High Mechanical Energy Storage Capacity of Ultranarrow Carbon Nanowires Bundles by Machine Learning Driving Predictions

Energy storage and renewable energy sources are critical for addressing the growing global energy demand and reducing the negative environmental impacts of fossil fuels. Carbon nanomaterials are extensively explored as high reliable, reusable, and high‐density mechanical energy storage materials. In this context, machine learning techniques, specifically machine learning potentials (MLPs), are employed to explore the elastic properties of 1D carbon nanowires (CNWs) as a promising candidate for mechanical energy storage applications. The study focuses on the elastic energy storage properties of these CNWs, utilizing MLPs trained with data from first‐principles molecular dynamics simulations. It is found that these materials exhibit an exceptionally high tensile elastic energy storage capacity, with a maximum storage density ranging from 2262 to 2680 kJ kg−1. Furthermore, it is discovered that some CNWs exhibit a superior torsional energy storage capacity compared to their tensile energy storage capacity. Overall, this research demonstrates the effectiveness of machine learning‐based computational approaches in accelerating the exploration and optimization of novel materials. It also highlights the potential of CNWs as promising candidates for future energy storage applications.

DOI: 10.1002/aesr.202300112Energy storage and renewable energy sources are critical for addressing the growing global energy demand and reducing the negative environmental impacts of fossil fuels.Carbon nanomaterials are extensively explored as high reliable, reusable, and high-density mechanical energy storage materials.In this context, machine learning techniques, specifically machine learning potentials (MLPs), are employed to explore the elastic properties of 1D carbon nanowires (CNWs) as a promising candidate for mechanical energy storage applications.The study focuses on the elastic energy storage properties of these CNWs, utilizing MLPs trained with data from first-principles molecular dynamics simulations.It is found that these materials exhibit an exceptionally high tensile elastic energy storage capacity, with a maximum storage density ranging from 2262 to 2680 kJ kg À1 .Furthermore, it is discovered that some CNWs exhibit a superior torsional energy storage capacity compared to their tensile energy storage capacity.Overall, this research demonstrates the effectiveness of machine learning-based computational approaches in accelerating the exploration and optimization of novel materials.It also highlights the potential of CNWs as promising candidates for future energy storage applications.
[35][36][37] This approach facilitates the exploration of materials behaviors under various conditions, such as temperature, pressure, and strain, while maintaining computational efficiency.Moreover, these simulations can aid in understanding the underlying mechanisms governing materials properties, offering insights that can drive the design and optimization of novel materials with tailored performance characteristics.
In this study, we deeply investigated the elastic energy storage performance and intrinsic mechanism of CNWs during the elastic energy storage process by combining DFT and MLPs.The stretching elastic energy storage capacity of CNWs in comparison with CNTs, as well as the elastic potential energy density of CNW bundles during torsion, is compared with different simulation methods.Our results reveal that CNWs demonstrate a remarkably high elastic energy storage capacity, comparable to that observed at very low temperatures.Moreover, CNWs possess a maximum elastic potential energy density ranging from 2262 to 2680 kJ kg À1 , which is 6-7 times than Li batteries.Furthermore, some CNWs exhibit superior torsional energy storage capacities compared to their tensile energy storage capacities.Comparing with DFT and force field, an efficient and rational workflow with superior machine learning simulation on elastic properties in nanowires was developed.It can gain a more comprehensive understanding of the mechanical and elastic properties of 1D nanomaterials and provides insights for their potential applications in a wide range of fields.

DFT Elastic Energy Storage Calculation
Furan (Figure 1a) is a heterocyclic organic compound characterized by five-membered rings with four carbon atoms and one oxygen atom.The carbon atoms can be classified into two types: A and B. Recently, a novel 1D CNW was synthesized experimentally through the modest-pressure polymerization of furan.According to experimental studies, [17] there are three possible CNW structures: syn/anti (Figure 1b), syn (Figure 1c), and anti (Figure 1d).These structures result from the polymerization of furan, with differences arising due to furan's arrangement during the reaction.
During polymerization, A-carbon atoms bond with B-carbon atoms, forming long chains.If oxygen atoms align on the same side of the chain, a syn configuration is formed, characterized by a 1D nanowire with a sawtooth structure and a single class of carbon atoms.In contrast, the anti configuration occurs when oxygen atoms are staggered on both sides of the chain.The syn/anti CNW features two oxygen atoms on one side of the chain followed by two on the opposite side.To understand the elastic properties of this new CNW, we investigated all three possible structures in this study.
To gain insightful understanding of the elastic properties of these three possible CNW structures, we first employed DFT calculations to systematically analyze the stress-strain relationships and determine their elastic constants.The tensile strain progression of simulated CNT (10,0) and three types of CNWs were compared.The variation of energy during the stretching process for CNTs and CNWs is shown in Figure 2a.The maximum energy storage density of CNT (10,0) reaches 8508 kJ kg À1 , while the CNWs are only 2457 kJ kg À1 (syn/anti), 3453 kJ kg À1 (syn), and 3139 kJ kg À1 (anti).
According to the theory of elasticity, the relationship between tensile stress and energy storage density can be expressed as σ ¼ ρ dEd dε , where σ is the stress, ρ is the density of the material, Ed is the energy storage density, and ε is the strain.The variation of the stresses in the CNW during the stretching process is shown in Figure 2b.The maximum strains limit of syn/anti CNW, syn CNW, and anti CNW are 28.2%, 21.6%, and 29.3%, respectively, and the ultimate tensile strength of them is 30, 39, and 38 GPa respectively.Although the ultimate tensile strength of CNTs (10,0) reaches 96 GPa, the superlubricity of CNT surface limits the force loading on surrounding media.Young's modulus is the slope of the stress-strain curve at the zero point.By linearly fitting the stress-strain curves, their Young's modulus are obtained as 155 GPa (syn/anti), 313 GPa (syn), and 372 GPa (anti), respectively.The ultimate tensile strength of these CNW is about 10 times over than that of maraging steel, which is only about 2.6 GPa.

MLPs Constructions
After analyzing the elastic properties of three possible CNW structures using DFT, we aim to employ molecular dynamics simulations to model the more realistic and diverse elastic behavior of CNWs in large scale.As shown in Figure 2c, traditional reactive force field methods not only fail to accurately describe the elastic energy storage capabilities but also provide inaccurate predictions of the fracture strain for CNWs.
Ab initio molecular dynamics (AIMD) simulations are employed to obtain the energy structure-related data of CNWs under different strain states as training data for the MLP.Initial structures of single CNWs were built under various tensile strains (0%-32%).Then they are simulated with NVT ensemble at 300 K and last for 500 steps to allow them to thoroughly traverse the state space.We also constructed bundles composed of multiple CNWs and performed similar molecular dynamics simulations to obtain structures with interactions between nanowires.This provided rich structural information for the MLP.We collected the energies and forces of the different structures obtained during the AIMD simulations.A large amount of training data ensures that the MLP can effectively learn the structural features and physical properties of CNWs and bundles under various strains.Using this sampling method, we obtained a training dataset comprising approximately 40 000 structures.These data provide the foundation for developing high-precision MLP to describe the physical properties of CNWs.
As shown Figure 3, the MLP we trained exhibited excellent accuracy on the test set.Figure 3a illustrates the relationship between the energies obtained at DFT level and those predicted by our trained MLP.Within the energy range of À6.3 to À5.7 eV atom À1 , the MLP performs well, with an energy mean absolute error (MAE) of only 1 meV atom À1 and a root mean square error (RMSE) of 1.6 meV atom À1 .
Figure 3b depicts the relationship between the atomic forces obtained from DFT calculations and those predicted by our trained MLP.Across the entire test set, the MLP demonstrates good performance.Additionally, the atomic forces in the test set exhibit a wide distribution, ranging from À15 to 15 eV Å À1 , which bolsters our confidence in employing this approach to investigate the elastic behavior under various extreme conditions.We also calculated the MAE of the predicted atomic forces, obtaining a value of 17 meV Å À1 and an RMSE of 29 meV Å À1 .Furthermore, we computed the average cosine error for the atomic forces predicted by the MLP, which was only 0.000446.This indicates that the forces predicted by the MLP are nearly perfectly aligned with the forces obtained from DFT calculations.

Comparing Simulation on Elastic Properties of CNWs using DFT and MLPs
By employing MLPs to calculate the elastic properties of the three types of CNWs, we found the maximum elastic energy storage capacities for syn/anti, syn, and anti CNWs.The syn/anti CNWs reached their energy storage capacity of 2497 kJ kg À1 at a strain of 28.6%, the syn CNWs at 21.2% with a capacity of 3332 kJ kg À1 , and the anti CNWs at 29.8% with a capacity of 3283 kJ kg À1 .Figure 4a displays the elastic energy storage capacity calculated using DFT and MLP.We observed significant consistency between MLP and DFT results, indicating the reliability and accuracy of MLP in predicting CNWs' energy storage capacity.
We then used machine learning to directly simulate the stretching process.Thanks to the powerful computational capabilities of MLP, we were able to calculate larger models and simulate a more realistic stretching process.We found that using this simulation method resulted in lower energy storage densities.The energy storage densities of syn and anti CNWs decreased substantially, mainly due to the smaller fracture strains calculated using the new method compared to previous results.The fracture strain of syn CNWs dropped to 18.8%, with the energy storage density decreasing to 2585 kJ kg À1 , while the  fracture strain of anti CNWs decreased to 27.0%, with the energy storage density dropping to 2680 kJ kg À1 .In contrast, the fracture strain of syn/anti CNWs remained nearly unchanged at 28.7%, with a slight reduction in energy storage density, decreasing by 234 kJ kg À1 (Figure 4b).
This could be attributed to stress concentration in specific regions of the longer structures, causing the material to reach its stress limit more quickly in the simulation.This observation suggests that the elastic properties calculated using DFT might not be entirely reliable due to the size limitations of the DFT computational model and the constraints of the simulation methods.Longer structural models, closer to real material structures, can more accurately describe the physical properties of CNWs.This finding demonstrates the value of MLP as a tool to accelerate the study of the elastic properties of new materials by significantly reducing the computational cost and time required to study the elastic properties of substances.

Bundle Twist Energy Storage with MLPs Simulations
Torsional energy storage is another important type of mechanical energy storage in elastic systems.The CNWs were built into bundles composed of 1-19 CNWs to investigate the impact of different bundle sizes on torsional energy storage performance.By conducting torsional tests on these nanowire bundles of varying sizes, we can assess the influence of different bundle scales on torsional energy storage performance.This will help us better understand the potential and limitations of torsional energy storage and optimize nanowire bundle designs for improved torsional energy storage efficiency.To accurately study these aspects, MLP are indispensable, further highlighting the significant role of MLP in this field.
Figure 5a-c displays the energy storage density variation with strain during torsion, where strain is defined as ε ¼ ϕD l , with ϕ being the angle of rotation, D is the bundle diameter, and l is the length of the torsional part of the bundle.This dimensionless quantity unifies the torsion effect for different bundle lengths, making our simulations applicable to longer cases as well.
For the syn/anti CNW, the torsional energy storage density reaches approximately 2299 kJ kg À1 , while its tensile energy storage density is about 2262 kJ kg À1 .This indicates that the syn/anti CNW has a slightly higher capacity for torsional energy storage compared to tensile storage.In contrast, the syn CNW exhibits a torsional energy storage density of 2968 kJ kg À1 and a tensile storage density of 2584 kJ kg À1 .This reveals that the syn CNW's torsional energy storage capacity is significantly higher than its tensile capacity.Similarly, the anti CNW demonstrates a torsional energy storage density of 2860 kJ kg À1 , which is also notably higher than its tensile storage density of 2680 kJ kg À1 .
For bundles composed of multiple CNWs, the maximum torsional energy storage capacity decreases significantly.Even in a bundle with just two CNWs, the maximum torsional energy storage capacity is only about 60% of that of a single CNW.This is because the interaction between the two CNWs results in a notable increase in the strain exerted on the carbon atoms.Figure 5c displays the atomic von Mises strain of bundle-2 with a syn/anti structure during the torsional deformation process.When one of the CNWs breaks due to excessive local strain, the other CNW quickly reduces its strain as it is no longer constrained and compressed by the other CNW.The images of the other two configurations can be found in Figure S1 and S2, Supporting Information.
For larger diameter bundles, the outer CNWs experience earlier breakage due to greater tensile deformation compared to the inner layers.This uneven strain distribution prevents larger bundles from fully utilizing the energy storage capacity of CNWs.This phenomenon can be clearly observed in Figure S3-S5, Supporting Information.Consequently, the maximum energy storage density tends to decrease with increasing bundle diameter.For example, the maximum torsional energy storage density for bundle-19 of all three CNW types is only about one-quarter of that of a single CNW.Additionally, we discovered that for bundles with fewer than 19 CNWs, the maximum torsional energy storage density is reached at the first breakage event in the entire nanowire bundle.However, for bundle-19, the maximum torsional energy storage density is achieved only after multiple breakages occur throughout the entire nanowire bundle.

Conclusions
In summary, we have developed MLP-based molecular dynamics simulations to investigate the elastic energy storage properties of CNWs.Our results reveal that individual CNWs, particularly those with syn and anti configurations, exhibit exceptionally high elastic energy storage capacities under torsional loading.Moreover, we discovered that the torsional energy storage density of CNWs decreases with increasing bundle diameter, which can be attributed to the uneven strain distribution within larger bundles.
However, despite the exciting mechanical energy storage characteristics of this new material, there are still significant challenges to overcome before practical applications can be realized.Currently, CNWs are synthesized only in laboratory settings, and the process of fabricating them into fibers, improving their utilization efficiency, and establishing the necessary infrastructure for energy storage and conversion involves complex procedures.Additionally, the issue of material fatigue over long-term usage also requires attention.
More broadly, this research underscores the effectiveness of MLPs in accelerating the study of various mechanical properties in new materials.The integration of machine learning techniques with traditional computational methods, such as density functional theory and molecular dynamics simulations, has the potential to greatly enhance our understanding of material behavior and drive the discovery of innovative materials for a sustainable future.

Experimental Section
DFT: The first-principles calculations in this work were performed using the Vienna Ab initio Simulation Package (VASP). [38]The density generalized function theory was employed with the Perdew-Burke-Ernzerh (PBE) exchange-correlation functional [39] under the generalized gradient approximation (GGA) to calculate the electronic structure and energy of the CNWs.The plane wave basis with a cutoff energy of 400 eV was used in the calculations.A convergence criterion of 10 À6 eV for the total energy and 0.02 eV Å À1 for the atomic forces was adopted.In order to avoid interactions between adjacent layers, a vacuum region of about 25 Å was introduced along the nonperiodic direction.
ReaxFF and MLP: To consider larger scale, a molecular dynamics approach based on both the reaction force field and MLP was used.The ReaxFF potential, [23] developed by van Duin, Goddard and co-workers, was employed in conjunction with the MLP to study the elastic properties of the CNWs.The ReaxFF potential uses a distance-dependent bond order function to represent the contribution of chemical bonds to the potential energy and includes hydrogen bonding interactions.However, to further improve the accuracy and efficiency of the simulations, we trained a MLP for the CNWs.The MLP was developed using DFT data and the corresponding implementation of E(3)-equivariant graph neural networks provided in the open-source code of NequIP [40] was used.The entire model consisted of a total of 171 640 parameters, divided into four interaction blocks.The neighbor list was constructed with a truncation radius of 4.0 Å, and eight basis functions were utilized in the radial basis.The maximum angular momentum for interatomic edges was set to 1, and each irreducible representation in the output features of the hidden layer convolution contained 32 features, including those with odd mirror parity.The neural network for computing the spherical harmonic convolution weights consisted of two hidden layers, each containing 64 neurons.An equivariant nonlinear activation function employing a gating mechanism was used, and node features were self-connected.The loss function considered both energy and atomic forces, and the total energy error was quantified using the mean squared error (MSE) calculated per atom.The overall loss was a combination of energy and force losses, with equal weights assigned to each (1:1).During the model training process, a learning rate of 0.005 was chosen, and a batch size of 5 was used.The training procedure consisted of 51 epochs, with a total of 8,400 steps performed per epoch, resulting in a cumulative total of 428 400 steps.This method allowed us to capture the complex and nonlinear interactions among the atoms in the CNWs and to incorporate the data-driven learning into the simulations.
Molecular Dynamics Simulation: In the simulation of the elastic properties, a low temperature of 2 K was used to reduce thermal disturbance.The temperature was equilibrated using a Nosé-Hoover thermostat. [41,42]A time step of 0.5 fs was adopted for all simulations.Prior to all simulations, a conjugate gradient (CG) algorithm was used for structural optimization, and the NVT ensemble was used to relax structure for 10 ps.All simulations were performed using the LAMMPS software package. [43]

Figure 2 .
Figure 2. a) Compared tensile elastic energy storage curves of CNTs and CNWs using DFT simulations.b) Stress-strain curves for CNTs and CNWs calculated using DFT methods.c) Comparison of tensile elastic energy storage calculated using reactive force field methods and DFT methods.

Figure 3 .
Figure 3. a) Relationship between the energy predicted by the machine learning model and the energy calculated using DFT on the test set.b) Relationship between the atomic forces predicted by the machine learning models and those based on DFT level.

Figure 4 .
Figure 4. a) Elastic energy storage capacity calculated using DFT and MLP.b) Elastic energy storage curves obtained from molecular dynamics stretching simulations using MLP.

Figure 5 .
Figure 5. a-c) Torsional elastic energy storage curves for CNW bundles composed of syn/anti, syn, and anti structures.d) Atomic von Mises strain of bundle-2 with syn/anti structure during torsional deformation process.