Advancing tribological simulations of carbon-based lubricants with active learning and machine learning molecular dynamics

Abstract

The need to move toward more sustainable lubricant materials has sparked an ever growing interest on the tribological performances of additives based on environmentally friendly molecules, such as carbon-based compounds, and green liquid media as aqueous solutions. The prediction of the solubility of the additives into the liquid and the tribochemistry of decomposition and polymerization of the additive molecules under harsh conditions is essential for understanding the atomistic mechanisms leading to the formation in situ of the carbon-based lubricious tribofilms so effective in reducing friction and wear at solid interfaces. To this extent, the application of tools like ab initio molecular dynamics based on first-principle density functional theory is severely hindered by the size of the systems of interests and the need to simulate their dynamics over relatively long times. To enable tribological simulations with quantum accuracy for a first time, we develop a workflow for smart configuration sampling in active learning, to obtain machine learning interatomic potentials which are shown to be sufficiently robust and accurate also in the characteristic harsh conditions generated by high loads and shear rates. Focusing on glycerol rich lubricants, we apply this active learning strategy to generate a neural network potential to simulate the formation and behavior of nanometer thick molecular tribofilms. The simulations reveal the superior accuracy of the machine learning approach with respect to classical molecular dynamics with reactive force fields, and pave the way for more promising in depth exploration of novel carbon-based lubricants.

1 Introduction

The development and application of machine learning interatomic potentials for atomistic simulations (MLIP) have recently witnessed rapid growth, offering diverse approaches to tackle the trade-off between accuracy and efficiency in predicting atomic properties [1,2,3]. This dilemma revolves around achieving accurate predictions comparable to ab initio methods at a computational cost closer to classical force field methods. Currently there is no straightforward consensus about which kind of ML approach is best suited in describing atomic potential and many different MLIP are being developed, a situation often referred to as “zoo of MLIP.” This large variety of MLIP can be classified in two main categories: kernel methods and neural network or neural network potentials (NNP) [4, 5]. Kernel method MLIP leverage similarity functions to fit the atomic dataset, while NNP use neural network for the same purpose. The training of artificial neural networks (NN) scales more effectively with dataset size than kernel methods, making them the preferred choice for handling large datasets [6].

In atomistic simulations, predicting the system energy and consequently the forces on atoms is paramount to any other property prediction. Neural network potentials are often constructed under the assumption of local energy decomposition, such that the total energy of the system is expressed as a sum of atomic energies depending on the central atom and its neighbors [7]. To ensure the correct structural symmetries, atomic coordinates are not fed directly to the NNP. They are first transformed into symmetry-preserving descriptors which are subsequently mapped to local atomic energies by the NNP [8]. When the descriptor has an analytical form and can be computed, a priori one speaks about fixed descriptor NNP [9, 10]. Instead, if the descriptor is found using a neural network (called descriptor or embedding neural network) one speaks of trainable descriptor or end-to-end NNP [11, 12]. Although fixed descriptors have a known analytical form, often having a straightforward physical interpretation, they can sometimes have problems when multiple chemical environments are present in the systems of interest and they must be ad hoc tuned. On the other hand, trainable descriptors have highly adaptive nature with minimal a priori intervention even if their physical interpretations become obscured [5].

Prediction of friction coefficients and energy dissipation at the sliding interfaces requires a proper description of the vibrational degrees of freedom of the semi-infinite bulks in contact and the surface chemistry [13, 14]. The relative sliding motion between the surfaces induces vibrations at a characteristic frequency, also called washboard frequency, lying in the low-frequency part of the atomic vibration spectrum [15]. To properly sample these frequencies in Fourier space, simulation times of several hundred picoseconds and simulation cell sizes of several nanometers are required. Chemical accuracy required by the load-enhanced reactivity and the need of large time and space scale simulations to capture dissipative phenomena form the perfect playground to exploit neural network potentials. Constructing a comprehensive dataset is pivotal for a good NNP training, especially in a tribological setup where different compounds are subjected to a broad range of temperatures and stresses. It is strongly suggested to sample system configurations that go beyond the ones expected, in order to consider unavoidable thermodynamic fluctuations during the simulation. Tribochemistry adds another layer of complexity due to the transformation of the chemical compounds during the in silico experiment, leading to the need of sampling new chemical conformers which can be not known a priori [16]. The chemical reactivity of confined molecules at the interface is often increased due to high pressure and shear conditions. This enhanced chemical reactivity is often responsible for the formation of a tribofilm providing resistance to surface wear and helping in reducing friction between the contacts [17].

In this work, we develop for the first time in our knowledge, a NNP robust enough to be applied for tribological simulations. We show how to efficiently train a NNP by employing an in-house developed active learning workflow and how this procedure can enhance the scope and the capabilities of pure ab initio methods. When the NNP is used to simulate the behavior of carbon-based lubricant molecules, it is able to predict realistic coefficient of friction in agreement with nanoscale experiments. We also identify two distinct sliding behaviors for the same liquid lubricant confinement: A solid-like behavior at high confining loads and a viscous fluid for lower loads. We also show that by using a classical ReaxFF force field the magnitude of the friction coefficients is overestimated due to poorly described interaction between the lubricant and the substrate and the differences in the liquid behavior at high and low loads cannot be captured. In this way, we highlight the efficacy of NNP models with respect to classical force field. The former can be trained in a straightforward manner without the need of a meticulous fitting of ad hoc parameters for a functional form fixed a priori and with poor transferability.

2 Methods

2.1 Dataset for tribochemistry simulations

A standard setup for the simulation of a tribological interface, as shown in Fig. 1, comprises two surfaces that are squeezed against each other creating an effective load acting on the interfacial medium which is composed by a liquid containing some lubricant additives. Figure 2 presents an overview of the different systems included in the dataset. Each of these systems represents a small, characteristic, sub-system of the realistic nano-interface of Fig. 1. This sub-system decomposition and cut in size allows for ab initio calculations at the DFT level to obtain predictions on the interaction for the representative variety of local atomic environments. We included the diamond substrate both as a crystalline bulk and as a passivated surface in vacuum and with adsorbed molecules. The lubricant is composed by a glycerol solvent pure and with hypericin molecules dispersed in it. Graphene and water molecules have been also included in the training set. Graphene in order to make the NNP more aware of sp$^2$ bonds and because graphitization is one of the key known mechanism for hypericin lubricity in glycerol [18], while water has been added because it is believed to be one of the possible by-product of hypericin graphitization. The dataset includes also clusters of pristine and randomly defected hypericin molecules and a glycerol/water mixture with solvated hypericins to explore possible local environments occurring inside the interfacial medium. All these systems have been simulated across a wide range of temperature and load conditions in order to sample high energy configurations typically encountered in a tribological setup.

Fig. 1

Schematic view of the tribological interface simulated by the NNP. The tribological conditions consist in the application of an external sliding velocity and external load which may promote chemical reactions

Fig. 2

Systems used for training the NNP. System sizes and cell dimensions must be compatible with ab initio calculations, and they typically do not exceed few nanometers per side

2.2 Dataset construction with active learning method: smart configuration sampling

The most time consuming part in the development of the NNP is the construction of the dataset due to the computational bottleneck of ab initio simulations. To efficiently generate the dataset, we have developed an active learning workflow called smart configuration sampling (SCS). Active learning approaches are well known within ML, and their usage has also emerged within MLIP [19, 20]. The key idea behind these methods is to exploit the computational efficiency of the NNP to iteratively generate new candidate configurations to be calculated ab initio and included in the dataset [21].

The workflow scheme of SCS is reported in Fig. 3. As for other active learning workflows, SCS consists of repeated iterations which includes three phases: ensemble training, exploration and ab initio sampling. During the training phase at the beginning of the iteration, several NNP (4 in our case) are trained on the available dataset, differentiating the training sequence with a different seed for each NNP initialization. This creates an ensemble of trained NNP which is used in the exploration phase to generate new atomic configurations. One NNP is used to perform a molecular dynamics simulation on the systems of interest, while the remaining NNPs are used to predict energies and forces on the same configurations from the trajectory generated with the first NNP. This procedure creates an ensemble of energies and forces for the same atomic configurations which can then be used as a metric to quantify the uncertainty of the NNP predictions on each atomic configuration [22]. In particular, because we are interested in molecular dynamics, we focus on measuring the force deviations within the ensemble. For each atomic configuration, we define the ensemble deviation $\sigma$ as the maximum of the root mean square error (RMSE) of the ensemble [21]:

$$\sigma ^2:= \max _{i, \alpha } \frac{1}{N_{\mathrm{NNP}}}\sum _{\mathrm{NNP}} \left( \overline{F}_{i, \alpha } - F_{i, \alpha }^{\mathrm{NNP}}\right) ^2$$

(1)

where the indexes i and $\alpha$ identify the atomic force components. The ensemble deviation provides the metric to select relevant configurations to be later computed ab initio. We select a configuration if its deviation $\sigma$ exceeds a lower threshold $\sigma _{\mathrm{low}}$. A higher threshold $\sigma _{\mathrm{high}}$ is also used to avoid nonphysical configurations where reaching self-consistency may be problematic with ab initio methods. The idea behind this selection criteria is that configurations that have $\sigma > \sigma _{\mathrm{low}}$ are probably “distant” from the configurations already present in the dataset. When the exploration phase is terminated, energies and forces for the selected frames are computed using ab initio methods and these new frames are included in the dataset. At the end of the iteration, the dataset contains more relevant configurations and a new SCS iteration can begin. The $\sigma$ defined in Eq. 1 does not take into account the magnitude of the force component. Due to the large variation in temperatures and stresses present in tribological conditions, atomic forces can span several order of magnitudes and the selection criteria $\sigma > \sigma _{\mathrm{low}}$ may lead to an over sampling of high energy configurations and to an under sampling of low energy configurations. For this reason, we used the normalized deviation $\tilde{\sigma }:= \sigma /|F|$ when applying the selection criteria [23]. In our active learning workflow SCS, we have also implemented a refinement of the selection procedure which can save a lot of computational time by uncorrelating the collected configurations. Before proceeding directly to the ab initio sampling, the candidate configurations are first listed with decreasing $\sigma$, then they are selected after making sure that there are no configurations with a time separation smaller than $\tau _{\mathrm{S}}$ time steps. A proper choice of the correlation time $\tau _{\mathrm{S}}$ allows to collect fewer configurations, the independent ones, reducing the number of ab initio calculations required at each iteration.

Fig. 3

Workflow of smart configuration sampling (SCS). The software consists of an in-house developed pipeline for the optimal construction of the training set

SCS relies on three different, open source, software packages for the training, the exploration and the sampling phases. It uses DeePMD-kit [24, 25] software to train and develop the neural network potential, LAMMPS [26] with DeePMD-kit plug-in to perform NNP molecular dynamics during exploration and Quantum Espresso [27,28,29] package is used for ab initio calculations.

2.2.1 Enhancing ab initio calculations with active learning

One of the systems used for the training contains several glycerol molecules confined at 5 GPa. This systems has been explored using SCS by performing ensemble NNP molecular dynamics with temperatures ranging from 50 to 1000 K. During the exploration phase of one SCS iteration, the dynamical trajectory produced with the reference NNP predicts water formation from glycerol activated by the high confinement load and high temperatures.

In Fig. 4 the blue curve shows the energy profile predicted by the reference NNP during such SCS iteration, while the histogram plot at the bottom shows the ensemble deviation $\tilde{\sigma }$ for the twelve selected atomic configurations sampled over a 1.1 ps timeframe during the SCS exploration phase. Frame number 10, encircled in the red oval, has the biggest deviation $\sim \!\!68\%$ and as such, accordingly to the refinement sampling procedure in SCS, its configuration was sampled to be added to the training dataset after calculating energy and forces at the DFT level. An interesting observation is related to the effectiveness of SCS and the interpolation power of NNP: Adding just this one frame to the dataset makes the next SCS iteration NNPs already able to predict the whole energy profile (orange, green, red and violet curves/points) in much closer agreement to the energies recalculated ab initio with single-point DFT calculations (black curve/points) for the configurations selected to represent the event. This example shows the usefulness of the active learning procedure to enhance the scope of pure ab initio simulations [30, 31]. Thanks to the generative capabilities of NNPs, one can gain insights into possible chemical processes and, once a trajectory is obtained, one can re-compute properties ab initio to analyze and validate the training process. Needless to say that, without the capability of efficiently achieving trajectories of several hundred picoseconds provided by the NNP, these processes would be practically impossible to predict using purely ab initio methods. With one sampled frame every 1.5 ps on average, we emphasize the effectiveness of SCS and its refined selection method for the optimization of the training dataset.

Fig. 4

Energy profiles for the water formation event. Colored curves represent the energies predicted with the reference NNP during the SCS exploration phase (SCS, blue curve) at a given iteration and the recalculated energies with NN0 (orange), NN1 (green), NN2 (red) and NN3 (violet), i.e., the four newly trained models at the SCS iteration. Black curve (DFT) refers to the ab initio calculated energies for the same atomic configurations

2.3 Training results and testing

At the end of our active learning workflow, we were able to build a dataset containing a total of 30,904 frames. Table 1 summarizes all the systems contained in the dataset, specifying the number of configurations, SCS usage and the range of pressure explored. This dataset was used to train an end-to-end NNP having the seAe2 architecture employed in the DeePMD-kit software [25]. We trained our model using $\left[ 25, 50, 100\right]$ and $\left[ 120, 120, 120\right]$ neurons per layer for the descriptor and the fitting net, respectively. Figure 5 shows the performance of the model on the atomic force prediction over the validation dataset. The validation dataset amounts to the $10\%$ of the whole dataset which was not used for training. Atomic forces are the most important quantity to perform molecular dynamics, and their accurate prediction is more difficult to achieve than atomic energies. Our validation set contains $\sim$ 3 million force components over the different systems shown in Fig. 2.

Table 1 Systems used for the training, number of configuration computed ab initio, inclusion of active learning scheme SCS and pressure range explored

Fig. 5

Root mean square error (RMSE) on force components over validation dataset. On the left the dataset corresponding to different physical systems is highlighted by different colors. On the right a density plot on a close-up of forces between − 10 and 10 eV/Å

The majority of the data points are clustered near equilibrium configurations, where the atomic forces are close to zero. However, there is an extensive sampling of high force configurations, as evident in Fig. 5 where the atomic forces reach values as high as $\sim$ 60 eV/Å. The harsh conditions present at the nano-asperities requires a sufficient sampling of such high force configurations where the average distance between the molecules shortens. Our trained model is able to reach a root mean square error of 0.16 eV/Å on such diverse dataset.

2.3.1 Comparing fixed descriptor and end-to-end NNP

End-to-end NNP can provide exceptional fitting performance even if a clear physical interpretation is often obscured within the layers of the neural network. More physical approaches are provided by fixed descriptor NNP. In these cases, one provides an analytical form for the atomic description which is directly computable from the atomic configuration [4]. SOAP is one of those approaches [8]. Following the implementation of SOAP descriptors in the Dscribe package, its analytical expression is given by [32]:

$$\begin{aligned} p_{nn'l}^{Z_1Z_2} = \pi \sqrt{\frac{8}{2l+1}}\sum _mc_{nlm}^{Z_1*}c_{n'lm}^{Z_2} \end{aligned}$$

(2)

where $c_{nlm}$ is the expansion coefficients of the atomic density on the spherical harmonics. On the other hand, the DP-seAe2 model is an end-to-end NNP and its atomic descriptor is given by [25]:

$$\begin{aligned} D = \mathcal {G}^T\tilde{\mathcal {R}}\tilde{\mathcal {R}}^T\mathcal {G}^< \end{aligned}$$

(3)

where $\tilde{\mathcal {R}}$ is the local environment matrix which is directly computable from atomic coordinates, while $\mathcal {G}$, $\mathcal {G}^<$ are the outputs from the last neuron layer of the embedding neural network. Naively we can say that while SOAP descriptors are built based on our physical knowledge, the embedding neural network has no a priori knowledge and it must learn a suitable expression for the atomic descriptors during the training.

Figure 6 shows the comparison of the atomic descriptors between the DeePMD descriptors (left) at the end of the production training and the SOAP descriptors (right). In particular, we calculate the first two PCA components of their respective carbon atom descriptors on the same dataset using the scikit-learn package [33]. We observe that the topology of the clusters looks similar so that different clusters in DP correspond to different clusters in SOAP. Interestingly we notice that the latent space for the diamond bulk (black cluster) and for the diamond surface (rightmost clusters) are overlapping in DP descriptor while they are far apart in SOAP. This evidence may indicate a more refined clustering made by the DP model which is able to identify some bulky feature in the surface. This fact highlights the efficacy of the interpolative power of an end-to-end model which best adapts to the dataset even if it is not constructed from a clear physical basis.

Fig. 6

Comparison of DeePMD (left) an SOAP (right) carbon atom descriptors over the dataset. The topology of the clusters looks similar in both cases even if DP descriptor seems better in recognizing the similarity between diamond bulk (black cluster) and diamond surface (rightmost colored clusters)

3 Results

3.1 Application to tribological interfaces

We deployed our trained NNP to perform large-scale atomistic simulations of the tribological interface depicted in Fig. 1. The interface is composed of two mating diamond (111) surfaces which are completely passivated by hydrogen atoms. The interfacial region is filled with a lubricant composed by a mixture of glycerol and hypericin. Glycerol is well known for its tribological properties, and it is commonly used as a lubricant additive. Hypericin is a new discovered eco-friendly additive which can enhance the lubricant property of glycerol at higher temperatures [34]. Load and sliding are imposed to the interface by applying a normal force and a shear velocity to the group of hydrogen atoms that passivate the outermost surface of each slab.

3.2 Friction measurement of the interfaces

Panel a of Fig. 7 shows the behavior of the friction force per unit area for the lubricant composed of pure glycerol with a mass density reproducing the experimental one of 1.26 g/cm$^3$. We repeated the in silico tribological test for different applied loads, ranging from 0.1 to 2 GPa, and different sliding velocities. We calculated the friction force as the time average of the lateral component of the force acting on the dragged hydrogen atoms over 200 ps of sliding molecular dynamics. Before collecting the data, we performed 150 ps of thermalization at 300 K under the applied external load and we executed 300 ps of sliding dynamics to reach a steady state. We observe two general trends for the friction force: It increases with the external load and external sliding velocity. This result is evident from Fig. 7 even if unavoidable fluctuations in the averages arise at such small scales. The measured friction force per unit area, in the range of few MPa, indicates a coefficient of friction (CoF) ranging between 0.004 and 0.09 which is comparable to experimentally measured CoFs for systems lubricated with glycerol [35].

Fig. 7

Behavior of friction force per unit area versus applied external load and imposed sliding velocity (Panel a). Glycerol density profile along the direction perpendicular to the interface for different applied external loads (Panel b). Vertical dashed line highlights the increase in glycerol thickness for decreasing loads

3.3 Tribological behavior of confined glycerol

Panel b of Fig. 7 shows the mass density profile of glycerol along the direction perpendicular to the interface for different applied loads. When the load is increased, glycerol is confined within narrower regions as expected. In particular, we notice the formation of layers in glycerol due to load and shear stresses induced by the external forces acting on the interface. Figure 8 highlights the layered structure of confined glycerol molecules. Each layer corresponds to one of the four density peaks as illustrated by the use of different colored areas under the density profile curve in Fig. 8. As expected for a liquid thin film, the glycerol outer layers in contact with the diamond surfaces peak at a higher density due to adhesion. To investigate how this layered structure of glycerol affects sliding, we performed a velocity profile analysis based on the position of the glycerol molecules and, as in the density profiles in Fig. 7b, we can identify four distinct glycerol layers. We calculated the center of mass velocity for each of them, averaging over the last 50 ps of sliding dynamics at steady state. To establish the layer to which each atom belongs, we clustered the atoms with the Kmeans algorithm using 4 means, as implemented in the scikit-learn package [33].

Fig. 8

Layering division of the glycerol under 1 GPa of applied load at the interface using the Kmeans algorithm. Atoms belonging to different layers are colored as their corresponding peaks in the density profile plot

Within the investigated range of external loads and sliding velocities applied to the diamond interface, we find that all layers move with the same sliding velocity, and that glycerol flows resembles the motion of an amorphous layer with a speed which is approximately half of the external relative driving velocity. Apparently, as the low measured coefficient of friction testifies, the interactions between glycerol and the passivated diamond surfaces produce an interfacial shear strength which is not high enough to overcome the internal viscosity of glycerol, and the thin liquid film moves as a whole, performing in a way similar to that of a layered solid lubricant.

3.4 Comparison with classical force field

In this section, we compare the behavior and the friction properties obtained using our trained NNP and a ReaxFF reactive force field, recently reparametrized for systems with the same atomic compositions [36].

Table 2 shows the comparison between the measured friction force and the interfacial separation predicted by our trained NNP and those predicted by the classical force field ReaxFF. We notice how the differences in the predicted friction shear amount to more than one order of magnitude, with much larger, somewhat unrealistic, friction coefficients predicted by ReaxFF. While in both cases the trends of friction with respect to applied external load are equivalent, the values predicted by ReaxFF for the CoFs turn out to be out of scale for an interface formed by fully passivated carbon lubricated with glycerol [37, 38].

Table 2 Comparison of friction shear and interface separation between NNP and classical ReaxFF force field

Furthermore, we notice that the interfacial separation for the ReaxFF force field seems not as sensitives to the external applied load, as for the NNP case where its decrease with increasing load is more pronounced. The differences between ReaxFF and the NNP are made evident by comparing the mass density of glycerol in Fig. 9. The lower diamond surface is held in place by the constrains on the positions of the bottommost hydrogen atoms (cfr Fig. 1) and both potential models predict a very similar structure for the diamond slab, with the lower interface at the same height. On the contrary, in the inner regions the glycerol density behaves rather differently: NNP predicts the presence of ordered layers which is accompanied by a $\sim$ 25% reduced interfacial separation with respect to the ReaxFF result, where there are no sharp peaks and the density profile is more akin to that of a homogeneous bulk, with some additional random fluctuation from the uniform value. The differences in the predicted interfacial separations, reported in Table 2, are clearly depicted in Fig. 9 where the upper diamond surface, both at 0.1 GPa and at 1 GPa load, is clearly in a much higher position for ReaxFF than for NNP.

Fig. 9

Density profile of glycerol during sliding conditions predicted by the NNP and ReaxFF force field at 1 GPa load and at 0.1 GPa load. NNP predicts a layered structure for glycerol, while ReaxFF does not distinguish different layers, apart from some ordering close to the diamond surfaces

The formation of ordered layers in glycerol appears to favor sliding motion at the interface presenting diamond with a smoother, less corrugated surface, responsible for the lower friction coefficient. This kind of layering is missed in the ReaxFF drive simulations, and the predicted friction coefficients are two orders of magnitude larger than those predicted by the NNP.

3.5 Simulation of a realistic lubricant composed by a liquid containing additives

The NNP model was trained on a variety of systems and different conditions to allow us to simulate more realistic tribological interfaces with an accuracy equivalent to pure ab initio calculations. Indeed, quantum chemical simulations of the tribological behavior of molecular additives in real liquid lubricants have been so far hindered by the prohibitive computational cost of pure ab initio simulations for systems with large number of particles. However, real-world lubricants are not pure substances as the glycerol compound described in the previous section, but complex mixtures composed by a solvent and often more than one lubricant additive. As a preliminary proof of concept, we used the NNP model to simulate the dynamics of the same hydrogen passivated diamond interface this time lubricated using a mixture of glycerol and hypericin molecules. In fact, thanks to the high solubility of hypericin in glycerol, which is very well reproduced by the NNP potential model, the latter can be used in place of less environmentally friendly oils, as base lubricant with hypericin as additive. We used a hypericin concentration in glycerol which amounts to approximately $\sim \, 2\%$ in weight, a value which is comparable to typical additive concentrations actually used in experiments. The average friction forces measured for the hypericin–glycerol mixture are shown in panel a of Fig. 10.

Fig. 10

Comparison a of the friction force per unit area versus the external driving velocity at 1 GPa of applied external load for lubricants composed by pure glycerol and a mixture of glycerol and hypericin. Snapshot of the sliding interface, sliding motion along x-axis (b). The layered structure of the confined amorphous glycerol is not disrupted by the solvated hypericin molecules. Glycerol–hypericin O–O and O–H radial distribution functions c Showing the characteristic first peaks associated to hydrogen bonding

The values of the friction coefficients are very much close to those previously obtained with pure glycerol, only marginally lower at high sliding velocities. As shown in Fig. 10b, the few hypericin molecules do not strongly interact with the passivated surface: They prefer to remain independently solvated in the glycerol without disturbing much its layered structure, remaining embedded inside the thin liquid film and sliding consistently with it. Each hypericin molecule persistently forms a number of hydrogen bonds with glycerol molecules, as it is clearly showed by the two peaks at around 1.9 Å for O–H and H–O atomic pairs and by the peak at 3.0 Å for O–O atomic pairs of the radial distribution functions calculated by selecting the first atom from glycerol molecules and the second one from hypericin molecules and shown in Fig. 10c. We actually find that, during the simulation under sliding, a hypericin molecule can have at the same time from 3 to 6 hydrogen bonds with glycerol molecules, 4.1 each on average, by applying a standard geometric criteria, i.e., we considered that one hydrogen bond is formed between one hypericin and one glycerol molecule if the oxygen–oxygen separation is less than 4 Å and the angle ${\mathrm{O}}\ldots {\mathrm{O}}{-}{\mathrm{H}} \le 30^\circ$.

The NNP model will make it be possible to directly simulate larger systems for the much longer times needed to explore the processes induced by the mechanochemical reactions which are at the basis of tribofilm formation in lubricants that can play a fundamental role to explain the superlubricant properties of aromatic molecules like hypericin in diluted glycerol solutions.

4 Summary and conclusions

In this work, we show for the first time application of machine learning to develop a model potential for atomistic simulations of a tribological interface. We train a neural network potential (NNP) to perform such simulations using SCS, an in-house developed active learning workflow that allows for the efficient generation of the training dataset, by speeding up the calculation time devoted to pure ab initio methods. We show also some preliminary results obtained by using our production NNP, trained on the final dataset obtained with SCS. We are able to compute friction coefficients of glycerol for very low film thickness. We find that the magnitude of the friction coefficients is very close to those experimentally observed for such systems and that, as expected in the adopted sliding conditions, the computed CoFs increase with applied external load and with sliding velocity. By studying the mass density and the velocity profiles of glycerol in the interfacial region, we also find that achievement of very low CoFs is correlated to a behavior which resembles that of a layered solid lubricant. On the contrary, the dynamics predicted with a ReaxFF reactive force field produces results which do not compare well with experiment, predicting unreasonable large friction coefficients, pointing to the necessity of undergoing throughout an expensive ad hoc reparametrization specific for each system. With the wide range of structures included in our training, we show that our NNP can be used to perform large-scale atomistic simulation of tribological interfaces with interactions close to quantum accuracy for system containing glycerol as a solvent, plus lubricant additive as complex as hypericin. The advent of MLIP with the adoption of smart active learning approaches makes it now possible to develop NNPs capable of handling the harsh conditions present in tribological experiments. In the near future, we plan to apply the NNP model described in this paper to investigate in details by means of large-scale molecular dynamics simulations the formation of carbon-rich tribofilm induced by mechanochemistry using different lubricant additives.

Appendix

1.1 Computational setup

All the ab initio calculations have been performed using the Quantum Espresso software [27] at DFT level employing the PBE form of the exchange-correlation functional [39]. PAW pseudopotentials [40] of the SSSP library [41] are used to represent the ionic potential near the nuclei. The cutoffs of the plane wave expansions are fixed at 60 and 540 Rydberg for the electron wave function and electron density, respectively, employing $\Gamma$ point sampling since the dimensions of the cell are large enough.

The training procedure is carried out using the DeePMD-kit package [24]. All the neural network potentials in this work use the Smooth Edition 2 of the DeeP Potential dpse2 [23] with angular feature in the local environment matrix. The NNP contains $\left[ 25, 50, 100\right]$ and $\left[ 120, 120, 120\right]$ neurons for the three layers of the embedding and fitting network, respectively, with 16 axis neurons for descriptor embedding dimension. During the active learning of the SCS workflow, we train 4 NNPs for 1 million training steps at each iteration. Once the final dataset is obtained, we train the production NNP for a total of 5 million training steps.

Machine learning molecular dynamics is performed using the LAMMPS code [26] interfaced with DeePMD-kit plug-in, while classical molecular dynamics uses LAMMPS with the Kokkos package [42] employing the ReaxFF force field [36]. The integration time step for the Verlet algorithm is set to 0.1 fs, and hydrogen has the deuterium mass. The large-scale simulations shown in Sect. 3 have a simulation cell with roughly 53Å $\times$ 52Å of lateral dimensions where periodic boundary conditions are applied. These systems contain 345 glycerol molecules, and the diamond surfaces are composed by 6 bilayers of C atoms passivated by hydrogen atoms at the boundary regions. Nosé–Hoover thermostat at 300 K is applied to the C atoms of the diamond surfaces; sliding velocity and loads are imposed on the topmost and the bottommost hydrogen layers, while standard NVE integration is used for the atoms at the interfacial regions.

1.2 Smart configuration sampling

In this appendix, we discuss in more details the active learning workflow used for the dataset construction. SCS is written in python 3 and bash scripting. It requires numpy and dpdata python packages as well as a bash release version $\ge$ 4.0.

The workflow operates within two main folders: Iterations and Reference. The first folder contains the data generated at each SCS iteration, and it is progressively updated during the active learning cycles. Specifically, it contains one folder for each iteration number which includes: training and exploration directory. The Reference folder contains reference input files for the training, ML-MD and ab initio input files, job-scripts and geometries for every system used in the active learning cycles. If one system has to be added or removed from the SCS iterations, it is sufficient to add or remove its corresponding directory inside the reference folder. One main advantage of keeping the different systems independent from each other is that the systems can be propagated in parallel within the active learning workflow because each system has its own scheduling and different computational resources can be independently allocated for different systems. Furthermore, the adding or removal of one system can be done between one iteration and another without altering the workflow functioning. The Reference folder contains also the initial and the updated dataset at each iteration so that one can easily retrieve it to perform tests while the active learning workflow keeps running.

The main program runs using minimal computational resources, either in background or in a compute node. It merely serves as automatic scheduler for the various computationally expensive tasks. In particular, the main program synchronizes the scheduling of training, ML-molecular dynamics and ab initio jobs. It keeps track of the status of the iteration by querying the output of the scheduled jobs, and it launches intermediary routines which are mainly involved in input file construction, sampling frame selection and dataset handling. Its output is redirected to a LOG file which can also be used to restart an interrupted run. The workflow can run in principle on any architecture, from HPC to local workstations: The user has to provide for the specific batch scheduler in use the job-scripts that perform the training, ML-molecular dynamics and ab initio single-point calculations.