I develop a coupled electrochemical-mechanical framework to determine a rational axial pressure window for solid electrolyte cells. The central problem is that a solid electrolyte cell does not behave like a conventional liquid-electrolyte cell. Its electrode-electrolyte interfaces are solid-solid contacts, and the real contact area is usually smaller than the nominal area. This contact loss changes interfacial charge transfer, interfacial polarization, capacity utilization, and local mechanical loading. At the same time, axial pressure is one of the most direct external variables available for recovering contact and regulating stack pressure. However, increasing axial pressure also increases interfacial stress mismatch and pressurization cost. Therefore, I treat the axial pressure window as a multi-criteria optimization problem rather than a single-variable electrochemical tuning problem.
In my model, the solid electrolyte cell is represented as a one-dimensional layered structure through the thickness direction. The positive electrode, solid polymer electrolyte, and negative electrode are treated as continuous homogeneous media. I neglect in-plane transport and edge effects under the ideal stacked condition. The solid electrolyte cell is assumed to operate isothermally. The model includes lithium transport, interfacial charge transfer, pressure-dependent contact area, chemo-elastic stress, and interfacial stress difference. The pressure-dependent contact state is coupled to both interfacial reaction kinetics and interfacial mechanical response. This coupling path allows me to examine how axial pressure influences a solid electrolyte cell through contact recovery, polarization reduction, capacity improvement, and mechanical penalty accumulation.
1. Electrochemical Model for the Solid Electrolyte Cell
I describe lithium transport in the positive and negative electrodes by mass conservation and a stress-coupled flux expression. For each electrode region, the governing equation is
$$
\frac{\partial c_k}{\partial t}+\frac{\partial j_k}{\partial x}=0,\quad k=p,n
$$
where \(c_k\) is lithium concentration, \(j_k\) is lithium flux, \(x\) is the through-thickness coordinate, and \(t\) is time. The flux includes concentration gradient and hydrostatic stress gradient:
$$
j_k=-D_k\left(\frac{\partial c_k}{\partial x}-\frac{\Omega_k c_k}{RT}\frac{\partial \sigma_k}{\partial x}\right),\quad k=p,n
$$
Here \(D_k\) is the lithium diffusion coefficient, \(\Omega_k\) is the partial molar volume, \(\sigma_k\) is hydrostatic stress, \(R\) is the gas constant, and \(T\) is temperature. This expression captures the fact that mechanical stress in a solid electrolyte cell is not a passive output; it can modify the driving force for lithium transport.
For the solid polymer electrolyte, I use a modified Nernst-Planck form:
$$
\frac{\partial c_e}{\partial t}+\frac{\partial j_e}{\partial x}=0
$$
$$
j_e=-D_e\left(\frac{\partial c_e}{\partial x}-\frac{\Omega_e c_e}{RT}\frac{\partial \sigma_e}{\partial x}\right)
$$
The effective diffusion coefficient and partial molar volume of the solid electrolyte cell are written as
$$
D_e=\frac{2D_+D_-}{D_++D_-}
$$
$$
\Omega_e=\Omega_++\Omega_-
$$
where \(D_+\) and \(D_-\) are cation and anion diffusion coefficients, and \(\Omega_+\) and \(\Omega_-\) are the corresponding partial molar volumes. Charge conservation in the electrolyte is enforced by
$$
\frac{\partial i_e}{\partial x}=0
$$
The interfacial electrochemical reaction is described by a Butler-Volmer equation. For the positive and negative interfaces, I write
$$
i_k=i_{k,0}\left[\exp\left(\frac{\beta_k F\eta_k}{RT}\right)-\exp\left(-\frac{(1-\beta_k)F\eta_k}{RT}\right)\right]
$$
For small to moderate overpotentials, this can be represented by the sinh form:
$$
i_k=2i_{k,0}\sinh\left(\frac{F\eta_k}{2RT}\right),\quad k=p,n
$$
The reference exchange current densities for the solid electrolyte cell are
$$
i_{p,0}=Fk_p\left(c_{p}^{\max}-c_{p,s}\right)^{1-\beta_p}c_{p,s}^{\beta_p}\left(c_{e,s}\right)^{1-\beta_p}
$$
$$
i_{n,0}=Fk_n\left(c_{n}^{\max}-c_{n,s}\right)^{1-\beta_n}c_{n,s}^{\beta_n}\left(c_{e,s}\right)^{1-\beta_n}
$$
where \(c_{k}^{\max}\) is maximum lithium concentration, \(c_{k,s}\) is surface concentration, \(c_{e,s}\) is electrolyte lithium concentration at the interface, and \(\beta_k\) is the charge transfer coefficient. The state of charge of the solid electrolyte cell is defined as
$$
\mathrm{SOC}(t)=\frac{\int_{0}^{l_p+l_e+l_n}c_n(x,t)\,dx}{c_n^{\max}l_n}
$$
The electrochemical boundary conditions are summarized in Table 1.
| Boundary | Condition |
|---|---|
| Positive electrode outer boundary | $$j_p(x=0)=0$$ |
| Positive electrode/SPE interface | $$j_p(x=l_p)=i_p/F$$ |
| Negative electrode outer boundary | $$j_n(x=l_p+l_e+l_n)=0$$ |
| Negative electrode/SPE interface | $$j_n(x=l_p+l_e)=i_n/F$$ |
| SPE at positive interface | $$j_e(x=l_p)=\frac{D_-}{D_++D_-}\frac{i_p}{F}$$ |
| SPE at negative interface | $$j_e(x=l_p+l_e)=\frac{D_-}{D_++D_-}\frac{i_n}{F}$$ |
| SPE potential at positive interface | $$i_e(x=l_p)=i_p$$ |
| SPE potential at negative interface | $$i_e(x=l_p+l_e)=i_n$$ |
I decompose the terminal voltage of the solid electrolyte cell into reference, concentration, stress, overpotential, and electrolyte potential contributions:
$$
V_{\mathrm{cell}}=V_{\mathrm{ref}}+V_{\mathrm{con}}+V_{\mathrm{str}}+V_{\mathrm{ove}}+V_{\mathrm{ele}}
$$
The voltage components are listed in Table 2.
| Voltage component | Expression |
|---|---|
| Reference voltage | $$V_{\mathrm{ref}}=3.9\,\mathrm{V}$$ |
| Concentration contribution | $$V_{\mathrm{con}}=\frac{RT}{F}\ln\left(\frac{c_{e,s_1}(c_{p}^{\max}-c_{p,s_1})c_{n,s_2}}{c_{e,s_2}(c_{n}^{\max}-c_{n,s_2})c_{p,s_1}}\right)$$ |
| Stress contribution | $$V_{\mathrm{str}}=\frac{1}{F}\left(\Omega_p\sigma_{p,s_1}-\Omega_n\sigma_{n,s_2}+\Omega_e\sigma_{e,s_2}-\Omega_e\sigma_{e,s_1}\right)$$ |
| Overpotential contribution | $$V_{\mathrm{ove}}=\eta_p-\eta_n$$ |
| Electrolyte potential contribution | $$V_{\mathrm{ele}}=\Phi_{e,s_1}-\Phi_{e,s_2}$$ |
2. Mechanical Model for the Solid Electrolyte Cell
I assume quasi-static mechanical equilibrium in the solid electrolyte cell:
$$
\nabla\cdot\boldsymbol{\sigma}+\mathbf{b}=0
$$
The external axial pressure is applied along the thickness direction. I further assume that the solid electrolyte cell is strongly constrained in the in-plane directions, so the in-plane strains are zero:
$$
\varepsilon_{yy}=\varepsilon_{zz}=0
$$
The through-thickness stress is set by the external axial pressure:
$$
\sigma_{xx}=-P
$$
The total strain is decomposed into elastic and chemical contributions:
$$
\varepsilon=\varepsilon^{\mathrm{elas}}+\varepsilon^{\mathrm{chem}}
$$
Using linear elasticity and chemical strain from concentration change, the stress state in each layer can be expressed as
$$
\sigma_k=\frac{E_k}{(1+\nu_k)(1-2\nu_k)}\left[(1-\nu_k)\varepsilon_k+\nu_k(\varepsilon_{yy}+\varepsilon_{zz})\right]-\frac{E_k\Omega_k(c_k-c_{k,0})}{3(1-\nu_k)}
$$
where \(E_k\) is Young’s modulus and \(\nu_k\) is Poisson’s ratio. The layer strain under axial pressure and concentration change is
$$
\varepsilon_k=-\frac{(1+\nu_k)(1-2\nu_k)}{E_k(1-\nu_k)}P+\frac{1}{3}\Omega_k(c_k-c_{k,0})
$$
The total thickness change of the solid electrolyte cell is
$$
\Delta l=l_p\varepsilon_p+l_e\varepsilon_e+l_n\varepsilon_n
$$
3. Interfacial Contact Model for the Solid Electrolyte Cell
The solid-solid interfaces in a solid electrolyte cell are rough. Therefore, the real contact area is smaller than the nominal contact area. I define the interfacial contact factor as
$$
\gamma=\frac{A}{A_0}
$$
where \(A\) is the real contact area and \(A_0\) is the nominal contact area. I use a Persson-type contact model to relate \(\gamma\) to axial pressure:
$$
\gamma(P)=\frac{A}{A_0}=\int_{q_L}^{\infty}\mathrm{erf}\left(\frac{q}{q_L}\sqrt{\frac{E^*}{2C(q)}}\right)\frac{dq}{q}
$$
The equivalent modulus is
$$
\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}
$$
The surface roughness power spectrum is represented as
$$
C(q)=10^{mq+I}
$$
The lower wavenumber is
$$
q_L=\frac{2\pi}{\sqrt{A_0}}
$$
I take the nominal contact area as \(A_0=1\,\mathrm{cm^2}\) and the characteristic diffusion time as \(t=3600\,\mathrm{s}\). Different interface types are assigned different roughness parameters. The positive electrode/SPE interface is treated as film-type, while the negative electrode/SPE interface is treated as bulk-type because of its more pronounced particulate morphology.
I couple the contact factor to the interfacial reaction kinetics by modifying the exchange current density:
$$
i_{p,0}^{\mathrm{eff}}=\gamma_1 i_{p,0}
$$
$$
i_{n,0}^{\mathrm{eff}}=\gamma_2 i_{n,0}
$$
I also couple the contact factor to the effective mechanical stress at the interface. Because the external nominal pressure is carried mainly by the real contact spots, the local elastic load is amplified. The effective elastic stress terms are written as
$$
\sigma_{p,s_1}^{\mathrm{eff,elas}}=\frac{P}{\gamma_1}\frac{1+\nu_p}{1-\nu_p}
$$
$$
\sigma_{e,s_1}^{\mathrm{eff,elas}}=\frac{P}{\gamma_1}\frac{1+\nu_e}{1-\nu_e}
$$
$$
\sigma_{e,s_2}^{\mathrm{eff,elas}}=\frac{P}{\gamma_2}\frac{1+\nu_e}{1-\nu_e}
$$
$$
\sigma_{n,s_2}^{\mathrm{eff,elas}}=\frac{P}{\gamma_2}\frac{1+\nu_n}{1-\nu_n}
$$
The chemical stress contribution remains
$$
\sigma_{k}^{\mathrm{chem}}=-\frac{E_k\Omega_k(c_k-c_{k,0})}{3(1-\nu_k)}
$$
Thus, the total effective stress at an interface is
$$
\sigma_{k,s_i}^{\mathrm{eff}}=\sigma_{k,s_i}^{\mathrm{eff,elas}}+\sigma_{k,s_i}^{\mathrm{chem}}
$$
I define the interface stress difference as
$$
\Delta\sigma_{s_1}^{\mathrm{eff}}=\sigma_{p,s_1}^{\mathrm{eff}}-\sigma_{e,s_1}^{\mathrm{eff}}
$$
$$
\Delta\sigma_{s_2}^{\mathrm{eff}}=\sigma_{n,s_2}^{\mathrm{eff}}-\sigma_{e,s_2}^{\mathrm{eff}}
$$
This quantity measures the mismatch in mechanical response across the solid-solid interface of the solid electrolyte cell. A larger interfacial stress difference indicates a stronger mismatch between the electrode and the solid electrolyte under the same axial pressure and electrochemical state.
4. Parameters, Numerical Solution, and Validation
The main parameters used in my solid electrolyte cell model are given in Table 3.
| Parameter | Positive electrode | SPE | Negative electrode |
|---|---|---|---|
| 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\(^{-1}\)) | \(1.53\times10^{-11}\) | — | \(5.03\times10^{-11}\) |
| Diffusion coefficient (m\(^2\) s\(^{-1}\)) | \(1.76\times10^{-15}\) | \(D_+=2.5\times10^{-13}\), \(D_-=3\times10^{-13}\) | \(8\times10^{-14}\) |
| Partial molar volume (m\(^3\) mol\(^{-1}\)) | \(1.92\times10^{-6}\) | \(\Omega_+=3.95\times10^{-6}\), \(\Omega_-=1.46\times10^{-4}\) | \(4.17\times10^{-6}\) |
| Charge transfer coefficient | 0.5 | — | 0.5 |
| Young’s modulus (GPa) | 70 | 0.5 | 10 |
| Poisson’s ratio | 0.20 | 0.40 | 0.30 |
I solve the coupled system with a fully coupled Newton method and a direct linear solver. The relative tolerance is set to 0.005. The pressure scan interval is 0.3 MPa below 4 MPa and 1 MPa above 4 MPa. The solid electrolyte cell is charged at 1 A m\(^{-2}\) to 4.03 V and then discharged at the same current density until the state of charge reaches 0.05.
I validate the electrochemical part by setting both contact factors to unity and comparing the terminal voltage versus state of charge response with a reference thin-film solid electrolyte cell model. The coefficient of determination is \(R^2=0.9931\), and the maximum absolute voltage error is 0.0310 V. I also compare the pressure sensitivity of charge transfer resistance with available experimental trends. For a pressure increase from 0.43 MPa to 1.2 MPa, my model predicts a normalized resistance ratio of 0.7041, while the experimental trend is about 0.7326. These comparisons support the use of my contact-coupled interfacial kinetics for analyzing a solid electrolyte cell under axial pressure.
5. Electrochemical Response of the Solid Electrolyte Cell under Axial Pressure
I first examine how axial pressure changes the interfacial contact factor. For both the positive electrode/SPE interface and the negative electrode/SPE interface, the contact factor increases rapidly at low pressure and then approaches saturation. The positive interface recovers faster because its roughness parameters correspond to a film-type contact. The negative interface requires higher pressure to approach complete contact. This difference is important because the solid electrolyte cell is limited by the poorer of the two interfaces.
Table 4 summarizes the qualitative pressure response of the solid electrolyte cell.
| Pressure range | Contact response | Electrochemical response | Mechanical response |
|---|---|---|---|
| Low pressure | Poor real contact area; strong contact loss | High interfacial polarization; low capacity utilization | Lower interfacial stress difference but limited SOC range |
| Medium pressure | Rapid contact recovery; diminishing improvement | Reduced overpotential; improved capacity and voltage | Interfacial stress difference increases steadily |
| High pressure | Near saturation; small additional contact gain | Electrochemical gain becomes marginal | Stress difference and pressurization cost continue to accumulate |
I calculate discharge voltage versus capacity for different contact factors. As the contact factor increases, the discharge curve shifts to higher voltage and higher capacity. When the contact factor is small, the effective reaction area is small. Under galvanostatic discharge, the same applied current must pass through a smaller interfacial area, so the interfacial charge transfer polarization increases. This increases the overpotential loss and causes the terminal voltage to drop earlier. Therefore, a solid electrolyte cell with poor contact cannot fully use its nominal capacity.
I also decompose the terminal voltage into concentration, stress, overpotential, and electrolyte potential contributions. The electrolyte potential contribution is very small. The concentration contribution changes mainly with state of charge and is not strongly affected by axial pressure. The stress contribution increases with state of charge, but its magnitude remains on the order of tens of millivolts. The overpotential contribution is the most pressure-sensitive term. At low pressure, contact loss increases the interfacial overpotential. At higher pressure, contact recovery increases the effective exchange current density, reduces charge transfer polarization, and lowers the overpotential contribution. This is the main reason why axial pressure improves the terminal voltage and capacity utilization of the solid electrolyte cell.
Table 5 gives the qualitative ranking of voltage component sensitivity to axial pressure.
| Voltage component | Sensitivity to SOC | Sensitivity to axial pressure | Role in the solid electrolyte cell |
|---|---|---|---|
| Concentration contribution | Strong | Weak | Mainly controlled by lithium concentration |
| Stress contribution | Moderate | Moderate | Corrective term from chemo-elastic stress |
| Overpotential contribution | Strong | Very strong | Dominant pressure-dependent loss term |
| Electrolyte potential contribution | Weak | Weak | Small correction in the present solid electrolyte cell |
When pressure increases from 0.1 MPa to 10 MPa, the improvement in contact and electrochemical output is significant. Beyond that, the contact factor approaches saturation, and the reduction in overpotential becomes smaller. This shows a clear marginal diminishing return. From an electrochemical viewpoint, moderate axial pressure is necessary for a solid electrolyte cell, but very high pressure does not guarantee proportional performance improvement.
6. Mechanical Response of the Solid Electrolyte Cell under Axial Pressure
I analyze the through-thickness stress distribution at different states of charge and different axial pressures. As the state of charge increases, the positive electrode tensile stress increases, and the negative electrode compressive stress magnitude increases. The stress in the SPE remains relatively small. This behavior comes from lithium concentration changes and chemical strain. Because the positive electrode, SPE, and negative electrode have different elastic moduli and partial molar volumes, their chemical strains and constraint responses are different. Therefore, a clear layer-to-layer stress difference appears in the solid electrolyte cell.
At a fixed state of charge, the direct effect of axial pressure on the layer stress distribution is relatively limited. When pressure increases from 0.1 MPa to 1 MPa, the stress distribution changes are small. When pressure increases to 10 MPa or 50 MPa, the compressive stress magnitude in the SPE and negative electrode increases slightly, while the positive electrode tensile stress remains nearly unchanged. This means that the internal stress of a solid electrolyte cell is not controlled by external pressure alone. It is jointly determined by external pressure, concentration-induced strain, and material property mismatch.
However, axial pressure has an indirect mechanical effect. Higher pressure improves interfacial contact and reduces polarization, allowing the solid electrolyte cell to operate over a wider SOC range. At higher SOC, chemical strain develops further, and the stress difference between the positive and negative electrodes becomes more pronounced. Thus, the mechanical response under axial pressure includes both the direct compressive effect and the indirect effect of extending the accessible SOC range.
I calculate the maximum interfacial stress difference during operation for both interfaces. As axial pressure increases from 0.1 MPa to 50 MPa, the maximum interfacial stress difference increases overall. The growth is fast at low pressure and gradually slows at higher pressure. This occurs because the real contact area increases with pressure, which reduces local load concentration caused by incomplete contact. At the same time, the external pressure itself continues to increase, and the solid electrolyte cell can reach higher SOC states, where chemical strain is larger. As a result, the maximum interfacial stress difference still increases with pressure. The positive electrode/SPE interface shows a higher stress difference than the negative electrode/SPE interface, indicating that the positive side is the more critical mechanical location in my model.
I also examine the maximum thickness variation, defined as
$$
\Delta H(P)=\max_t l(P,t)-\min_t l(P,t)
$$
The maximum thickness variation increases with axial pressure, mainly in the low-pressure range. When pressure increases from 0.1 MPa to about 4 MPa, the thickness variation increases from about 4 nm to about 8 nm. At higher pressure, the curve becomes flatter. This is because higher pressure improves contact and expands the accessible SOC range, increasing concentration variation and chemical strain. Once contact and capacity improvements saturate, the thickness variation also tends to plateau. Although the thickness variation is sensitive to pressure in the low-pressure range, its absolute magnitude is still on the nanometer scale relative to the total thickness of the solid electrolyte cell. Therefore, I do not treat the maximum thickness variation as the main limiting factor for the high-pressure range. The interfacial stress difference is a more meaningful mechanical constraint for pressure window selection.
7. Numerical Sensitivity and Parameter Robustness
I perform mesh independence, time-step independence, and roughness parameter sensitivity tests. Table 6 lists the mesh cases.
| Mesh case | Positive electrode | SPE | Negative electrode | Total elements |
|---|---|---|---|---|
| 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 |
Across these mesh cases, the maximum voltage deviation is only 0.274 mV, and the maximum deviation in the positive interface stress difference at 10 MPa is 0.111 MPa. These deviations are small relative to the corresponding quantities. I therefore use mesh M1 with uniform elements in each layer.
Table 7 lists the maximum time-step cases.
| Time-step case | Maximum time step |
|---|---|
| T1 | 2 s |
| T2 | 1 s |
| T3 | 0.5 s |
| T4 | 0.25 s |
| T5 | 0.125 s |
Across the time-step cases, the maximum voltage deviation is 0.53 mV, and the interfacial stress difference is nearly identical. I select a maximum time step of 2 s to balance accuracy and computational cost.
I also vary the roughness parameters over a wide range. Table 8 shows the parameter sets.
| Case | Positive interface \(m\) | Negative interface \(m\) | Positive interface \(I\) | Negative interface \(I\) |
|---|---|---|---|---|
| R1 | -2.77 | -3.08 | -9.35 | -6.61 |
| R2 | -2.67 | -2.98 | -10.35 | -7.61 |
| R3 | -2.72 | -3.03 | -9.85 | -7.11 |
| R4 | -2.82 | -3.13 | -8.85 | -6.11 |
| R5 | -2.87 | -3.18 | -8.35 | -5.61 |
When the interface becomes rougher, the contact factor recovers more slowly with axial pressure. The discharge curves differ more at low pressure, but the differences become smaller at medium and high pressure. The maximum interfacial stress difference decreases as roughness increases, but the trend of increasing stress difference with axial pressure remains. Therefore, roughness changes shift numerical values, especially in the low-pressure range, but they do not change the main qualitative conclusions for the solid electrolyte cell.
8. Multi-Criteria Optimization of the Axial Pressure Window
The axial pressure of a solid electrolyte cell has a dual role. It improves interfacial contact and electrochemical output, but it also increases interfacial stress mismatch and pressurization cost. Therefore, I define four evaluation metrics:
$$
Q(P)=\frac{i_{\mathrm{app}}t_{\mathrm{cut}}(P)}{3600}
$$
$$
\gamma_{\min}(P)=\min\left(\gamma_{s_1}(P),\gamma_{s_2}(P)\right)
$$
$$
S(P)=\max\left(\left|\sigma_{p,s_1}^{\mathrm{eff}}-\sigma_{e,s_1}^{\mathrm{eff}}\right|,\left|\sigma_{n,s_2}^{\mathrm{eff}}-\sigma_{e,s_2}^{\mathrm{eff}}\right|\right)
$$
$$
P
$$
Here \(Q\) is usable areal capacity, \(\gamma_{\min}\) is the poorer interfacial contact factor, \(S\) is the maximum interfacial stress difference, and \(P\) is the axial pressure itself, which represents the pressurization cost. I normalize these quantities between the minimum and maximum pressures considered:
$$
Q_n(P)=\frac{Q(P)-Q_{\min}}{Q_{\max}-Q_{\min}}
$$
$$
\gamma_n(P)=\frac{\gamma_{\min}(P)-\gamma_{\min,\min}}{\gamma_{\min,\max}-\gamma_{\min,\min}}
$$
$$
S_n(P)=\frac{S(P)-S_{\min}}{S_{\max}-S_{\min}}
$$
$$
P_n(P)=\frac{P-P_{\min}}{P_{\max}-P_{\min}}
$$
I then define a composite loss function:
$$
J(P)=J_Q+J_\gamma+J_S+J_P
$$
with
$$
J_Q=\omega_Q\left[1-Q_n(P)\right]^2
$$
$$
J_\gamma=\omega_\gamma\left[1-\gamma_n(P)\right]^2
$$
$$
J_S=\omega_S S_n(P)
$$
$$
J_P=\omega_P P_n(P)
$$
The weights represent different engineering preferences. I consider three schemes: performance priority, balanced, and conservative. Table 9 lists the weights.
| 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 |
The optimal pressure is
$$
P^*=\arg\min_{P_{\min}\le P\le P_{\max}} J(P)
$$
Because the loss function is produced by the coupled solid electrolyte cell model, I cannot obtain an explicit analytical gradient. I therefore use a derivative-free bounded search. I first apply particle swarm optimization, where each particle represents a candidate axial pressure. The velocity and position updates are
$$
v_i^{k+1}=\omega v_i^k+c_1r_1\left(p_i^k-P_i^k\right)+c_2r_2\left(g^k-P_i^k\right)
$$
$$
P_i^{k+1}=P_i^k+v_i^{k+1}
$$
I use 15 particles and 9 iterations. The inertia weight is 0.65, and both learning factors are 1.45. The speed limit is 8 MPa. After the swarm search, I perform a local scan near the best pressure to refine the minimum.
To verify that the result is not algorithm-specific, I also use a deterministic method: a full-range coarse scan followed by one-dimensional bounded optimization on candidate intervals. I use shape-preserving piecewise cubic Hermite interpolation for the normalized capacity, contact factor, and stress difference. The pressure cost is computed analytically. The coarse scan covers 0.1 MPa to 50 MPa with a maximum spacing of 1 MPa. For any node satisfying
$$
J(P_i)\le \min\left(J(P_{i-1}),J(P_{i+1})\right)
$$
I form a candidate interval and refine it with a bounded one-dimensional optimizer. The convergence tolerance is \(10^{-3}\) MPa.
Table 10 compares the optimal pressures from the two search methods.
| Scheme | Particle swarm result | Deterministic scan result |
|---|---|---|
| Performance priority | 15.8 MPa | 15.7 MPa |
| Balanced | 12.2 MPa | 12.4 MPa |
| Conservative | 10.0 MPa | 9.8 MPa |
The difference between the two methods is no more than 0.2 MPa in all schemes. This confirms that the optimal pressure is not an artifact of a particular search algorithm. I use the particle swarm results in the following discussion.
9. Optimal Axial Pressure Window and Trade-Off Analysis
Across the three weighting schemes, the optimal pressure moves from 15.8 MPa under performance priority to 12.2 MPa under balanced weighting and to 10.0 MPa under conservative weighting. The balanced scheme represents a compromise among capacity, contact recovery, interfacial stress difference, and pressurization cost. Table 11 lists the key indicators at these optima.
| 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 |
In the balanced scheme, the loss function components show the trade-off clearly. At low pressure, the capacity loss and contact loss dominate. As pressure increases, both terms decrease rapidly. However, the stress difference cost and pressurization cost continue to rise. The composite loss function reaches a minimum near 12.2 MPa. This pressure gives a good balance between electrochemical benefit and mechanical or engineering penalty for the solid electrolyte cell.
I define marginal electrochemical benefit as
$$
B_n(P)=\frac{Q_n(P)+\gamma_n(P)}{2}
$$
and marginal pressure-related cost as
$$
C_n(P)=\frac{S_n(P)+P_n(P)}{2}
$$
Then I compute the changes per 1 MPa increase:
$$
\Delta B=B_n(P+\Delta P)-B_n(P)
$$
$$
\Delta C=C_n(P+\Delta P)-C_n(P)
$$
At low pressure, the marginal benefit is high. As pressure increases, the marginal benefit falls quickly. The marginal cost remains positive over a wide pressure range. Near 12 MPa, the marginal benefit and marginal cost become comparable. Beyond about 16 MPa, the marginal cost exceeds the marginal benefit. This means that additional pressure mainly increases interfacial stress difference and pressurization cost without a proportional gain in capacity or contact. This is the central trade-off for axial pressure design in a solid electrolyte cell.
Based on the indicator trends, the marginal benefit-cost relation, and the three weighting schemes, I obtain a trade-off axial pressure window of
$$
10.0\ \mathrm{MPa}\le P \le 15.8\ \mathrm{MPa}
$$
with a balanced optimum of
$$
P^*=12.2\ \mathrm{MPa}
$$
This window means that, for the material system and ideal layered structure considered here, a solid electrolyte cell should be operated with moderate axial pressure. Lower pressure cannot maintain sufficient solid-solid contact. Higher pressure gives limited electrochemical improvement but continues to increase interfacial stress mismatch and pressurization cost.
10. Robustness of the Pressure Window
I test the robustness of the pressure window under random weights, different normalization intervals, different loss function forms, and different current densities.
For random weights, I sample the four weights from a Dirichlet distribution and retain 5000 valid sets. The resulting optimal pressures have 5%, 25%, 50%, 75%, and 95% quantiles of 7.12 MPa, 9.10 MPa, 11.10 MPa, 13.81 MPa, and 17.92 MPa. The median is 11.10 MPa, close to the balanced optimum of 12.2 MPa. About 63.56% of the samples fall inside the window from 10.0 MPa to 15.8 MPa. Thus, the window is not caused by a single weight choice.
Table 12 shows the effect of changing the normalization interval.
| Normalization interval | Trade-off pressure window |
|---|---|
| Original: 0.1-50 MPa | 10.0-15.8 MPa |
| Upper bound extended by 20%: 0.1-60 MPa | 10.8-17.1 MPa |
| 5%-95% quantile interval: 2.6-47.5 MPa | 11.8-17.9 MPa |
Although the endpoints shift, all windows remain in the medium-pressure range and overlap substantially with the original window. Therefore, the pressure window is reasonably robust to the normalization interval.
Table 13 shows the effect of different loss function forms.
| Loss function form | Trade-off pressure window |
|---|---|
| Original form | 10.0-15.8 MPa |
| Full quadratic form | 11.5-16.7 MPa |
| Normalized Huber form with threshold 0.5 | 12.8-17.5 MPa |
| Exponential form | 11.0-18.6 MPa |
All forms produce windows in the medium-pressure range. The endpoints shift, but the overall pressure region does not move far. This supports the robustness of the proposed pressure window for the solid electrolyte cell.
Table 14 shows the effect of different charge-discharge current densities.
| Current density | Trade-off pressure window |
|---|---|
| 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 |
As the current density decreases, the pressure window shifts slightly toward lower pressure. However, the change is not large, and all windows remain in the medium-pressure range. Therefore, the original pressure window is stable enough for the present solid electrolyte cell analysis.
11. Engineering Load Implication
To interpret the pressure window at the engineering level, I convert the axial pressure into a total preload force for a large-format cell. For an effective cell area of \(120\,\mathrm{mm}\times80\,\mathrm{mm}\), the force is
$$
F=PA
$$
With \(A=120\times80=9600\,\mathrm{mm^2}=9.6\times10^{-3}\,\mathrm{m^2}\), the pressure window from 10.0 MPa to 15.8 MPa corresponds to a total preload force from 96.0 kN to 151.7 kN. The balanced optimum of 12.2 MPa corresponds to about 117.1 kN. Table 15 summarizes the conversion.
| Pressure | Force on \(120\,\mathrm{mm}\times80\,\mathrm{mm}\) area |
|---|---|
| 10.0 MPa | 96.0 kN |
| 12.2 MPa | 117.1 kN |
| 15.8 MPa | 151.7 kN |
This force level is substantial. It means that the pressure window of a solid electrolyte cell cannot be specified only from electrochemical contact considerations. The end plates, tie rods, frames, and clamping structures must be designed to sustain these loads while maintaining pressure uniformity. If the fixture is not sufficiently stiff or if the pressure distribution is nonuniform, local pressure may deviate from the nominal value. The local contact state, interfacial stress difference, and degradation behavior may then differ from the ideal layered model. Thus, the pressure window I obtain should be interpreted as a model-based trade-off for the chosen material parameters and ideal layered structure. It provides a starting point for fixture design and stack pressure control, but it still requires module-level structural verification and experimental validation.
12. Conclusions
I have developed a coupled electrochemical-mechanical model for a solid electrolyte cell with pressure-dependent interfacial contact. The model couples lithium transport, interfacial Butler-Volmer kinetics, chemo-elastic stress, Persson-type contact mechanics, and interfacial stress difference. I use the model to investigate axial pressure effects and to perform multi-criteria optimization of the axial pressure window.
The main conclusions are as follows.
First, axial pressure improves the electrochemical performance of a solid electrolyte cell mainly by increasing the real contact area at the electrode/SPE interfaces. As the contact factor increases, the effective exchange current density increases, the interfacial charge transfer polarization decreases, and the overpotential contribution falls. This shifts the discharge curve to higher voltage and higher capacity. However, the improvement has strong marginal diminishing returns. After the contact factor approaches saturation, additional pressure gives only a small electrochemical gain.
Second, axial pressure affects the mechanical response of a solid electrolyte cell in several ways. It changes the layer stress state indirectly by extending the accessible state-of-charge range, increases the maximum interfacial stress difference, and alters the thickness variation during operation. The positive electrode/SPE interface shows a higher stress difference than the negative electrode/SPE interface in my model. The maximum thickness variation increases mainly at low pressure and then tends to plateau. Because its absolute magnitude is small, I do not treat it as the main high-pressure constraint. The interfacial stress difference is a more important mechanical cost for pressure selection.
Third, considering usable areal capacity, minimum interfacial contact factor, maximum interfacial stress difference, and pressurization cost, the trade-off axial pressure window for the present material system and ideal layered structure is 10.0-15.8 MPa. The balanced optimum is 12.2 MPa. This window maintains good capacity output and contact recovery while limiting further accumulation of interfacial stress difference and pressurization cost.
Fourth, sensitivity tests show that the pressure window is robust to reasonable changes in weights, normalization intervals, loss function forms, and current density. Random weight sampling gives a median optimal pressure of 11.10 MPa, and about 63.56% of random samples fall inside the 10.0-15.8 MPa window. The window also overlaps strongly with results from alternative normalization and loss definitions. This suggests that the medium-pressure region is a stable design target for the solid electrolyte cell under the assumptions used here.
Fifth, the model is limited to an ideal one-dimensional layered solid electrolyte cell under isothermal conditions. It does not include interface damage evolution, debonding, local contact failure, temperature gradients, or module-level structural compliance. Future work should incorporate interface strength, fracture energy, allowable stress thresholds, end-plate stiffness, pressure uniformity, and temperature effects. It should also validate the predicted interfacial mechanical response experimentally and extend the pressure window optimization to real rough interfaces, multilayer integrated solid electrolyte cells, and automotive module-level stack pressure control.
In summary, my results indicate that axial pressure in a solid electrolyte cell should be treated as a multi-objective design variable. A moderate pressure window can improve solid-solid contact and electrochemical utilization, but excessive pressure mainly increases interfacial stress mismatch and engineering cost. The model and optimization framework provide a method for pressure regulation and interfacial reliability analysis of solid electrolyte cells and can be extended to more realistic stack architectures.