3D-Microbattery Architectural Design Optimization Using Automatic Geometry Generator and Transmission-Line Model.

Optimization of the 3D battery architecture is crucial to improve the performance of microbatteries. However, such optimization is difficult and time consuming by hand for even experts. In this article, we propose a battery optimization system, which consists of an automatic geometry generator and performance simulators. The geometry generator creates feasible 3D batteries without using any human intuition and experience for the spatial arrangement of positive and negative electrodes. For quick evaluation of the internal resistance, which relates power and energy densities, we propose the transmission line model, the so-called 3D porous electrode model, as one of the performance simulators. To show the effectiveness of the optimization system, we designed the lithium-ion microbatteries. In the trade-off frontier for the internal resistance and the capacity, we successfully found a new battery architecture that has higher power and energy densities over the conventional interdigitated plates configuration.

Introduction

The rapid development of information technology since the end of the twentieth century and the technological progress of sensors and devices have familiarized the concept of the Internet of Things (IoT). In particular, the development of high-speed and large-capacity communication infrastructure on a global basis, drastic reductions in sensor unit prices, and the advent of smart devices such as smartphones and tablet computers have led to the explosive spread of IoT services since 2010. The number of internet-connected devices surpassed the number of internet users in 2008 (Swan, 2012), and, at present, further spread of IoT in various fields, including medical care, automotive and transportation, and agriculture, is expected (Markit, 2017).

So far, the miniaturization of the sensors and devices has significantly contributed to the rapid growth of the IoT ecosystem. However, compared with such technological progress on the sensors and devices, the miniaturization technology of the power sources for the off-grid operation of the electronics has not been well established, which hinders further prevalence of the IoT (Swan, 2012; Hur et al., 2018). Especially, for the integration with microelectronics, the high areal energy density is recognized as one of the key properties for such power sources (Long et al., 2004; Zhang et al., 2017; Zhu et al., 2017; Wei et al., 2017).

The lithium-ion batteries, due to their exceptional performance represented by high energy density, high operating voltage, as well as long lifetime, have been used as the power sources in a variety of fields, such as electronics, e.g., smartphones and laptop computers, and electric and hybrid vehicles, and their miniature batteries, the so-called lithium-ion microbatteries, have been considered to be one of the promising candidates for the IoT-enabled power sources. Indeed, thin-film microbatteries were already commercialized, and their applications to the IoT devices, such as sensors, renewable energy storage devices, RFID (radio-frequency identification) tags, and smart cards, have been explored (Dudney, 2008). However, the thin-film microbatteries have a problem with their low areal energy densities (1 mWh/cm2) (Pikul and Ning, 2018). Although the use of thicker electrodes improves the areal energy density, it leads to lower power density due to the increase of the internal resistance.

To improve the areal energy density while maintaining high power density, the three-dimensional design of the microbattery at the full cell level has attracted attention (Long et al., 2004). One example of such 3D batteries is shown in Figure 1A. The benefit of the geometrical modification is the lowering of the internal resistance by changing the ion and electron transport distances without reducing the battery capacity. The recent advancement of a variety of 3D manufacturing technologies (Sun et al., 2013; Wei et al., 2017; Hur et al., 2018) has allowed the realization of the 3D microbattery (Ning et al., 2015; Mazor et al., 2012; Pikul et al., 2013; Sun et al., 2013; Fu et al., 2016; Izumi et al., 2012; Hur et al., 2018). For example, Ning et al. fabricated the on-chip high-energy-density 3D lithium-ion microbatteries by combining 3D holographic lithography and conventional photolithography (Ning et al., 2015). Using template-assisted electrodeposition, Pikul et al. achieved the high-power lithium-ion microbatteries where nanoporous electrodes were used as the electrodes in the 3D interdigitated battery (Pikul et al., 2013). Very recently, Hur et al. proposed the full 3D battery fabrication process based on semiconductor processing technologies, and their battery showed excellent performance, e.g., more than two times higher discharge capacities (up to 1.8 mAh cm−2) compared with the highest thin-film batteries and attainment of 100 cycles (Hur et al., 2018).

Figure 1

Examples of 3D Battery Geometries and the Battery Optimization System

(A) Comparison of conventional battery (left) and one example of the 3D designed batteries (right: interdigitated anode and cathode plates configuration).

(B) The framework of the battery optimization system. For simplicity, the current collectors and separator are removed from the battery geometries.

Although such drastic improvements of a variety of the fabrication processes realized the 3D microbatteries having both high power and energy densities, the electrode structures are limited to simple geometries such as interdigitated plate (Sun et al., 2013; Pikul et al., 2013; Fu et al., 2016; Izumi et al., 2012) or cylindrical (Hur et al., 2018) electrodes, and, therefore, the battery performance is restricted by such electrode structures. To design the 3D battery architecture beyond the interdigitated configurations, the 3D battery optimization method at the full cell level is vital. Also, the 3D battery optimization method is crucial from the point of view of industrial demand because there are a variety of IoT devices that have different functions, and it is demanding to optimize the battery following the requirements of each IoT device by hand. However, there are no reports regarding the 3D battery optimization methods at the full cell level.

In this article, we propose a novel battery optimization system at the full cell level, which consists of an automatic geometry generator and performance simulators (Figure 1B). So far, battery optimizations have been carried out using continuum simulations (McKelvey et al., 2020). However, most of such methods investigate only the influence of simple structural change (such as width, length, pitch, and end shape) of the interdigitated electrodes to the battery performance (Hart et al., 2003; McKelvey et al., 2017; Zadin et al., 2010, 2011; Zadin and Brandell, 2011; Priimägi et al., 2016; Miranda et al., 2016; Pikul et al., 2017; Li et al., 2018). Although Zadin et al. optimized the 3D electrode geometry using the level-set method and obtained the complicated-shaped electrodes, the design space was too simple as the practical battery cell (one tooth per electrode) (Zadin et al., 2013). In addition, all the previous works start optimizations from the interdigitated configurations. Contrary to this, our geometry generator designs the full cell based on random sampling, without using any human intuition and experience for the spatial arrangement of positive and negative electrodes, by only receiving the size and resolution of the 3D battery cell and the volume ratio of the positive and negative electrodes as the input data. For compatibility with the geometry generator, as one of the performance simulators, we newly propose the transmission line model, the so-called 3D porous electrode model, to compute the internal resistance of the 3D battery quickly. As shown in Figure 1B, our optimization system designs the 3D battery by generating a large number of battery geometries, evaluating battery performances of each geometry quickly, and, then, creating the trade-off frontier (a set of data points that are not dominated by other data points in a given dataset). We show the effectiveness of our method by creating the trade-off frontier for the internal resistance and capacity that relate to the power and energy densities of the 3D battery and finding the new battery geometry that has higher power and energy densities over the interdigitated plates configuration.

Results and Discussion

Automatic Geometry Generator

We generate the battery cell using the newly proposed randomized algorithm. In this article, for simplicity, we generate the 3D battery having two degrees of freedom for electrode designs, e.g., the interdigitated anode and cathode plates configuration shown in Figure 1A and geometries illustrated in Figure 1B. However, we should emphasize that it is straightforward to extend our algorithm to the 3D electrode designs such as the full cell shown in Hur et al. (2018). Also, we assume that the current collectors are placed as shown in Figure 1A, i.e., the current collector plate for the positive electrode is located in the left-hand side of the cell and that for the negative electrode in the right-hand side. Here, we call the mixture of active material, a polymer binder, and an electrolyte filling the pores as the positive and negative electrodes because we assume porous electrodes. When necessary, the conductive agents also become components of the electrode.

The central idea of the battery generation method is illustrated in Figure 2A. At first, the vacant cell with predefined width (W), height (H), and depth (D) is prepared and the space is divided into finite elements. Then, the space is filled in with the cuboid-shaped electrodes one by one after determining the numbers of the (cuboid-shaped) positive and negative electrodes using the volume ratio of the positive and negative electrodes (which is specified in the input). In this article, we call the cuboid-shaped electrode as an electrode element. Here, it is noteworthy that we need to determine only the locations of positive and negative electrode elements, i.e., the position of the separator is automatically determined once the positions of all electrode elements were determined. The fineness of the finite element mesh may be determined by following the resolution of fabrication methods such as the 3D printers.

Figure 2

The Battery Generation Method

(A) The central idea of the battery generation. The battery cell is divided into finite elements, and the divided space is filled in with the electrode elements.

(B) The matrix to represent the battery geometry shown in (A), where “1” and “3” represent the positive and negative electrode elements, respectively.

(C) One example of the battery geometry generated by the random sampling with no restrictions, and examples of problems as a battery. The battery cell is divided into 30 30 elements.

(D) A list of the possible positions for the next positive electrode element. Blue and red squares represent positive and negative electrode elements, respectively. For simplicity, the separator at the interface between two electrodes is neglected in the figures.

In our code, the battery geometry and the spatial positions of the positive and negative electrodes are represented as a matrix and its elements, respectively. For example, the interdigitated geometry shown in Figure 2A is represented by Figure 2B, where “1” and “3” represent the positive and negative electrode elements, respectively. We call this matrix as the geometry matrix. Also, the vacant cell is represented by a zero matrix. Thus, the geometry generation process shown in Figure 2A corresponds to the one-by-one determination of each element value (“1” or “3”) in the matrix from the zero matrix. The actual 3D battery can be constructed from the geometry matrix and the information about the cell size (W, H, and D).

Each geometry-matrix-element value is determined randomly. However, if the geometry matrix is created by simply filling in with “1” or “3” randomly without any restrictions, a huge number of unacceptable geometries are generated, as exemplified in Figure 2C. The geometry in the figure has two kinds of clear problems as a battery: “the negative/positive electrode comes next to the current collector for the positive/negative electrode (the short circuit problem)” and “isolation or surrounding of the positive/negative electrode (the isolation problem).” One of the key points of our randomized algorithm is the circumvention of these problems.

The detailed protocol to generate the geometry matrix is as follows:

The battery size (), the thickness of the separator, the resolution of the finite element mesh, and the volume ratio of the positive and negative electrodes are specified in the input data.

The numbers of the positive and negative electrode elements are computed using the input data.

The geometry matrix (the vacant cell) is prepared.

All the elements in the first column of the geometry matrix are set to “1,” and all the elements in the last column of the geometry matrix are set to “3.”

The positions of the positive electrode elements are determined one by one, where a list of next candidate positions in the matrix is created each time a positive electrode element is placed, and the next location is picked up randomly from the list. The list is created to avoid the isolation of the electrodes and based on the rule that the positive electrode element can be placed only next to the existing positive electrode elements, but, if that causes the isolation of the vacant elements (“0s” in the matrix), such candidates are removed from the list (Figure 2D). The isolation (connectivity) of the vacant elements is checked using the breadth-first search algorithm (Cormen et al., 2009).

After placing all the positive electrode elements, all the “0s” in the geometry matrix are replaced by “3s,” i.e., the negative electrode elements.

In step 4, the short circuit problem is avoided, and we avoid the isolation problem in step 5.

Our geometry generation method is quite useful for the actual 3D battery design because the method can generate the geometry by following the required battery size and the resolution of the fabrication methods. Also, it is important to emphasize that we can evaluate the volumes of the electrodes and the separator directly from the geometry, which relate to the capacity of the battery.

The 3D Porous Electrode Model

One of our motivations in this article is to improve both the energy and power densities of the battery by changing the geometry of the electrodes, which corresponds to the reduction of the internal resistance () while maintaining the capacity of the battery in this context. Although battery simulations using the continuum models are popular methods to evaluate , the methods are known to have a problem with their computational cost (McKelvey et al., 2020), especially when they are combined with the randomized algorithm because it generates a huge number of battery geometries, and values need to be evaluated for all. Therefore, in our approach, we evaluate using a computationally less-expensive equivalent circuit model.

As the lithium-ion battery generally uses the porous electrodes for both positive and negative electrodes, the equivalent circuit model has to describe the electrode reactions in the porous electrode to evaluate of the lithium-ion battery. The transmission line model (TLM) is known to be useful to capture such nature of the porous electrode (Itagaki et al., 2007a; Ogihara et al., 2012). The example of the TLM is illustrated in Figure 3A. As can be seen in the figure, TLM is constructed by the following three steps. At first, the battery cell is divided into some elements and, then, centers of the elements (which are represented by the circles in the figure) are connected by two kinds of resistances separately, i.e., ionic and electronic resistances. Finally, the two kinds of the series circuits (the series circuits of the ionic and electronic resistances) are connected by the interfacial impedances (the orange-colored rectangle). As both ion and electron paths exist inside the electrodes, the reactions inside the porous electrodes can be represented. When the TLM is used for electrochemical analyses of the conventional battery cells, the electronic resistance is often neglected and the analytical formula, i.e., the equation when the system is divided into infinite number of elements, is used to investigate the cell impedance (Ogihara et al., 2012, 2015). It is noteworthy that, however, we do not use such an equation in our model, i.e., we construct the TLM from the finite number of elements.

Figure 3

Comparison of the Conventional Transmission Line Model and the 3D Porous Electrode Model

(A) The conventional battery and the corresponding transmission line model when the system is divided into six elements including the current collectors.

(B and C) The 3D porous electrode model for the conventional battery (B) and the 3D battery (C) when the system is divided into six by two elements including the current collectors. The green- and black-colored resistances represent ionic and electronic resistances, respectively. The orange-colored rectangle represents the interfacial impedance and corresponds to the parallel circuit of the charge transfer resistance and the electric double-layer capacitance if the Faradaic process is assumed (Itagaki et al., 2007a).

We extend the TLM for use in three-dimensionally designed batteries, which we call the 3D porous electrode model. One of the key characteristics of the 3D porous electrode model is the introduction of one more degrees of freedom, as is clear from the comparison of Figures 3A and 3B. This allows representation of the geometrical difference of the electrodes, e.g., comparison of Figures 3B and 3C. The construction of the TLM is analogous to the conventional TLM. In our model, the electronic resistances in the current collectors are also considered in order for the stability of the calculations. The electronic resistance of the current collector is much smaller than other resistances, and, therefore, the influence of this treatment on is negligibly small. Here, we should note that the resolution of the 3D porous electrode model, i.e., the fineness of the finite element mesh, is not necessarily equal to that used for the battery generation, e.g., Figure 2A. To avoid any confusion, we call the element for the TLM as the TLM element. As mentioned previously, in this article, we allow 2D geometrical change to the electrodes, and, therefore, the corresponding TLM is for 2D design. However, it is straightforward to extend this model to the 3D batteries with the 3D designed electrodes.

The Electrical Resistance and Capacitance

To construct the 3D porous electrode model, the ionic and electronic resistances between the TLM elements and the charge transfer resistance in the TLM element have to be defined. Also, the electric double-layer capacitance () in the TLM element is important to carry out the impedance analysis using our model. Ionic and electronic resistances ( and ) between the same electrode elements (and the same current collector elements) are computed using the common equation given bywhere ρ is the ionic or electronic resistivity and l and a are the length and cross-sectional area of the resistance that connects the centers of two TLM elements, respectively. Detailed definitions of ionic and electronic resistances (as well as the resistance between the electrode and the current collector) are given in the Transparent Methods section of the Supplemental Information.

The ionic resistance at the interface between the positive and negative electrodes is defined bywhere s is the thickness of the separator and , , and are the ionic resistivities of the positive electrode, the negative electrode, and the separator, respectively. It is noteworthy that , , and are different from the ionic resistivity of the bulk electrolyte because it changes with the structural properties of the electrodes and the separator (e.g., the porosity of the separator). Also, we should note that we included the contributions of the positive electrode and negative electrode as well as that from the separator in to remove the resolution dependence (the dependence of the fineness of the TLM element mesh) on . However, due to this treatment, only when , agrees with the electrolyte bulk resistance () (Ogihara et al., 2012) and the simulated impedance spectrum can be compared with the experimental one.

The charge-transfer resistances of the positive and negative electrodes are defined bywhere and are the reaction resistivity ( cm2) of the interface between the electrolyte and the active material in the electrode and the reaction surface area, respectively. In this article, the reaction surface area is assumed to be computed bywhere is the volume of the electrode in one TLM element and c is the conversion factor. Although usually corresponds to the volume of the TLM element (), is not equal to at the interface between the two electrodes due to the existence of the separator () and is given by

By substituting Equation 5 into Equations 3 and 4, we obtainwhere we call as the charge-transfer resistivity and the unit is cm3.

The corresponding electric double-layer capacitances are defined bywhere and are the permittivity and the thickness of the electric double layer, respectively. By substituting Equation 5 into Equations 9 and 10, we arrive atwhere the unit of the is . Here, we should note that the use of in Equations 7, 8, 11, and 12 removes the resolution dependence on the cell impedance from these terms.

Among the necessary parameters to compute the internal resistance or the cell impedance, the parameters related to the geometry, such as l, a, s, and , are determined from the specifications of the battery and the resolution of the TLM. The other parameters are obtained from the AC (alternating current) impedance analysis of the symmetric cells (Ogihara et al., 2012, 2015; 2019; Itou et al., 2020). The necessary parameters to compute and are the electronic and ionic resistivities ( and ). and have to be determined to compute and .

We determine and from the AC impedance analysis of the symmetric cell. As the symmetric cell has two identical electrodes, the relationship between the experimentally measured charge-transfer resistance () and is given bywhere W, D, and H represent the width, depth, and height of the battery cell, respectively, and s represents the thickness of the separator. Here, we assume the separator is placed perpendicular to the height direction. Thus, is obtained by

is determined using the relationship between the top of the capacitative semicircle () and the product of and that

By substituting Equations 7 and 11 (or Equations 8 and 12) into Equation 15, we obtain

Electrodes' Volume Effect on

In this article, we optimize the battery under the condition that the volume () and shape of the battery cell are fixed and, therefore, the volume ratio between the separator () and electrodes () changes with the electrode structures. The decrease of indicates the reduction of the total number of the lithium cation, leading to the increase of . As the 3D porous electrode model cannot describe this effect, we introduce this effect approximately using and aswhere denotes the internal resistance obtained using the 3D porous electrode model. We distinguish between and hereafter. It is noteworthy that although the charge-transfer resistance and the electric double-layer capacitance of the battery cell also change with because the reaction surface area () changes, this effect is included in Equation 6.

Optimization of the Microbattery

We look for the 3D lithium-ion microbatteries that have both low internal resistance () and high capacity by combining the automatic geometry generator and the 3D porous electrode model. Details of the computational conditions, including specification and resolution of the 3D lithium-ion microbattery and material property parameters, are given in the Transparent Methods section of the Supplemental Information (Figures S1–S3 and Table S1). Throughout this section, unless otherwise stated, we use the same resolution as the geometry generation for constructing the equivalent circuits, i.e., TLM elements. Here, because the values are determined using the symmetric cells at the SOC (state of charge) of 50%, we simulate at this SOC. It is noteworthy that the battery performance at the SOC of 50% is considered to be important for practical applications. As for the capacity of the battery, as we design 3D electrodes under the constraint that the battery volume () is constant and only the geometrical change of the electrodes is considered during the optimization, the ratio of electrode volume () to corresponds to the capacity of the battery. Thus, we evaluate instead of the battery capacity. Also, even if constraints are introduced to the random geometry generation to produce only the acceptable geometries as the battery, still the search space is quite huge. Thus, we introduced the four kinds of periodicity conditions for the random geometry generations as shown in panel (A) of Figures S4–S7 in the Supplemental Information, where units of periodicity are , , , and elements, respectively. At each condition, we generate 50,000 geometries and evaluate and at each geometry. Even under such conditions, we can obtain complicated battery geometries beyond the interdigitated plates configuration as exemplified in Figure S7A.

All the data points (200,000 data points) are plotted as a function of in Figure 4A. To understand the scaling effect introduced by Equation 17, data points are plotted in Figure 4B. For both figures, ideal batteries should come to the lower right edge. However, and has a trade-off relation, and, therefore, there are no such batteries. From the comparison of Figures 4A and 4B, the volume effect (Equation 17) influences the shape of the trade-off frontier at around , when the thickness of the separator is 20 μm. However, in this region, the difference of the internal resistance on the trade-off frontier is small even in Figure 4A, i.e., the is in the range between around 60.0 and 70.0 . This means that the volume effect is not so significant to design the 3D batteries at the current specification. However, this effect is important from the point of view of battery design for the developers.

Figure 4

Results of Random Geometry Optimizations

(A) as a function of the volume ratio of electrodes ().

(B) as a function of the volume ratio of electrodes ().

(C) The characteristic geometries picked up from the results, where A–D correspond to A–D in panels (A) and (B).

(D) Nyquist plots of geometries A–D. To draw the Nyquist plots, the resolution of the 3D porous electrode model is set to be 150 30 elements. Also, the volume effect, which is shown in Equation 17, is not included.

We pick up four characteristic batteries as shown in Figure 4C. Geometry A is the conventional parallel plate configuration and shows the largest capacity (), whereas it has the largest of 171.9 . Although the geometrical difference between geometries A and B seems to be small, the drastic reduction of by 33.0 is confirmed for geometry B. This indicates that even small modifications of the electrode structure strongly affect . Geometry C is the so-called interdigitated anode and cathode plates configuration and shows the smallest capacity ( = 71.1%) among all the generated geometries. However, indeed, this configuration is quite effective to reduce (= 69.1 ). Geometry D has the smallest (62.9 ) as well as good capacity ( = 82.6%), and we can expect that geometry D is superior to geometry C in terms of both energy and power densities.

From Figure 4A, there are a variety of geometries that have similar and in the low (e.g., of 60–70 ) and high (e.g., of 80–85%) regions. This is an encouraging result from the point of view of the microbattery design because this implies that there is a space to improve other battery performances such as lifetime and thermal stability while maintaining good power and energy densities.

To clarify the 3D geometrical effect on , the Nyquist plots of geometries A, B, C, and D are compared in Figure 4D. In this analysis, we used 150 30 TLM elements to create the equivalent circuit. In this condition, l equals to s in Equation 2, meaning agrees with the electrolyte bulk resistance () (Ogihara et al., 2012). In this resolution, values slightly reduce to 159.23, 119.31, 45.29, and 49.31 for geometries A, B, C, and D, and the corresponding values are 164.72, 131.58, 63.72, and 59.51 , respectively. However, it is noteworthy that the order of the magnitudes of the values does not change regardless of the resolution of the transmission line model. Details of the resolution dependence are discussed in the following section. As the electronic resistance of the electrodes is small (Table S1 in the Supplemental Information), may be decomposed into five components aswhere and represent contributions from the ionic resistance of the 3D positive and 3D negative electrodes and where and represent the charge transfer resistance of the positive and negative electrodes, respectively. Also, denotes the electrolyte bulk resistance of the 3D battery. Here, we should note that increases in proportion to the inverse of the interface area; and increase if the distance between the electrode element and the separator increase, which corresponds to the ion transport distance, and and reduce if the electrode volume increases.

For all the geometries, the Nyquist plots show two distorted semicircles. The semicircle at the lower Z′ region corresponds to the impedance of the negative electrode, and the width of the distorted semicircle corresponds to . The semicircle at the higher Z′ region corresponds to the impedance of the positive electrode. Also, the width between Z’ = 0 and the starting point of the semicircle of the negative electrode corresponds to . As the width of the semicircle derived from the positive electrode of geometry A is almost the same as that of geometry B, the difference of the between geometries A and B comes from , , and . Geometries C and D show much smaller semicircles as well as than geometry A.

The main difference of between geometries C and D stems from the electrolyte bulk resistance ( of geometry C is 5.55 and that of geometry D is 8.92 ). This is understandable by considering the fact that increases with the inverse of the interface area between the two electrodes. Geometries C and D have similar electrode resistance () because of the balance between the ionic and charge transfer resistances. Namely, geometry C has lower ionic resistance than geometry D because the number of electrode elements at the interface is higher, where the ion transport distance is the shortest. Contrary to this, geometry C has larger and because of the smaller electrode volume. Although geometry C has smaller than geometry D, of geometry D is smaller due to the volume effect (Equation 17). As a result, our method predicts that geometry D has better performance than geometry C by around 7% for and by 16% for the capacity.

We finalize this section by summarizing the advantages of our optimization system over the previously proposed 3D battery optimization methods based on the continuum model, where only simple structural change of the interdigitated electrodes is considered (Zadin et al., 2010, 2011; Zadin and Brandell, 2011; Priimägi et al., 2016; Miranda et al., 2016; Pikul et al., 2017). First, the use of the low-cost transmission line model allows evaluations of a variety of 3D battery geometries (e.g., 200,000 geometries), which is difficult with the continuum model. Here, we should emphasize that although the main benefit of the 3D electrode design is recognized to be the reduction of the ion transport distance (ionic resistance of the electrode) while maintaining the battery capacity (volume of the electrodes, which also relates to ) (Pikul and Ning, 2018), the 3D porous electrode model can evaluate such effects. One of the main advantages of our geometry generator over the previously proposed methods is automatic battery design along with allowing much more flexibility in the electrode design. Also, in actual 3D battery design, the 3D battery has to be designed in a specified footprint area, e.g., Fu et al. (2016); Sun et al. (2013). In such a situation, the volume fraction of the electrodes and the separator changes with the electrode design. Our geometry generator takes into account such effect for the first time, whereas the previous works did not consider it. Therefore, our method is useful for the actual battery design. Effectiveness of the combination of the novel transmission line model and automatic geometry generator is demonstrated by finding new 3D battery architecture, which has better performance over the conventional interdigitated anode and cathode plates configuration.

Resolution Dependence of the 3D Porous Electrode Model

The internal resistance () has the resolution dependence of the 3D porous electrode model. As shown in Figures 3B and 3C, the error in the internal resistance comes from three kinds of sources, i.e., the discretization of the system (the battery cell) into the finite number of TLM elements, the ionic resistances of the TLM elements next to the current collectors, and the electronic resistances of the TLM elements at the interface between two electrodes.

The first error source is intrinsic of our approach and the error comes from the approximation (discretization) of the system, which can be reduced by increasing the resolution (Itagaki et al., 2007b). As for the second problem, at the current implementation, the 3D porous electrode model does not compute the ionic resistances at the edge of the battery, which is shown as the white double arrows in Figures 3B and 3C. This error also can be reduced by improving the resolution of the 3D porous electrode model. Similar to the second problem, at the current implementation, our model does not compute the electronic resistance at the interface. However, it is noteworthy that this error is small because the contribution from the electronic resistance to the cell impedance itself is small as is clear from values in Table S1 in the Supplemental Information.

In Table 1, we summarize the resolution dependence of the 3D porous electrode model on for geometries A–D in Figure 4C. As an overall trend, the order of the error is almost the same magnitude regardless of the geometry at the same resolution. Also, the size of the error is small even if the coarsest resolution (50 10 TLM elements) is used. Namely, the error is at most 7 and the order of s among geometries A–D does not change. Although we further investigate the convergence of using higher resolution and thinner separator as shown in Table S2 in the Supplemental Information, the error from the coarsest resolution is small and decreases monotonically with the resolution. Therefore, we conclude that optimization using 50 10 TLM elements is reasonable and that the important conclusions that are drawn from Figure 4 do not change with the use of the finer resolutions, i.e., geometry D is superior to C in terms of power and energy densities and there is a possibility to improve other properties such as the lifetime while keeping high power and energy densities by changing the 3D geometry.

Table 1

Resolution Dependence on []

Resolution	A	B	C	D
50 × 10	166.21 (6.98)	125.95 (6.64)	49.14 (3.84)	51.93 (2.80)
100 × 20	158.43 (−0.80)	119.08 (−0.23)	45.33 (0.04)	49.05 (−0.08)
150 × 30	159.23 (0.00)	119.31 (0.00)	45.29 (0.00)	49.13 (0.00)

The error from the high-resolution result (150 30 TLM elements) is shown in parenthesis. A–D correspond to the geometries A–D in Figure 4C.

Conclusions

In this article, for the first time, we have proposed the battery optimization system at the full cell level, which consists of the automatic geometry generator and performance simulators. As geometry generator creates the battery automatically by only receiving the size and the resolution of the 3D battery cell, and volume ratio of the positive and negative electrodes as input data, any human intuition and experience are unnecessary for the spatial arrangement of the positive and negative electrodes. Also, as the resolution used in the generator can be set by following the resolution of the battery fabrication technologies, the direct design of the practical 3D battery is feasible.

As the battery design is a multiobjective optimization problem, quick performance evaluation methods are essential. As one of such methods, we proposed the 3D porous electrode model for the evaluation of the internal resistance of the 3D battery. The 3D porous electrode model can evaluate the internal resistance of the battery having 3D designed electrodes by simply introducing one more degree of freedom to the conventional transmission line model for the porous electrodes.

To demonstrate the effectiveness of our battery optimization system, we looked for the lithium-ion microbatteries that have both low internal resistance and high capacity. We created the trade-off frontier for the internal resistance and the capacity by generating 200,000 data points and successfully found that the 3D battery that has lower internal resistance and higher capacity over the conventional interdigitated battery. The new 3D battery showed better performance over the interdigitated geometry by around 7% for the internal resistance and 16% for the capacity. Also, we found that there were a variety of 3D batteries that have small internal resistance and high capacity. This is an encouraging result because this indicates that there is a possibility to improve other battery properties including cycle life and safeness, while maintaining good power and energy densities.

We strongly believe that our optimization system is compatible with 3D printing technologies. Especially, combination with direct ink writing (Fu et al., 2016; Sun et al., 2013) is expectable for the actual fabrication of the 3D battery.

As future work, we will improve the performance of the geometry generator by introducing sophisticated search algorithms such as Monte Carlo Tree Search methods and evolutionary algorithms (Browne et al., 2012; Wang and Sebag, 2013; Obayashi et al., 2007). Also, we will develop quick performance simulators for the other important battery properties, including thermal and mechanical properties as well as electrochemical properties.

Limitations of the Study

As our electrochemical model neglects ion diffusion and non-linear effects, influences of the lithium-ion depletion and the solid-state diffusion are not considered in the geometry optimization. Current in-house python code can optimize only 3D batteries having two degrees of freedom for electrode design, and, therefore, extensions of both the geometry generator and the 3D porous electrode model are necessary for the full 3D battery design.

Resource Availability

Lead Contact

Further information and requests for resources should be directed to and will be fulfilled by the Lead Contact, Kaito Miyamoto (kaito.miyamoto@toyota.com).

Materials Availability

This study did not generate new materials.

Data and Code Availability

The dataset generated during this study (the battery geometries and associated electrode volumes and internal resistances) is available (https://doi.org/10.17632/c7zvggmh78.1).

Methods

All methods can be found in the accompanying Transparent Methods supplemental file.