As electrified transportation continues to demand higher safety, energy density, and service reliability, the electrochemical and structural margins of conventional liquid-electrolyte cells are being pushed toward their physical limits. A solid state battery replaces the flammable organic liquid electrolyte with a solid electrolyte, which offers intrinsic safety benefits, improved thermal stability, and a credible pathway toward high energy density. However, the transition from a liquid–solid to a solid–solid interface fundamentally changes the mechanics of the cell. The solid electrolyte cannot spontaneously wet the electrode surface, so the real contact area at the electrode/electrolyte interface is usually smaller than the nominal geometric area. This contact loss restricts the effective reaction zone, raises interfacial polarization, and limits the accessible capacity of a solid state battery. At the same time, the volumetric change of the electrode during lithiation and delithiation generates chemical strain that propagates through mechanically mismatched layers, producing interfacial stress differences, potential delamination, and electrolyte cracking.
Because a solid state battery cannot rely on liquid wetting to maintain interfacial integrity, an external stack pressure becomes a functional operating parameter rather than a mere packaging constraint. Axial pressure compacts the electrode/electrolyte interface, restores the real contact area, lowers charge-transfer resistance, and improves capacity utilization. Nevertheless, the same pressure also amplifies the elastic load carried by the contact spots and increases the mechanical mismatch between adjacent layers. A pressure that is too low leaves the interface partially separated, whereas a pressure that is too high accumulates mechanical and assembly penalties without a proportional electrochemical return. Identifying a defensible axial pressure window therefore requires a coupled electrochemical–mechanical description in which the pressure-dependent contact state is explicitly linked both to interfacial reaction kinetics and to interfacial mechanical response.
In this work I develop a coupled electrochemical–mechanical model of a solid state battery that treats the pressure-dependent contact factor as the central coupling variable. The contact factor modifies the reference exchange current density at both the positive-electrode/solid-electrolyte and negative-electrode/solid-electrolyte interfaces, and it simultaneously modifies the elastic bearing term of the interfacial effective stress. In this way, a single physical quantity carries the influence of axial pressure into both the reaction kinetics and the interfacial stress state. On this basis I construct a multi-criteria evaluation that combines usable areal capacity, interfacial contact factor, interfacial stress difference, and pressurization cost, and I search for the axial pressure that minimizes a weighted composite loss. The result is a trade-off pressure window that balances electrochemical gain against mechanical and engineering penalties, which I intend as a methodological reference for pressure regulation and interfacial reliability assessment of multilayer-integrated solid state battery architectures.
1 Coupled Electrochemical–Mechanical Model
Because the in-plane dimensions of an automotive solid state battery are far larger than its thickness, edge effects can be neglected under ideal stacking conditions, and the cell can be reduced to a one-dimensional layered domain along the thickness direction. The domain consists of a positive electrode, a solid polymer electrolyte (SPE), and a negative electrode, with natural thicknesses denoted by lp, le, and ln, respectively. In-plane transport and in-plane potential gradients are neglected, and only lithium transport along the thickness coordinate x, interfacial charge transfer, and the resulting electrochemical response are resolved. All three regions are treated as continuous homogeneous media, and the cell is assumed to operate isothermally.
1.1 Electrochemical Model
Lithium transport in the positive and negative electrodes obeys mass conservation combined with a stress-coupled flux expression:
$$\frac{\partial c_k}{\partial t}+\frac{\partial j_k}{\partial x}=0,\qquad k=p,n$$
$$j_k=-D_k\left[\left(1-\frac{c_k}{c_k^{\max}}\right)\frac{\partial c_k}{\partial x}-\frac{\Omega_k c_k}{RT}\frac{\partial \sigma_k}{\partial x}\right],\qquad k=p,n$$
where cp and cn are the lithium concentrations in the positive and negative electrodes, cpmax and cnmax are the corresponding maximum host capacities, jp and jn are the lithium fluxes, Dp and Dn are the lithium diffusivities, Ωp and Ωn are the partial molar volumes, and σp and σn are the hydrostatic stresses. The gas constant and absolute temperature are denoted by R and T.
Ion transport inside the solid electrolyte is described by a modified Nernst–Planck formulation:
$$\frac{\partial c_e}{\partial t}+\frac{\partial j_e}{\partial x}=0$$
$$j_e=-D_e\left(\frac{\partial c_e}{\partial x}+\frac{\Omega_e c_e}{RT}\frac{\partial \sigma_e}{\partial x}\right)$$
with the effective diffusivity and effective partial molar volume of the electrolyte given by
$$D_e=\frac{2D_+D_-}{D_++D_-},\qquad \Omega_e=\Omega_++\Omega_-$$
Charge conservation in the electrolyte requires that the ionic current density be divergence-free:
$$\frac{\partial i_e}{\partial x}=0$$
$$i_e=-FD_e\frac{\partial c_e}{\partial x}-\frac{F^2D_ec_e}{RT}\frac{\partial \phi_e}{\partial x}+\frac{FD_ec_e\left(\Omega_+-\Omega_-\right)}{RT}\frac{\partial \sigma_e}{\partial x}$$
Interfacial charge transfer at the electrode/electrolyte boundaries follows Butler–Volmer kinetics:
$$i_k=i_{k,0}\,2\sinh\left(\frac{F\eta_k}{2RT}\right),\qquad k=p,n$$
$$i_{p,0}=Fk_p\left(c_p^{\max}-c_{s,p1}\right)^{\beta_p}c_{s,p1}^{1-\beta_p}c_{e,s1}^{1-\beta_p}$$
$$i_{n,0}=Fk_n\left(c_n^{\max}-c_{s,n2}\right)^{\beta_n}c_{s,n2}^{1-\beta_n}c_{e,s2}^{1-\beta_n}$$
The state of charge is defined with respect to the lithium inventory of the negative electrode:
$$SOC(t)=\frac{\displaystyle\int_{l_p+l_e}^{l_p+l_e+l_n}c_n(t)\,dx}{\displaystyle\int_{l_p+l_e}^{l_p+l_e+l_n}c_n^{\max}\,dx}$$
The boundary conditions applied to the flux and current variables are summarized in Table 1.
| Boundary | Condition |
|---|---|
| Positive electrode outer boundary | $$j_p(x=0)=0$$ |
| Positive electrode / SPE interface | $$j_p(x=l_p)=\frac{i_p}{F}$$ |
| Negative electrode outer boundary | $$j_n(x=l_p+l_e+l_n)=0$$ |
| Negative electrode / SPE interface | $$j_n(x=l_p+l_e)=\frac{i_n}{F}$$ |
| Positive electrode / SPE interface | $$j_e(x=l_p)=\frac{i_pD_-}{\left(D_++D_-\right)F}$$ |
| Negative electrode / SPE interface | $$j_e(x=l_p+l_e)=\frac{i_nD_-}{\left(D_++D_-\right)F}$$ |
| Positive electrode / SPE interface | $$i_e(x=l_p)=i_p$$ |
| Negative electrode / SPE interface | $$i_e(x=l_p+l_e)=i_n$$ |
To attribute the terminal voltage response to individual physical contributions, I decompose the cell voltage into a reference potential, a concentration term, a stress term, an interfacial overpotential term, and an electrolyte potential term:
$$V_{cell}=V_{ref}+V_{con}+V_{str}+V_{ove}+V_{ele}$$
| Component | Expression |
|---|---|
| Reference potential | $$V_{ref}=3.9\ \mathrm{V}$$ |
| Concentration contribution | $$V_{con}=\frac{RT}{F}\ln\left(\frac{c_{e,s2}\left(c_p^{\max}-c_{s,p1}\right)c_{s,n1}}{c_{e,s1}c_{s,p1}\left(c_n^{\max}-c_{s,n1}\right)}\right)$$ |
| Stress contribution | $$V_{str}=\frac{1}{F}\left(\Omega_p\sigma_{p,s1}-\Omega_n\sigma_{n,s2}+\Omega_e\sigma_{e,s2}-\Omega_e\sigma_{e,s1}\right)$$ |
| Interfacial overpotential contribution | $$V_{ove}=\eta_p-\eta_n$$ |
| Electrolyte potential contribution | $$V_{ele}=\phi_{e,s1}-\phi_{e,s2}$$ |
1.2 Mechanical Model
The mechanical response satisfies the equilibrium condition
$$\nabla\cdot\sigma+b=0$$
Axial pressure is applied along the thickness direction, and because an encapsulated solid state battery is strongly constrained in plane, the in-plane strains are set to zero, while the through-thickness stress is prescribed by the external load:
$$\varepsilon_{yy}=\varepsilon_{zz}=0,\qquad \sigma_{xx}=-P$$
Here P represents the equivalent nominal axial pressure transmitted by the stack structure to the layered cell, which is appropriate for the central region of a cell under approximately uniform compression with negligible edge effects and assembly tolerances. Each material region is treated as a homogeneous, isotropic, linearly elastic solid. The total strain is the sum of an elastic part and a chemical part:
$$\varepsilon=\varepsilon^{elas}+\varepsilon^{chem}$$
Chemical strain is expressed through the partial molar volume and the concentration change, and the resulting hydrostatic stress under combined axial loading and chemical expansion is written as
$$\sigma_{k,s}^{eff}=-\frac{\nu_k}{1-\nu_k}\frac{P}{\gamma_s}+\frac{E_k\Omega_k\left(c_k-c_{k,0}\right)}{9\left(1-\nu_k\right)},\qquad k=p,e,n$$
where Ek and νk denote the Young modulus and Poisson ratio of each layer, and γs is the interfacial contact factor introduced in the next subsection. The corresponding layer strains are obtained from the same constitutive relation, and the total thickness variation is accumulated as
$$\Delta l=l_p\varepsilon_p+l_e\varepsilon_e+l_n\varepsilon_n$$
The maximum thickness variation over a full cycle at a given pressure is defined as
$$H(P)=\max_t l(P,t)-\min_t l(P,t)$$
1.3 Interfacial Contact Model
At a solid–solid interface, the real contact area is smaller than the nominal area because of surface roughness and stiffness mismatch. This incomplete contact affects two distinct processes. First, it reduces the electrochemically active area and therefore raises interfacial polarization. Second, the applied nominal pressure is not distributed uniformly over the nominal area but is carried preferentially by the contacting asperities, producing a local elastic load amplification. To quantify the pressure dependence of the real contact area, I introduce the interfacial contact factor
$$\gamma=\frac{A}{A_0}$$
where A is the real contact area and A0 is the nominal contact area. Following a Persson-type contact description, the contact factor is evaluated as an integral over the roughness spectrum:
$$\gamma=\frac{A}{A_0}=\int_{q_L}^{\xi q_L}\mathrm{erf}\left(\frac{P}{\sqrt{2\pi}\,E^*\Lambda(q)}\right)\frac{dq}{q}$$
$$\Lambda(q)=\left[\int_{q_L}^{q}C(q’)\,q’^3\,dq’\right]^{1/2}$$
where the equivalent modulus of the contacting pair is
$$\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}$$
and the roughness power spectral density, the lower wave-number bound, and the magnification factor are respectively
$$C(q)=10^{m}q^{I},\qquad q_L=\frac{2\pi}{\sqrt{A_0}},\qquad \xi=\sqrt{\frac{A_0}{Dt}}$$
In the present work the characteristic diffusion time t is taken as 3600 s, the nominal contact area A0 is taken as 1 cm2, and the roughness exponents m and I are assigned separately for the positive-electrode interface and the negative-electrode interface. Because the positive electrode is treated as a dense continuous film while the negative electrode exhibits a pronounced particulate morphology, the two interfaces are classified as film-type and bulk-type respectively, and the corresponding exponents differ accordingly.
The contact factor enters the electrochemical model through the reference exchange current density:
$$i_{p,0}^{eff}=\gamma_{s1}i_{p,0},\qquad i_{n,0}^{eff}=\gamma_{s2}i_{n,0}$$
At the same time, because the external load is carried by the real contact spots, the elastic bearing term of the interfacial stress is amplified by the inverse of the contact factor, while the chemical term remains unchanged. The effective interfacial stresses at the two electrode/electrolyte boundaries therefore become
$$\sigma_{p,s1}^{eff}=-\frac{\nu_p}{1-\nu_p}\frac{P}{\gamma_{s1}}+\frac{E_p\Omega_p\left(c_p-c_{p,0}\right)}{9\left(1-\nu_p\right)}$$
$$\sigma_{e,s1}^{eff}=-\frac{\nu_e}{1-\nu_e}\frac{P}{\gamma_{s1}}+\frac{E_e\Omega_e\left(c_e-c_{e,0}\right)}{9\left(1-\nu_e\right)}$$
$$\sigma_{e,s2}^{eff}=-\frac{\nu_e}{1-\nu_e}\frac{P}{\gamma_{s2}}+\frac{E_e\Omega_e\left(c_e-c_{e,0}\right)}{9\left(1-\nu_e\right)}$$
$$\sigma_{n,s2}^{eff}=-\frac{\nu_n}{1-\nu_n}\frac{P}{\gamma_{s2}}+\frac{E_n\Omega_n\left(c_n-c_{n,0}\right)}{9\left(1-\nu_n\right)}$$
The interfacial stress difference, which measures the in-plane stress mismatch caused by dissimilar elastic properties and chemical strains on the two sides of the interface, is then defined for each boundary:
$$\Delta\sigma_{s1}^{eff}=\sigma_{p,s1}^{eff}-\sigma_{e,s1}^{eff},\qquad \Delta\sigma_{s2}^{eff}=\sigma_{n,s2}^{eff}-\sigma_{e,s2}^{eff}$$
1.4 Model Parameters
The material properties, initial conditions, and operating conditions used in the computation are listed in Table 3. The operating temperature is 300 K and the reference concentration is 1 mol m-3. The cell is charged at 1 A m-2 up to a cutoff voltage of 4.03 V and then discharged at the same current density until the state of charge reaches 0.05.
| Parameter | Positive electrode (LiCoO2) | Solid polymer electrolyte (PEO–LiPF6) | Negative electrode (graphite) |
|---|---|---|---|
| Thickness (nm) | 550 | 2000 | 200 |
| Maximum concentration (mol m-3) | 23300 | 1500 | 30555 |
| Initial concentration (mol m-3) | 22135 | 1200 | 1527.75 |
| Reaction rate constant (m2.5 mol-0.5 s-1) | 1.53 × 10-11 | – | 5.03 × 10-11 |
| Diffusion coefficient (m2 s-1) | 1.76 × 10-15 | 2.5 × 10-13 (cation), 3 × 10-13 (anion) | 8 × 10-14 |
| Partial molar volume (m3 mol-1) | 1.92 × 10-6 | 3.95 × 10-6 (cation), 1.46 × 10-4 (anion) | 4.17 × 10-6 |
| Charge transfer coefficient | 0.5 | – | 0.5 |
| Young modulus (GPa) | 70 | 0.5 | 10 |
| Poisson ratio | 0.20 | 0.40 | 0.30 |
The coupled system is solved numerically with a fully coupled solver and a sparse direct linear solver, using a relative tolerance of 0.005 and a tolerance factor of 0.1. For the pressure sweep, the increment is 0.3 MPa below 4 MPa and 1 MPa above 4 MPa.
1.5 Relation to Existing Modeling Frameworks
To position the present formulation, Table 4 contrasts its main modeling features with the principal capabilities of previously reported frameworks.
| Modeling framework | Principal feature |
|---|---|
| Bending-load electro-chemo-mechanical model | Couples a hydrostatic-stress term into mass transport and interfacial kinetics and investigates the effect of bending loads |
| Contact-loss electrochemical model | Introduces Persson contact mechanics into an electrochemical model and defines a contact factor entering interfacial kinetics |
| Composite-electrode multi-physics model | Couples stack pressure to composite-electrode microstructure, porosity, and mechanical parameters |
| Present model | Derives the axial-pressure-dependent hydrostatic stress and couples the Persson contact factor simultaneously into the interfacial exchange current density, the hydrostatic stress, and the interfacial stress difference |
1.6 Model Verification
To verify the electrochemical description, the contact factors at both interfaces are set to unity and the resulting terminal voltage versus state-of-charge curve is compared with a reference simulation performed on the same cell chemistry under a mechanically unconstrained baseline condition. The two curves agree closely in plateau voltage, accessible state-of-charge range, and the position of the voltage rise at high state of charge. The coefficient of determination is 0.9931 and the maximum absolute deviation is 0.0310 V, confirming that the present model reproduces the essential electrochemical behavior of the solid state battery.
The pressure sensitivity of the interfacial impedance provides a second verification path. If the interfacial impedance is written in terms of the contact factor,
$$R_{int}=\frac{RT}{Fi_0^{eff}}=\frac{RT}{\gamma Fi_0}$$
then the area exponent defined by
$$n=\frac{d\ln R_{int}}{d\ln\gamma}$$
takes the value n = −1 in the present formulation, because the contact factor enters the exchange current density to the first power. Independently reported experimental measurements of ceramic solid electrolytes place this exponent in the range −1.13 to −1.05, so the linear coupling adopted here reproduces both the order of magnitude and the sensitivity of the experimentally observed response.
For the pressure response of the whole-cell charge-transfer resistance, I define the normalized quantity
$$R_{ct}=\frac{RT}{F}\left(\frac{1}{\gamma_{s1}i_{p,0}}+\frac{1}{\gamma_{s2}i_{n,0}}\right),\qquad R_{norm}=1-\frac{R_{ct}(P)}{R_{ct}(P_0)}$$
For a pressure change from 0.43 MPa to 1.2 MPa, the present simulation yields Rnorm = 0.7041, while the corresponding experimental value reported for a polymer-based lithium metal configuration is 0.7326. The close agreement indicates that the model captures the pressure sensitivity of the charge-transfer impedance of the solid state battery with acceptable fidelity.
2 Electrochemical and Mechanical Response under Axial Pressure
With the coupled model established, I now examine how axial pressure shapes the operating response of the solid state battery. The discussion proceeds from interfacial contact and electrochemical output, through internal stress and interfacial stress difference, to the overall thickness response.
2.1 Effect of Axial Pressure on Interfacial Contact
Increasing axial pressure raises the contact factor at both electrode/electrolyte interfaces, but the two interfaces respond on markedly different pressure scales. The positive-electrode interface recovers rapidly and approaches essentially complete contact at a comparatively low pressure, on the order of a few megapascals. The negative-electrode interface, by contrast, requires substantially higher pressure to approach the fully contacted state, reflecting its rougher particulate morphology and the larger asperity spectrum that must be flattened. In both cases the contact factor saturates at high pressure, so the marginal benefit of further pressurization diminishes steadily.
2.2 Effect of Axial Pressure on the Electrochemical Response
Improving the contact factor shifts the discharge curve toward both higher voltage and higher capacity. A small contact factor means a small effective reaction area, so a fixed applied current must be accommodated by a smaller interfacial region, which raises the charge-transfer overpotential and accelerates the approach to the cutoff voltage. A large contact factor enlarges the effective reaction area, reduces the overpotential loss, and allows the discharge to proceed further before the cutoff is reached. Interfacial contact improvement therefore enhances capacity utilization primarily by suppressing interfacial polarization.
Analysis of the voltage components clarifies the underlying mechanism. The concentration contribution is governed mainly by the state of charge and varies little with pressure. The stress contribution grows with state of charge and rises slightly at higher pressure, but its magnitude remains on the order of tens of millivolts and therefore acts as a correction term rather than a dominant factor. The overpotential contribution is by far the most pressure-sensitive component. Under low pressure the limited contact area forces a large interfacial overpotential, producing a pronounced polarization loss; as pressure increases, the contact factor rises, the effective exchange current density increases, and the overpotential loss shrinks, allowing the cell to sustain a higher terminal voltage over a wider state-of-charge range.
When the absolute magnitudes of the three contributions are compared as relative shares of the terminal voltage, the overpotential share falls sharply with increasing pressure while the concentration share rises. This shift does not mean that pressure directly strengthens the concentration contribution; rather, it reflects the diminishing weight of the overpotential once interfacial polarization has been suppressed. At a higher state of charge the overpotential share and the stress share are both larger than at a lower state of charge, indicating that interfacial polarization plays a more influential role in the electrochemical and transport state at higher state of charge, and that chemically induced strain and stress grow as lithiation proceeds.
Overall, the electrochemical benefit of axial pressure is transmitted mainly through the interfacial contact state. When the pressure is raised from 0.1 MPa to about 10 MPa, the improvement in interfacial contact and electrochemical output is substantial. Beyond that level the contact factor approaches saturation, and the reductions in overpotential and the gains in terminal voltage become progressively smaller, exhibiting clear diminishing marginal returns.
2.3 Effect of Axial Pressure on the Mechanical Response
The through-thickness stress distribution varies strongly with state of charge and much more weakly with axial pressure at a fixed state of charge. As the state of charge rises, the tensile stress in the positive electrode increases markedly and the compressive stress magnitude in the negative electrode rises correspondingly, while the stress magnitude in the solid electrolyte remains comparatively small. Because the elastic moduli and partial molar volumes of the three layers differ, and because the concentration changes in each layer are not identical, the chemically induced strains and the resulting constraint responses are inconsistent across the thickness, producing a pronounced layer-to-layer stress contrast. At a fixed state of charge, increasing the pressure from 0.1 MPa to 1 MPa barely alters the layer stress distribution, whereas increasing it to 10 MPa and 50 MPa modestly raises the compressive stress magnitudes in the electrolyte and the negative electrode while leaving the tensile stress in the positive electrode almost unchanged.
The interfacial stress difference behaves differently. As axial pressure rises from 0.1 MPa to 50 MPa, the maximum interfacial stress difference at both electrode/electrolyte boundaries increases overall, but the growth is staged: it rises rapidly in the low-pressure range and then decelerates. Two mechanisms explain this behavior. On one hand, under incomplete contact the external nominal pressure is borne mainly by the real contact spots, so the local bearing level is elevated; as pressure increases, the contact factor grows and the load concentration caused by incomplete contact is alleviated, which slows the growth of the interfacial stress difference in the medium- and high-pressure ranges. On the other hand, the applied pressure itself continues to increase, and higher pressure also widens the accessible state-of-charge range by improving contact and reducing polarization, so the chemically induced strain at higher state of charge continues to develop. The net result is that the maximum interfacial stress difference still increases monotonically with pressure, though at a decreasing rate. The positive-electrode interface exhibits a systematically larger stress difference than the negative-electrode interface and retains a steeper growth trend at high pressure, identifying it as the more mechanically critical location in the present solid state battery configuration.
The maximum thickness variation increases with pressure and is concentrated in the low-pressure range. Between 0.1 MPa and about 4 MPa it grows from roughly 4 nm to roughly 8 nm, after which the curve flattens and the differences among high-pressure cases become marginal. This behavior is consistent with the contact and capacity trends: at low pressure the limited accessible state-of-charge range restricts the concentration swing, so the thickness variation is small; as pressure rises, the widened operating window enlarges the concentration change and the associated chemical strain; once contact improvement and electrochemical performance saturate, the further expansion of the operating window is limited and the thickness fluctuation levels off. Although the low-pressure growth is pronounced, the absolute magnitude remains on the nanometer scale relative to the total cell thickness, so thickness breathing is a useful indicator of pressure-induced operating deformation but is not the dominant constraint at high pressure.
3 Multi-Criteria Evaluation and Optimization of the Axial Pressure Window
Because axial pressure exerts a dual influence on a solid state battery, its value cannot be chosen on the basis of electrochemical improvement alone, nor on the basis of a single mechanical constraint. I therefore construct a multi-criteria evaluation that simultaneously accounts for interfacial contact restoration, electrochemical performance gain, interfacial stress difference control, and pressurization cost.
3.1 Evaluation Indicators and Normalization
Four indicators are adopted: the usable areal capacity Q, the minimum interfacial contact factor γmin, the maximum interfacial stress difference S, and the pressurization cost P. The usable areal capacity under galvanostatic discharge is
$$Q(P)=\frac{i_{app}\,t_{cut}(P)}{3600}$$
where iapp is the applied discharge current density and tcut is the discharge duration. The minimum contact factor is
$$\gamma_{\min}(P)=\min\left\{\gamma_{s1}(P),\gamma_{s2}(P)\right\}$$
and the governing interfacial stress difference is
$$S(P)=\max\left\{\left|\Delta\sigma_{s1}^{eff}\right|,\left|\Delta\sigma_{s2}^{eff}\right|\right\}$$
The four indicators are rendered dimensionless through min–max normalization over the pressure range from Pmin = 0.1 MPa to Pmax = 50 MPa:
$$Q_n(P)=\frac{Q(P)-Q_{\min}}{Q_{\max}-Q_{\min}}$$
$$\gamma_n(P)=\frac{\gamma_{\min}(P)-\gamma_{\min,\min}}{\gamma_{\min,\max}-\gamma_{\min,\min}}$$
$$S_n(P)=\frac{S(P)-S_{\min}}{S_{\max}-S_{\min}}$$
$$P_n(P)=\frac{P-P_{\min}}{P_{\max}-P_{\min}}$$
The maximum thickness variation is treated as a supplementary indicator rather than a primary optimization objective. Because its absolute magnitude is very small relative to the total cell thickness, forcing it into the normalized objective would artificially amplify a numerically minor structural response and make the composite loss excessively sensitive to an auxiliary quantity. Retaining it as a supplementary measure ensures that the loss function primarily reflects the trade-off among capacity gain, contact restoration, interfacial stress difference control, and pressurization cost.
3.2 Composite Loss Function
The composite loss function is defined as the sum of four weighted contributions:
$$J(P)=J_Q+J_\gamma+J_S+J_P$$
$$J_Q=w_Q\left[1-Q_n(P)\right]^2$$
$$J_\gamma=w_\gamma\left[1-\gamma_n(P)\right]^2$$
$$J_S=w_S S_n(P)$$
$$J_P=w_P P_n(P)$$
The weights wQ, wγ, wS, and wP encode the engineering preference attached to capacity, interfacial contact, interfacial mechanical risk, and pressure constraint. The optimal axial pressure at a given weighting scheme is
$$P^{*}=\arg\min_{P_{\min}\leq P\leq P_{\max}}J(P)$$
Three representative weighting schemes are considered, corresponding to a performance-oriented preference, a balanced preference, and a conservative preference. Their weights are listed in Table 8.
| Scheme | $$w_Q$$ | $$w_\gamma$$ | $$w_S$$ | $$w_P$$ |
|---|---|---|---|---|
| Performance-oriented | 0.35 | 0.40 | 0.15 | 0.10 |
| Balanced | 0.30 | 0.35 | 0.20 | 0.15 |
| Conservative | 0.25 | 0.30 | 0.25 | 0.20 |
3.3 Solution Strategy
Because the composite loss depends on the outputs of a fully coupled electrochemical–mechanical solid state battery model, no closed-form expression is available and no analytical gradient can be evaluated. I therefore solve the bounded optimization problem with a derivative-free particle swarm search, in which every candidate pressure is evaluated by re-solving the coupled model without interpolation surrogates. The velocity and position of particle i at iteration k are updated according to
$$v_i^{k+1}=\omega v_i^{k}+c_1r_1\left(p_i^{k}-P_i^{k}\right)+c_2r_2\left(g^{k}-P_i^{k}\right)$$
$$P_i^{k+1}=P_i^{k}+v_i^{k+1}$$
where pik is the historical best position of the particle and gk is the current global best. Each updated position is constrained within the pressure bounds, and a velocity cap is imposed to prevent overly large single-step excursions. The swarm uses 15 particles over 9 iterations, with an inertia weight of 0.65, individual and social learning factors of 1.45 each, and a velocity cap of 8 MPa. After the swarm search identifies a provisional optimum, a local sweep is performed in its neighborhood to confirm the location of the minimum.
To confirm that the result is not an artifact of the chosen algorithm, I also apply a deterministic strategy that combines a coarse full-range scan with partitioned one-dimensional bounded optimization, using identical loss functions and weights. Shape-preserving piecewise cubic Hermite interpolation is used to construct continuous representations of the normalized capacity, contact, and stress-difference indicators, which avoids the overshoot that ordinary cubic splines can produce in the low-pressure region where the contact factor varies rapidly. The pressure range from 0.1 MPa to 50 MPa is then scanned with a maximum interval of 1 MPa, local minima are identified from the scan nodes, and each candidate interval is refined by a one-dimensional bounded search combining golden-section reduction with parabolic interpolation. The axial pressure convergence tolerance is set to 10-3 MPa, with a maximum of 200 iterations and 500 objective evaluations.
4 Optimization Results and Sensitivity Analysis
4.1 Normalized Indicator Trends
The normalized capacity and contact indicators both increase with pressure, but their growth rates differ. The capacity indicator rises rapidly at low pressure and levels off beyond roughly 10 MPa, whereas the contact indicator continues to rise over a wider pressure range, reflecting the slower recovery of the poorer of the two interfaces. The stress-difference and pressurization-cost indicators, being cost-type quantities, increase monotonically. The pressure cost grows essentially linearly, while the stress difference continues to rise in the medium- and high-pressure ranges. Consequently, increasing pressure alone cannot simultaneously deliver electrochemical benefit and control of mechanical and engineering penalties.
4.2 Optimal Pressures under Different Weighting Schemes
The optimal axial pressures obtained with the swarm search are reported in Table 9, and the corresponding results from the deterministic scan-and-refine strategy are reported in Table 10.
| Scheme | Trade-off optimal pressure |
|---|---|
| Performance-oriented | 15.8 MPa |
| Balanced | 12.2 MPa |
| Conservative | 10.0 MPa |
| Scheme | Trade-off optimal pressure |
|---|---|
| Performance-oriented | 15.7 MPa |
| Balanced | 12.4 MPa |
| Conservative | 9.8 MPa |
The two strategies agree to within 0.2 MPa in every scheme, confirming that the result is not sensitive to the specific search algorithm. The deterministic approach requires a dense computational grid when high accuracy is demanded, whereas the swarm approach avoids interpolation surrogates and objective gradients and scales more naturally if additional design variables such as end-plate stiffness, preload displacement, or material parameters are introduced and the problem becomes multi-variable or multi-objective. I therefore adopt the swarm results in the subsequent discussion.
The key indicators at the optimal pressure of each scheme are summarized in Table 11.
| Scheme | $$Q$$ (Ah m-2) | $$\gamma_{\min}$$ | $$S$$ (MPa) | $$P$$ (MPa) |
|---|---|---|---|---|
| Performance-oriented | 0.1368 | 0.8079 | 579.25 | 15.8 |
| Balanced | 0.1342 | 0.7471 | 574.93 | 12.2 |
| Conservative | 0.1316 | 0.6971 | 569.51 | 10.0 |
The composite loss function exhibits a clear minimum in the medium-pressure range for all three schemes. At low pressure the capacity and contact contributions dominate the loss because interfacial contact is insufficient; as pressure rises, these contributions fall rapidly. Beyond a certain level the capacity term approaches its floor, the contact term declines more slowly, and the stress-difference and pressurization-cost terms continue to accumulate, so the total loss begins to rise again. Shifting the preference from electrochemical performance toward mechanical reliability and pressure constraint moves the optimum toward lower pressure.
4.3 Marginal Benefit and Marginal Cost
To make the diminishing-return and accumulating-cost character explicit, I define a normalized electrochemical benefit and a normalized mechanical and engineering cost:
$$B(P)=\frac{Q_n(P)+\gamma_n(P)}{2},\qquad C(P)=\frac{S_n(P)+P_n(P)}{2}$$
The marginal quantities ΔB and ΔC are evaluated for a 1 MPa pressure increment. At low pressure the marginal electrochemical benefit of additional pressure is high, but it falls rapidly and approaches zero beyond the medium-pressure range. The marginal mechanical and pressure cost, in contrast, persists over a wide pressure range because the maximum interfacial stress difference and the pressurization requirement accumulate continuously. Near 12 MPa the two marginal quantities are comparable, indicating that this pressure represents a reasonable compromise between performance gain and pressure penalty. Beyond that point the additional electrochemical return is limited while the cost term continues to grow, so the cost consistently exceeds the benefit.
Once the axial pressure exceeds roughly 10 MPa, the improvement in usable areal capacity begins to level off and the contact factor, although still increasing, does so at a decreasing rate, while the maximum interfacial stress difference and the pressurization cost continue to rise. Between roughly 10 MPa and 16 MPa the marginal benefit approaches the marginal cost. Above about 16 MPa the marginal cost exceeds the marginal benefit, meaning that the additional capacity and contact improvement no longer offset the accumulated stress difference and pressure penalty. Since the three representative schemes yield trade-off optima of 15.8 MPa, 12.2 MPa, and 10.0 MPa, all lying within this medium-pressure interval, I identify 10.0–15.8 MPa as the trade-off axial pressure window for the present material system and idealized layered configuration, with 12.2 MPa as the balanced optimum.
4.4 Robustness of the Pressure Window
I examine the stability of the window with respect to the weighting coefficients, the normalization interval, the loss-function form, and the charge–discharge current density.
For the weighting coefficients, I sample a broad admissible weight space subject to the constraints
$$0.18\leq w_Q\leq 0.45,\quad 0.20\leq w_\gamma\leq 0.50,\quad 0.08\leq w_S\leq 0.35,\quad 0.05\leq w_P\leq 0.30$$
and compute the optimal pressure for each admissible combination. A Dirichlet distribution with parameter vector equal to a fixed set of positive values is used to generate candidate weight vectors, which are then accepted or rejected according to the constraints above until 5000 valid samples are obtained. The optimum concentrates in the medium-pressure range. The 5th, 25th, 50th, 75th, and 95th percentiles of the optimal pressure are 7.12 MPa, 9.10 MPa, 11.10 MPa, 13.81 MPa, and 17.92 MPa respectively. The median of 11.10 MPa lies close to the balanced optimum of 12.2 MPa, and 63.56 % of the samples fall inside the identified trade-off window. The medium-pressure window is therefore not an artifact of a single weighting choice.
For the normalization interval, I compare the original range, an upper limit extended by 20 %, and a range defined by the 5th and 95th percentiles of the sampled pressure distribution, while keeping the weights fixed. The results are reported in Table 12.
| Normalization interval | Trade-off pressure window |
|---|---|
| Original scheme | 10.0–15.8 MPa |
| Upper limit extended by 20 % | 10.8–17.1 MPa |
| 5th–95th percentile interval | 11.8–17.9 MPa |
Although the endpoints shift somewhat, all three windows remain concentrated in the medium-pressure range and overlap substantially with the original window, indicating that the result is reasonably robust to the choice of normalization interval.
For the loss-function form, I examine three alternatives in addition to the original formulation. A fully quadratic form replaces the stress-difference and pressure-cost terms by their squared normalized values. A normalized Huber form with threshold parameter 0.5 applies a piecewise quadratic–linear penalty,
$$J(P)=w_QH\left(Q_n\right)+w_\gamma H\left(\gamma_n\right)+w_S H\left(S_n\right)+w_P H\left(P_n\right)$$
$$H(x)= \begin{cases} \dfrac{4}{3}x^2, & 0\leq x\leq 0.5\\[6pt] \dfrac{4}{3}x-\dfrac{1}{3}, & 0.5<x\leq 1=""
and an exponential form applies
$$J(P)=w_QE\left(Q_n\right)+w_\gamma E\left(\gamma_n\right)+w_SE\left(S_n\right)+w_PE\left(P_n\right),\qquad E(x)=\frac{1-e^{-2x}}{1-e^{-2}}$$
The results are summarized in Table 13.
| Loss-function form | Trade-off pressure window |
|---|---|
| Original formulation | 10.0–15.8 MPa |
| Fully quadratic | 11.5–16.7 MPa |
| Normalized Huber | 12.8–17.5 MPa |
| Exponential | 11.0–18.6 MPa |
All four windows lie in the medium-pressure range and overlap appreciably with the original window. Although the endpoints move, the overall pressure range does not shift dramatically, indicating acceptable robustness with respect to the functional form of the composite loss.
For the operating condition, I repeat the optimization at several charge–discharge current densities. The results are listed in Table 14.
| Charge–discharge current density | Trade-off pressure window |
|---|---|
| 1 A m-2 (original) | 10.0–15.8 MPa |
| 0.135 A m-2 | 8.8–14.0 MPa |
| 0.270 A m-2 | 9.3–14.9 MPa |
| 0.406 A m-2 | 9.6–15.4 MPa |
| 0.541 A m-2 | 9.8–15.6 MPa |
| 0.744 A m-2 | 9.9–15.8 MPa |
Lower current densities shift the window modestly toward lower pressure, but the numerical variation is small and every window remains within the medium-pressure range, so the original window is reasonably stable across operating conditions.
4.5 Engineering Implications of the Pressure Window
To translate the pressure window into an engineering load, consider a large-format pouch-type solid state battery with an effective electrode area of 120 mm × 80 mm. Using the relation
$$F=pA$$
the trade-off window of 10.0–15.8 MPa corresponds to a total preload force of approximately 96.0–151.7 kN, and the balanced optimum of 12.2 MPa corresponds to approximately 117.1 kN. This range falls inside the clamping pressure range commonly adopted for large-format solid state battery fixtures, and a well-designed end plate and clamping mechanism can keep the pressure distribution reasonably uniform across the cell surface. Nevertheless, a preload in the hundreds-of-kilonewtons range places substantial demands on end-plate strength and fastener load capacity. Because no module-level structural model or yield safety-factor check is included here, the present window should be interpreted as a trade-off result under the assumed material parameters, idealized layered geometry, and stated evaluation preferences. Its application to an automotive cell or module still requires further design and verification against the specific cell area and constraint structure.
5 Conclusions
I have developed a coupled electrochemical–mechanical model of a solid state battery that explicitly incorporates axial pressure and the pressure-dependent interfacial contact state, and I have used it to optimize the axial pressure window on a multi-criteria basis. The principal findings are as follows.
First, in terms of the electrochemical response, axial pressure improves the performance of a solid state battery by restoring the electrode/electrolyte contact state, but the benefit exhibits diminishing marginal returns. At low pressure the limited effective reaction area raises the charge-transfer overpotential, lowering the terminal voltage and the accessible capacity. As pressure increases, the contact factor rises, the overpotential contribution falls, and the terminal voltage and capacity output improve. In the medium- and high-pressure range the contact factor approaches saturation and the additional reduction in overpotential and gain in capacity become small, indicating that a moderate axial pressure is a necessary condition for maintaining effective solid–solid contact but that pressurization beyond a reasonable range yields little further electrochemical improvement.
Second, in terms of the mechanical response, axial pressure influences the layer stresses, amplifies the interfacial stress difference, and alters the thickness response. The layer stress distribution is governed primarily by the state of charge rather than by the applied pressure, because the chemically induced strain associated with lithiation and delithiation is the dominant source of stress evolution. With increasing pressure, the maximum interfacial stress difference at both boundaries grows overall, and the positive-electrode interface consistently exhibits a larger stress difference, identifying it as the more mechanically critical location. The maximum thickness variation increases noticeably in the low-pressure range but remains small in absolute terms and levels off at high pressure, so it is not the dominant constraint on further pressurization. The interfacial stress difference is a more informative indicator of the mechanical penalty at high pressure.
Third, combining usable areal capacity, interfacial contact state, interfacial stress difference, and pressurization cost in a composite loss function, the trade-off axial pressure window for the present material system, idealized layered structure, and isothermal condition is 10.0–15.8 MPa, with a balanced optimum of 12.2 MPa. This range maintains good capacity output and interfacial contact while limiting the further accumulation of interfacial stress difference and pressurization cost, and therefore represents a practical balance between electrochemical performance gain and interfacial mechanical risk control in a solid state battery.
The window reported here applies to the assumed material parameters and idealized layered geometry. Multi-layer thin-film integration offers a credible route toward combining energy and power in a solid state battery architecture that can be packaged and manufactured at scale, and the modeling and optimization methodology presented here is intended to support pressure regulation and interfacial reliability analysis for such multilayer-integrated automotive cells. Future work should address realistic rough interfaces, interfacial damage evolution, debonding, and local contact failure, and should incorporate physically meaningful mechanical constraints based on interfacial strength, fracture energy, and allowable stress thresholds. Extending the framework to include end-plate stiffness, structural mass, pressure uniformity, and thermal effects, and evaluating high-rate, overcharge, overdischarge, and abnormal-pressure conditions, would further strengthen the design relevance of the pressure window. Quantitative experimental validation of the interfacial mechanical response across different material systems and constraint conditions would provide the final step toward a validated axial pressure window for a solid state battery at the cell or module level.