I investigate the axial pressure window for solid electrolyte cells by developing a coupled electrochemical–mechanical model that accounts for pressure-dependent interfacial contact. The solid electrolyte cell replaces conventional liquid electrolytes with solid electrolytes, offering high safety, high stability, and high energy density potential. However, the solid–solid interfaces between electrodes and solid electrolytes suffer from limited wetting and poor self-adaptation, making interfacial contact critical for ion transport, charge transfer, and capacity utilization. During charge and discharge, volume changes in electrode materials induce internal stresses and interfacial stress evolution, which can lead to contact loss, solid electrolyte cracking, and impedance rise. Therefore, the performance of a solid electrolyte cell depends not only on material ionic conductivity and interfacial reaction kinetics but also on internal stress distribution, interfacial mechanical states, and external constraint conditions. Axial pressure is a key external parameter for improving solid–solid interfacial contact and regulating stack pressure. However, increasing axial pressure simultaneously raises mechanical risks. Thus, the axial pressure window for a solid electrolyte cell must be determined by comprehensively considering interfacial contact, electrochemical output, mechanical stability, and pressurization cost.
I establish a one-dimensional electrochemical–mechanical model for a solid electrolyte cell, considering lithium transport, interfacial charge transfer, and stress coupling. The model couples pressure-dependent interfacial contact state to interfacial reaction kinetics and interfacial mechanical response. I then perform multi-criteria optimization of axial pressure by considering usable areal capacity, interfacial contact factor, interfacial stress difference, and pressurization cost. The results show that increasing axial pressure improves interfacial contact, reduces interfacial polarization, and enhances capacity utilization, but the electrochemical gains gradually diminish, while interfacial stress difference and pressurization cost continue to increase. Under isothermal conditions and for the given material system and ideal layered structure, the trade-off axial pressure range is 10.0–15.8 MPa, and the balanced scheme yields a trade-off optimum of 12.2 MPa. These results provide methodological guidance for pressure regulation and interfacial reliability analysis of multilayer-integrated solid electrolyte cells for automotive applications.
I begin with the electrochemical model. The solid electrolyte cell is simplified as a one-dimensional structure along the thickness direction because the in-plane dimensions are much larger than the thickness. I denote the natural thicknesses of the positive electrode, solid polymer electrolyte (SPE), and negative electrode as \(l_p\), \(l_e\), and \(l_n\), respectively. I neglect in-plane mass transport and potential variations and consider only lithium-ion transport along the thickness direction \(x\), interfacial charge transfer, and the corresponding electrochemical responses. The positive electrode, negative electrode, and SPE are treated as continuous homogeneous media, and the cell operates under isothermal conditions. The lithium transport in the positive and negative electrodes is described by
$$ \frac{\partial c_k}{\partial t} + \frac{\partial j_k}{\partial x} = 0, \quad k = p, n $$
$$ j_k = -D_k \left( \frac{\partial c_k}{\partial x} – \frac{\Omega_k c_k}{RT} \frac{\partial \sigma_k}{\partial x} \right), \quad k = p, n $$
where \(c_p\) and \(c_n\) are the lithium concentrations in the positive and negative electrodes, \(c_p^{\max}\) and \(c_n^{\max}\) are the maximum lithium concentrations, \(j_p\) and \(j_n\) are the lithium fluxes, \(D_p\) and \(D_n\) are the lithium diffusion coefficients, \(\Omega_p\) and \(\Omega_n\) are the partial molar volumes, and \(\sigma_p\) and \(\sigma_n\) are the hydrostatic stresses. \(R\) is the gas constant, and \(T\) is the temperature. For the SPE, the modified Nernst–Planck equation is used:
$$ \frac{\partial c_e}{\partial t} + \frac{\partial j_e}{\partial x} = 0 $$
$$ j_e = -D_e \left( \frac{\partial c_e}{\partial x} – \frac{\Omega_e c_e}{RT} \frac{\partial \sigma_e}{\partial x} \right) $$
where \(c_e\) is the lithium concentration in the SPE, \(j_e\) is the lithium flux, \(D_e\) is the lithium diffusion coefficient, \(\Omega_e\) is the partial molar volume, and \(\sigma_e\) is the hydrostatic stress. The diffusion coefficient and partial molar volume of the SPE are expressed as
$$ D_e = \frac{2 D_+ D_-}{D_+ + D_-} $$
$$ \Omega_e = \Omega_+ + \Omega_- $$
where \(D_+\) and \(D_-\) are the cation and anion diffusion coefficients, and \(\Omega_+\) and \(\Omega_-\) are the cation and anion partial molar volumes. The reaction current in the SPE satisfies charge conservation:
$$ \frac{\partial i_e}{\partial x} = 0 $$
$$ i_e = -\frac{F D_+ D_- c_e}{RT} \frac{\partial \Phi_e}{\partial x} + \frac{F (D_+ – D_-)}{RT} \frac{\partial c_e}{\partial x} + \frac{F D_+ D_- c_e}{RT} \frac{\partial \sigma_e}{\partial x} (\Omega_+ – \Omega_-) $$
At the electrode/electrolyte interfaces, the Butler–Volmer equation describes the electrochemical reaction:
$$ i_k = i_{k,0} 2 \sinh \left( \frac{F}{2RT} \eta_k \right), \quad k = p, n $$
$$ i_{p,0} = F k_p (c_{p}^{\max} – c_{p,s1})^{\beta_p} c_{p,s1}^{1-\beta_p} c_{e,s1}^{1-\beta_p} $$
$$ i_{n,0} = F k_n (c_{n}^{\max} – c_{n,s2})^{\beta_n} c_{n,s2}^{1-\beta_n} c_{e,s2}^{\beta_n} $$
where \(i_p\) and \(i_n\) are the reaction current densities at the positive/SPE and negative/SPE interfaces, \(i_{p,0}\) and \(i_{n,0}\) are the reference exchange current densities, \(\eta_p\) and \(\eta_n\) are the interfacial overpotentials, and \(\beta_p\) and \(\beta_n\) are the charge transfer coefficients. The state of charge (SOC) is defined as the ratio of the current lithium content in the negative electrode to the maximum lithium capacity:
$$ SOC(t) = \frac{\int_{l_p + l_e}^{l_p + l_e + l_n} c_n(t) dx}{\int_{l_p + l_e}^{l_p + l_e + l_n} c_n^{\max} dx} $$
The electrochemical boundary conditions are summarized in Table 1.
| Boundary | Boundary condition |
|---|---|
| Positive electrode boundary at \(x=0\) | \(j_p = 0\) |
| Positive/SPE interface at \(x=l_p\) | \(j_p = \frac{i_p}{F}\) |
| Negative electrode boundary at \(x=l_p+l_e+l_n\) | \(j_n = 0\) |
| Negative/SPE interface at \(x=l_p+l_e\) | \(j_n = \frac{i_n}{F}\) |
| Positive/SPE interface at \(x=l_p\) | \(j_e = \frac{i_p D_-}{D_+ + D_- F}\) |
| Negative/SPE interface at \(x=l_p+l_e\) | \(j_e = \frac{i_n D_-}{D_+ + D_- F}\) |
| Positive/SPE interface at \(x=l_p\) | \(i_e = i_p\) |
| Negative/SPE interface at \(x=l_p+l_e\) | \(i_e = i_n\) |
To analyze the contributions of different physical factors to the cell voltage, I decompose the terminal voltage of the solid electrolyte cell into five parts: reference potential term \(V_{ref}\), concentration contribution \(V_{con}\), stress contribution \(V_{str}\), interfacial overpotential contribution \(V_{ove}\), and electrolyte potential contribution \(V_{ele}\). The expressions are given in Table 2.
| Voltage component | Expression |
|---|---|
| Cell voltage | \(V_{cell} = V_{ref} + V_{con} + V_{str} + V_{ove} + V_{ele}\) |
| Reference voltage | \(V_{ref} = 3.9\) V |
| Concentration contribution | \(V_{con} = \frac{RT}{F} \ln \left( \frac{c_{e,s1} c_{n,s2} (c_{p}^{\max} – c_{p,s1})}{c_{e,s2} c_{p,s1} (c_{n}^{\max} – c_{n,s2})} \right)\) |
| Stress contribution | \(V_{str} = \frac{1}{F} [ \Omega_p \sigma_{p,s1} – \Omega_n \sigma_{n,s2} + \Omega_e (\sigma_{e,s2} – \sigma_{e,s1}) ]\) |
| Interfacial overpotential contribution | \(V_{ove} = \eta_p – \eta_n\) |
| Electrolyte potential contribution | \(V_{ele} = \Phi_{e,s1} – \Phi_{e,s2}\) |
Next, I describe the mechanical model. The mechanical response of the solid electrolyte cell obeys the equilibrium equation:
$$ \nabla \cdot \sigma + b = 0 $$
External pressure is applied along the thickness direction \(x\). Considering that the solid electrolyte cell is strongly constrained in the in-plane direction by packaging or fixtures, I assume zero in-plane strains, i.e., \(\epsilon_{yy} = \epsilon_{zz} = 0\). The thickness-direction stress is determined by the external axial pressure, i.e., \(\sigma_{xx} = -P\). Here, \(P\) is the equivalent nominal axial pressure transferred from the stack structure to the cell, which is valid for the central region of the cell under relatively uniform compression and small edge effects and assembly deviations. Each material region is treated as a homogeneous, isotropic linear elastic solid. During charge and discharge, lithium insertion and extraction cause chemical expansion or contraction. The total strain consists of elastic strain \(\epsilon_{elas}\) and chemical strain \(\epsilon_{chem}\):
$$ \epsilon = \epsilon_{elas} + \epsilon_{chem} $$
I use partial molar volume and concentration change to describe the chemical strain. Through the linear elastic constitutive relation, the stress state under the combined action of axial pressure and concentration change can be calculated as
$$ \sigma_k = \frac{E_k}{(1+\nu_k)(1-2\nu_k)} \left[ (1-\nu_k) \epsilon_{elas} + \nu_k (\epsilon_{yy} + \epsilon_{zz}) \right] – \frac{E_k \Omega_k}{3(1-2\nu_k)} (c_k – c_{k,0}), \quad k = p, e, n $$
where \(E_p, E_n, E_e\) and \(\nu_p, \nu_n, \nu_e\) are the Young’s moduli and Poisson’s ratios of the positive electrode, negative electrode, and SPE, respectively. The strains of each material layer during operation are calculated as
$$ \epsilon_p = \frac{(1+\nu_p)(1-2\nu_p)}{E_p} \frac{P}{1-\nu_p} + \frac{\Omega_p}{3} (c_p – c_{p,0}) \frac{1+\nu_p}{1-\nu_p} $$
$$ \epsilon_e = \frac{(1+\nu_e)(1-2\nu_e)}{E_e} \frac{P}{1-\nu_e} + \frac{\Omega_e}{3} (c_e – c_{e,0}) \frac{1+\nu_e}{1-\nu_e} $$
$$ \epsilon_n = \frac{(1+\nu_n)(1-2\nu_n)}{E_n} \frac{P}{1-\nu_n} + \frac{\Omega_n}{3} (c_n – c_{n,0}) \frac{1+\nu_n}{1-\nu_n} $$
The thickness change is calculated as
$$ \Delta l = l_p \epsilon_p + l_e \epsilon_e + l_n \epsilon_n $$
For the interfacial contact model, the electrode/SPE interfaces in a solid electrolyte cell are solid–solid contacts. Due to surface roughness and material stiffness differences, the real contact area is usually smaller than the nominal contact area. Incomplete interfacial contact affects two processes: first, it reduces the effective contact area and increases interfacial polarization; second, the applied nominal axial pressure is not uniformly distributed over the entire nominal area but is mainly borne by the real contact regions, leading to local elastic load amplification at contact spots. To quantitatively describe the effect of axial pressure on the real contact area at the electrode/SPE interfaces, I introduce an interfacial contact factor \(\gamma\), defined as the ratio of the real contact area \(A\) to the nominal contact area \(A_0\):
$$ \gamma = \frac{A}{A_0} = \frac{1}{\pi} \int_{q_L}^{\infty} \text{erf} \left( \frac{2 E^* q}{\sigma_0} \right) C(q) dq $$
where \(E^*\) is the equivalent modulus determined by the Young’s moduli and Poisson’s ratios of the two materials on both sides of the contact interface:
$$ \frac{1}{E^*} = \frac{1-\nu_1^2}{E_1} + \frac{1-\nu_2^2}{E_2} $$
\(C(q)\), \(q_L\), and \(\xi\) represent the power spectral density of surface roughness, the lower cutoff wavenumber, and the magnification factor, respectively:
$$ C(q) = 10^{m} q^{I} $$
$$ q_L = \frac{2\pi}{\sqrt{A_0}} $$
$$ \xi = \frac{4}{\sqrt{D t}} $$
where \(t\) is the characteristic diffusion time, taken as 3600 s. \(D\) is the diffusion coefficient of Li\(^+\) in the solid electrolyte. The nominal contact area \(A_0\) is taken as 1 cm\(^2\). Since the positive electrode is a continuous dense LiCoO\(_2\) film and the graphite negative electrode has a more pronounced particle morphology, I classify them as “film-type” and “bulk-type” interfaces, respectively. The parameters \(m\) for the positive/SPE and negative/SPE interfaces are \(-2.77\) and \(-3.08\), and the parameters \(I\) are \(-9.35\) and \(-6.61\), respectively. \(\sigma_0\) is the nominal contact stress.
To describe the effect of axial pressure on the interfacial contact state, I introduce the contact factor \(\gamma\) into the reference exchange current density:
$$ i_{p,0}^{eff} = \gamma_1 i_{p,0} $$
$$ i_{n,0}^{eff} = \gamma_2 i_{n,0} $$
In addition to the effect of contact on electrochemistry, because the real contact area is usually smaller than the nominal contact area, the applied nominal axial pressure is mainly borne by the real contact regions, resulting in local elastic load amplification. I modify the interfacial reaction area and the external pressure elastic stress term using the pressure-dependent contact factor. The effective stresses at the interfaces for the electrodes and SPE are calculated as
$$ \sigma_{p,s1}^{eff} = \sigma_{p,s1}^{elas} + \sigma_{p,s1}^{chem} = \frac{P}{\gamma_1} \frac{1}{1-\nu_p} – \frac{E_p \Omega_p}{3(1-\nu_p)} (c_{p,s1} – c_{p,0}) $$
$$ \sigma_{e,s1}^{eff} = \sigma_{e,s1}^{elas} + \sigma_{e,s1}^{chem} = \frac{P}{\gamma_1} \frac{1}{1-\nu_e} – \frac{E_e \Omega_e}{3(1-\nu_e)} (c_{e,s1} – c_{e,0}) $$
$$ \sigma_{e,s2}^{eff} = \sigma_{e,s2}^{elas} + \sigma_{e,s2}^{chem} = \frac{P}{\gamma_2} \frac{1}{1-\nu_e} – \frac{E_e \Omega_e}{3(1-\nu_e)} (c_{e,s2} – c_{e,0}) $$
$$ \sigma_{n,s2}^{eff} = \sigma_{n,s2}^{elas} + \sigma_{n,s2}^{chem} = \frac{P}{\gamma_2} \frac{1}{1-\nu_n} – \frac{E_n \Omega_n}{3(1-\nu_n)} (c_{n,s2} – c_{n,0}) $$
The interfacial stress difference is defined as
$$ \Delta \sigma_{s1}^{eff} = \sigma_{p,s1}^{eff} – \sigma_{e,s1}^{eff} $$
$$ \Delta \sigma_{s2}^{eff} = \sigma_{n,s2}^{eff} – \sigma_{e,s2}^{eff} $$
This stress difference characterizes the in-plane stress mismatch between the two sides of the interface due to different elastic parameters and chemical strains. Similarly, the hydrostatic stresses at the electrode/SPE interfaces are modified as
$$ \sigma_{k,s1}^{eff} = \frac{1-2\nu_k}{3} \frac{P}{\gamma_1} + \frac{E_k \Omega_k}{9(1-\nu_k)} (c_k – c_{k,0}), \quad k = p, e $$
$$ \sigma_{k,s2}^{eff} = \frac{1-2\nu_k}{3} \frac{P}{\gamma_2} + \frac{E_k \Omega_k}{9(1-\nu_k)} (c_k – c_{k,0}), \quad k = n, e $$
The main material parameters, initial conditions, and operating conditions are listed in Table 3. The temperature \(T\) is 300 K, and the reference concentration \(c_{ref}\) is 1 mol/m\(^3\). The cell is charged at 1 A/m\(^2\) to a cutoff voltage of 4.03 V and then discharged at the same current density until the SOC reaches 0.05. The model is solved numerically in COMSOL Multiphysics using a fully coupled solver and the MUMPS linear solver, with a relative tolerance of 0.005 and a tolerance factor of 0.1. The pressure scan interval is 0.3 MPa for pressures up to 4 MPa and 1 MPa for pressures above 4 MPa.
| Parameter | Positive electrode (LiCoO\(_2\)) | SPE (PEO-LiPF\(_6\)) | 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 (m\(^{2.5}\) mol\(^{-0.5}\) s\(^{-1}\)) | \(1.53 \times 10^{-11}\) | – | \(5.03 \times 10^{-11}\) |
| Diffusion coefficient (m\(^2\)/s) | \(1.76 \times 10^{-15}\) | \(2.5 \times 10^{-13}\) (\(D_+\)), \(3 \times 10^{-13}\) (\(D_-\)) | \(8 \times 10^{-14}\) |
| Partial molar volume (m\(^3\)/mol) | \(1.92 \times 10^{-6}\) | \(3.95 \times 10^{-6}\) (\(\Omega_+\)), \(1.46 \times 10^{-4}\) (\(\Omega_-\)) | \(4.17 \times 10^{-6}\) |
| Charge transfer coefficient | 0.5 | – | 0.5 |
| Young’s modulus (GPa) | 70 | 0.5 | 10 |
| Poisson’s ratio | 0.20 | 0.40 | 0.30 |
To clarify the extension of my model relative to previous studies, I compare the main innovations in Table 4.
| Model | Main innovation |
|---|---|
| Previous study A | Coupled stress terms with hydrostatic stress into mass transport and interfacial kinetics, and studied the effect of bending loads. |
| Previous study B | Introduced Persson contact mechanics theory into the electrochemical model, proposed the contact factor, and applied it to interfacial kinetics. |
| Previous study C | Established a coupled electrochemical–mechanical model for composite electrode solid electrolyte cells, incorporating the effect of pressure on composite electrode microstructure into electrochemical and mechanical coupling. |
| My model | Provides hydrostatic stress expressions under axial pressure and couples the Persson contact factor into the interfacial exchange current density, hydrostatic stress, and interfacial stress difference to study the effect of axial pressure. |
To verify the model, I set the contact factors at both the positive/SPE and negative/SPE interfaces to 1 and compare the terminal voltage–SOC curve with a reference model under the same cell composition and a baseline condition. The results show that the charge–discharge curves are basically consistent in voltage plateau, SOC variation range, and high-SOC voltage change position. The coefficient of determination \(R^2\) is 0.9931, and the maximum absolute error is 0.0310 V. These results indicate that my model can effectively capture the key electrochemical laws of a solid electrolyte cell. In addition, the area factor describing the relationship between interfacial impedance and contact area ratio is \(-1\) in my model, while experimental measurements report values in the range of \(-1.13\) to \(-1.05\), which is close to my model. This indicates that introducing the contact factor linearly into the exchange current density makes the interfacial impedance exhibit an inverse contact area scaling, and this treatment has public experimental support. For the pressure response of interfacial impedance, I use the normalized charge transfer impedance \(R_{norm}(P) = R_{ct}(P) / R_{ct}(P_0)\). For example, at \(P = 1.2\) MPa and \(P_0 = 0.43\) MPa, the experimental result is \(R_{norm} = 0.7326\), while my simulation gives \(R_{norm} = 0.7041\). The difference is small, indicating that my model predicts the pressure sensitivity of the charge transfer impedance of the whole cell reasonably well. In summary, my model can describe the electrochemical behavior under the given material system parameters and supports the rationality of the interfacial contact–interfacial kinetics coupling form and the impedance pressure sensitivity.
Based on the coupled electrochemical–mechanical model and the pressure-dependent interfacial contact model, I analyze the effects of axial pressure on the operating response of the solid electrolyte cell. I first examine the effect of axial pressure on the interfacial contact state. As axial pressure increases, the contact factors at both interfaces first increase rapidly and then gradually stabilize, indicating that axial pressure can increase the effective contact area between the electrodes and the SPE and improve the solid–solid interfacial contact state. When the axial pressure further increases, the contact factor gradually approaches saturation, and further pressure increase brings limited contact improvement. For the positive/SPE interface, the contact factor increases rapidly in the low-pressure range and approaches 100% when the pressure increases to about 4 MPa, after which further pressure increase has little effect on the interfacial contact state. In contrast, the contact factor of the negative/SPE interface responds more slowly to pressure, and a higher axial pressure is required to gradually approach the complete contact state.
I then analyze the effect of axial pressure on the electrochemical response. The terminal voltage of the solid electrolyte cell under different negative electrode contact factors is shown. As the contact factor \(\gamma\) increases, the discharge curve shifts toward higher voltage and higher capacity. According to the contact factor definition, a smaller \(\gamma\) means a smaller effective contact area at the electrode/SPE interface, and the effective reaction region available for charge transfer is limited. Under constant-current discharge, if the effective reaction area is insufficient, the same applied current must be borne by a smaller interfacial region, the interfacial charge transfer polarization increases, the overpotential loss increases, and the terminal voltage drops faster. Therefore, the releasable capacity of the solid electrolyte cell is limited under low contact factor conditions. Conversely, a larger \(\gamma\) can increase the effective interfacial reaction area, reduce the interfacial overpotential loss, and enable the cell to maintain a higher terminal voltage at the same discharge capacity, thereby extending the discharge process before reaching the cutoff voltage and increasing the releasable capacity. This result indicates that improved interfacial contact can enhance capacity utilization by reducing interfacial polarization, which is an important pathway by which axial pressure affects electrochemical output.
I further analyze the terminal voltage and its main components as functions of SOC under different axial pressures. The electrolyte potential contribution is very small, so I mainly discuss the concentration contribution \(V_{con}\), stress contribution \(V_{str}\), and overpotential contribution \(V_{ove}\). Under low-pressure conditions, the terminal voltage is generally lower, the voltage drop during discharge is more obvious, and the operable SOC range is smaller. As axial pressure increases, the terminal voltage curve extends to a wider SOC range, indicating that pressurization can improve the capacity utilization of the solid electrolyte cell. Among the components, \(V_{con}\) mainly varies with SOC, and the differences between different pressures are small, indicating that the direct effect of axial pressure on the concentration term is limited. \(V_{str}\) increases with increasing SOC, indicating that the concentration change during lithium insertion and extraction affects the terminal voltage through chemical strain and stress evolution. At higher pressures, \(V_{str}\) is slightly higher overall, indicating that external pressure enhances the stress-related voltage contribution. However, in terms of magnitude, \(V_{str}\) is only tens of millivolts, and its role is still a correction term in the terminal voltage. In contrast, \(V_{ove}\) is the most sensitive to axial pressure. Under low pressure, insufficient interfacial contact leads to a small effective interfacial reaction area, and under constant current, a larger interfacial overpotential is required to maintain the charge transfer process. Therefore, the overpotential contribution is larger in magnitude, causing a more obvious polarization loss in the terminal voltage. As axial pressure increases, the interfacial contact factor increases, the modified exchange current density increases, the interfacial charge transfer polarization weakens, the overpotential loss decreases, and the solid electrolyte cell can maintain a higher terminal voltage over a wider SOC range. Thus, the improvement in terminal voltage and capacity utilization by axial pressure mainly originates from the reduction in overpotential caused by improved interfacial contact, while the responses of concentration and stress contributions to pressure changes are relatively weak.
I also examine the absolute magnitude proportions of the concentration, stress, and overpotential contributions in the terminal voltage components at SOC = 0.2 and SOC = 0.45. The proportions are calculated as \(100\% \times V_i / (V_{con} + V_{str} + V_{ove})\), where \(i = con, str, ove\). At both SOC values, as axial pressure increases, the overpotential contribution proportion decreases significantly, while the concentration contribution proportion gradually increases. However, combined with the previous analysis, this does not mean that axial pressure significantly enhances the concentration contribution itself. Instead, improved interfacial contact increases the effective interfacial reaction area, weakens the interfacial charge transfer polarization, and reduces the overpotential loss. When the overpotential contribution decreases, the relative proportion of the concentration contribution increases. Therefore, the increase in the concentration contribution proportion mainly reflects the relative contribution change after interfacial polarization is weakened. Comparing different SOC values, the overpotential contribution proportion at SOC = 0.45 is larger than that at SOC = 0.2, indicating that under higher SOC conditions, interfacial polarization has a greater impact on the electrochemical reaction and transport state of the solid electrolyte cell. At the same time, the stress contribution proportion at SOC = 0.45 is higher than that at SOC = 0.2, because the chemical strain and stress response caused by concentration changes during lithium insertion and extraction increase with SOC. However, the stress contribution is significantly lower than the concentration and overpotential contributions at both SOC values, indicating that the stress voltage term mainly plays a correction role and is not the dominant factor in terminal voltage changes. Overall, the effect of axial pressure on the terminal voltage components is mainly reflected in the reduction of the overpotential contribution. As pressure increases, improved interfacial contact weakens interfacial polarization, and the feature of terminal voltage response dominated by interfacial overpotential gradually diminishes, while the relative role of the concentration contribution increases. This further indicates that the main pathway by which axial pressure improves electrochemical output is to improve the interfacial contact state and reduce the interfacial overpotential, rather than directly changing the concentration or stress contributions.
In summary, the effect of axial pressure on the electrochemical response of a solid electrolyte cell is mainly transmitted through the interfacial contact state. Under low pressure, the effective contact area at the electrode/SPE interface is small, the interfacial reaction region is limited, and the interfacial charge transfer polarization is enhanced during constant-current discharge, increasing the overpotential loss, lowering the terminal voltage, and limiting the operable SOC range and capacity utilization. As axial pressure increases, the contact conditions at the positive/SPE and negative/SPE interfaces improve, the effective interfacial reaction area increases, the modified exchange current density increases, and the interfacial overpotential contribution decreases significantly, shifting the terminal voltage curve toward higher voltage and higher capacity. The voltage component analysis further shows that the concentration contribution is mainly controlled by SOC changes and responds weakly to pressure changes; the stress contribution increases slightly with SOC but is small in magnitude and mainly acts as a correction; in contrast, the overpotential contribution is the most sensitive to axial pressure and is the main source of pressure-induced improvement in terminal voltage and capacity utilization. When the pressure increases from 0.1 MPa to 10 MPa, the improvement in interfacial contact and electrochemical output is relatively obvious; when the pressure continues to increase, the contact factor gradually approaches saturation, and the reduction in overpotential and the increase in terminal voltage diminish, showing a clear marginal decreasing characteristic. Therefore, from an electrochemical perspective, appropriately increasing axial pressure is beneficial for improving interfacial reaction conditions and capacity utilization, but further pressure increase brings limited electrochemical gains.
I then examine the effect of axial pressure on the mechanical response. The in-layer stress distribution along the thickness direction of the solid electrolyte cell under different axial pressures and SOC values is analyzed. The stress state of each material layer varies significantly with SOC: as SOC increases, the tensile stress in the positive electrode region increases significantly, the compressive stress amplitude in the negative electrode region increases synchronously, and the stress amplitude in the SPE region is relatively small. This phenomenon indicates that the chemical strain caused by concentration changes during lithium insertion and extraction is an important source of in-layer stress evolution. Because the elastic moduli and partial molar volumes of the positive electrode, SPE, and negative electrode are different, and the lithium concentration changes in each layer differ during operation, the chemical strains and constraint responses of each layer are not consistent, resulting in obvious interlayer stress differences along the thickness direction. Compared with the effect of SOC changes, the direct effect of axial pressure on the in-layer stress distribution at the same SOC is relatively limited. When the pressure increases from 0.1 MPa to 1 MPa, the stress distribution in each layer changes little; when the pressure further increases to 10 MPa and 50 MPa, the compressive stress amplitudes in the SPE and negative electrode regions increase slightly, while the tensile stress amplitude in the positive electrode region remains almost unchanged. These results indicate that the in-layer stress response is not simply controlled by axial pressure directly but is determined by the combined action of external pressure and concentration-induced strain. It is worth noting that the operable SOC range of the solid electrolyte cell is narrow under low pressure, while higher pressure allows the cell to enter a higher SOC range by improving interfacial contact and reducing polarization loss. Under high SOC, the chemical strain caused by concentration changes further develops, and the difference between the positive electrode tensile stress and negative electrode compressive stress becomes more obvious. Therefore, the effect of axial pressure on mechanical response is not only reflected in the direct effect of external pressure on in-layer stress but also in the enhancement of stress state at high SOC after the operable SOC range is expanded. These results provide a basis for analyzing the variation of interfacial stress difference with pressure.
I analyze the maximum interfacial stress difference at the positive/SPE and negative/SPE interfaces during operation under different axial pressures. Due to the differences in elastic moduli, partial molar volumes, and lithium concentration changes between the electrodes and the SPE, the stress responses on both sides of the interface are significantly different. As the axial pressure increases from 0.1 MPa to 50 MPa, the maximum interfacial stress difference at both interfaces increases overall, but the growth trend has obvious stages: the interfacial stress difference rises rapidly in the low-pressure range and then gradually slows down. This result indicates that the interfacial stress difference does not increase simply linearly with pressure. Combined with the previous interfacial contact model and in-layer stress analysis, this change can be further explained from two aspects: the actual bearing state of the interface and the development of chemical strain at high SOC. On the one hand, under incomplete interfacial contact, the real contact area at the electrode/SPE interface is smaller than the nominal contact area, and the applied nominal pressure is mainly borne by the real contact regions, increasing the actual bearing level in the contact regions. As axial pressure increases, the contact factor increases, the effective interfacial contact area increases, and the local bearing concentration caused by incomplete contact is alleviated, so the growth of interfacial stress difference in the medium- and high-pressure ranges gradually slows down. On the other hand, the external pressure itself is still increasing, and higher pressure allows the solid electrolyte cell to enter a higher SOC range by improving interfacial contact and reducing polarization loss. Under high SOC, the chemical strain caused by concentration changes further develops. Therefore, the maximum interfacial stress difference still increases overall with increasing axial pressure. Comparing the two interfaces, the maximum interfacial stress difference at the positive/SPE interface is overall higher than that at the negative/SPE interface and maintains a more obvious growth trend in the high-pressure range, indicating that the positive electrode side interface is the more significant interfacial mechanical response difference location in my model. This result indicates that although axial pressure can improve solid–solid interfacial contact and enhance electrochemical output, excessive pressure will continue to increase the interfacial stress difference. Therefore, in the subsequent pressure window evaluation, the interfacial stress difference should be used as an important mechanical cost indicator limiting further pressure increase, rather than relying solely on contact improvement and electrochemical performance enhancement.
To analyze the overall thickness fluctuation during operation, I further analyze the relationship between the maximum thickness change \(\Delta H\) and the axial pressure \(P\). The maximum thickness change \(\Delta H\) is defined as the difference between the maximum and minimum total thickness of the solid electrolyte cell during operation under a given pressure:
$$ \Delta H(P) = \max_t l(P,t) – \min_t l(P,t) $$
where \(l(P,t)\) is the cell thickness at time \(t\) under applied pressure \(P\). As axial pressure increases, \(\Delta H\) shows an overall upward trend, but the change is mainly concentrated in the low-pressure range. When the pressure increases from 0.1 MPa to about 4 MPa, \(\Delta H\) increases from about 4 nm to about 8 nm, a relatively obvious increase; after further pressure increase, the curve gradually flattens, and the difference in thickness change amplitude between different pressure conditions in the high-pressure range decreases significantly. This change is related to the effect of axial pressure on interfacial contact and capacity utilization. Under low pressure, insufficient interfacial contact limits the operable SOC range of the solid electrolyte cell, and the concentration change amplitude caused by lithium insertion and extraction in the positive and negative electrodes is small, so \(\Delta H\) is low. As pressure increases, the interfacial contact state improves, the overpotential loss decreases, and the cell can operate in a wider SOC range. The material concentration change and the corresponding chemical strain amplitude increase, so \(\Delta H\) increases significantly. When the pressure further increases, the improvement in interfacial contact and electrochemical performance gradually saturates, the expansion of the operable SOC range is limited, and the thickness fluctuation caused by concentration changes also tends to be flat. Overall, although \(\Delta H\) increases significantly in the low-pressure range, its absolute magnitude is still on the nanometer scale, which is small relative to the total thickness of the solid electrolyte cell, indicating that the thickness breathing response can reflect the pressure-induced operating deformation change but is not the main factor limiting further pressure increase in the high-pressure range.
In summary, the effect of axial pressure on the mechanical response of a solid electrolyte cell is mainly reflected in three aspects: in-layer stress distribution, interfacial stress difference, and overall thickness change. First, the in-layer stress state is closely related to SOC. As SOC increases, the tensile stress in the positive electrode region increases, and the compressive stress amplitude in the negative electrode region increases, indicating that the concentration change and the resulting chemical strain during lithium insertion and extraction are important sources of in-layer stress evolution. In contrast, at the same SOC, the direct effect of axial pressure on the in-layer stress distribution is relatively limited, indicating that the internal stress of the solid electrolyte cell is not simply controlled by external pressure but is jointly affected by external pressure constraint, concentration change, and material parameter differences. Second, as axial pressure increases, the maximum interfacial stress differences at the positive/SPE and negative/SPE interfaces both increase overall, with the positive/SPE interface stress difference being higher, indicating that the positive electrode side interface is the more important interfacial mechanical response difference location in my model. This result indicates that high pressure can improve solid–solid interfacial contact but also further enhances the stress response difference between the two sides of the interface. Finally, the maximum thickness change \(\Delta H\) increases overall with increasing pressure and increases significantly in the low-pressure range; when the pressure continues to increase, \(\Delta H\) gradually flattens, indicating that axial pressure can change the thickness response during operation, but the absolute magnitude of this change is small and is not the main limiting factor in the high-pressure range. Thus, the effect of axial pressure on mechanical response cannot be understood only as the direct compression effect of external pressure; it should also consider the indirect effect on high-SOC stress states after improving interfacial contact and expanding the operable SOC range. Compared with the overall thickness change, the interfacial stress difference continues to increase during pressure increase, which better reflects the problem of inconsistent mechanical responses on both sides of the electrode/SPE interface under high pressure, and should therefore be regarded as an important mechanical factor limiting further pressure increase.
To further verify the reliability of my simulation results, I conducted grid independence verification, maximum time step independence verification, and parameter sensitivity analysis. For grid independence, I divided the model into five different mesh schemes, and the number of mesh elements in each solid domain is listed in Table 5.
| Scheme | Positive electrode | SPE | Negative electrode | Total |
|---|---|---|---|---|
| M1 | 14 | 18 | 10 | 42 |
| M2 | 21 | 27 | 15 | 63 |
| M3 | 35 | 45 | 25 | 105 |
| M4 | 70 | 90 | 50 | 210 |
| M5 | 140 | 180 | 100 | 420 |
Under different mesh schemes, the maximum voltage deviation of the charge–discharge curve is only 0.274 mV, and the maximum deviation of the maximum stress difference at the positive/SPE interface under 10 MPa is only 0.111 MPa. These deviations are low relative to the magnitude of the original data. Therefore, different mesh schemes have little effect on the simulation results. I adopt mesh scheme M1 with uniform mesh distribution in each layer. For time step independence, I used five different maximum time steps. The maximum voltage deviation of the charge–discharge curve is 0.53 mV, and the maximum interfacial stress difference is almost identical under different time steps. To balance computational cost and accuracy, I set the maximum time step to 2 s. The time step schemes are listed in Table 6.
| Scheme | Maximum time step |
|---|---|
| T1 | 2 s |
| T2 | 1 s |
| T3 | 0.5 s |
| T4 | 0.25 s |
| T5 | 0.125 s |
I also performed a sensitivity analysis on the parameters \(m\) and \(I\) over a wide range to explore the model behavior under different interfacial roughness parameters. The parameter schemes are listed in Table 7. As the parameter \(m\) decreases and the parameter \(I\) increases, the interface becomes rougher, and the contact factors at both the positive/SPE and negative/SPE interfaces recover more slowly with increasing applied pressure.
| Positive/negative interface \(m\) | Positive/negative interface \(I\) |
|---|---|
| \(-2.77 / -3.08\) | \(-9.35 / -6.61\) |
| \(-2.67 / -2.98\) | \(-10.35 / -7.61\) |
| \(-2.72 / -3.03\) | \(-9.85 / -7.11\) |
| \(-2.82 / -3.13\) | \(-8.85 / -6.11\) |
| \(-2.87 / -3.18\) | \(-8.35 / -5.61\) |
Under different roughness parameters, the charge–discharge curves of the solid electrolyte cell under different applied pressures show differences in the low-pressure range but are similar in the medium- and high-pressure ranges. As the parameter \(m\) decreases and the parameter \(I\) increases, the interfacial roughness increases, and the operable SOC range under the same pressure becomes smaller. In addition, even with different roughness parameters, the conclusion that the operable SOC range of the solid electrolyte cell increases with increasing applied pressure still holds. Combined with the small effect of different roughness parameters on the charge–discharge curves in the medium- and high-pressure ranges, I conclude that different roughness parameters affect the contact response and related indicator values in the low-pressure range but do not change the core qualitative conclusions. For the maximum stress difference at the positive/SPE interface, as the parameter \(m\) decreases and the parameter \(I\) increases, the interfacial roughness increases, the overall maximum interfacial stress difference decreases, and the difference between different roughness parameters is relatively large in the low-pressure range and gradually decreases as the applied pressure increases. However, for different roughness parameters, the maximum interfacial stress difference increases monotonically with increasing external pressure and the growth rate slows down, indicating that the roughness parameters do not affect the general trend of the maximum interfacial stress difference with external pressure.
I establish a multi-criteria evaluation and optimization method for the axial pressure window. Axial pressure is an important operating parameter for regulating the solid–solid interfacial contact state of a solid electrolyte cell, and its reasonable value directly affects interfacial electrochemical reactions, ion transport, and interlayer mechanical reliability. According to the previous analysis, axial pressure has a dual effect on the solid electrolyte cell, so the selection of axial pressure cannot be based only on electrochemical performance improvement or a single mechanical constraint. It should be comprehensively balanced among interfacial contact improvement, electrochemical performance enhancement, interfacial stress difference control, and pressurization cost. I select usable areal capacity \(Q\), minimum interfacial contact factor \(\gamma_{\min}\), maximum interfacial stress difference \(S\), and pressurization cost \(P\) to construct the axial pressure multi-criteria evaluation method. \(Q\) measures the capacity utilization capability, \(\gamma_{\min}\) measures the contact recovery degree of the worse side among the two electrode/SPE interfaces, \(S\) measures the interfacial stress response difference and the corresponding local mechanical risk, and \(P\) measures the pressure constraint cost caused by applying and maintaining external pressure. The maximum thickness change \(\Delta H\) is mainly used to measure the overall thickness response during operation. From the previous mechanical response analysis, the absolute magnitude of \(\Delta H\) is extremely small relative to the total thickness of the solid electrolyte cell. Compared with indicators such as capacity utilization, interfacial contact recovery, and effective interfacial stress difference, it is difficult to serve as the dominant constraint determining the axial pressure window. If \(\Delta H\) is directly included as one of the optimization objectives and normalized, it may artificially amplify the thickness fluctuation difference with a small actual magnitude, causing the comprehensive loss function to become overly sensitive to this auxiliary structural response indicator. Therefore, I treat \(\Delta H\) as a supplementary analysis quantity for the overall thickness response under axial pressure, rather than including it in the main optimization objective function, to ensure that the comprehensive loss function mainly reflects the trade-off among capacity improvement, contact improvement, interfacial stress difference control, and pressurization cost.
Under constant-current discharge, the areal capacity of the solid electrolyte cell can be calculated from the discharge current density and cutoff time:
$$ Q(P) = \frac{i_{app} t_{cut}(P)}{3600} $$
where \(Q(P)\) is the areal capacity under pressure \(P\) in Ah/m\(^2\), \(i_{app}\) is the discharge current density of 1 A/m\(^2\), and \(t_{cut}(P)\) is the single discharge time in seconds. The normalized capacity indicator is
$$ Q_n(P) = \frac{Q(P) – Q_{\min}}{Q_{\max} – Q_{\min}} $$
The contact factor of the worse interface is
$$ \gamma_{\min}(P) = \min \{ \gamma_{s1}(P), \gamma_{s2}(P) \} $$
where \(\gamma_{s1}(P)\) and \(\gamma_{s2}(P)\) are the contact factors at the positive/SPE and negative/SPE interfaces under axial pressure \(P\). The normalized contact indicator is
$$ \gamma_n(P) = \frac{\gamma_{\min}(P) – \gamma_{\min}^{\min}}{\gamma_{\min}^{\max} – \gamma_{\min}^{\min}} $$
The maximum stress difference of the worse interface is
$$ S(P) = \max \{ | \sigma_{p,s1}^{eff} – \sigma_{e,s1}^{eff} |, | \sigma_{n,s2}^{eff} – \sigma_{e,s2}^{eff} | \} $$
The normalized maximum interfacial stress difference is
$$ S_n(P) = \frac{S(P) – S_{\min}}{S_{\max} – S_{\min}} $$
For automotive stack structures, higher axial pressure usually means higher preload requirements, stricter assembly tolerance control, and higher structural stiffness and constraint capability requirements. Therefore, I use the normalized pressure indicator as the pressurization cost:
$$ P_n(P) = \frac{P – P_{\min}}{P_{\max} – P_{\min}} $$
where \(P_{\min} = 0.1\) MPa and \(P_{\max} = 50\) MPa. Based on the above normalized indicators, I establish the axial pressure comprehensive loss function:
$$ J(P) = J_Q + J_\gamma + J_S + J_P $$
The components are
$$ J_Q = \omega_Q [1 – Q_n(P)]^2 $$
$$ J_\gamma = \omega_\gamma [1 – \gamma_n(P)]^2 $$
$$ J_S = \omega_S S_n(P) $$
$$ J_P = \omega_P P_n(P) $$
where \(\omega_Q, \omega_\gamma, \omega_S, \omega_P\) are the weight coefficients for areal capacity, interfacial contact, interfacial stress difference, and pressurization cost, respectively. Under a given weight scheme, the optimal axial pressure is
$$ P^* = \arg \min_{P_{\min} \leq P \leq P_{\max}} J(P) $$
To examine the influence of different engineering preferences on the optimal pressure, I set three weight schemes: performance-priority, balanced, and conservative. The performance-priority scheme increases the weights of capacity and interfacial contact recovery to match operating conditions that focus more on electrochemical performance improvement. The balanced scheme simultaneously considers capacity, contact, interfacial mechanical risk, and pressurization cost. The conservative scheme increases the weights of effective interfacial stress difference and pressurization cost to match operating conditions that focus more on mechanical reliability and assembly constraints. The weight coefficients are listed in Table 8.
| Scheme | \(\omega_Q\) | \(\omega_\gamma\) | \(\omega_S\) | \(\omega_P\) |
|---|---|---|---|---|
| Performance-priority | 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 |
Because the comprehensive loss function \(J(P)\) is composed of multiple outputs of the COMSOL electrochemical–mechanical coupling model, it is difficult to obtain an explicit analytical expression. To directly perform derivative-free bounded search based on the full model response, I use a particle swarm optimization algorithm to optimize the axial pressure variable. Each candidate pressure calls COMSOL for a new solution without using interpolation surrogates. In this study, the position of each particle represents a candidate axial pressure value. In the \(k\)-th iteration, the position and velocity of the \(i\)-th particle are denoted as \(P_i^k\) and \(v_i^k\), respectively. The particle swarm algorithm guides the search through the individual optimal position and the global optimal position. The velocity and position update formulas are
$$ v_i^{k+1} = \omega v_i^k + c_1 r_1 (p_i^k – P_i^k) + c_2 r_2 (g^k – P_i^k) $$
$$ P_i^{k+1} = P_i^k + v_i^{k+1} $$
where \(p_i^k\) is the individual optimal pressure of the \(i\)-th particle in the historical search process, and \(g^k\) is the current global optimal pressure of the particle swarm. \(\omega\) is the inertia weight, \(c_1\) and \(c_2\) are the individual and social learning factors, and \(r_1\) and \(r_2\) are random numbers between 0 and 1. In each iteration, the pressure corresponding to the particle \(P_i^k\) is input into the evaluation process to obtain \(Q_n(P_i^k)\), \(\gamma_n(P_i^k)\), \(S_n(P_i^k)\), and \(P_n(P_i^k)\), and then substituted into the comprehensive loss function to calculate \(J(P_i^k)\). If the loss function corresponding to the current pressure is smaller than the historical optimal value of the particle, the individual optimal pressure is updated. If its loss function is smaller than the current global optimal value, the global optimal pressure is updated. To avoid pressure search out of bounds, the updated particle position is limited within \([P_{\min}, P_{\max}]\), and a particle velocity upper limit is set to avoid an excessively large single-step search span. In this study, the number of particles is 15, the number of iterations is 9, the inertia weight \(\omega\) is 0.65, the individual learning factor \(c_1\) and the social learning factor \(c_2\) are both 1.45, and the velocity upper limit is 8 MPa. To further confirm the optimization result, after the particle swarm algorithm obtains the preliminary optimal pressure, I perform a local scan of the nearby pressure interval to determine the minimum position of the comprehensive loss function. The optimal pressure under each scheme is finally determined by the minimum value of the corresponding comprehensive loss function.
To verify that the optimization result does not change significantly with a specific algorithm, I further use a deterministic method combining full-range direct coarse scanning and partition one-dimensional bounded optimization for optimization, with the loss function and weight factors consistent with the particle swarm optimization. To obtain the objective function under continuous pressure, I use shape-preserving piecewise cubic Hermite interpolation for \(Q_n(P)\), \(\gamma_n(P)\), and \(S_n(P)\), while \(P_n(P)\) is still calculated analytically. This method reduces the possibility of interpolation overshoot in the low-pressure region where the contact factor changes rapidly, compared with ordinary cubic splines. Then, I perform a full-range coarse scan in the range of 0.1–50 MPa with a maximum interval of 1 MPa. The coarse scan pressure set is \(\Omega_c = \{0.1 \text{ MPa}, 1.1 \text{ MPa}, \ldots, 49.1 \text{ MPa}, 50 \text{ MPa}\}\). For any scan node, if \(J(P_i) \leq \min \{J(P_{i-1}), J(P_{i+1})\}\), the node is taken as a local minimum candidate point, and its adjacent two nodes form a candidate interval. For each candidate interval, a one-dimensional bounded optimization method is used for refined search:
$$ P_i^* = \arg \min_{P \in [P_{i-1}, P_{i+1}]} J(P) $$
In the calculation, the MATLAB function fminbnd is used to implement one-dimensional bounded optimization. This function uses a combination of golden section search and parabolic interpolation to find the minimum of the objective function within a given pressure interval. After completing all candidate interval searches, the optimization results of each interval are compared with the pressure upper and lower boundaries:
$$ P^* = \arg \min_{P \in \{P_{\min}, P_{\max}, P_1^*, P_2^*, \ldots, P_m^*\}} J(P) $$
where \(m\) is the number of local minimum candidate intervals obtained by the coarse scan. In this study, the axial pressure convergence tolerance of the fminbnd function is set to \(10^{-3}\) MPa, the maximum number of iterations is 200, and the maximum number of objective function evaluations is 500.
I now present the optimization results. The variation of each normalized evaluation indicator with axial pressure \(P\) shows that both \(Q_n\) and \(\gamma_n\) increase with increasing pressure, but their growth rates differ: \(Q_n\) rises rapidly in the low-pressure range and flattens after about 10 MPa, while \(\gamma_n\) continues to rise, and its overall recovery process is relatively slower, indicating that the contact improvement of the worse interface requires higher pressure to be gradually promoted. \(S_n\) and \(P_n\) as cost-type indicators continue to increase with increasing pressure, where \(P_n\) increases linearly, and \(S_n\) still maintains an upward trend in the medium- and high-pressure ranges. These results indicate that with increasing axial pressure, capacity utilization and interfacial contact improvement gradually show diminishing returns, while interfacial stress difference and pressurization cost continue to accumulate. Therefore, simply increasing pressure cannot simultaneously achieve the optimal result of electrochemical gain and mechanical/engineering cost control.
Based on the comprehensive loss function established above, I optimized the axial pressure for the performance-priority, balanced, and conservative schemes. The optimization results based on the particle swarm algorithm are listed in Table 9.
| Optimization scheme | Trade-off optimal pressure |
|---|---|
| Performance-priority | 15.8 MPa |
| Balanced | 12.2 MPa |
| Conservative | 10.0 MPa |
The optimization results obtained by combining direct coarse scanning and partition one-dimensional bounded optimization are listed in Table 10.
| Optimization scheme | Trade-off optimal pressure |
|---|---|
| Performance-priority | 15.7 MPa |
| Balanced | 12.4 MPa |
| Conservative | 9.8 MPa |
Comparing the optimization results of the two algorithms, the difference between the trade-off optimal pressure values obtained by the two algorithms under the three schemes does not exceed 0.2 MPa. Therefore, the optimization result does not change significantly with the specific algorithm used. If the direct scanning method is used, a large number of calculation nodes are required when high precision is demanded, resulting in a large computational load. Compared with the one-dimensional optimization method combining full-range direct coarse scanning and partition one-dimensional bounded optimization used in this study, the particle swarm algorithm does not have an advantage in computational cost and algorithm complexity, but as a derivative-free bounded population search method, it does not rely on interpolation surrogates and objective function gradients, making it suitable for models such as the one in this study where analytical gradients are difficult to obtain. In addition, if multiple design variables such as end plate stiffness, preload displacement, or material parameters are further introduced, and the problem is extended to multi-variable or even multi-objective problems, the applicability of particle swarm algorithms will become more obvious. Based on these considerations, the following discussion uses the optimization results obtained by the particle swarm algorithm.
The variation of the comprehensive loss function with axial pressure under different weight schemes shows that the comprehensive loss function has a relatively optimal result in the medium pressure range under all three weight schemes. This is because in the low-pressure range, insufficient interfacial contact leads to low capacity and contact recovery. As pressure increases, capacity and contact factor increase significantly, causing the comprehensive loss function to decrease rapidly. When pressure continues to increase, capacity and contact improvement gradually saturate, while the maximum interfacial stress difference and pressurization cost continue to increase, causing the comprehensive loss function to rise again. As the evaluation preference shifts from electrochemical performance to interfacial mechanical reliability and pressure constraint cost, the trade-off optimal pressure gradually moves toward the low-pressure side. The areal capacity, worse-interface contact factor, and maximum interfacial stress difference under each scheme are listed in Table 11.
| Scheme | \(Q\) (Ah/m\(^2\)) | \(\gamma_{\min}\) | \(S\) (MPa) | \(P\) (MPa) |
|---|---|---|---|---|
| Performance-priority | 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 contribution of the comprehensive loss function and its components under the balanced scheme shows that in the low-pressure range, the capacity term \(J_Q\) and the contact term \(J_\gamma\) occupy the main contribution, indicating that the comprehensive loss mainly originates from insufficient capacity output and insufficient interfacial contact. As axial pressure increases, \(J_Q\) and \(J_\gamma\) decrease rapidly. When pressure further increases, the capacity term is already close to a low level, and the decrease rate of the contact term gradually slows down, while the interfacial stress difference term \(J_S\) and the pressurization cost term \(J_P\) continue to increase, so the comprehensive loss function \(J\) no longer continues to decrease. Under the balanced scheme, the comprehensive loss function achieves a relatively optimal result near about 12.2 MPa, indicating that at this pressure, capacity improvement, contact recovery, interfacial stress difference, and pressurization cost reach a relative balance.
To further illustrate the diminishing returns and cost accumulation characteristics in axial pressure regulation, I construct marginal benefit and marginal cost based on normalized indicators. The capacity improvement and interfacial contact recovery are regarded as electrochemical benefits:
$$ B(P) = \frac{Q_n(P) + \gamma_n(P)}{2} $$
The maximum interfacial stress difference and pressurization demand are regarded as the cost of applying pressure:
$$ C(P) = \frac{S_n(P) + P_n(P)}{2} $$
On this basis, I calculate the marginal benefit \(\Delta B\) and marginal cost \(\Delta C\) when the applied axial pressure increases by 1 MPa. In the low-pressure range, the marginal electrochemical benefit brought by increasing axial pressure is relatively high, but as pressure continues to increase, the marginal electrochemical benefit decreases rapidly and gradually approaches zero after the medium pressure range. In contrast, the marginal mechanical/pressure cost persists over a wide pressure range, reflecting that the maximum interfacial stress difference and pressurization demand will accumulate continuously with increasing pressure. Near 12 MPa, the electrochemical benefit and pressure cost are close, indicating that this pressure can better reflect the trade-off between performance improvement and pressure cost. After further increasing pressure, the additional electrochemical benefit is limited, while the cost term continues to increase. At this time, the cost term is always greater than the benefit term, which further verifies the dual effect of axial pressure on the solid electrolyte cell.
From the normalized indicators and marginal analysis, when the applied axial pressure reaches about 10 MPa, the improvement in usable areal capacity gradually slows down, the interfacial contact factor continues to increase but the growth rate gradually decreases, while the maximum interfacial stress difference and pressurization cost continue to increase. In the medium pressure range of about 10–16 MPa, the marginal benefit \(\Delta B\) gradually decreases and approaches the marginal cost \(\Delta C\), indicating that the electrochemical benefit and the cost of applying pressure are relatively close in this range. When the pressure exceeds about 16 MPa, the marginal cost \(\Delta C\) is higher than the marginal benefit \(\Delta B\), indicating that the capacity improvement and interfacial contact improvement brought by further pressure increase are no longer sufficient to offset the increase in maximum interfacial stress difference and pressurization cost. The trade-off optimal pressures obtained by the three representative weight schemes are 15.8 MPa, 12.2 MPa, and 10.0 MPa, respectively, all located in the above medium pressure range. Combining the variation laws of each indicator, the relationship between marginal benefit and marginal cost, and the multi-scheme optimization results, I take 10.0–15.8 MPa as the trade-off optimal axial pressure window, with 12.2 MPa as the trade-off optimal pressure under the balanced scheme.
To further examine the stability of the above trade-off optimal pressure window under a wider range of weight changes, I performed sensitivity analysis on the weight of each component, the normalization interval, the loss function form, and the charge–discharge current density. First, I randomly generated multiple weight combinations within a wider weight range and calculated the corresponding optimal axial pressure. The weight ranges are \(0.18 \leq \omega_Q \leq 0.45\), \(0.20 \leq \omega_\gamma \leq 0.50\), \(0.08 \leq \omega_S \leq 0.35\), and \(0.05 \leq \omega_P \leq 0.30\). The random weight perturbation values were sampled from a Dirichlet distribution with parameter vector \((3,3,2,2)\) under the condition that the sum of the four weights is 1, and screened according to the above ranges until 5000 valid samples were obtained, with a random seed of 20260617. The random sampling results show that the optimal pressure is mainly distributed in the medium pressure range. The 5%, 25%, 50%, 75%, and 95% quantiles 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 the trade-off optimal pressure is 11.10 MPa, which is close to the trade-off optimal pressure of 12.2 MPa obtained under the balanced scheme. At the same time, 63.56% of the 5000 samples are distributed within the trade-off optimal pressure window, indicating that the medium pressure window obtained in this study is not accidentally determined by a single weight combination but has a certain robustness under weight perturbations.
Second, I optimized for three different normalization intervals: the original scheme, an upper limit extended by 20%, and a 5%–95% quantile interval. The original scheme normalization interval is [0.1 MPa, 50 MPa], the upper limit extended by 20% is [0.1 MPa, 60 MPa], and the 5%–95% quantile interval is [2.6 MPa, 47.5 MPa]. The corresponding weights for calculating the trade-off optimal axial pressure window remain unchanged. The optimization results are listed in Table 12.
| Normalization interval | Trade-off optimal pressure window |
|---|---|
| Original scheme | 10.0–15.8 MPa |
| Upper limit extended by 20% | 10.8–17.1 MPa |
| 5%–95% quantile | 11.8–17.9 MPa |
The results show that although the trade-off optimal pressure windows obtained under different normalization intervals have some offset, they are all concentrated in the medium pressure range and overlap to a large extent with the original scheme pressure window. Therefore, the pressure window obtained in this study has a certain robustness to changes in the normalization interval. Third, I optimized for different loss function forms. In addition to the original scheme loss function, I added three different loss functions: a fully quadratic function form, an exponential form, and a normalized Huber loss function with a threshold parameter \(\delta = 0.5\). The fully quadratic function form changes \(J_S\) and \(J_P\) in the original scheme to \(\omega_S S_n^2(P)\) and \(\omega_P P_n^2(P)\), respectively, while the other terms remain unchanged. The normalized Huber loss function with threshold parameter \(\delta = 0.5\) is
$$ J(P) = \omega_Q H(1 – Q_n(P)) + \omega_\gamma H(1 – \gamma_n(P)) + \omega_S H(S_n(P)) + \omega_P H(P_n(P)) $$
$$ H(x) = \begin{cases} \frac{2}{3} x^2, & 0 \leq x \leq 0.5 \\ \frac{4}{3} x – \frac{1}{3}, & 0.5 < x \leq 1 \end{cases} $$
The exponential form loss function is
$$ J(P) = \omega_Q E(1 – Q_n(P)) + \omega_\gamma E(1 – \gamma_n(P)) + \omega_S E(S_n(P)) + \omega_P E(P_n(P)) $$
$$ E(x) = \frac{1 – e^{-2x}}{1 – e^{-2}} $$
The optimization results for different loss function forms are listed in Table 13.
| Loss function form | Trade-off optimal pressure window |
|---|---|
| Original scheme | 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 |
The results show that the trade-off optimal pressure windows obtained under different loss function forms are all in the medium pressure range and overlap significantly with the original scheme pressure window. Although the window endpoints change to some extent, the overall pressure range does not deviate significantly, indicating that the pressure window obtained in this study has a certain robustness to the loss function form. Finally, to explore the influence of different charge–discharge current magnitudes on the optimized pressure window, I optimized for different charge–discharge current densities. The optimization results are listed in Table 14.
| Charge–discharge current | Trade-off optimal pressure window |
|---|---|
| Original scheme, 1 A/m\(^2\) | 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 |
The results show that as the current density decreases, the trade-off optimal pressure window shifts toward lower pressure overall, but the numerical change of the window is small and all are in the medium pressure range. Therefore, the pressure window obtained by the original scheme has a certain stability. After confirming that the obtained pressure interval has a certain robustness to operating conditions and evaluation settings, and after examining the sensitivity of the obtained pressure interval to the evaluation method and the considered current densities, it is also necessary to further examine its engineering load implications for large-area cells. For a large-area pouch solid electrolyte cell with an effective area of 120 mm × 80 mm, the total preload force corresponding to the trade-off optimal pressure range of 10.0–15.8 MPa obtained by multi-scheme optimization is 96.0–151.7 kN, and the balanced scheme at 12.2 MPa corresponds to 117.1 kN. The trade-off optimal pressure range obtained in this study falls within the clamping pressure range of 10–30 MPa reported in previous studies. In addition, the pressure uniformity indicator \(\Delta \sigma_P / P_{op}\) is about 2.16% in the preload pressure range of 10–30 MPa, where \(\Delta \sigma_P\) is the standard deviation of the cell surface pressure distribution and \(P_{op}\) is the operating pressure. Therefore, under the premise of reasonable end plate and clamping mechanism design, the solid electrolyte cell can have good pressure uniformity. However, the preload force range of hundreds of kilonewtons corresponding to the pressure window obtained in this study does indeed impose high requirements on end plate strength and connector load-bearing capacity. I have not established a module structure or carried out maximum equivalent stress and yield safety factor checks. Therefore, the obtained pressure interval should be understood as a trade-off result under the material parameters, ideal layered structure, and evaluation preferences of this study, and its application in automotive cells or modules still requires further design and verification combined with specific cell area and constraint structures.
In summary, the reasonable pressure of a solid electrolyte cell lies in the medium pressure range where performance gain and mechanical cost are balanced. The initial selection and regulation of axial pressure should avoid simply pursuing high pressure for electrochemical performance improvement and should comprehensively consider interfacial contact recovery, electrochemical performance improvement, maximum interfacial stress difference, and pressurization cost. Based on the above optimization results and sensitivity analysis, I determine that the trade-off optimal axial pressure window for the given material system parameters is 10.0–15.8 MPa, and the trade-off optimal pressure under the balanced scheme is 12.2 MPa.
I draw the following conclusions. First, in terms of electrochemical response, axial pressure can improve the electrochemical performance of a solid electrolyte cell by improving the electrode/SPE interfacial contact state, but its benefit has a marginal decreasing characteristic. Under low pressure, the effective interfacial reaction area is limited, the interfacial charge transfer polarization is enhanced, and the terminal voltage and capacity utilization are reduced. As pressure increases, interfacial contact improves, the overpotential contribution decreases, and the terminal voltage and capacity output of the cell are improved. After entering the medium- and high-pressure range, the contact factor gradually approaches saturation, and the reduction in overpotential and the improvement in capacity are significantly reduced. This indicates that appropriate axial pressure is a necessary condition for maintaining effective solid–solid interfacial contact, but further pressure increase beyond a reasonable range cannot significantly improve electrochemical performance.
Second, in terms of mechanical response, axial pressure affects the in-layer stress level, enhances the interfacial stress difference, and changes the thickness change response. First, the change in in-layer stress level by axial pressure is mainly achieved indirectly by affecting the electrochemical reaction. Second, as pressure increases, the maximum interfacial stress differences at the positive/SPE and negative/SPE interfaces both increase overall, with the positive/SPE interface stress difference being higher, indicating that high pressure enhances the mechanical response difference between the two sides of the interface. Finally, the maximum thickness change \(\Delta H\) increases significantly in the low-pressure range, but its absolute magnitude is small and gradually flattens in the high-pressure range, indicating that the overall thickness change is not the main factor limiting further pressure increase, and the interfacial stress difference better reflects the mechanical constraint under high pressure.
Third, by comprehensively optimizing usable areal capacity, interfacial contact state, interfacial stress difference, and pressurization cost, the trade-off optimal axial pressure window under the material system parameters, ideal layered structure, and isothermal conditions of this study is 10.0–15.8 MPa, and the trade-off optimal pressure under the balanced scheme is 12.2 MPa. This pressure range can maintain good capacity output and interfacial contact state while reducing the further accumulation of interfacial stress difference and pressurization cost, reflecting the balance between electrochemical performance improvement and interfacial mechanical risk control.
I obtained the coupling law between axial pressure and interfacial contact state at the ideal thin-film layered unit scale, but the obtained pressure window is only applicable to the current material system parameters and ideal layered structure. Although the areal capacity of a single solid electrolyte thin-film cell unit is limited, previous studies have shown that multilayer thin-film unit integration has the potential to balance energy and power and is feasible for further packaging and manufacturing scale-up, providing a potential structural path for the expansion of such cells to automotive high-energy and high-power applications. The modeling and analysis method in this study can provide a reference for pressure regulation and interfacial reliability analysis of multilayer-integrated solid electrolyte cells for automotive applications. Future research can further target real rough interfaces and consider mechanisms such as interfacial damage evolution, debonding, and local contact failure. Combined with interface strength, fracture energy, and allowable stress thresholds, more physically meaningful mechanical constraints can be established. At the same time, engineering constraints of battery modules such as end plate stiffness, structural mass, and pressure uniformity can be included in the evaluation. Temperature effects can be introduced to analyze performance and safety risks under extreme conditions such as high rate, overcharge, overdischarge, and abnormal pressure. In addition, quantitative experimental verification of interfacial mechanical response can be further carried out, and the axial pressure window of cells or modules under different material systems and pressure constraint conditions can be verified and corrected.