Physical Model and Machine Learning Enabled Electrolyte Channel Design for Fast Charging

Thick electrode is highly effective to increase the specific energy of a battery cell, but the associated increase in transport distance causes a major barrier for fast charging. We introduce a bio-inspired electrolyte channel design into thick electrodes to improve the cell performance, especially under fast charging conditions. The effects of channel length, width, tapering degree and active material width on the electrochemical performance and mechanical integrity are investigated. Machine learning by deep neural network (DNN) is developed to relate the geometrical parameters of channels to the overall cell performance. Integrating machine learning with the Markov chain Monte Carlo gradient descent optimization, we demonstrate that the complicated multivariable channel geometry optimization problem can be efficiently solved. The results show that within a certain range of geometrical parameters, the specific energy, specific capacity and specific power can be greatly improved. At the same time, the maximum first principal stress which is in the cathode region next to the separator can be significantly reduced, giving better mechanical integrity. Comparing to conventionaldesigned cells without electrolyte channels, we show a 79% increase in specific energy using channel design optimization. This study provides a design strategy and optimization method to achieve significantly improved battery performance. © 2020 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/ 1945-7111/aba096]

Eigenstrain of lithium intercalation e ij Electrode level strain E Effective Young's modulus of the porous electrode (GPa) E eq Equivalent potential (V) E p Young's modulus of the solid particle (GPa) f ± Electrolyte active coefficient F Faraday's constant (C mol −1 ) i Reaction current density as the surface of the particle (A m −2 ) i 0 Exchange current density at the surface of the particle (A m −2 ) i app Applied charging or discharging current density (A m −2 ) I Discharging current (A) k Reaction rate constant (m 2.5 mol −0.5 s −1 ) L n , L p Current collector thickness for the anode and cathode (μm) L EA , L EC Length of the channel in anode and cathode (μm) L s Separator thickness (μm) m Mass of the full cell (kg) m Al Aluminum current collector mass (kg) m Cu Copper current collector mass (kg) Electrolyte potential (V) Φ s Solid potential (V) Ω Lithium ion partial molar volume (m 3 mol −1 ) The lithium-ion battery (LIB) is widely used in many applications from consumer electronics to electric vehicles. 1,2 LIBs with thick electrode design have grained a lot of attention recently, where the increase in the volume fraction of active material comparing to current collector and other non-active components can effectively improve the specific energy and energy density. However, the thick electrode also increases the transport distance of ions and electrons. 3 As a result, the capacity utilization of active materials is reduced due to the associated polarization. The negative impact arising from longer diffusion distance is particularly significant under high current or fast charging conditions, while fast charging is increasingly demanded by many applications. In fact, the usable specific energy will decrease when the electrode is thicker than a threshold for any given current density requirement. Therefore, improving capacity utilization and rate performance for thick electrodes becomes highly important.
Several particle-level electrode designs have been proposed to increase the LIB performance, such as creating distributed porosity along the electrode thickness direction, 4 or aligning decorated graphite particles with magnetic field to reduce the diffusion tortuosity. 5 Besides, 3D electrodes have been explored for better performance, such as corrugating the current collectors, electrodes and separator together to form a trench architecture, [6][7][8] or forming the electrode into pillars arranged in interdigitated patterns. 9,10 These design strategies can reduce the ion diffusion distance and increase the contact area between the active material and the electrolyte. As a result, the battery performance in terms of energy, capacity or power density is improved. However, these kinds of strategies often require significant changes of the battery structure from the conventional configuration.
Introducing some forms of internal electrode-level architectures into the 2D planar electrode to increase its effective porosity is another way to improve the battery performance, which has been receiving increasing attention currently. [10][11][12][13][14] These architectures can further be used to affect tortuosity (quantification of ion conduction rate through the electrolyte of porous electrodes 15,16 ). Besides, adding internal architectures into a planar electrode does not change the conventional sandwich battery structure, and there are many ways to manufacture architectures such as channels in an electrode. A channel design is much easier to be commercialized than complicated 3D structures. 17 The improvement in performance of electrodes containing architectures has been observed in several studies. Clancy et al. 18 numerically investigated the behavior of a Liion micro-battery with a core-shell structure so that the cathode rod (core) is surrounded by electrolyte (shell) on the side. By considering various types of electrolytes including solid-state, polymer, polymer-gel and liquid, they found overall improvement of battery capacity for all the electrolytes used as a result of the geometric effect of the structure. Ye et al. 17 considered a pore-network structure in the anode. The discharge energy density was calculated for selected structural parameters, electrode thickness, porosity and discharging C rate, showing that the performance improvement was dependent on the structural parameters. Reale et al. 19 added macroscopic pores into the electrode to reduce the effective tortuosity, and showed that the areal capacitance could be significantly improved. They also suggested that a bi-tortuous electrode design could be helpful for fast charging/discharging. Li et al. 20 brought grooves to the cathode and derived an analytical expression to estimate the improvement in specific capacity.
These studies suggest that introducing internal electrode-level architectures to the electrode can improve the performance without any change in the electrode material or the form factor of the electrode. The first aim of this paper is to systematically study the effect of adding tapered electrolyte channels in electrodes, quantifying both the electrochemical performance and mechanical integrity of the channel design during fast charging. The tapered channel design allows investigating a wide range of geometrical parameters not studied before, which will help understand the potential of performance improvement brought by electrolyte channels.
Electrochemical models [21][22][23][24] can be used to calculate the battery characteristics for any given sets of channel geometry by numerical approaches such as the finite element method (FEM). However, conducting parametric optimization using FEM is challenging due to the huge computational cost associated with the highly non-linear electrochemical model. 25 It is critical to find a method to effectively connect the geometrical parameters of an electrolyte channel to the battery cell performance, to be able to effectively conduct a parametric optimal design. The second aim of this paper is to introduce machine learning for channel design optimization.
Machine learning is gaining increasing attention in solving emerging problems, which can calculate the output with high efficiency after proper training. This approach overcomes the difficulties and computational cost of solving complex differential equations. Artificial neural network (ANN), an important machine learning technique, has been used to estimate the state of charge (SOC). Dang et al. 26 employed a dual network setup, with the first network to identify the electrochemical model parameters and the second one to capture the relationship between the open circuit voltage and SOC. Weigert et al. 27 applied ANN to predict the SOC for both battery and battery-supercapacitor systems. For LIB performance, Wu et al. 25 developed an ANN system that can automatically generate electrode design maps for any given specific energy and specific power requirement, providing design parameters such as electrode thickness and porosity. Their work highlighted that ANN can handle the non-linear, complex and computationally expensive problem of battery design and optimization. Takagishi et al. 28 utilized ANN to predict the charging and discharging resistance based on process parameters including active material porosity, particle radius, electrolyte conductivity, compacting pressure and binder/additive volume ratio. The predicted resistance by the ANN regression model matched well the simulation results. Parthiban et al. 29 demonstrated ANN to estimate and predict the specific capacity change of a LIB with a CoO anode for 50 cycles. These studies demonstrated the capability and accuracy of ANN in capturing complex, non-linear physical relationships. Here we develop ANN to perform analysis and design optimization of electrolyte channels.
In this paper we introduce a bio-inspired electrolyte channel design into thick electrodes to improve the battery performance during fast charging. The effects of channel length, width, tapering degree and width of active material on electrochemical performance and mechanical integrity are investigated by finite element simulations. These results provide a fundamental understanding of how the electrolyte channel parameters affect the battery performance, the local state of charge and the stress level of the active material. With the simulation results as the training data, a deep neural network (DNN) is developed to directly relate the electrolyte channel parameters to the specific energy, specific capacity and specific power. A Markov chain Monte Carlo method is used together with DNN to find the Ragone planes and the optimized channel designs. For the purpose of this study, we only consider the electrochemical performance in terms of specific energy, specific capacity and specific power. Side reactions such as lithium plating or cell degradation are not considered.

Modelling
Electrode with channels.-We first present a full cell model with bio-inspired channels in both the anode and cathode, as shown in Fig. 1. The channel is filled with electrolyte. The geometry of a channel is characterized by its length (L EA and L EC , with subscript "A" denoting the anode channel and "C" denoting the cathode channel), root width (W EA and W EC ) and tip width (W EA-b and W EC-b ). The half-widths of active electrode material between two neighboring channels at their roots are W A and W C . The structure is periodic along the in-plane direction (y-axis) of the electrode, and the periodic width is denoted by W H = W EA + 2W A = W EC + 2W C . In this study, the anode thickness L n and cathode thickness L p are fixed as 250 μm. The separator has a thickness of L s = 30 μm. The thickness of current collector for the anode (L cn ) and for the cathode (L cp ) are both 15 μm. For demonstration, here we use a graphite (Li x C 6 ) anode and a Li y Mn 2 O 4 (LMO) cathode. The current collector on the anode side is copper while on the cathode side is aluminum. The electrolyte is 1 M LiPF 6 in EC:DEC (1:1 wt%). It should be noted that while the channels appear aligned in Fig. 1, the result of this paper is applicable when the channels are not aligned. As shown later, the separator region will homogenize any y-axis variation of lithium ion concentration in the electrolyte, so that the lithium ion concentration is only a function of x before the ions reach the other electrode. In other words, the anode interacts with the separator region and then the separator region interacts with the cathode. The anode and cathode do not interact with each other directly. In practical applications, the channels of the two electrodes do not need to be aligned to benefit from the channel design.
The cell performance with the electrolyte channel design will be compared to that with the conventional electrode design having no channel. Our electrolyte channel design is motivated by the catheter structure in the xylem of a trunk (Fig. 1, bottom figure). With the catheter, nutrition can be easily transported from the root to the tree crown. Our hypothesis is that electrolyte channels in the electrodes can facilitate lithium ion transport in a similar way to improve the cell performance with thick electrodes during fast charging.
Electrochemical and mechanical modeling.-Governing equations.-We developed a pseudo three-dimensional electrochemical model and implemented it using COMSOL Multiphysics 5.4. The two electrode-level dimensions, as shown in Fig. 1, are the x-axis along the electrode thickness direction and the y-axis along the inplane direction of the electrode. The third dimension is the r-axis along the radial direction of the active material particles.
The solid phase potential, F , s and electrolyte phase potential, F , e are given by 1.5 respectively. The active surface area per electrode volume is given by a s = 3ε s /r p . Following most literatures, here we assume that the electrolyte active coefficient f ± is a constant so that dlnf ± /dlnc e = 0. In the electrode region, the reaction current density, i, is given by the Bulter-Volmer equation shown later. In the separator region, i is zero. The lithium ion concentration in the electrolyte is given by The current density between the solid phase and the electrolyte phase, i, is governed by the Bulter-Volmer equation, and the exchange current density is given by s surf e s s surf , 1 , max , ( ) The overpotential, η, is given by Here E eq is the equilibrium potential of the cathode or anode material (See Fig. A·1 in Appendix).
The lithium ion concentration in a spherical active material particle is given by The values of various parameters in the electrochemical equations are given in Table A·I Appendix.
The mechanics of the electrodes are calculated using the solid structure module in COMSOL Multiphysics 5.4. In this paper we aim to provide the electrode-level stress induced by lithium intercalation as an additional indicator to the typical electrochemical performance, rather than devoting to a study of fully coupling electrochemistry and mechanics. Therefore, like many literatures, we consider one-way coupling of the result of electrochemical process on mechanical stress generation. For simplicity, the reverse effect of stress on electrochemistry is neglected which is typically small. Lithium intercalation into an active particle causes its volume to expand. The corresponding strain, known as the eigenstrain of lithium intercalation, is related to the average lithium concentration in the particle given by 30 The value of c s ave , is obtained by averaging the lithium concentration inside a particle obtained by Eq. 6 of the electrochemical model. The partial molar volume, Ω, of LiMn 2 O 4 cathode material and graphite anode material are 3.497 × 10 −6 m 3 mol −1 and 3.646 × 10 −6 m 3 mol −1 , respectively. 31,33 Similar as thermal stress, the eigenstrain of lithium intercalation causes stress generation in the electrode.
For a porous electrode, its effective elastic modulus, E, and Poisson's ratio, ν, can be calculated based on the following fitting The bulk elastic modulus, E p , and Poisson's ratio, ν p , of the cathode and anode material are given in

/ /
Here d ij is the Kronecker delta. The elastic field is solved by the equilibrium equation in the solid structure module of COMSOL.
Simulation conditions.-In our simulations the battery was first fast charged by the constant current, constant voltage (CCCV) strategy, and then discharged until the terminal voltage decreased to the cut-off voltage. The applied charge current density was set to be 160 A m −2 during the CC stage. When the terminal voltage reached 4.35 V, the CC stage was terminated and the CV stage was invoked. In the CV stage, the terminal voltage was set to be a constant 4.35 V. The lower cut-off current density was set to be 80 A m −2 , and the charging process was stopped when the charge current density dropped below this threshold. For discharging, the applied discharge current density was set to be constant at 25 A m −2 , and the cut-off terminal voltage was set to be 3.5 V. The initial SOC was set to be uniformly 0.995 for the cathode (3.6 V) and uniformly 0.030 for the anode (0.6 V). Note that side reactions such as SEI formation and lithium loss are not considered in this study. We assume that an SEI is already formed and further lithium loss in one charge/discharge cycle is negligible.
The local SOC of each active particle is calculated by For the mechanical calculations, fixed boundary conditions were applied to the two current collectors. The stress was set to be zero at the interface between the electrode and the separator since the separator was fairly soft. These give In addition, the stress was set to be zero at the interface between the electrode and the electrolyte (i.e. surface of the channel).
Quantification of electrochemical performance and mechanical stress.-The specific energy, specific power and specific capacity during discharging are chosen to quantify the battery performance. For simplicity, we will later refer them without specifically mentioning "discharge." The specific energy (SE) and specific power (SP) are calculated by where t ds is the start time of discharging, t de the end time, and V t the terminal voltage during the discharging process. The discharge current is I = i app A where A denotes the cross-sectional area of the electrode. The cell mass is given by The subscripts "p", "n" and "sep" denote the cathode, anode and separator regions, respectively. The values are given in Table A·I Appendix. The specific capacity (SC) of the cell is calculated by.

= -
The negative to positive capacity ratio (N/P ratio) is an important cell parameter. Various battery characterizes, such as lithium plating and electrode degradation, are related to this ratio. 35,36 With electrolyte channel design, the N/P ratio is given by

[ ] /
where c s,max,n denotes the maximum lithium concentration in the anode particle and c s,max,p denotes the maximum lithium concentration in the cathode particle.
To quantify the mechanical integrity, we employ the first principle stress which is directly related to electrode-level crack formation.

Simulation Results And Discussions
Effect of electrolyte channel length.-To investigate the effect of channel length on battery performance, two groups of simulations were performed. In the first group, the cathode channel length L EC was set to be 25 μm, 50 μm, 100 μm, 150 μm and 200 μm, respectively. The anode channel length L EA was set to be 200 μm. In the second group, L EA was set to be 25 μm, 50 μm, 100 μm, 150 μm and 200 μm, respectively while L EC was set to be 200 μm. The cathode and anode electrolyte channel width, W EC and W EA , were both set to be 20 μm. W EA-b was set to be the same as W EA , and W EC-b was set to be same as W EC . The cathode and anode active material half-width, W C and W A , were both set to be 10 μm. The N/P ratio of the first group of cells are 0.81, 0.86, 0.96, 1.10 and 1.29, respectively. The N/P ratio of the second group of cells are 2.04, 1.93, 1.72, 1.50 and 1.29, respectively. The curves of terminal voltage V t during CCCV charging and of the areal capacity are shown in Fig. 2. Figure 2a1 shows that with L EA being fixed at 200 μm, the CC charging time decreases slightly with L EC , causing the areal capacity of the cell to decrease slightly at the end of the CC charging process; but the CV charging time increases significantly with L EC , causing the areal capacity of the cell to increase significantly with L EC as the CV charging process ends. Figure 2b1 shows that the areal capacity increases monotonically from 17.81 Ah m −2 to 20.02 Ah m −2 when L EC increases from 25 μm to 200 μm. The conventional-designed cell has the shortest CC and CV charging time, indicating the smallest areal capacity (14.93 Ah m −2 as the CCCV charging ends). Figure 2a2 shows that with L EC being fixed at 200 μm, the CC charging time increases with L EA while the CV charging time decreases with L EA . Figure 2b2 shows that the areal capacity at the end of CCCV charging first decreases and then increases with L EA . The largest areal capacity is 20.02 Ah m −2 as L EA reaches 200 μm. Besides, the CC charging time of the conventional-designed cell is close to that of the cell with the shortest channel of L EA = 25 μm, but its CV charging time is significantly shorter than the cell with the channel.
The local state of charge of the cathode and anode material is analyzed at the end of CCCV charging. For the conventionaldesigned cell, the local SOC of the anode material decreases monotonically from the separator toward the anode current collector along the electrode thickness direction, as shown in Fig. 3a. The material adjacent to the separator has the highest SOC and the high SOC region (SOC ⩾ 0.8) is fairly narrow. For the cathode, the material adjacent to the separator has the lowest SOC, while a large amount of material far from the separator has high SOC. These results indicate that for the thick electrode, only a small amount of cathode material adjacent to the separator gets deintercalated during fast charging. Similarly, only a small amount of anode material adjacent to the separator gets well intercalated (SOC ⩾ 0.8). A large amount of the anode material far from the separator is poorly charged (SOC < 0.4). Thus the conventional-designed cell has poor performance as shown later. In the following we refer SOC ⩾ 0.8 as the high charging state and SOC < 0.4 as the low charging state when talking about anode. Figure 3a1 shows that with L EA being fixed at 200 μm, the amount of anode material with high charging state first increases (blue line moves left) and then decreases slightly (blue line moves right) with L EC . The amount of anode material with low charging state decreases monotonically, indicating that the anode charging state becomes more uniform. For the cathode, the SOC at each position decreases with L EC , suggesting that longer channels make it easier for the cathode particles to de-intercalate lithium ions. Figure 3a2 shows that with L EC being fixed at 200 μm, the amount of anode material with high charging state first increases and then decreases slightly with L EA , while the amount of anode material with low charging state decreases significantly with L EA . These trends indicate that the anode charging state becomes more uniform. For the cathode, the SOC at each position first decreases slightly then increases slightly with L EA , suggesting that the lithium ion deintercalation at the cathode is not affected much by the anode electrolyte channel.
The lithium ion concentration distribution in the electrolyte at the end of CCCV charging is also shown in Fig. 3b. With L EA being fixed, the lithium ion concentration in the electrolyte at each position in the cathode region decreases significantly with L EC (bottom figure in Fig. 3b1) while the concentration at each position in the anode region increases slightly. The concentration gradient in the electrolyte along the electrode thickness direction decreases with L EC , suggesting that longer channels make lithium ions distribute more uniformly in the electrolyte. With L EC being fixed, the lithium ion concentration in the electrolyte at each position of the anode region increases with L EA (bottom figure in Fig. 3b2), and so does the concentration in the electrolyte at each position of the cathode region. Figure 4 shows the discharge SE, SP and SC obtained by calculating the discharge process after the completion of CCCV charging. It can be seen from Fig. 4a that with L EA being fixed, SE, SP and SC all increase monotonically with L EC . With the shortest channel, the SE, SP and SC values are 19.48%, 9.93% and 18.98% lower than those with the longest channel. However, even with the shortest channel, the SE, SP and SC values are 32.90%, 5.40% and 31.95% higher than those of the conventional-designed cell. With L EC being fixed, the SE and SC values first decrease slightly and then increase with L EA , as shown in Fig. 4b. The SP value increases monotonically with L EA . Even with the shortest channel, the SE, SP and SC values are 50.21%, 13.34% and 49.01% higher than those of the conventional-designed cell. With the shortest channel, the SE, SP and SC values are 6.60%, 3.52% and 6.21% lower than those with the longest channel. These observations indicate that by changing the electrolyte channel length alone, the battery specific energy, specific power, and specific capacity can significantly exceed the conventional-designed battery cell. The cell performance appears to be more sensitive to the cathode electrolyte channel length than to the anode electrolyte channel length.
The first principal stress inside the battery electrodes during charging was analyzed to investigate the effect of electrolyte channel length on mechanical integrity. The first principal stress is directly related to electrode-level crack formation in-between active material particles. Figure 5a shows the contours of first principal stress distribution in the conventional-designed cell at the end of the CCCV charging. We can observe that the first principal stress in the cathode is significantly larger than that in the anode. In addition, the first principal stress is positive in the cathode while negative in the anode. This suggests that during charging, the cathode undergoes tensile loading, which is consistent with Li-ion deintercalation at cathode and that the associated shrinkage of particle volume is constrained by the current collector. The anode material undergoes compression by the constraint of current collector against particle expansion due to Li-ion interaction. Simulations have shown that the maximum first principal stress occurs in the cathode at the location of the cathode/separator interface for both conventionaldesigned cells and cells with electrolyte channels, as a result of more Li-ion deintercalation at this location. Therefore, we compare the first principal stress at the cathode/separator interface for various channel design to evaluate the mechanical integrity. Figure 5b shows the first principal stress in the cathode at the location of the cathode/separator interface as a function of SOC for the conventional-designed cell and cells with different anode channel length, L EA . Here the average SOC is calculated on the entire cathode volume to reflect the overall state. We can observe that the maximum first principal stress in the cathode is significantly lower with electrolyte channels than that in the conventionaldesigned cell. In addition, the first principal stress increases rapidly with SOC in the conventional-designed cell, while the trend is less steep in the cell with channels. When the average cathode SOC reduces to 0.78 (which is the level that a conventional-designed cell would research the end of charging), the first principal stress of the conventional-designed cell is 3.06 MPa larger than that of the cell with electrolyte channels of L EA = 25 μm and L EC = 200 μm, indicating that electrolyte channels can significantly reduce the cathode stress near the separator. Thus, the introduction of electrolyte channels has a positive effect on the mechanical integrity of the cell.
Effect of electrolyte channel width.-Three groups of simulations were performed to investigate the effect of the electrolyte channel width. In the first group, W EA and W EC were set to be the same and their values were 5μm, 10 μm, 20 μm, 40 μm and 80 μm. In the second group, the cathode had no electrolyte channel while W EA was set to be 5 μm, 10 μm, 20 μm, 40 μm and 80 μm. In the third group, the anode had no electrolyte channel while W EC was set to be 5 μm, 10 μm, 20 μm, 40 μm and 80 μm. L EA and/or L EC were set to be 200 μm. W EA-b was set equal to W EA , W EC-b was set equal to W EC . W C and/or W A were set to be 20 μm. The N/P ratio of the first group of cells are all 1.29. The N/P ratio of the second group of cells are 1.53, 1.75, 2.14, 2.76 and 3.57, respectively. The N/P ratio of the third group of cells are 1.08, 0.94, 0.77, 0.60 and 0.46, respectively.  The simulation results of terminal voltage curves during the CCCV charging process are presented in Fig. 6a.
For the cell with concurrent cathode and anode channels, the CC and CV charging time first increases and then decreases with W EA and W EC (Fig. 6a1). The maximum CCCV charging time occurs when W EA and W EC are 10 μm. The CCCV charging time is even shorter than that of the conventional-designed cell when W EA and W EC reach 80 μm. Similarly, as shown in Fig. 6b1, the areal capacity at the end of CCCV charging first increases to a maximum of 20.95 Ah m −2 as W EA and W EC reach 10 μm. Then, the areal capacity decreases to a minimum of 8.85 Ah m −2 as W EA and W EC reach 80 μm. This minimum is 40.72% lower than the areal capacity of the conventional-designed cell (14.93 Ah m −2 ). When only one of the electrode has channels, the maximum CCCV charging time occurs when W EA = 10 μm or W EC = 20 μm (Figs. 6a2 and 6a3). When W EA reaches 40 μm or W EC reaches 80 μm, the CCCV charging time is shorter than that of the conventional-designed cell. Figures 6b2 and 6c2 show that the areal capacity for a cell with a flat cathode reaches its maximum of 17.65 Ah m −2 as W EA reaches 10 μm, while the areal capacity for a cell with a flat anode reaches its maximum of 19.49 Ah m −2 as W EC reaches 20 μm. These indicate that the maximum areal capacity of a cell is larger with electrolyte channels in both the anode and cathode, than that with channels in only one of them. Now consider the local SOC after CCCV charging. For the cell with both cathode and anode channels, the amount of anode material with high charging state first increases and then decreases rapidly with W EA and W EC (Fig. 6c1). The amount of anode material with low charging state first decreases and then increases, suggesting that the charging state of the anode material becomes more uniform with appropriate channel width. For the cathode, the SOC at each position first decreases and then increases rapidly with W EA and W EC , indicating that it is more difficult for Li-ion de-intercalation when W EA and W EC exceed 20 μm.
For the cell with only anode channels, the amount of anode material with high charging state increases rapidly with W EA (Fig. 6c2), and the amount of anode material with low charging state decreases monotonically and rapidly with W EA . In the cathode, the SOC at each position increases slightly with W EA , indicating that it is slightly getting more difficult for the Li-ion to deintercalate. For the cell with only cathode channel, the amount of anode material charging state is slightly affected by W EC . The anode material with high charging state decreases with W EC (Fig. 6c3), and the amount of anode material with low charging state first decreases slightly and then increases slightly. By contrast, the cathode SOC at each position decreases dramatically with W EC . These results show that the local SOC has more complicated dependence on the channel width.
The dependence of SE, SP and SC on channel width is presented in Fig. 7. For the cell having both anode and cathode electrolyte channels of the same size, SE and SC first increase with W EA and W EC when they increase from 5 μm to 10 μm, and then decrease rapidly when they change from 20 μm to 80 μm (Fig. 7a). The maximum SE and SC are 65.99% and 64.32% larger than those of the conventional-designed cell. However, the SE and SC are 27.36% and 27.61% lower than those of the conventional-designed cell when W EA and W EC are 80 μm. The SP increases monotonically with W EA and W EC , and is 27.74% larger than that of the conventionaldesigned cell when they are 80 μm.
For the cell with only anode electrolyte channel, the changing pattern of SE, SC and SP with W EA is similar to that with concurrent cathode and anode electrolyte channels. The maximum SE, SP and SC of the cell with anode channel are 25.42%, 24.72%, and 5.78% larger than those of the conventional-designed cell, respectively (Fig. 7b). For the cell with only cathode electrolyte channel, SE and SC first increase with W EC till about 10 μm or 20 μm, and then decrease rapidly (Fig. 7c). The SE and SC are still larger than those of the conventional-designed cell when W EC reaches 80 μm. The SP increases monotonically with W EC . The maximum SE, SC and SP of the cell with cathode channel are 55.80%, 54.39%, and 20.36% larger than those of the conventional-designed cell, respectively.
These results suggest that with changing electrolyte channel width, the cell performance can be significantly higher or lower than the conventional-designed cell. There exists optimal channel widths for maximum specific energy and specific capacity, while the specific power tends to change monotonically with the channel width.
The first principal stress in the cathode at the location of the cathode/separator interface as a function of SOC were investigated for various electrolyte channel width. Figure 8 shows that with increasing electrolyte channel width, the first principal stress decreases rapidly. The magnitude is much smaller than that in the conventional-designed cell. When the cathode average SOC reduces to 0.78, the first principal stress in the conventional-designed cell is 2.15 MPa larger than that of the cell with W EA =W EC = 5 μm, and 3.82 MPa larger than that of the cell with W EA = W EC = 80 μm. These suggest that increasing the electrolyte channel width can significantly reduce the cathode stress.
Effect of active electrode material width.-To investigate the effect of active electrode material width on the cell performance, three groups of simulations were performed. In the first group, the active anode and cathode material half-width W A and W C were set to be the same, with values of 10 μm, 20 μm, 40 μm, 80 μm and 160 μm. In the second group, the cathode had no channel, while the active anode material half-width W A was set to be 10 μm, 20 μm, 40 μm, 80 μm and 160 μm. In the third group, the anode had no channel, while the active cathode material half-width W C was set to be 10 μm, 20 μm, 40 μm, 80 μm and 160 μm. L EA and/or L EC were set to be 200 μm. W EA and/or W EC were set to be 20 μm. W EA and W EA-b were set to be same. W EC and W EC-b were set to be the same. The N/P ratio of the first group of cells are all 1.29. The N/P ratio of the second group of cells are 2.76, 2.14, 1.75, 1.53 and 1.41, respectively. The N/P ratio of the third group of cells are 0.60, 0.77, 0.94, 1.08 and 1.17, respectively. The terminal voltage curves during the CCCV charging process are presented in Fig. 9a. The CC and CV charging time both first increase then decrease with the active electrode material half-width W A and W C for concurrent channels in the anode and cathode (Fig. 9a1). Similarly, Fig. 9b1 shows that the areal capacity first increases and then decreases with W A and W C . The maximum areal capacity reaches 20.42 Ah m −2 when W A and W C are both 20 μm. When there is only channel in the anode, the CC charging time decreases monotonically with W A , while the CV charging time first increases rapidly and then decreases with W A (Fig. 9a2). The CCCV charging time reaches its maximum when W A is 20 μm. The areal capacity also reaches its maximum of 17.36 Ah m −2 when W A is 20 μm, as shown in Fig. 9b2. When there is only channel in the cathode, the CC charging time increases slightly with W C , while the CV charging time first increases and then decreases with W C (Fig. 9a3). The CCCV charging time reaches its maximum when the W C is 10 μm. The areal capacity also reaches its maximum of 19.50 Ah m −2 when W C is 10 μm, as shown in Fig. 9b3. With the electrolyte channel width fixed and keeping increasing the active electrode material width, the areal capacity will approach to that of the conventional-designed cell.
For the local the local SOC after CCCV charging, the amount of anode material with high charging state first increases and then decreases with W A and W C for concurrent channels in the anode and cathode (Fig. 9c1). The amount of anode material with low charging state increases monotonically. In the cathode, the SOC at each position increases monotonically with W A and W C , indicating that it is getting increasingly difficult for the cathode to release Li-ions.
When there is only anode channel, the amount of anode material with high charging state decreases rapidly with W A (Fig. 9c2). The amount of anode material with low charging state increases dramatically. The SOC at each position of the cathode only increases slightly, indicating that the anode active electrode material width does not affect much the deintercalation at the cathode. When there is only cathode channel, the amount of anode material with high charging state first increases slightly and then decreases slightly with W C (Fig. 9c3), indicating that the cathode active electrode material width does not affect the anode charging state much. The SOC at each position of the cathode increases rapidly with W C . The results suggest that changing the active electrode material width can bring a complex effect on the local charging state. It should be noted that when the active electrode material width is large enough (e.g., larger than 80 μm), the SOC distribution along the x-axis is similar to that the conventional-designed cell. Figure 10 shows the SE, SC and SP for various active electrode material width. For the cell having concurrent anode and cathode electrolyte channels of the same size, SE and SC first increase with W A and W C when they change from 5 μm to 10 μm, and then decrease when they change from 10 μm to 80 μm (Fig. 10a). The SP decreases monotonically with W A and W C . The maximum SE, SC and SP are 65.05%, 62.87% and 23.17% larger than those of the conventional-designed cell, respectively. For the cell with only anode or electrode electrolyte channel, the changing patterns of SE, SC and SP with W A or W C are similar as those of concurrent channels (Figs. 10b and 10c). The maximum SE, SC and SP of the cell with only anode channel are 24.45%, 23.57% and 5.03% larger than those of the conventional-designed cell, respectively. The maximum SE, SC and SP of the cell with only cathode channel are 55.83%, 54.49% and 17.11% larger than those of the conventional-designed cell, respectively.
These results indicate that by changing the active electrode material width, improved specific energy and specific capacity can be obtained. It should be noted that when the active electrode material width is large enough, the cell performance becomes close to that of the conventional-designed cell since the electrolyte channels are too sparse to affect the cell performance.
The first principal stress in the cathode at the location of the cathode/separator interface as a function of SOC were investigated for various active electrode material width. Figure 11 shows that with increasing active electrode material width, the principal stress increases to approach that in the conventional-designed cell. When the cathode SOC reduces to 0.78, the first principal stress in the conventional-designed cell is 3.62 MPa larger than that of the cell with W A = W C = 5 μm.
Effect of simultaneously changing electrolyte channel width and active electrode material width.-The previous two sections show that changing the electrolyte channel width and the active electrode material width has a significant effect on the cell performance. Here we consider the effect of changing them simultaneously to investigate the design option. The ratio of the electrolyte channel width to the cell periodic width (W EA /W H and W EC /W H ) was introduced. Two groups of simulations were performed. W EA was set to be constant at 20 μm in the first group, while W EC was set to be 5 μm, 10 μm, 20 μm, 40 μm and 80 μm respectively. The ratio W EC /W H was 0.13, 0.25, 0.50, 0.67 and 0.80, respectively. The ratio W EA /W H was 0.50, 0.50, 0.50, 0.33 and 0.20 respectively. In the second group, W EC was set to be constant at 20 μm, while W EA was set to be 5 μm, 10 μm, 20 μm, 40 μm and 80 μm, respectively. The ratio W EA /W H was 0.13, 0.25, 0.50, 0.67 and 0.80, respectively. The ratio W EC /W H was 0.50, 0.50, 0.50, 0.33 and 0.20 respectively. L EC and L EA remained constant at 200 μm in both groups. W EA /W EA-b and W EC /W EC-b were set to be 1. The SE, SP and SC of the cells were calculated and are presented in Fig. 12.
By simultaneously increasing W EA /W H and decreasing W EC /W H , SE, SP and SC first increase slightly and then decrease rapidly (Fig. 12a). The largest SE, SP and SC are 80.14%, 17.03% and 77.56% larger than those of the conventional-designed cell, respectively. The minimum SE and SC are 4.68% and 5.17% lower than that of the conventional-designed cell, respectively, suggesting that design optimization can play a large role here. By increasing W EC /W H and decreasing W EA /W H simultaneously, SE and SC first increase then decrease (Fig. 12b), while SP increases monotonically. The largest SE, SP and SC are 65.09%, 22.52% and 62.87% larger than those of the conventional-designed cell, respectively. These phenomena indicate that by simultaneously changing the electrolyte channel width and the active electrode material width, the battery performance can be significantly changed. Besides, the optimized specific energy, specific power, and specific capacity are much higher compared to the case  when only one of the two is changed (e.g. sections "Effect of electrolyte channel width" and "Effect of active electrode material width" ). Thus simultaneously tuning electrolyte channel width and active electrode material width provides better opportunity in improving the battery performance.
Another way to change the electrolyte channel width and the active electrode material width simultaneously is by the taper design of the electrolyte channel to make W EA-b and W EC-b smaller than W EA and W EC , respectively. The tapering degrees of the cathode and anode electrolyte channels are quantified by W EC /W EC-b and W EA /W EA-b , respectively. A group of simulations was introduced to explore the effect of electrolyte channel tapering degree on battery performance.  Fig. 13. Figure 13 shows that SE and SC first increase and then decrease slightly as W EA /W EA-b and W EC /W EC-b increase simultaneously from 1.0 to 4.0 and from 4.0 to 10.0, respectively. The maximum SE and SC are 16.38% and 16.53% larger than those of the cell without electrolyte channel tapering design, respectively, and 49.34% and 47.98% larger than those of the conventional-designed cell, respectively. SP decreases monotonically with W EA /W EA-b and W EC /W EC-b . These phenomena indicate that by introducing the electrolyte channel tapering design, the battery specific energy and specific capacity are improved, while the specific power is reduced.
Next we consider how the tapering design affects the maximum first principal stress. W EA /W EA-b and W EC /W EC-b were set to be the same, taking values of 1.0, 2.0, 4.0, 8.0 and 10.0, respectively. Figure 14 shows that the first principal stress in the conventionaldesigned cell increases more rapidly with SOC than that of the cell with electrolyte channels. The first principal stress increases with the tapering degree, but at a reduced pace. When the average cathode SOC reduces to 0.78, the first principal stress in the conventionaldesigned cell is 2.51 MPa larger than that of the cell with channels of  channel length, width, tapering degree and the active electrode material width all have significant effects on the battery cell performance. We have only considered their individual effects and demonstrated preliminary yet rich effect of combined electrolyte channel width and active electrode material width. In Li-ion battery design, the geometrical parameters of the electrolyte channel offer a huge design space of various combination of the parameter values. Therefore, it is necessary to develop an approach to efficiently evaluate the cell performance for arbitrary electrode geometrical parameter combinations to search for the optimized design. Using finite element simulations to evaluate the cell performance for all these scenarios are impractical due to the large amount of computations. Here we first develop a machine learning system based on DNN to predict the cell performance for arbitrary combination of electrolyte channel parameters. Then we use the system for design optimization. Figure 15a shows a schematic of the deep artificial neural network to predict cell performance with 7 electrolyte channel design parameters as input, including L EC , L EA , W H , W EA /W H , W EC /W H , W EA /W EA-b and W EC /W EC-b . These parameters can completely determine the geometry of a tapered electrolyte channel. Figure 15b shows the implementation using the machine learning toolbox in MATLAB, which includes 40, 20, and 10 neurons in the first, second, and third hidden layer, respectively. This network has three outputs: specific discharge energy, specific discharge power and specific discharge capacity of the cell. For the DNN training process, we generated 20000 groups of dataset based on FEM of the equations in the section "Modelling" using COMSOL. We used 16000 groups to train the neural network, 2000 groups for validation, and 2000 groups for testing. In MATLAB, the validation groups are used as a threshold of the training iterations. When the mean square error within the validation groups reaches its minimum and does not decrease in the following six epoches, the training process ends. The testing groups are used to monitor the training accuracy of the DNN during the epoches. The transfer function was chosen to be a sigmoid function within the hidden layers, and the transfer function in the output layer was chosen to be a linear function. Because there are orders of magnitude difference among the input values and output values, we performed normalization for the input and output parameters to make the input and output values having the same order of magnitude. This can improve the training accuracy of the DNN.
For the seven input parameters, the range of L EC and L EA are both from 3 μm to 248 μm. The range of W H is from 6 μm to 300 μm. The range of W EA /W H and W EAC /W H are both from 0.05 to 0.95. The range of W EA /W EA-b and W EC /W EC-b are both from 1.0 to 10.0. The DNN output, to balance the orders of different physical quantities for better performance, is a three-element normalized output vector (SE, SP, SC). Each element is normalized by its corresponding maximum and minimum values among the 20000 groups of dataset. For instance, the normalized SE is calculated by (SE−SE min )/(SE max −SE min ). The physical value is easily calculated from the normalized output value. After training with the 16000 groups, the DNN gave a testing error of 0.0013 on the 2000 testing groups. Here the testing error is the mean squared error of the normalized output vector vs the ground truth vector, as is widely used for quantifying machine learning performance and defined in MATLAB. In the following sections, we will also show comparison of DNN predicted physical SE, SP, SC values and the results of FEM calculations on the same inputs. The relative error of DNN results to  the FEM results will provide direct indication of the DNN performance in practical applications.
Approach of design optimization.-The gradient descent algorithm was employed to optimize the electrolyte channel parameters for optimal cell performance. The trained DNN was used as a function evaluator in the optimization process. Since it is difficult to identify the concavity and convexity of the trained DNN, using the gradient descent algorithm alone may cause the algorithm to get trapped in a local optimum instead of the global optimum. We used the Markov chain Monte Carlo (MCMC) approach to address this problem. The optimization process (using the specific energy SE as an example) is shown in the flowchart in Fig. 16. We first generate N groups of input parameters randomly with k input parameters in each group (k =7 in this research). For the nth group at the jth iteration, the trained DNN function is first used to calculate the SE n value. Denote the value of the ith input parameter in the nth group byIn .
Then the ith input parameter value is varied by a sensitive factor ε (chosen to be a very small number, 10 −6 , which has been widely used in self-adjusting optimization processes [37][38][39] ) to e + In 1 .
The corresponding branch gradient, B , After calculating the branch gradients for all the k input parameters, the total gradient within this jth iteration is given by This total gradient is compared with 10 −4 . If Grad n j ( ) is less than 10 −4 , we regard that the SE value has reached its local maximum. Then we record the local maximum SE value (denoted by SE n ) and the corresponding optimized input parameters. It should be noted that when an input parameter reaches its upper or lower bound, indicating that the bound value is the optimized value of the corresponding input parameter, the corresponding branch gradient is assigned with 0. Otherwise (Grad n j ( ) ⩾ 10 −4 ), we continue to the next iteration with the input parameters given by Here μ denotes the learning rate, which is chosen to be 0.2 in this research, and α denotes the direction adjustment factor, which is 1 when ( ) and −1 otherwise for finding the optimized maximum SE value. To show the window of SE, we also search for the minimum value. In finding the minimum SE, α is taken to be 1 when    It should be noted that for an input In ,  If the SDR value is smaller than 10 −2 , the maximum SE value within these N groups of SE values is regarded as the global optimized SE value, and the corresponding input parameters are regarded as the global optimized values. If the SDR value is larger than 10 −2 , then the top 10% SE values among the N groups of SE values, and their corresponding optimized input parameters are used to form a candidate pool. A new N groups of input parameters are randomly generated, with each input parameter within the range of the upper and lower bounds of the parameters in the candidate pool. The gradient descent calculation and random generation of N groups of input parameters are repeated until the SDR value is smaller than 10 −2 .
Design maps.-We first conducted three case studies after the DNN training to investigate the combined effect of adjusting two electrolyte channel geometrical parameters on the cell performance. The results are organized into design maps, which can provide guidance for the parametric design of the electrolyte channel geometrical parameters.
As shown in the section "Effect of electrolyte channel length", changing the cathode and anode electrolyte channel length individually can significantly affect the cell performance. Thus it is interesting to see how freely changing both the cathode and anode electrolyte channel length affects the cell performance. The ranges of L EA and L EC were both set to be from 5 μm to 245 μm. W H was set to be 100 μm. W EA /W H , W EC /W H , W EA /W EA-b and W EC /W EC-b were set to be 0.50, 0.50, 1.5 and 1.5, respectively. The trained DNN function was utilized to calculate SE, SP and SC. The design maps are presented in Fig. 17. The maximum SE, SP and SC value was found by the MCMC gradient descent method introduced in the section "Approach of design optimization". To examine the prediction accuracy of the trained DNN, we imported the corresponding electrolyte geometrical parameter values into FEM calculations to obtain the corresponding SE, SP and SC values. The relative error, err = |V i,D -V i,F |/V i,F (i = SE, SP, SC), is employed to quantify the DNN accuracy where V i,D denotes the value calculated by DNN and V i,F denotes the value calculated by FEM.
The patterns of SE or SC as a function of L EC and L EA appears similar (Figs. 17a and 17c). Increasing L EC and L EA simultaneously causes SE and SC to increase. SE and SC can exceed those of the conventional-designed cell by more than 60% when L EC and L EA are larger than 200 μm. The maximum SE and SC values are 61.05 Wh kg −1 and 16.17 Ah kg −1 , respectively, at L EC = 230 μm and L EA = 245 μm. These are 69.07% and 67.22% larger than the SE and SC of the conventional-designed cell (SE: 36.11 Wh kg −1 , SC: 9.67 Ah kg −1 ), respectively. With increasing L EA and L EC , SP increases monotonically (Fig. 17b), indicating that increasing cathode and anode electrolyte channel length can continuously increase the specific power. The maximum SP (84.08 W kg −1 ) is reached when L EA and L EC are both 245 μm. This SP value is 18.14% larger than that of the conventional-designed cell (71.17 W kg −1 ). It can also be seen that changing the cathode electrolyte channel length has a stronger effect on the cell performance.
We further created the Ragone plane for showing the window of all possible specific energy and specific power of the cell with changing L EC and L EA based on DNN and MCMC. It can be seen from Fig. 17d that when SE is relatively small (below 47.50 Wh kg −1 ), the SP span is narrow. The difference between the maximum and minimum SP is approximately 0.6 W kg −1 under a constant SE value. When SE is within 47.50 Wh kg −1 and When SE is within 54.80 Wh kg −1 and 61.05 Wh kg −1 , the slope of the upper bound of SP decreases, while the slope of the lower bound of SP increases. Eventually, the SP span reduces to zero when SE reaches its maximum. We selected 34 data points at the upper and low bounds obtained by DNN and MCMC, extracted the corresponding seven electrolyte channel input parameters for each of the data point, and input them into FEM calculations. The FEM results are shown as circles. It is seen that the SE and SP values predicted by DNN and those by FEM agree very well. The average of the relative error of the 34 points are 1.08% for SE and 0.17% for SP, indicating that the DNN has a high accuracy. Finally, it can be seen from Figs. 17e that using the electrolyte channel geometrical parameters corresponding to the maximum SE, SP and SC, the values calculated by DNN are very close to those calculated by FEM simulations. The relative errors are all far below 5%, again indicating that the accuracy of the trained DNN is high enough for conducting this first case study. Figure 18 shows the design maps of changing cathode and anode electrolyte channel width as the second case study. The ranges of W EA /W H and W EC /W H were both chosen to be from 0.05 to 0.95. L EA and L EC were both set to be 200 μm. W H was set to be 100 μm. W EC /W EC-b and W EA /W EA-b were both set to be 1.5. The maximum SE, SP and SC were found by the MCMC gradient descent method. To examine the prediction accuracy of the trained DNN, we imported the corresponding electrolyte geometrical parameter values into FEM calculations to obtain the corresponding SE, SP and SC values. The relative error is employed to quantify the DNN accuracy.
The patterns of SE or SC as a function of cathode and anode electrolyte channel width appear similar (Figs. 18a and 18c). SE and SC both first increase and then decrease with W EA /W H and W EC /W H , and show a peak in the middle region of map. When W EA /W H is in the range of 0.20 to 0.45 and W EC /W H is in the range of 0.50 to 0.65, the cell shows more than 65% increase in specific energy and specific capacity than the conventional-designed cell. The maximum SE and SC are 62.17 Wh kg −1 and 16.46 Ah kg −1 when W EA /W H and W EC /W H are 0.33 and 0.54, respectively. These are 72.17% and 70.22% higher than the SE and SC of the conventional-designed cell, respectively. It can also be seen that when W EA /W H and W EC /W H are approximately above 0.84 and 0.92, respectively, SE and SC are lower than those of the conventional-designed cell. These indicate that changing the cathode and anode electrolyte width can significantly influence both the specific energy and specific capacity. However, inappropriate design of the electrolyte channel width may reduce the cell performance. SP increases almost monotonically with W EA /W H and W EC /W H (Fig. 18b), indicating that increasing cathode or anode electrolyte channel width can continuously increase the specific power. The maximum SP is 90.75 W kg −1 when W EA /W H and W EC /W H are 0.80 and 0.95, respectively. This is 27.52% larger than the SP of the conventional-designed cell. Changing the cathode electrolyte channel width has a stronger effect on SP than changing the anode electrolyte channel width.
We further created the Ragone plane for showing the window of all possible specific energy and specific power of the cell with changing W EA /W H and W EC /W H by DNN and MCMC. It is seen from Fig. 18d that as SE increases from 17.90 Wh kg −1 to 42.00 Wh kg −1 , the upper bound of SP shows two increase-decrease curves with SE. The lower bound of SP decreases sharply and then more slowly. These cause the SP span to show a bird head shape. When SE is within 42.00 Wh kg −1 and 62.17 Wh kg −1 , the upper bound of SP decreases gradually, while the lower bound of SP increases almost linearly with SE. Eventually, the SP span reduces to zero when SE reaches its maximum. We selected 31 data points at the upper and low bounds obtained by DNN and MCMC, extracted the corresponding seven electrolyte channel input parameters for each of the data point, and input them into FEM calculations. The FEM results are shown as circles. It is seen that the SE and SP values predicted by DNN and those by FEM agree very well. The average of the relative error of the 31 points are 3.86% for SE and 0.65% for SP. The average of the relative error is 1.41% for SE and 0.64% for SP when SE is larger than 40 Wh kg −1 . These indicate that the DNN has a high accuracy. Finally, it can be seen in Fig. 18e that the errors of the maximum SE, SP and SC calculated by DNN are all far below 5%, again indicating that the trained DNN has a high enough accuracy in conducting this second case study. Figure 19 shows the design maps of changing cathode and anode electrolyte channel tapering degree as the third case study. The ranges of W EA /W EA-b and W EC /W EC-b were both set to be from 1.0 to 10.0. L EA and L EC were both set to be 200 μm. W H was set to be 100 μm. W EA /W H and W EC /W H were both set to be 0.60.
As can be seen in Figs. 19a and 19c, the patterns of SE or SC as a function of cathode and anode electrolyte channel tapering degree appear similar. When W EC /W EC-b is below 1.5 and W EA /W EA-b is between 2.0 and 3.4, the specific energy and specific capacity are improved for more than 55% compared with the conventionaldesigned cell. The maximum SE and SC are 57.32 Wh kg −1 and 15.22 Ah kg −1 when W EC /W EC-b and W EA /W EA-b are 1.0 and 2.5, respectively. These are 58.74% and 57.40% larger than the SE and SC of the conventional-designed cell, respectively. SP decreases monotonically with W EC /W EC-b and W EA /W EA-b . When W EC /W EC-b and W EA /W EA-b are both above 3.5, any increase of W EC /W EC-b and W EA /W EA-b does not affect SP much.
We further created the Ragone plane for showing the window of all possible specific energy and specific power of the cell with changing cathode and anode electrolyte channel tapering degree based on DNN and MCMC. It can be seen from Fig. 19d   decreases gradually with some minor fluctuations as SE increases from 54.15 Wh kg −1 to 57.32 Wh kg −1 . These cause the SP span to have a relatively large value initially and then decrease slowly after SE reaches 55.50 Wh kg −1 . We selected 30 data points at the upper and low bounds obtained by DNN and MCMC, extracted the corresponding seven electrolyte channel input parameters for each of the data point, and input them into FEM calculations. The FEM results are shown as circles. It is seen that the SE and SP values predicted by DNN and those by FEM agree well. The average of the relative error of the 30 points are 1.46% for SE and 0.38% for SP, indicating that the DNN has a high accuracy. Finally, it can be seen in Fig. 19e that the errors of the maximum SE, SP and SC calculated by DNN are all far below 5%, again indicating that the trained DNN has a high enough accuracy in conducting this third case study.
Optimal design by freely changing 6 geometrical parameters.-The design maps in the previous section shows that by freely changing two electrolyte channel geometrical parameters, the cell performance can be significantly improved. The optimized specific energy, specific power, and specific capacity are much higher than those of the conventional-designed cell. In this section, more freely changeable electrolyte channel geometrical parameters are taken into consideration to further explore the potential of performance improvement. W H was kept to be constant at 100 μm. The other electrolyte channel geometrical parameters were all set to be variable. The range of L EA and L EC were both from 5 μm to 245 μm. The range of W EA /W H and W EC /W H were both set to be from 0.05 to 0.95. The range of W EA /W EA-b and W EC /W EC-b were both set to be from 1.0 to 10.0. The MCMC gradient descent was utilized to find the global optimized values. The specific energy was chosen to be the optimization goal. After optimization, the maximum SE value was 64.54 Wh kg −1 , which is 78.73% larger than that of the conventional-designed cell (36.11 Wh kg −1 ). The corresponding optimized L EA , L EC , W EA /W H , W EC /W H , W EA /W EA-b and W EC /W EC-b values were 240.00 μm, 231.40 μm, 0.3313, 0.7496, 1.05 and 9.18 respectively, indicating that changing 6 electrolyte channel geometrical parameters can give even larger specific energy than adjusting only 2 geometrical parameters. The results again indicate that with the introduction of bio-inspired electrolyte channels inside the thick electrode, the battery performance can be significantly improved during fast charging. Finally, FEM calculations were employed to examine the accuracy of the optimized SE. Using the corresponding optimized electrolyte channel parameters, the FEM calculation gave SE to be 66.02 Wh kg −1 . The relative error is only 2.24%, which is below the typical engineering requirement of 5%. This suggests that the trained DNN has a high enough accuracy for conducting the optimization.

Conclusions
In this research, we introduced a bio-inspired Li-ion battery cell design strategy by adding electrolyte channels inside the thick electrode and investigated the effect of electrolyte channel geometrical parameters on the electrochemical performance and mechanical integrity of battery cells during fast charging. First, the impact of electrolyte channel length, width, tapering degree, and active electrode material width on cell performance were investigated. Then, a DNN machine learning system was built to correlate the electrolyte geometrical parameters and cell performance. The MCMC method was applied for optimization. Finally, we obtained three groups of design maps and demonstrated optimization of specific energy with 6 freely adjustable electrolyte channel geometrical parameters. The conclusions are summarized below (1) Changing the electrolyte channel geometry causes the battery cell performance, including specific energy, specific power, and specific capacity, to vary significantly. The performance can be greatly improved comparing to the conventional-designed cell.
On the other hand, inappropriate design can decrease the performance. The local state of charge of the electrode during CC charging can also be significantly affected by the electrolyte channel geometrical parameters. (2) During charging, the cathode material undergoes tensile stress, while the anode material undergoes compressive stress. The stress level in the cathode is larger. The maximum first principal stress in the cathode occurs at the location of cathode/separator interface. With electrolyte channel design, the maximum first principal stress can be significantly reduced comparing to that of the conventional-designed cell. (3) After training, the developed DNN machine learning system can correlate the electrolyte geometrical parameters with the battery performance with a high accuracy, which can assist the parametric design of electrolyte channels. The MCMC gradient descent method and the trained DNN evaluation function have provided optimized solution without the need to know the concavity or convexity of the DNN function. The accuracy of the predicted optimized solution such as specific energy has been confirmed by FEM simulations. (4) The design maps show that the pattern of specific energy on electrolyte channel geometrical parameters (cathode and anode electrolyte channel length, width, tapering degree) appears similar to the pattern of specific capacity on these parameters. The specific power increases almost monotonically with these parameters. By freely changing two electrolyte channel geometrical parameters, the specific energy and specific capacity can both be improved by more than 70%, while the specific power can be improved by more than 20%. By freely changing six electrolyte channel geometrical parameters, even more cell performance improvement can be achieved. The optimized specific energy is 78.73% larger than that of the conventionaldesigned cell. All these suggest that the introduction of bioinspired electrolyte channel with appropriate geometrical parameters inside the thick electrode can bring great performance improvement.
Lithium plating can occur when the charging current density is very high such that the anode over-potential is below 0 V vs Li + /Li. Lithium plating is more likely to happen on a thick electrode at the anode/separator interface because of the slow lithium ion transport through the thick electrode toward the interior of the anode next to the anode/current collector interface. While lithium plating is outside the scope of this paper, we would like to highlight that a key benefit of electrolyte channels is to enhance lithium ion transport. These channels can facilitate lithium ion transport toward the interior of the anode, reducing the possible accumulation of lithium ions at the anode surface next to the separator. As a result, electrolyte channels can potentially help to significantly reduce lithium plating. Using channel structures in an electrode to reduce lithium plating is an interesting topic for further study.