathrm{ion}}$ as follows,

Interface conditions for this advanced model are eqs. (29), accounting for identity (33) for the non-faradaic current contribution. The faradaic current emanates from charge transfer kinetics, as proposed in [20]. In view of the splitting of lithium flux in interstitial and hopping, constitutively specified by eq. (43), faradaic interface conditions split, too: with the interstitial and hopping contributions to charge transfer current clearly identified. They can be inferred from Butler-Volmer eqs. (7) or (34). The exchange current read: where ${\overline{c}}_{{\mathrm{Li}}^{+}}$, ${\overline{c}}_{{{\mathrm{Li}}^{+}}_{\mathrm{int}}}$ are average bulk concentrations of species ${\mathrm{Li}}^{+}$ and ${{\mathrm{Li}}^{+}}_{\mathrm{int}}$, respectively. Note that, differently from [20], those averages are not time independent. Lacking more clear understanding, we assume that the non faradaic current $i_{\alpha }^{dl}\left(t\right)$ as in eq. (33) is proportional to the faradaic splitting, i.e. where the jump $〚\varphi 〛$ of the electric potential at the electrolyte/electrode interface is always defined as the electrode potential minus the electrolyte potential and $s=a/e,e/c$. The Neumann conditions on fluxes at the left and right boundaries of the electrolyte eventually read:

Continuity of displacements and normal tractions are interface conditions for the mechanical governing equation (36f). The electrodes governing equations and boundary conditions do not differ from Section 2.3.