Maxime
van der Heijden
,
Gabor
Szendrei
,
Victor
de Haas
and
Antoni
Forner-Cuenca
*
Electrochemical Materials and Systems, Department of Chemical Engineering and Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, Netherlands. E-mail: a.forner.cuenca@tue.nl; m.v.d.heijden@tue.nl; szendrei.gabor09@gmail.com; v.d.haas@student.tue.nl
First published on 25th April 2024
Porous electrodes are performance-defining components in electrochemical devices, such as redox flow batteries, as they govern the electrochemical performance and pumping demands of the reactor. Yet, conventional porous electrodes used in redox flow batteries are not tailored to sustain convection-enhanced electrochemical reactions. Thus, there is a need for electrode optimization to enhance the system performance. In this work, we present an optimization framework to carry out the bottom-up design of porous electrodes by coupling a genetic algorithm with a pore network modeling framework. We introduce geometrical versatility by adding a pore merging and splitting function, study the impact of various optimization parameters, geometrical definitions, and objective functions, and incorporate conventional electrode and flow field designs. Moreover, we show the need for optimizing geometries for specific reactor architectures and operating conditions to design next-generation electrodes, by analyzing the genetic algorithm optimization for initial starting geometries with diverse morphologies (cubic and a tomography-extracted commercial electrode), flow field designs (flow-through and interdigitated), and redox chemistries (VO2+/VO2+ and TEMPO/TEMPO+). We found that for kinetically sluggish electrolytes with high ionic conductivity, electrodes with numerous small pores and high internal surface area provide enhanced performance, whereas for kinetically facile electrolytes with low ionic conductivity, low through-plane tortuosity and high hydraulic conductance are desired. The computational tool developed in this work can further expanded to the design of high-performance electrode materials for a broad range of operating conditions, electrolyte chemistries, reactor designs, and electrochemical technologies.
Genetic algorithms (GAs) are promising for exploring a broad geometrical design space for the optimization of porous electrodes. GAs are probabilistic global optimization techniques inspired by the theory of evolution that enable heuristic optimization of a given design space based on a fitness function.14–17 Therefore, GAs require only one objective function, can be parallelized, and have a large solution space. These unique features have motivated the application of GAs to a wide variety of research fields, including the integration of electrochemical numerical frameworks for the optimization of RFB parameters and conditions18,19 and electrode structures.11–13 For the optimization of the electrode geometry, a GA should be coupled to pore-scale simulations to capture the relationship between the battery performance and electrode microstructure in a computationally light manner. To this end, pore network models (PNMs) can be used as they capture microstructural effects at the mesoscale whilst being computationally light, with a limited loss in computational accuracy,20 owing to geometrical simplifications of the pore network.8,21–23 In PNMs, the pore space is captured by a combination of pores and throats, allowing the porous structure to be represented as a set of finite discrete points, enabling the mapping of the pores into a population in the GA. The coupling of PNMs and GAs has been used in other research fields including the extraction of pore networks of porous rock samples for petroleum recovery.11,24–26 However, the coupling of GAs with PNMs for the bottom-up design and optimization of electrode microstructures for RFBs, without the requirement of large datasets for optimization, remains largely unexplored.
The concept of combining a genetic algorithm with PNMs to optimize three-dimensional microstructures for flow battery electrodes was demonstrated in our previous work for cubic networks with fixed pore positions.13 The performance of the individual networks was evaluated by a fitness function that maximizes the electrochemical power and minimizes the pumping power of the networks. Here, we extend the optimization framework to include more design flexibility by extending beyond fixed pore positions through the integration of merging and splitting of pores (Fig. 1). In addition, we evaluate the optimization algorithm for structures with diverse morphologies such as X-ray tomography-extracted networks of commercial electrodes translated into pore networks, which have been proven to represent the microstructural properties of the electrodes reasonably well.21 By optimizing a commercial electrode, we show that we can obtain high-performing electrodes with improved electrochemical performance and lower pumping losses than benchmark materials. Furthermore, we investigate the effect of operation conditions by evaluating two redox chemistries, VO2+/VO2+ and TEMPO/TEMPO+, and two prevailing flow field geometries, a flow-through flow field (FTFF) and an interdigitated flow field (IDFF).
In this work, we first describe the modeling framework including the network generation, the coupling of the GA with the electrochemical algorithm, the genetic operations included in the GA, and the operating parameters investigated. Second, we show the geometrical evolution for the addition of pore merging and splitting as an additional mutation operation. Third, we deliberate on the influence of the network structure of the initial population on the fitness evolution and assess the impact of the flow field geometry on the structure evolution. Fourth and last, we perform the electrode optimization for two redox chemistries to investigate the importance of the starting network and the specific reactor architectures and operating conditions. This study, although applied to redox flow batteries here, shows the potential of optimization by genetic algorithms to design and optimize porous materials for a wide variety of convection-enhanced electrochemical applications. Furthermore, this work emphasizes the importance of co-designing electrodes and flow fields and assesses the sensitivity of genetic algorithms to optimization definitions.
The presented coupled optimization routine consists of nine steps: the network generation, initialization, volume scaling, the electrochemical PNM, parent selection, crossover, mutation, merging and splitting, and termination (see Fig. 2 for the schematic overview of the GA with the integrated PNM). In the network generation stage, the type of network was selected (cubic or extracted) of which a random set was generated in the initialization step defining the first population. In the first step of the iterative GA-PNM, volume scaling was performed to ensure a meaningful comparison at constant electrode porosity between the different individuals in a population and was repeated for each generation. Thereafter, the networks were evaluated based on the electrochemical PNM and a fitness function. Successively, the fittest individuals (i.e., parents) in the population were selected and defined as the next population by inheriting the information of two parent networks with a crossover step. Afterward, stochastic changes were made in the networks of the new population by pore mutation, after which the pores in the network had a chance to merge and split, resulting in the next generation. The iterative GA-PNM was repeated until the termination criterion was achieved based on the total number of generations. The population size, number of parents, mutation range and probability, and merging and splitting probability and ratio were initially chosen arbitrarily or inspired by the work of Grefenstette.29 The number of generations and network size strongly impact the required computational time, which was set to only 1000 generations and a small network size (∼500 × 500 × 200 μm3) to showcase the principle of the GA-PNM without being computationally intensive and requiring high computing resources. Finally, the parameters defining the evaluation criteria of the PNM were based on our previous works.13,21 All reference parameters used in this work for the GA can be found in Table 1.
Parameter | Value |
---|---|
Number of generations | 1000 |
Number of offspring | 50 |
Number of parents | 10 |
Mutation range | 0.1 |
Mutation probability | 0.05 |
Merging and splitting probability | 0.1 |
Merging and splitting ratio | 0.5 |
After the generation of the network structures with diverse morphologies, geometrical properties were attributed to the pores and throats in the network with geometry objects. The geometry objects are a subclass within OpenPNM that can be assigned to parts of the modeling domain. In this work, the geometric StickAndBall approach was applied to all networks, which handles pores as spheres and throats as cylinders in the generated networks for which geometrical properties can be calculated. A detailed description of the artificial and extracted network generation can be found in Section A1 in the Appendix.†
The geometrical property worth mentioning here is the pore internal surface area (Ap), described by the OpenPNM geometry functions, defined by subtracting the throat cross-sectional area (ST) of NT number of neighboring throats from the pore surface area obtained with the pore diameter dp (eqn (1)). This definition, used for cubic networks unless stated otherwise, is a simplification of the pore internal surface area as the curvature of the intersection between the pore and throat was not considered. To this end, this definition cannot be applied to the extracted electrode as a negative internal surface area is obtained (connectivity >6 resulting in a large throat cross-sectional area subtraction, Section A4.1†) resulting in the optimization of small pores. Hence, eqn (2) was used for the extracted electrodes, which results in a underestimation of the internal surface area in comparison with literature values (i.e., 2.2 × 104 m−1 using eqn (2)vs. 7.2 × 104 m−1 obtained in the literature for the Freudenberg H23 electrode6,21). As the surface area of the extracted electrode is in the same order of magnitude as measured experimentally, we believe that for this computational work eqn (2) is a reasonable approximation of the surface area for the extracted electrode. However, we must point out that the simplified geometrical properties are a limitation of the PNM when capturing the exact properties, such as the electrochemically active surface area, compared to practical systems. Nevertheless, the distinct electrode structures should mainly be compared in terms of their optimization trends rather than quantitatively. The reader is referred to Section A4.1† for a detailed discussion of the surface area definition.
![]() | (1) |
Ap = π × dp2 | (2) |
The geometrical definitions used strongly affect the optimization as they directly impact the electrochemical and pumping power. Therefore, in Section A4,† the sensitivity of the optimization depending on the definitions of the internal surface area, throat diameter, and electrode size were analyzed. Changing the definitions affected the electrode optimization, showing the importance of selecting the appropriate geometrical definitions for the optimization. However, all studies did result in optimized structures with a bimodal pore size distribution with longitudinal transport pathways in the flow direction that reduced the pumping power. As simplified descriptions can strongly over/underestimate the performance as well as alter the optimization, geometry definitions need to be identified that consider the manufacturing method of the final electrodes. For example, when optimizing structures that will be 3D printed, the surface area definition could consider all pore and throat walls as internal surface area, but this is beyond the scope of this work.
The network properties of the cubic and extracted structures are given in Table 2, where, in this work, small three-dimensional electrode structures were optimized with electrode geometrical areas of approximately 500 × 500 μm2. To allow comparison between the two network structures, the network shape and spacing of the cubic structure were based on the microstructural properties of the off-the-shelf Freudenberg H23 paper electrode (Fuel Cell Store, 80% porosity) with a median pore size of 20 μm and a measured thickness of 210 μm.6,10,13,21
Parameter | Cubic | Extracted | Unit |
---|---|---|---|
Porosity | 54 | 51 | % |
Network shape | [13, 13, 4] | — | — |
Number of pores | 676 | 3348 | — |
Number of throats | 1755 | 10![]() |
— |
Network size | 580 × 580 × 220 | 500 × 500 × 198 | μm3 |
![]() | (3) |
The individuals with the highest fitness value (ξ → 0.603) were selected as parent networks and were subjected to crossover in the next step. Finally, the total percentual fitness increase over the generations can be calculated according to (ξn−ξ1)/ξ1 × 100%, with ξn the fitness of the best-performing network in the last evaluated generation, and ξ1 the fitness of the best network of the first generation.
Because the objective of the GA is to optimize electrode structures based on a defined fitness function, a suitable definition needs to be determined that considers the objective of the optimization. In Section A4.4† the effect of the fitness function on the optimization was investigated by performing optimization studies with only the pumping power or only the electrochemical power considered. From this study, we found that when only optimizing for the electrochemical power, the pumping power remained unoptimized. On the contrary, when optimized for the pumping power alone, the optimization was comparable to that with eqn (3), suggesting that this fitness definition steers towards the optimization of the pumping power over the electrochemical power. Furthermore, future work could investigate alternative optimization parameters, such as the overpotential and flow rate, or the pumping-corrected voltage efficiency and areal power density, instead of the current density and the pumping power. To change the objective of the optimization, only the electrochemical PNM must be adapted to solve for the desired parameters, which thereafter should be incorporated in the fitness function definition. Finally, the form of the fitness function could be reconsidered, for example by defining the fitness function based on the difference in performance between the new and initial generation.
dMp = cM × dop. | (4) |
Pore merging was performed based on an built-in OpenPNM function in which two or more pores can be combined at the center of the selected pores. In this GA, only two neighboring pores were allowed to merge, where the neighbor of the selected pore with the smallest pore diameter was chosen for merging. Furthermore, the pore volume was defined as the summation of the two pores, ensuring a constant total pore volume. Thereafter, the throat connections were reestablished between the neighbors of the merged pores and the new pore, decreasing the number of throats by one.
With pore splitting, on the other hand, the selected pore was split into two pores with equal pore volume. The new pore locations were stochastically determined within half a maximum pore diameter distance in each direction and a new throat was formed between the two new pores. The old throat connections with the neighboring pores of the split pore were reestablished to the closest of the two new pores, increasing the total number of throats in the network by one. Finally, it must be noted that for both pore merging and splitting, the new pore locations must be checked for pore overlap with nearby pores and if the new pores are within the specified network dimensions. In case of pore overlap, the pore locations were updated by stochastically altering their coordinates.
Parameter | Vanadium | TEMPO | Unit |
---|---|---|---|
Density of the electrolyte, ρ | 992 | 852 | kg m−3 |
Viscosity of the electrolyte, μ | 8.9 × 10−4 | 3.4 × 10−4 | Pa s |
Diffusion coefficient, D1 | 2.11 × 10−10 | 1.3 × 10−9 | m2 s−1 |
Diffusion coefficient, D2 | 2.11 × 10−10 | 1.3 × 10−9 | m2 s−1 |
Inlet concentration, c1,in | 100 | 100 | mol m−3 |
Inlet concentration, c2,in | 100 | 100 | mol m−3 |
Supporting electrolyte concentration | 1.0 M H2SO4 | 1.0 M TBAPF6 | |
Bulk electrolyte conductivity, σ | 28 | 1.99 | S m−1 |
Cathodic transfer coefficient, αc | 0.42 | 0.5 | — |
Anodic transfer coefficient, αa | 0.42 | 0.5 | — |
Exchange current density, j0 | 0.39 | 460 | A m−2 |
Electrolyte velocity, u | 1.5 | 1.5 | cm s−1 |
Cell potential, Ecell | 1.26 | 1.25 | V |
Overpotential, Elosses | 0.5 | 0.5 | V |
The results in Section A5.1† show that the mutation step results in the main fitness improvement by allowing the formation of the electrolyte transport pathways in the flow direction consisting of large pores (36–40 μm), connected by large throats (29–32 μm, Fig. 3a), enhancing both the electrical and pumping power. Merging and splitting alone, on the other hand, results in a fitness improvement (21%), but the pores do not form well-defined transport pathways because of the randomized locations of merging and splitting and due to the absence of mutation. When combined, transport pathways13 are formed (visualized in Fig. A4 in Section A3†), decreasing the pumping power and increasing the electrical performance. However, the total fitness improvement is lower (31% vs. 42%) compared to the case with only mutation. To this end, the effect of the merging and splitting ratio is investigated in Section A5.2† where we found that the ratio between merging and splitting is an important parameter that can steer the electrode optimization toward the formation of networks with more or fewer pores than the starting network.
The network properties of the cases with mutation and combined with merging and splitting are compared in Fig. 3. In Fig. 3a the pore diameter evolution is visualized, presenting the randomization of the merging and splitting operator and the formation of the longitudinal transport pathways for both cases.13 These transport paths consist of interconnected large pores, which, by geometrical definition, have throats with a large diameter connecting the pores. The large throats feature a high absolute velocity (Fig. 3b) and are driving the transport of the electrolyte through the electrode, decreasing the pressure drop. Alongside the electrolyte transport pathways, isolated large pores are present with a large surface area (eqn (1)) that allow more redox reactions to take place, decreasing the concentration locally (Fig. 3c) and resulting in a high current output (Fig. 3d). The outlet concentration of the reactant in Fig. 3c is high because of the small network sizes (∼500 μm). If we run a network-in-series approach on the networks to simulate a larger electrode size (>10 cm2), the species outlet concentration is much lower as discussed in our previous work.21 When comparing Fig. 3 for the two cases, the main difference is in the randomization of the structures as the overall optimization trends remain comparable, such as the formation of transport pathways and higher reaction rates near the membrane interface. To conclude from this comparison, we find that mutation is necessary to speed up the optimization of the networks, whereas merging and splitting adds an additional mode of randomization, allowing for more realistic networks to be formed, but at the cost of a slower fitness optimization.
The initial network structure strongly impacts the structure optimization in terms of the starting performance (fitness value and electrochemical and pumping power) and structure optimization flexibility (number of pores, pore locations, and maximum pore diameter), especially when bound to fixed lattice positions. Therefore, we compare the networks without merging and splitting to identify how distinct fixed lattice positions impact the electrode optimization. Furthermore, to show the system dependency of the optimization framework and thus the need to engineer electrodes depending on their application requirements, we incorporated another flow field design into the framework, i.e., the interdigitated design. In our previous works,30,40 as well as in other literature,41,42 the interplay between the electrode and flow field was proven to be imperative to the RFB performance. For example, Muñoz-Perales et al. found that for Fe2+/Fe3+ single-electrolyte flow cells in combination with an FTFF, woven bimodal electrodes are favorable. Whereas when using IDFFs, unimodal paper electrodes result in enhanced flow cell performance.41 Hence, in this work, we elect to investigate the GA-PNM structure optimization with the FTFF and IDFF in combination with cubic and extracted networks. The optimized structures are shown in Fig. 4 and the structure evolution and optimization values are presented in Sections A6 and A7.†
For the FTFF, both the cubic and extracted networks show an improvement in fitness over the generations, starting from different fitness values, where the fitness evolution is unique for each optimization case with diminishing returns after 100 and 400 generations, respectively. Moreover, both networks evolve towards a bimodal structure with large interconnected pores in addition to small pores (Fig. 4a and A21†). The bimodal structures consist of large connected pores in the flow direction, linked by throats of a large diameter and a high absolute velocity (Fig. 4b), responsible for the electrolyte transport and thus the decrease in pumping power. The decrease in pumping power is the most prominent for the extracted network (65% compared to 54% for the cubic network) related to the higher absolute pumping power required because of the less ordered throats (in-plane) compared to the cubic network. Moreover, the extracted network has the highest increase in electrical power upon comparison with the same surface area definition (22% vs. 3.6% for the cubic network).
Furthermore, with the FTFF the extracted network shows a higher species conversion in the smaller pore segments (Fig. 4c) compared to the cubic network. Especially for electrolytes with sluggish kinetics but high ionic conductivity, the large number of small pores in the extracted network is beneficial. In the electrochemical PNM used, the mass transfer coefficient is a function of the diffusion coefficient and the pore radius and is thus velocity-independent. Hence, smaller pores have a higher mass transfer coefficient and thus a higher species conversion per unit volume (Fig. 4c). Even though small pores have a lower surface area, the extracted network has the same total surface area compared to the cubic network (1.4 × 10−6 m2vs. 1.5 × 10−6 m2) because of the large number of small pores, resulting in enhanced mass transfer in the extracted network. The high number of small pores results in a significantly higher limiting current density, as seen in Fig. 4d. Moreover, as a result of the structure optimization by the formation of a bimodal pore size distribution with small (2–20 μm) and large pores (40–60 μm) and large throats (20–40 μm), there is a strong reduction in the activation (7% at ∼60 mA cm−2, due to the increase in internal surface area of 31%), concentration (24% at ∼60 mA cm−2), and ohmic overpotentials (41% at ∼60 mA cm−2). For the cubic network, the increase in performance is caused by the reduction in the activation and concentration overpotentials (at ∼20 mA cm−2 a 4% reduction in activation overpotential and a 41% reduction in concentration overpotential) as a result of the increase in internal surface area (34%), yet the ohmic overpotential is not significantly reduced. The optimization of the ohmic overpotential in the extracted network is expected to come from an increased ionic flux towards the membrane due to the formation of large throat segments (20–40 μm) with higher electrolyte velocity in the through-plane direction (Fig. 4b). The larger throats result in a greater penetration of the reaction front into the electrode for reactions, corroborated by the high hydraulic conductance of these larger throats through the network in the last generation21 (Section A6†). Combined with the high conductivity of the electrolyte, this provides the optimized extracted network with the largest current output at 1 V. Thus, it is anticipated that due to a large number of pores and their random orientation, the extracted network has a greater optimization chance for both the electrical power and pumping power compared to the cubic network with fixed lattice positions.
The fitness, electrical, and pumping power evolutions with the IDFF portray similar trends compared to those with the FTFF, as well as the percentual increases over the generations. When comparing the networks, the electrical power and surface area of the networks are about 2x greater with the IDFF because of the twice as wide network size. Moreover, noteworthy dissimilarities can be observed in the absolute pumping power required. For both the cubic and extracted networks, the pumping power increases when utilizing IDFFs because of the twice-as-wide electrode. The required pumping power for a larger electrode (>10 cm2) however would result in a much higher pressure drop for the FTFF because of the longer electrolyte pathway through the electrode compared to that for the IDFF. Where the cubic structure has a lower pumping power compared to the extracted network combined with an FTFF (5.2 μW vs. 7.8 μW), the extracted structure shows a reduced pumping power after optimization with the interdigitated design (9.0 μW vs. 6.0 μW). In our previous works, we observed that the pressure drop through carbon paper electrodes is strongly reduced when using an IDFF,30 whereas the pressure drop was even increased through 3D-printed model grid electrodes compared to FTFFs.40 The findings in this work are in line with our previous works as the optimized cubic structure with in-plane high-velocity pathways is favorable with FTFFs, whereas the (optimized) paper electrode results in a lower pressure drop combined with IDFFs because of the combined in-plane and through-plane fluid flow over the rib.
Furthermore, because of the distinct flow distributions of the flow fields, the GA-PNM optimizes the electrodes to significantly different structures (Fig. 4a). For both the cubic and extracted networks, large pores connected by throats with a large diameter are formed in the flow direction (i.e., from the inlet to the outlet channel over a rib). These electrolyte transport pathways result in throats with a high velocity from the inlet to the outlet (Fig. 4b), decreasing the pressure drop. In addition to the electrolyte transport pathways, with the IDFF stagnant zones are formed near the membrane under the inlet and outlet channels with low electrolyte velocities (Fig. 4b) and high species conversion (Fig. 4c).30 It must be noted that the species conversion is higher for the IDFF compared to the FTFF because of the electrode width and electrolyte pathway and because the results are shown without the network-in-series approach for the FTFF.21 Moreover, as shown in the polarization curves (Fig. 4d), the ohmic overpotential is again reduced for the extracted network as the ionic conductance in the flow direction is improved for the optimized structure because of the formation of large throats (Section A6†), enhancing the ionic flux towards the membrane. When comparing the polarization curves in Fig. 4d, we find that the cubic structure shows similar performance in terms of the current output with both flow fields (29 mA cm−2 at 1 V), whereas the extracted structure shows a 2.5 times lower performance with the interdigitated design (∼58 mA cm−2vs. ∼145 mA cm−2 at 1 V). The formation of the stagnant zones under the inlet and outlet channels with high species conversion, for both the initial and final networks, is anticipated to cause increased activation overpotential with the IDFF compared to the FTFF. These regions occupy a significant part of the internal surface area but feature a low electrolyte velocity (Fig. 4b) which could lead to inferior electrolyte replenishment.
In conclusion, when comparing the electrode optimization for FTFFs and IDFFs, similar trends can be observed. The optimization with both flow fields results in structures with a bimodal pore size distribution with large pores connected by throats with a large diameter, responsible for the electrolyte transport and causing a reduction in the pumping requirements of 55–77% for all investigated systems. Moreover, the electrical power is improved by 22–39% caused by an increase in the internal surface area and improved ionic conductance. In practice, utilizing IDFFs is expected to reduce the pressure drop through larger electrodes (>10 cm2), especially for structures with small pores. On the other hand, the current output depends on the electrode length, as for FTFFs the species outlet concentration is strongly correlated to the electrode length, which is not the case for the IDFF. Therefore, when translating the optimization results to larger electrode sizes, it must be kept in mind that the tradeoff between the pumping power and electrochemical power will shift, as especially the pumping power will have a larger contribution when using an FTFF. We find that, when optimizing for only the pumping power (Section A4.4†), the optimization results in similar structures and we thus expect this electrode design also to be favorable for RFBs with larger electrode sizes. Finally, the distinct flow fields introduce unique electrolyte flow profiles through the electrode with an in-plane flow in the length direction for the FTFF and combined in- and through-plane flow in the width and thickness direction for the IDFF. Thus, the coupling of electrode design with specific flow field architectures is crucial and should be considered in future RFB electrode design.
For both flow fields, the fitness of the networks with the TEMPO electrolyte is close to the theoretical maximum (ξ → 0.603) as both the pumping power and electrical power are improved. The pumping power is lower than for the vanadium electrolyte because of the lower electrolyte viscosity (3.4 × 10−4vs. 8.9 × 10−4 Pa s), yet the same percentual decrease in pumping power is obtained for both electrolytes (52–55% for the cubic structures with both flow fields and 65–66% with the FTFF, and 77% with the IDFF for the extracted networks). The electrical power is enhanced with the TEMPO electrolyte at 0.5 V applied potential (for the FTFF: 100 mA cm−2vs. 5 mA cm−2 for the cubic network and 35 mA cm−2vs. 23 mA cm−2 for the extracted network) because of the negligible activation overpotential due to the facile kinetics (Fig. 5d). For the FTFF to this end, the percentual increase in the internal surface area is lower for the TEMPO electrolyte (18% vs. 34% for the cubic and 25% vs. 31% for the extracted network), which translates to a lower percentual increase in the electrical power for the cubic (7.5% vs. 30%) and extracted networks (11 vs. 22%). Furthermore, facile kinetics give rise to a higher species conversion for the cubic networks as can be seen in the concentration profiles in Fig. 5c.
When evaluating the performance at 1 V applied potential in Fig. 4d and 5d for both flow fields, it is found that the cubic network has a considerably higher current output with the TEMPO electrolyte (140 mA cm−2) compared to the vanadium electrolyte (29 mA cm−2) because of the strong decrease in activation overpotential. Whereas the extracted network has a lower performance with the TEMPO electrolyte at 1 V (70 mA cm−2vs. 145 mA cm−2 for the vanadium electrolyte with the FTFF, and 36 mA cm−2vs. 58 mA cm−2 with the IDFF) due to an increased ohmic overpotential. These results support our claim on the importance of the operating conditions, reactor design, and electrolyte chemistry on the electrode selection and optimization, as the cubic network that performed significantly worse with the vanadium chemistry, outperforms the extracted network with the TEMPO chemistry. Even though the internal surface area is 1.6 x lower for the cubic network, the activation (0.04 V vs. 0.009 V at ∼70 mA cm−2 with the FTFF) and concentration overpotentials (0.04 vs. 0.006 V at ∼70 mA cm−2 with the FTFF) are larger, while the membrane resistance and ionic conductivity are the same for both networks, and thus the ohmic overpotential is notably higher for the extracted network at a fixed current density (0.09 V vs. 0.85 V at ∼70 mA cm−2 with the FTFF, not considering the membrane overpotential which is 0.13 V for both networks). The stark difference can be explained by the hydraulic transport through the networks. As the ionic conductivity is low for the TEMPO electrolyte, the hydraulic conductance of the networks becomes imperative to the electrode performance as it dictates the penetration of the reaction front from the membrane towards the current collector and is thus optimized for during the structure evolution (Fig. 5b). Due to the cubic structure with large pores and throats in the flow direction, but additionally in the through-plane direction causing a low resistance to flow, the ionic flux towards the membrane is high compared to the extracted network. The random orientation of the throats in the extracted network in combination with their small diameter, results in a low hydraulic conductance in the through-plane direction, and thus a high ohmic overpotential, resulting in lower species conversion compared to the cubic network (Fig. 5c, species conversion takes place predominantly near the membrane). The GA-PNM optimizes for the ionic conductance towards the membrane over the generations for both networks as is shown in Fig. A5b and A30,† resulting in a decrease in the ohmic overpotential of 13% (evaluated at ∼55 mA cm−2 for the extracted network and at ∼120 mA cm−2 for the cubic network, both with the FTFF). To this end the internal surface area near the membrane is increased in the cubic network to form throats with a large diameter, increasing the ionic conductance near the membrane to counterbalance the low ionic conductivity of the TEMPO electrolyte.
The 3x greater current output obtained with the cubic network compared to the extracted electrode at 0.5 V for the TEMPO electrolyte (even with surface area definition 1 for the cubic network which is an underestimation compared to definition 2) is substantial and should guide researchers for the design of next-generation electrodes for organic electrolytes where the ionic conductivity is generally the performance-limiting factor. However, a note must be made that the electrode dimensions and fluid flow rate most likely impact these results. Finally, we observed that with the TEMPO electrolyte, the trends for the IDFF between the extracted and cubic networks are similar compared to the vanadium electrolyte, suggesting that the influence of the flow field is not very different for the two investigated electrolytes at the evaluated electrode dimensions.
To conclude the chemistry-dependent optimization, we propose certain manufacturing guidelines for the fabrication of next-generation electrodes, specific to a given redox pair and electrolyte chemistry. For electrolytes with sluggish kinetics, electrodes with a high surface area are advantageous to decrease the activation overpotential. Where, especially for electrolytes with sluggish kinetics and high ionic conductivity, electrodes with numerous small pores and throats are beneficial, yet such structures result in a high pressure drop. Therefore, if this electrode structure would be combined with some in-plane electrolyte transport pathways along the flow direction, this would result in the best performance trade-off. Whereas for electrolytes with low ionic conductivity, electrodes with large throats in the through-plane direction (low through-plane tortuosity) result in increased performance. Especially for electrolytes with fast kinetics and low ionic conductivity, electrodes with large in- and through-plane electrolyte transport pathways result in increased current output and decreased pressure drop. However, the internal surface area of the electrodes should not be too low to be able to sustain a low activation overpotential.40 Furthermore, the presented guidelines for electrode design should go together with the selection of a suitable fabrication method and approach to translate the pore network to a solid structure as is often required for manufacturing purposes. While 3D printing is promising for optimized structures for electrolytes with low ionic conductivity and/or facile kinetics due to the high geometrical order that can be achieved, fabrication methods such as electrospinning or non-solvent induced phase separation might be more beneficial for kinetically-sluggish electrolytes as greater internal surface areas can be realized. Conventional manufacturing techniques to fabricate fibrous electrodes (e.g., paper-making, hydroentangling, weaving), which are commercially used, do not offer the design space required to manufacture the geometrically controlled microstructures obtained in this work. Nonetheless, not only could these electrodes be manufactured with the optimized design by techniques such as 3D printing (by for example printing the solid part of the optimized pore networks13), but the learnings from the optimization (e.g., bimodal electrodes with electrolyte transport pathways in the in- and through-plane directions) could also guide conventional electrode manufacturing towards enhanced electrodes for specific reactor architectures and operating conditions. Moreover, this work shows that the current GA-PNM framework can only optimize the performance of the electrodes to a certain degree. Consequently, with the current optimization approach, the selection of the starting network for the optimization is crucial. To diminish the importance of the starting network, mutation operators such as merging and splitting, or pore migration to induce freedom in the location of pores, should go together with the ability to change the number of pores, which could be investigated in depth in future work.
This work is a first step for the bottom-up optimization of electrode structures for flow electrochemical devices. By extending the presented tool, electrode optimization can be further improved and coupled to electrode manufacturing techniques. Possible extensions could include the incorporation of manufacturing constraints depending on the manufacturing technique of choice (e.g., mechanical stability, shrinkage upon carbonization, converting to the solid structure), redefining the fitness function, adding more mutation parameters such as pore migration and changing the number of pores, and incorporating various pore and throat geometries (beyond spheres and cylinders) or refined geometrical definitions (e.g., a better representation of the internal surface area). Other extensions could include reconsidering the network scaling step (as constraining the porosity limits the internal surface area enhancement), the optimization of larger electrode sizes, and optimizing the electrode and flow field designs in tandem. Finally, the translation to other electrochemical technologies, the coupling of the framework to computational fluid dynamic models (for higher detailed optimization or the simulation of advanced flow field geometries), and the full parallelization of the algorithm could be investigated. In conclusion, the optimization freedom in a genetic algorithm with many variables and constraints has both advantages and disadvantages. If the constraints are well understood and the variables are selected with care (e.g., appropriate geometrical definitions), this method can be very powerful for the optimization of electrode structures from the bottom-up. If appropriate optimization conditions cannot be found, this approach can become inefficient where optimal solutions might not be obtained.43
We found that mutation is the key operation driving the optimization, whereas pore merging and splitting can act as an additional mutation operation by inducing randomization and optimization beyond fixed pore coordinates. Moreover, the influence of system parameters, including the choice of electrolyte and flow field design, on the electrode optimization and performance is shown. For all analyzed systems, the genetic algorithm enhances the fitness by a strong reduction in pumping power of ∼51–77% and an improvement in electrical power of ∼8–39% by the formation of longitudinal flow pathways in the flow direction of large pores and throats with high hydraulic conductance, connected to regions with higher species conversion. We found that for the vanadium electrolyte, real electrodes extracted from X-ray tomographic images as starting structures speed up the electrode optimization, opening a path for the optimization of commercially available electrodes. Moreover, the structure evolution is strongly impacted by the flow field design because of the induced fluid path through the electrode. Furthermore, the electrode optimization was analyzed for two redox chemistries (VO2+/VO2+ and TEMPO/TEMPO+) for which substantial differences were observed in the current output with the cubic and extracted networks. For the vanadium chemistry with sluggish kinetics and high ionic conductivity, the extracted network results in the highest current output due to the presence of small pores with high mass transfer rates per unit volume, increasing the species conversion and thus the resulting current output. For the TEMPO electrolyte with fast kinetics and low ionic conductivity, electrodes with large in- and through-plane electrolyte transport pathways with low through-plane tortuosity have a higher current output because of the enhanced ionic conductance. Accordingly, we recommend that next-generation electrodes are optimized and manufactured tailored to the required reactor architectures and operating conditions.
In this study, we show the successful application of the genetic algorithm and that the obtained results can have a significant impact on the design of electrode structures for redox flow batteries. Hence, the developed tool can guide the design of next-generation electrodes for a broad range of operating conditions, electrolyte chemistries, reactor designs, and electrochemical technologies. Nevertheless, we encourage researchers to further extend the optimization framework by for example coupling the algorithm to suitable manufacturing techniques to further increase the impact and potential of this framework.
FTFF | Flow-through flow field |
GA | Genetic algorithm |
IDFF | Interdigitated flow field |
PNM | Pore network model |
RFB | Redox flow battery |
ΔP | Pressure drop, Pa |
A p | Pore internal surface area, m2 |
A T | Throat internal surface area, m2 |
c | Concentration, mol m−3 |
c M | Mutation value |
d | Diameter, m |
D | Diffusion coefficient, m s−1 |
E | Potential, V |
I | Generated total current, A |
j 0 | Exchange current density, A m−2 |
L T | Throat length, m |
N T | Number of throats |
P | Power, W |
Q | Electrolyte flow rate, m3 s−1 |
S p | Pore seed |
S T | Throat cross-sectional area, m2 |
u | Electrolyte velocity, m s−1 |
α | Charge transfer coefficient |
η p | Pumping energy efficiency |
μ | Viscosity, Pa s |
ξ | Fitness |
ρ | Density, kg m−3 |
σ | Conductivity, S m−1 |
σ M | Mutation range |
M | Mutated |
o | Old |
1 | Generation 1 |
1,2 | Pore number, active species |
a | Anodic |
c | Cathodic |
cell | Open circuit cell |
el | Electrochemical |
in | Inlet |
losses | Losses |
max | Thermodynamic maximum, maximum |
n | Last generation |
p | Pore |
pump | Pumping |
T | Throat |
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3dd00247k |
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