Molecular Dynamics Simulations of Asphaltene Aggregation: Machine-Learning Identification of Representative Molecules, Molecular Polydispersity, and Inhibitor Performance

Abstract

Molecular dynamics simulations have been employed to investigate the effect of molecular polydispersity on the aggregation of asphaltene. To make the large combinatorial space of possible asphaltene blends accessible to a systematic study via simulation, an upfront unsupervised machine-learning approach (clustering) was employed to identify a reduced set of model molecules representative of the diversity of asphaltene. For these molecules, single asphaltene model simulations have shown a broad range of aggregation behaviors, driven by their structural features: size of the aromatic core, length of the aliphatic chains, and presence of heteroatoms. Then, the combination of these model molecules in a series of mixtures have highlighted the complex and diverse effects of molecular polydispersity on the aggregation process of asphaltene. Simulations yielded both antagonistic and synergistic effects mediated by the trigger or facilitator action of specific asphaltene model molecules. These findings illustrate the necessity of accounting for molecular polydispersity when studying the asphaltene aggregation process and have permitted establishing a robust protocol for the in silico evaluation of the performance of asphaltene inhibitors, as illustrated for the case of a nonylphenol resin.

1. Introduction

As a consequence of global governmental policies, the increasing popularity of electric vehicles, and the momentum for hydrogen as a clean source of energy, the consumption of gasoline and other fuels is set to steadily decline. On the other hand, the proportion of the average oil barrel dedicated to petrochemicals will grow by an estimated 20% by 2040.¹ Therefore, ensuring a sustainable production of fossil resources will continue to be an objective of paramount importance. A predominant challenge for the oil and gas industry is the deposition of asphaltene,² a class of compounds defined as the fraction of crude oil that is soluble in toluene but not in n-heptane.³ Asphaltenes, considered to be among the heaviest and most polar components of crude oils, are generally described as a very polydisperse class of organic solids made of a variety of polyaromatic structures with aliphatic chains or heteroatoms, either organic or metallic.² Their interaction with water, clay, and between themselves can result in critical issues in oil fields. Overall, they can precipitate in the reservoir and plug production and transportation flowlines, risking economic loss due to flow interruption and environmental damage.⁴ An efficient and economical mitigation strategy consists of injecting chemical additives, referred to as asphaltene inhibitors, to stabilize asphaltene in crude oil. Hence, the development of these additives is of high industrial relevance. They are generally surfactants or polymers,⁵⁻⁸ but the potential of more exotic chemistries, such as amphiphilic macromolecules⁹ or deep eutectic solvents¹⁰ (a class of products formed by the hydrogen bonding between cheap and safe components, e.g. an amine and a carboxylic acid, which represents an alternative to the expensive traditional ionic liquids), has also been evaluated. The development of efficient asphaltene inhibitors is often hindered because of the oil field dependence of asphaltene stability and aggregation behavior. Therefore, it seems critical to understand and rationalize the underlying mechanisms of the asphaltene aggregation process to better address its inhibition.

After decades of research, two main interpretations of the asphaltene aggregation process are usually presented. On the one hand, the Yen–Mullins model argues a hierarchical description in which asphaltenes are predicted to form dense nanoaggregates of less than 10 molecules, driven by interactions between aromatic centers, which then aggregate less strongly into larger clusters.^11,12 Such macroaggregates end up being too heavy and produce solid deposits on the pipeline wall. On the other hand, as the Yen–Mullins model fails to predict or reconcile a series of experimental observations, such as the complexity of the asphaltene molecular structure or the heterogeneous distribution of nanoaggregate sizes, Gray et al.¹³ proposed an alternate supramolecular model to better capture the high complexity of the aggregation process. In this paradigm, aromatic π–π stacking is not considered to be the dominant aggregation driving force but a contributing factor alongside other interactions relevant to petroleum such as acid–base interactions, hydrogen bonding, metal coordination complexes, and interactions between cycloalkyl and alkyl groups.¹⁴ In this model, strongly bound nanoaggregates can continue to grow beyond 10 asphaltenes. As encouraged in the research article from Gray et al.,¹³ this paradigm has been put to the test both experimentally and computationally over the past decade.

Historically, asphaltene research has been mostly experimentally led,¹⁵ but in the context of the global digital transition, there is undoubtedly momentum for in silico approaches, thanks to their contributions to the understanding and the rationalization of complex chemical and physical processes, as well as their relative affordability in comparison to laboratory experiments. Seminal works such as those by Headen et al.^16,17 and Seghdi et al.¹⁸ have established robust molecular dynamics (MD) simulation protocols capable of providing valuable insights into the first stage of the asphaltene aggregation process. Moreover, many works, in particular from the University of Illinois,¹⁹⁻²¹ have focused on studying the influence of a variety of factors, such as salinity and interactions with solvent on the onset of asphaltene precipitation. More recently, via a series of MD studies,²²⁻²⁷ Santos Silva et al. have worked on decoding the complex relationships between asphaltene molecular structures and their aggregation, studying the role of heteroatoms positioned either on the core or on the lateral chains,^22,23 the role of metalloporphyrins and demulsifier molecules,^24,27 and the effect of variations in the size of the aromatic core and lateral chain length.²⁵ Overall, they have shown that the formation of nanoaggregates depends on the size of the conjugated core and on the possible presence of an H-bond-forming polar group, whereas macroaggregation is determined by the length of the lateral chains and their possible terminal polar group.²⁶ Given that their observations lay outside the domain of the Yen–Mullins model, they consequently argued that this colloidal model, even though it is capable of describing the aggregation process of standard asphaltenes, might be a particular case of the more general supramolecular model, as proposed by Gray et al.,¹³ which is better suited to address the complete chemical diversity of asphaltenes. Furthermore, in a few studies, MD simulations have also been deployed to investigate the inhibitor action of chemical additives such as dodecylbenzenesulfonic acid,²⁸ limonene,¹⁷n- octylphenol,²⁹⁻³¹ and a series of polymers (two succinimide-based structures and one maleic anhydride)³² on asphaltene aggregation. Finally, Headen et al. have demonstrated that MD aggregation simulations for single asphaltene model systems qualitatively reproduce neutron total scattering data.³³

However, two types of limitations have been identified for MD simulations of asphaltene:^34,35 (i) the size and time scale limitations of MD simulations restrain this approach to the first stage of aggregation, and coarse-grained simulations would be necessary to simulate the second and later stages of assembly (e.g., flocculation), and (ii) polydispersity in the asphaltene molecular structures is of great importance for the prediction of aggregation structures and should be included in any simulation. Indeed, in order to thoroughly investigate the effect of functional groups on the aggregation process, many previous works—with the exception of that of Javanbakht et al.³⁵—had to be limited to series of asphaltene model molecules with similar aromatic cores^{18,22,23,25,36} to prevent the complexity of the systems from hindering the decoding of interaction mechanisms that underlie such a process. To provide a different perspective on the asphaltene aggregation process, one objective of this work is to account for the global asphaltene molecular polydispersity.

The molecular structure of asphaltene has been a longstanding subject of debate and an important focus during decades of investigation; thus over time the molecular weight has been estimated at values spanning 6 orders of magnitude.³⁷ The development of asphaltene model molecules, which is not the focus of this work, designed for performing MD simulations has been addressed and reviewed in a series of studies.^34,38−41 In recent years, two experimental techniques have provided insights of unprecedented quality into the molecular structures of asphaltenes. On the one hand, atomic force microscopy studies have permitted the visualization of more than 100 asphaltene motifs,⁴² confirming the presence of structures made of polynuclear aromatic hydrocarbons with alkyl side chains, usually referred to as the “island” or “continental” model. On the other hand, extrographic fractionation and ultrahigh-resolution mass spectrometry studies advocate that in petroleum island-type asphaltenes coexist with less generally accepted “archipelago” motifs, in which multiple aromatic cores are bridged together and include multiple functionalities.^43,44 Such works argue that, while the island motifs are readily accessible, asphaltene purification is required to detect and characterize archipelago asphaltenes.

In this work, we have performed a series of MD simulations to investigate the aggregation of different systems of asphaltene. After identifying, via unsupervised machine-learning (ML) approaches, a set of asphaltenes that are representative of the diversity of the catalogue of Law et al.,⁴¹ we have focused first on the aggregation of single asphaltene model systems, i.e., involving only one type of asphaltene model molecule. Then, the selected model molecules have been combined in systems mixing different types of asphaltene to study the effect of the molecular polydispersity of asphaltene on the aggregation process. Finally, the most aggregating mixture of asphaltene has been selected as a test case to study the action of a nonylphenol resin asphaltene inhibitor at a reasonably low concentration of 1 wt%. In what follows, the results are reported and discussed while the details of the computational methods employed are presented in the final section.

2. Results and Discussion

2.1. Identification of Representative Asphaltene Models

The asphaltene molecular models used in this work have been selected from a catalogue of 100 plausible molecular models designed for MD simulations of asphaltenes (89 models) and resins (11 models) that comprises both island and archipelago motifs.⁴¹ To generate these models, the quantitative molecular representation approach implemented by Boek et al.⁴⁵ had been applied to elemental analysis and ¹H–¹³C nuclear magnetic resonance spectroscopy experimental data.⁴¹ Furthermore, as shown by Law et al.,⁴¹ their catalogue of molecules reasonably covers the same chemical space as previous asphaltene models collected from an in-depth literature review. Simulating all 89 asphaltene molecules and multiple combinations of these is currently beyond the reach of affordable MD simulations. Therefore, identifying representative asphaltene models out of this catalogue is a way of accounting for molecular polydispersity at a reasonable cost. After digitalization of the catalogue of molecules,⁴¹ an unsupervised machine learning strategy relying on unsupervised clustering, detailed in section 4.1, has been implemented to identify the 6 model molecules displayed in Figure 1. These molecules are representative of the large diversity of asphaltene structures, with 4 island and 2 archipelago asphaltenes. In particular, the island models have very different structural features: A3 has long side chains, A54 has many heterocycles, and A29 is very bulky. The detailed composition of each cluster is given in the Supporting Information. Finally, it is important to keep in mind that only a limited number of asphaltene molecules have been observed up to now. Therefore, despite the approach developed here, the possibility of having a bias in the initial pool of asphaltene molecular structures cannot be entirely discarded. Interestingly, our approach would be readily and seamlessly extendable to account for future findings regarding asphaltene molecular structures and would permit identification of additional relevant and different molecules to study via MD simulations.

Figure 1.

2D representations of the representative asphaltene model molecules selected after the cluster analysis. The molecule name is consistent with the original work from Law et al.,⁴¹ and cluster labels and colors are consistent with Figure 9. No molecule from cluster #3 has been selected, as they are resins.

2.2. Single Asphaltene Model Systems

The aggregation of representative asphaltene model molecules from Figure 1 has been studied by a series of MD simulations in toluene and heptane. The detail of the setup is given in section 4.2. To investigate the tradeoff between convergence and simulation cost, all single asphaltene model simulations were performed with 40, 100, and 200 asphaltene molecules, maintaining a concentration of 7 wt%. The aggregation state of asphaltenic systems is monitored consistently with previous works: the series of observables defined by Headen et al.³⁴ have been implemented in a homemade Python script using the package MDAnalysis version 2.0⁴⁶ and are defined in section 4.3. The aggregation number, g_n, which corresponds to the number of molecules per aggregate, also referred to as clusters, permits monitoring the equilibrium state of the system while estimating the size of a cluster of asphaltenes. In Figure 2, the evolution of g_n during the 240 ns simulation of 100 molecules of model A29 in heptane is represented along with 5 and 20 ns moving averages to guide the eye.

Figure 2.

Aggregation number, i.e., average asphaltene cluster size, from the simulation of 100 molecules of asphaltene A29 in heptane. 5 and 20 ns moving averages are used to filter short-term fluctuations of the aggregation dynamics.

For all other single asphaltene model simulations performed in this study, the evolution of g_n is displayed in Figures S5–S9 in the Supporting Information, and Table 1 reports the final values of the 20 ns average window of the aggregation number, thus filtered from short time oscillations. Additionally, intermediate values for the same observable after 120 ns are provided to evidence the necessity of extending these simulations up to 240 ns. As always in molecular modeling, identifying the reasonable system size and time scale to simulate is crucial to balance the tradeoff between simulation cost and accuracy. In recent studies, (i) Headen et al.³⁴ have performed simulations of 27 asphaltene molecules during 80 ns (even tough they occasionally extended up to 160 and 500 ns), (ii) Ghamartale et al.³⁰ with the same model molecules simulated 50 molecules of asphaltene during 120 ns, and (iii) Villegas et al.,³⁶ who focused on the aggregation of a subfraction of asphaltene in toluene, showed by comparison with simulations of system sizes up to 160 asphaltene molecules that in their case simulating 20 asphaltene molecules for 120 ns already yielded converged results. For the model molecules studied in this work, it appears that simulating systems of 100 asphaltenes during 240 ns, a system size and simulation time that is well in the top tier of the current state of the art, is the best compromise between cost and accuracy. Indeed, as shown in Table 1 and in Figures S7 and S8 in the Supporting Information, with 100 model molecules finite size effects are only observed for A29 in heptane, which arrives at full segregation within 240 ns (aggregate size of 100 molecules in Figure 2), whereas the aggregate size does not reach 200 when 200 asphaltenes are in the system. However, even at the largest system size studied (200 asphaltenes), aggregation is clearly much stronger with A29 than with any other model asphaltene, which is already well captured with 100 asphaltene molecules. When using only 40 asphaltene molecules, finite size effects are also observed with molecules A3, A54, and A85 in heptane. Although very insightful, the simulations performed with 200 asphaltene molecules are currently still too expensive (for instance up to 621,082 atoms for A3 in heptane) to be performed on a regular basis and must be reserved for case-by-case studies. Additionally, when the aggregation behavior of a system is uncertain, as for example with 40 molecules of A85 in heptane (see Figure S6 in the Supporting Information), extending the simulation time further than 240 ns can provide more trustworthy insights. In what follows, we focus our analysis and comments on simulations of 240 ns performed with 100 asphaltene molecules. While these system sizes and simulation time scales are accessible in high-performance computer facilities, though costly over an entire study, some initiatives that have been pushing the system size and time scale limits are worth mentioning. On the side of the simulation time scale, Glova et al.⁴⁷ performed simulations of 50 asphaltene molecules during 5 μs to identify the best partial charge method, namely AM1-BCC, to use in combination with the general AMBER force field, whereas from the system size aspect, Javanbakht et al.³⁵ studied the aggregation of asphaltene mixtures of size up to 1005 model molecules during 200 ns and concluded that, under their simulation conditions, 375 asphaltene molecules were enough to capture all possible nanoaggregate shapes.

Table 1. Values of the 20 ns Window Moving Average of the Aggregation Number of All Single Asphaltene Model Simulations Performed in This Study after Half of the MD production, g_n¹²⁰ at 120 ns, and at the End of the Simulations, g_n at 240 ns^a.

		40 asph		100 asph		200 asph
asph	solvent	gn120	gn240	gn120	gn240	gn120	gn240
A3	toluene	10.2 ± 0.9	7.6 ± 0.5	7.6 ± 0.2	8.3 ± 0.2	7.1 ± 0.1	6.7 ± 0.3
A3	heptane	20.6 ± 0.0	40.0 ± 0.0	12.6 ± 0.6	28.1 ± 1.7	12.0 ± 0.1	28.2 ± 1.5
A29	toluene	40.0 ± 0.3	29.2 ± 3.7	16.9 ± 0.7	60.2 ± 2.8	14.3 ± 0.2	40.8 ± 0.7
A29	heptane	22.1 ± 4.6	40.0 ± 0.0	45.6 ± 4.9	100.0 ± 0.0	37.9 ± 4.5	66.9 ± 0.3
A54	toluene	4.6 ± 0.1	4.0 ± 0.1	4.6 ± 0.1	5.3 ± 0.1
A54	heptane	15.2 ± 0.9	19.8 ± 1.3	16.5 ± 0.6	23.8 ± 1.4
A63	toluene	3.9 ± 0.1	3.7 ± 0.1	3.8 ± 0.1	3.9 ± 0.1	3.8 ± 0.0	3.5 ± 0.0
A63	heptane	5.4 ± 0.2	4.9 ± 0.2	6.4 ± 0.2	6.8 ± 02	5.4 ± 0.1	6.4 ± 0.1
A80	toluene	3.3 ± 0.2	3.1 ± 0.1	3.1 ± 0.1	3.4 ± 0.1
A80	heptane	6.1 ± 0.2	8.7 ± 0.8	10.2 ± 0.4	14.0 ± 0.4
A85	toluene	3.4 ± 0.1	3.7 ± 0.2	3.4 ± 0.2	3.5 ± 0.1
A85	heptane	7.8 ± 1.0	23.3 ± 4.2	10.0 ± 0.2	13.7 ± 1.0

^aThe uncertainties are the standard deviations of the moving average.

Further than the average number of asphaltene molecule per cluster, the size of these clusters can also be characterized via their radius of gyration, R_g, while an estimate of their density and their relative shape anisotropy κ², which takes values between 0 for a spherical cluster and 1 for a linear chain, provide information relative to their shape. Details of the implementation of these metrics are presented in section 4.3. Besides, as these observables have been defined to charaterize the equilibrated state of the asphaltenic systems that were simulated, we have focused on the last 40 ns of each simulation in order to compare what we consider equilibrated asphaltenic systems.

Along with the aggregation number, the different observables displayed in Figure 3, accumulated during the last 40 ns of a simulation with 100 asphaltenes A29, permit an analysis of the aggregation of the different single asphaltene model systems. Figures S10–S18 in the Supporting Information are available for all other simulations, and results (average values of radius of gyration, estimated density, and shape anisotropy) with 100 asphaltene model molecules are summarized in Table 2 (see Tables S3 and S4 in the Supporting Information for equivalent summaries for simulations with 40 and 200 asphaltene molecules, respectively). As could be expected, in toluene the dispersion of the model molecules studied in this work is generally very stable, with the exception of molecule A29, the aggregation numbers are very low and remain low along the entire trajectories. Overall, for simulations with 100 asphaltenes, aggregation numbers larger than 10 are only obtained for molecule A29 (see Table 1 and Figure S7). A29 is also very clearly the most strongly aggregating molecule in heptane. In heptane, Figure 2 shows different stages of aggregation of A29, with three plateaus around g_n = 33, g_n = 50, and g_n = 100, the last corresponding to a complete segregation of the asphaltenes from the solvent. As observed in other simulations,^18,26 the aggregation process of this system can be described hierarchically. Initially, in the stage that Sedghi et al.¹⁸ named nanoaggregation, the size of the aggregates smoothly increases up to g_n values of around 15–17 at 40 ns, which is larger than the definition of the Yen–Mullins model.^11,12 Then, between 40 and 120 ns, g_n starts exhibiting a stepwise increase characteristic of the beginning of a clustering stage, in which aggregation occurs between asphaltene clusters. As the simulation time increases, g_n steps increase, illustrating the merger into asphaltene clusters of increasing size until the event at 180 ns that results in the combination of the last two remaining clusters. In heptane, the asphaltene model molecules can be classified in two groups: (i) the stable A63, A80, and A85 that do not aggregate and (ii) the unstable A3, A29, and A54 that aggregate. The relative aggregation ranking of these systems is consistent with previous studies decoding the relationship between the structure of the asphaltenes and aggregation.^18,26,34 Indeed, as nanoaggregation has been shown to primarily depend on the size of the aromatic core of asphaltenes, it is to be expected that A29 reaches the highest level of aggregation in this study. Additionally, A29 contains polar aliphatic chains with sulfur heteroatoms that, in spite of their length, contribute favorably to macroaggregation. A29 is followed by A3, which also possesses long polar aliphatic chains, and A54, which is made up of many heterocycles and short apolar aliphatic chains. The relationship between the solubility of asphaltene molecules and the extension of their aromatic cores was already identified in early studies.⁴⁸ The extension of the aromatic core can be characterized by the aromatic condensation index, C_I/C₁, which is the ratio between the number of internal aromatic carbons (C_I) and the number of peripheral aromatic carbons (C₁) of the model molecules. Asphaltenes from deposits and unstable crude had shown extended aromatic cores and hence large values of C_I/C₁. In the single asphaltene model simulations performed in this work, we observe that the aggregation behavior of the island asphaltene molecules is indeed proportional to their aromatic condensation index.

Figure 3.

Properties of the asphaltene clusters from simulations of 100 molecules of A29 in toluene and heptane: (top) distributions of the cluster radius of gyration; (middle) distributions of the estimated cluster density; (bottom) distributions of the cluster relative shape anisotropy.

Table 2. Summary of the Monodisperse Simulations Performed with 100 Asphaltene Molecules^a.

asph	CI/C1	solvent	gn240	av Rg	av density	av κ2
A3	0.345	toluene	8.3 ± 0.2	14.5 ± 1.3	0.4 ± 0.0	0.2 ± 0.0
A3		heptane	28.1 ± 1.7	19.6 ± 2.3	0.4 ± 0.0	0.1 ± 0.0
A29	0.545	toluene	60.2 ± 2.8	30.0 ± 4.0	0.3 ± 0.0	0.2 ± 0.1
A29		heptane	100.0 ± 0.0	35.0 ± 0.3	0.4 ± 0.0	0.2 ± 0.0
A54	0.200	toluene	5.3 ± 0.1	10.3 ± 0.6	0.5 ± 0.0	0.2 ± 0.0
A54		heptane	23.8 ± 1.4	18.2 ± 1.6	0.4 ± 0.0	0.3 ± 0.0
A63	0.108	toluene	3.9 ± 0.1	11.6 ± 0.8	0.3 ± 0.0	0.3 ± 0.0
A63		heptane	6.8 ± 0.2	12.6 ± 0.8	0.4 ± 0.0	0.2 ± 0.0
A80	0.000	toluene	3.4 ± 0.1	15.6 ± 1.1	0.2 ± 0.0	0.3 ± 0.0
A80		heptane	14.0 ± 0.4	17.4 ± 1.6	0.3 ± 0.0	0.2 ± 0.0
A85	0.000	toluene	3.5 ± 0.1	13.1 ± 0.8	0.3 ± 0.0	0.3 ± 0.0
A85		heptane	13.7 ± 1.0	15.8 ± 1.6	0.3 ± 0.0	0.2 ± 0.0

^aCI/C₁ is the aromatic condensation index, i.e., the ratio between the number of internal aromatic carbons (C_I) and the number of peripheral aromatic carbons (C₁) of the model molecules. For the aggregation number, g_n²⁴⁰, we report the value of the 20 ns window moving average at the end of the simulation. For the radius of gyration, R_g (in Å), the estimated density (in g/mol/ų) and the relative shape anisotrpy, κ², we report the average value over the last 40 ns of the simulations. For g_n, the uncertainties are the standard deviations of the moving average, while for the other observables they are the standard deviation of the data acquired during the last 40 ns.

Over the 240 ns simulations, asphaltene aggregates of g_n ≥ 20 are formed in these the unstable systems A3, A29 and A54 in heptane. To illustrate their equilibrium states, snapshots of the simulations after 220 ns are displayed in Figure 4 and some full-page enlargements are provided in Figures S21–S23 in the Supporting Information. Even tough A3 long polar aliphatic chains do not seem to limit aggregation in comparison with A54, they govern the packing of the aggregates. Indeed, the aromatic cores of A3 and A54 are of similar size, but A3 exhibits ordered parallel stacks of 5–10 molecules, sometimes referred as “pancakes stacking”, whereas A54 aggregates in a more disordered manner with numerous T-shaped interactions between parallel stacks of smaller size, typically 2–5 molecules. It seems that even though T-shaped π–π interactions occur for A3, they are not as frequent as with A54. This can be attributed to a combination of factors: the heterocycles of A54 strengthen both parallel and T-shaped π–π interactions between aromatic cores, while the long aliphatic chains of A3 hinder T-shape π – π interactions. In the case of A29, in spite of the limitations of the static 2-dimensional view, we clearly see a single aggregate of 100 molecules, whereas many clusters are present in the two other systems. Consistently with its larger aggregation number, A29 in heptane also yields aggregates with a much larger radius of gyration: average R_g = 35.0 ± 0.3 Å in comparison with R_g = 19.6 ± 2.3 Å, R_g = 18.2 ± 1.6 Å, and R_g = 12.6 ± 0.8 Å, respectively, for A3, A54, and A63 in heptane (Table 2). Nevertheless, despite these differences, Figure S14 shows similar distributions for the estimated density of the aggregates produced by these island systems. The more peaked R_g and density distributions of A29 in comparison with A3, A54, and A63 confirm that A29 has reached a completely equilibrated segregation from the heptane solvent. With respect to the relative shape of the aggregates, as evidenced by the average values in Table 2, the clusters are more elongated than spherical (values close to 0). Among the three asphaltenes stable in heptane, A80 and A85 are of archipelago type, a type of molecule known for its stability, and A63, with its aromatic core of moderate size and absence of heteroatoms, do not present any structural feature favoring aggregation. As mentioned previously, the finite size effects of the simulations quantitatively affect the values of the observables for A29 in heptane. All details for simulations with 200 asphaltenes in heptane are provided in Figures S9, S12, S15, and S18 and Table S4 in the Supporting Information, and most of the qualitative conclusions on the differences between A29 and other model molecules still apply. However, it is worth mentioning that the differences in R_g and in density between A3 and A29 do not seem as important with simulations performed with 200 asphaltene molecules (Figures S12 and S15 and Table S4), even though the difference in final g_n is already striking with g_n = 66.8 ± 4.5 for A29 and g_n = 28.2 ± 0.1 for A3. This shows that with larger system sizes, more representative of the reality, the simulation time required for the system to settle into the clustering stage of the aggregation process and adopt its characteristics, beyond the stepwise increases visible with 200 asphaltenes A29 in heptane (Figure S9), can be longer than 240 ns. To obtain confirmation, we have extended up to 500 ns the simulations in heptane with 200 molecules of A3 and with 200 molecules of A29, which yields aggregation numbers g_n = 28.9 ± 0.6 and g_n = 200.0 ± 0.0, respectively. The observables for these simulations, plotted in Figures S19 and S20, confirm an important quantitative difference between these two asphaltene model molecules already observed in simulations with 100 asphaltene molecules in spite of the finite size effects. Consequently, the multistage aggregation processes observed in early simulation works may be artifacts from simulation finite size effects but could still be confirmed by longer or larger simulations. However, with the computational power currently available, gaining insights into the late stages of the aggregation process would probably require resorting to coarse-grained simulations.⁴⁹ Overall, the representative asphaltene model molecules identified by unsupervised ML show a diversity of aggregation behaviors that confirms their suitability for describing asphaltene molecular polydispersity once mixed together.

Figure 4.

Snapshots from single asphaltene model simulations of 100 model molecules in heptane for A3 (left), A29 (center), and A54 (right) after 220 ns. Carbon atoms are represented in gray, nitrogen atoms in blue, sulfur atoms in yellow, and hydrogen atoms in white. Full-page images for each system are available in Figures S21–S23 in the Supporting Information.

2.3. Mixtures of Asphaltene: Effect of Molecular Polydispersity

To study the effect of asphaltene molecular polydispersity on the aggregation process, a series of mixtures, made of 3–4 types of model molecules for a total concentration of 7 wt % of asphaltene, have been designed based on the results from single asphaltene model simulations (details in Table 3). While simulations of the first four quaternary mixtures (Q1–Q4) contain 25 molecules of each model, in ternary mixtures (T1–T4) 34 molecules of the first listed model and 33 of the other two are combined to reach 100 asphaltene molecules per simulation. Q5 and Q6 have been designed slightly differently from the other quaternary mixtures and contain respectively 23 molecules of A23, 33 molecules of A29, and 22 molecules of A54 and A85 for Q5 and 33 molecules of A29, 23 molecules of A63, and 22 molecules of A80 and A85 for Q6. The reasoning behind the specific selection of model molecules in the ternary mixtures is as follows: T1 and T4 were designed to mix together the asphaltene model molecules that are stable (A63, A80, and A85 in T1) and unstable (A3, A29, and A54 in T4) in heptane. Then, T2 mixes the two most unstable molecules (A3 and A29) with the less stable of the stable molecules, namely A85. Finally, T3 mixes two aggregating asphaltenes (A3 and A54) with A85 but does not include the very strongly aggregating molecule A29. Table 3 summarizes the results of the simulations, and all plots, in analogy to those of the single asphaltene model simulations, can be found in Figures S24–S33 in the Supporting Information. Furthermore, to enable a deeper understanding of the dynamics of the aggregation process and in particular the formation of the aggregates in heptane, additional observables are displayed in the Supporting Information. Indeed, Figures S34 and S35 show respectively for ternary and quaternary mixtures the percentage of each type of asphaltene molecule participating in the formation of aggregates along the trajectory. Furthermore, in Figures S36–S45, the composition of the aggregates for each system is represented in the form of bar charts at different points of the trajectory (10, 30, 50, 100, 150, and 220 ns) to compare their evolution.

Table 3. Summary of the Mixture Simulations Performed with 100 Asphaltene Molecules^a.

mixture	asph models	solvent	gn240	av Rg	av density	av κ2
T1	A63 A80 A85	toluene	3.3 ± 0.0	13.2 ± 0.8	0.3 ± 0.0	0.3 ± 0.0
T1	A63 A80 A85	heptane	8.8 ± 0.1	14.8 ± 1.3	0.4 ± 0.0	0.2 ± 0.0
T2	A3 A29 A85	toluene	9.5 ± 0.2	15.0 ± 1.5	0.4 ± 0.0	0.3 ± 0.0
T2	A3 A29 A85	heptane	52.6 ± 1.3	27.2 ± 3.7	0.3 ± 0.0	0.3 ± 0.1
T3	A3 A54 A85	toluene	6.2 ± 0.1	12.8 ± 1.0	0.4 ± 0.0	0.2 ± 0.0
T3	A3 A54 A85	heptane	20.0 ± 2.9	15.9 ± 2.8	0.4 ± 0.0	0.2 ± 0.0
T4	A3 A29 A54	toluene	8.3 ± 1.6	15.2 ± 2.7	0.4 ± 0.0	0.2 ± 0.1
T4	A3 A29 A54	heptane	18.8 ± 0.6	17.0 ± 1.1	0.4 ± 0.0	0.1 ± 0.0
Q1	A3 A29 A54 A85	toluene	7.0 ± 0.9	13.6 ± 1.3	0.4 ± 0.0	0.2 ± 0.0
Q1	A3 A29 A54 A85	heptane	18.2 ± 1.1	15.5 ± 1.4	0.4 ± 0.0	0.2 ± 0.0
Q2	A54 A63 A80 A85	toluene	3.9 ± 0.1	12.6 ± 0.8	0.3 ± 0.0	0.3 ± 0.0
Q2	A54 A63 A80 A85	heptane	12.1 ± 0.9	16.3 ± 1.9	0.4 ± 0.0	0.2 ± 0.1
Q3	A29 A63 A80 A85	toluene	4.5 ± 0.1	13.2 ± 1.0	0.3 ± 0.0	0.3 ± 0.0
Q3	A29 A63 A80 A85	heptane	18.2 ± 1.8	18.0 ± 3.3	0.4 ± 0.0	0.3 ± 0.1
Q4	A3 A63 A80 A85	toluene	4.1 ± 0.1	13.1 ± 0.8	0.3 ± 0.0	0.2 ± 0.0
Q4	A3 A63 A80 A85	heptane	10.4 ± 0.3	15.4 ± 1.2	0.4 ± 0.0	0.2 ± 0.0
Q5	A3 A29 A54 A85	toluene	9.0 ± 0.7	13.7 ± 1.3	0.4 ± 0.0	0.2 ± 0.0
Q5	A3 A29 A54 A85	heptane	35.3 ± 1.9	20.3 ± 2.8	0.5 ± 0.0	0.2 ± 0.0
Q6	A29 A63 A80 A85	toluene	4.6 ± 0.0	12.7 ± 0.8	0.4 ± 0.0	0.3 ± 0.0
Q6	A29 A63 A80 A85	heptane	20.0 ± 0.2	16.8 ± 1.1	0.4 ± 0.0	0.2 ± 0.0

^aIn ternary mixtures there are 34 molecules of the first listed model and 33 of the other two. Quaternary mixtures Q1–Q4 contain 25 molecules of each model. Q5 is a mixture of 23 molecules of A23, 33 molecules of A29, and 22 molecules of A54 and A85. Q6 is a mixture of 33 molecules of A29, 23 molecules of A63, and 22 molecules of A80 and A85. For the aggregation number, g_n²⁴⁰, we report the value of the 20 ns window moving average at the end of the simulation. For the radius of gyration, R_g (in Å), the estimated density (in g/mol/ų) and the relative shape anisotropy, κ², we report the average value over the last 40 ns of the simulations.For g_n, the uncertainties are the standard deviations of the moving average, while for the other observables they are the standard deviation of the data acquired during the last 40 ns.

As could be anticipated from the single asphaltene model simulations, aggregation in toluene is always quite low in mixture simulations (g_n always ≤11.0 in Table 3). The aggregation observed for T1 in heptane is quite low with g_n = 8.8 ± 0.1. Nevertheless, Figures S34 and S35 show that for all systems, already at the beginning of the production phase, more than 70% of the asphaltene molecules participate in aggregates. Moreover, from Figures S36–S45 it becomes evident that already after 10 ns all systems exhibit a least one large aggregate of 10 or more asphaltene molecules. Toward the end of the simulations, at 220 ns, even T1 (Figure S36) presents a very large aggregate of 44 asphaltene molecules, and 10 much smaller aggregates in this case. The final aggregation level of T1 is lower than the average aggregation from single asphaltene model simulations (g_n^av T1 = (0.34 × 6.8) + (0.33 × 14.0) + (0.33 × 13.7) = 11.5). Therefore, already in that case, we observe that molecular polydispersity can have an antagonistic effect on the aggregation process, i.e., the simulation yields an aggregation level inferior to the weighted average of the contributions from single asphaltene model simulations. This antagonistic effect is even more pronounced in T4, for which aggregation only reaches g_n = 18.8 ± 0.6, whereas individually, all single asphaltene model simulations of the constituents of this mixture exceed g_n = 23.8 (see Table 2). In T3, aggregation reaches g_n = 20.0 ± 2.9, which is in the same range as the weighted average of contributions from single asphaltene model simulations (g_n = (0.34 × 28.1) + (0.33 × 23.8) + (0.33 × 13.7) = 21.9) and in T2, aggregation reaches g_n = 52.6 ± 1.3, which even exceeds the respective average (g_n^av T2 = (0.34 × 28.1) + (0.33 × 100) + (0.33 × 13.7) = 47.1). Therefore, in the case of T2, asphaltene molecular polydispersity has a synergistic effect on the aggregation process, i.e., the simulation yields an aggregation level superior to the weighted average of the contributions from single asphaltene model simulations. Besides, it is worth mentioning that in Figures S34 and S35 the percentages of each model molecule involved in the formation of aggregates within mixture simulations confirm the aggregation strength of the asphaltene models already observed in single-component simulations. Indeed, A29, A3, and A54 appear as the most involved molecules in aggregates with levels generally close to 100%. Conversely, the three other models, A63, A80, and A85, only very rarely reach 100% participation in the aggregates.

Figure 5 displays the standard aggregation observables for T2, which exhibits the largest aggregation among the mixtures studied in this work. It is the only mixture to reach the clustering stage of the aggregation, as confirmed by the stepwise increases of its aggregation number (see Figure S25), early in the trajectory (see also aggregate composition in Figure S37). System Q5, discussed in more detail later, is also close to reaching the clustering stage toward the end of the simulation. Both systems have in common to show, with the exception of T4 (which is addressed further below), the largest total contribution of asphaltenes into aggregates (see black dashed lines in Figures S34 and S35), i.e., the lowest number of free asphaltene monomers. The tendency of T2 and Q5 systems to strongly aggregate can be attributed to the role of molecule A29, the most strongly aggregating molecule in this study, which act as a trigger to the aggregation process. Further than T2 and Q5, Figures S34 and S35 illustrate this general triggering role of A29 also for T4, Q1, Q3, and Q6, in which A29 is clearly leading the aggregation process at the beginning of the trajectories and is the first type of asphaltene to reach 100% participation in aggregates. Overall, the effect of molecular polydispersity is complex and the aggregation between molecules of different types is driven by a sum of correlated contributions from aromatic cores, aliphatic chains, or heteroatoms. Indeed, comparing T4 to T2 and Q5, one would be tempted to postulate a lower compatibility in the aggregation of molecules A29 with molecules A54 than in the aggregation between A29 and A85 that results in being more favorable to the aggregation process. A similar interpretation could be proposed for rationalizing the compatibility of A54 and A85 with molecule A3 comparing T4 to T3, which does not aggregate much more but is lacking the strong aggregation triggering action of molecule A29. However, for T4 in Figure S34, after 80 ns all asphaltene molecules are involved in aggregates, and in Figure S39, already at a very early stage, there are a number of medium-sized aggregates that are quite balanced and involve all 3 types of asphaltene models. Therefore, there is no incompatibility between the asphaltene model molecules present in T4, but these aggregates have difficulties merging into aggregates of larger size, despite the intrinsic good aggregation strength of all the asphaltene models involved. The difference between T4 and systems T2, T3, and Q5 is the lack of archipelago asphaltene molecules A85. Thus, these observations evidence that the archipelago nature of A85, even though usually associated with stable asphaltenic systems, seems to facilitate the aggregation—when combined with bulky asphaltenes such as A3 and A29 in T2 or Q5—more than the island nature of A54 with a very short aliphatic chain, as in T4, in particular for the transition from medium to large aggregates. It is interesting to point out that molecules A85 fulfill this facilitator role without reaching 100% of participation in aggregates (see Figures S34 and S35).

Figure 5.

Aggregation number g_n (top left) from simulations of mixture T2 in heptane. 5 and 20 ns moving averages are used to filter short-term fluctuations of the aggregation dynamics. The distributions of the properties of the asphaltene clusters from simulations of T2 in both heptane and toluene are also represented: namely, the radius of gyration (top right), the estimated density (bottom left), and the relative shape anisotropy (bottom right), accumulated over the last 40 ns of the simulations.

To further investigate the triggering action of molecule A29 and the synergistic action of A29 and A85 with respect to aggregation, mixtures Q5 and Q6 have been designed slightly differently from the other quaternary mixtures, namely with the same species as in Q1 and Q3, but in different concentrations. Conversely to T2, no synergistic effect is observed at the specific composition in Q5 or Q6 (aggregation does not exceed the weighted average contributions from the single asphaltene model simulations) even though molecule A29 is seen to trigger the general aggregation on comparing T1 with Q3 and Q6 and T3 with Q1 and Q5. However, this triggering effect is not linearly correlated with the composition of the systems. For example in Q3, there is the same number of each model molecule (25 molecules) A63, A80, A85, and A29, and an important increase in aggregation (g_n = 18.2 ± 1.1) is observed in comparison with T1 (g_n = 8.8 ± 0.1). However, in Q6 there are more molecules of the very strongly aggregating A29 (33 molecules) in comparison with Q3, but yielding a proportionally smaller increase in aggregation (g_n = 20.0 ± 0.2). Thus, it seems that in this case the trigger effect of molecule A29 reaches a saturation point. This is evidenced in Figures S34 and S35 by similar behaviors of the participation in aggregates in Q3, Q6, and even T1 (for the model molecules present in the last). Therefore, we conclude that, due to the weak aggregation strength of models A63, A80, and A85, the trigger action of adding molecule A29 in Q3 compared to T1 is saturating in Q6 and its number of molecules is not sufficient to reach higher aggregation levels. On the other hand, in Q1, which contains the same number of each model molecule (25 molecules) A3, A29, A54, and A85, the presence of 25 of the strongly aggregating A29 molecules does not lead to a larger aggregation (g_n = 18.2 ± 1.8) than in T3 (g_n = 20.0 ± 2.9), which only contains molecules A3, A54, and A85. Meanwhile, in Q5, where the number of molecules of A29 is set to 33, a clear increase in aggregation is observed (g_n = 35.3 ± 1.9), as if there were a concentration threshold to overcome within this mixture for the triggering effect of molecules A29 to be strong enough. This can be seen in Figure S35, in which molecule A29 reaches very quickly (already at 30 ns) and definitively 100% of involvement in aggregates, whereas 160 ns is needed in Q1. Then, molecule A54 participates slightly earlier in aggregates (70–100 ns) and a larger number of molecules A85 are involved in aggregates compared to Q1. Besides, comparing Q2 and Q4 with Q3 shows that A29 is clearly a stronger aggregation trigger than A3 and A54, but no synergistic effect is observed among the quaternary mixtures investigated in this work.

Finally, a word of caution must be mentioned here: even though the conclusions just presented above seem reasonable, it is important to keep in mind that some of them are only based on small differences obtained from single-run simulations. Ideally, one would like to perform many simulations per system in order to draw stronger conclusions. However, the computational cost of such MD simulations makes a systematic n-repetition process prohibitive. Overall, the investigation of the asphaltene aggregation process in these mixtures has shown a variety of complex and correlated effects: antagonistic and synergistic effects within mixtures, triggering effects of specific model molecules, A29 in particular, and the facilitator role of archipelago molecule A85 have been detected. These findings highlight once again the necessity of accounting for asphaltene molecular polydispersity even though the level of asphaltene aggregation in mixtures never reaches the largest aggregation level of the single asphaltene model simulation of A29. Moreover, as the ultimate goal is to contribute to the design of asphaltene inhibitors, it is important to ensure that the in silico evaluation of their performance is not biased by a specific asphaltene model molecule with which the inhibitor could interact a great deal and limit its aggregation, whereas it could interact more moderately with other molecules, depending on their chemistry. Accounting for molecular polydispersity in this characterization permits limiting this risk. Besides, in order to be able to capture the action of an inhibitor on a mixture of asphaltenes, it is necessary to use a mixture that reaches a significant level of aggregation. Therefore, T2 is the most suitable mixture of asphaltenes for this task and has been employed in the simulations presented in the next section.

2.4. Showcase of Inhibition Simulation

Beyond the investigation of asphaltene aggregation and the effect of molecular polydispersity on this process, another objective of this work is to set up a robust protocol for the in silico characterization of the action of asphaltene inhibitors. To illustrate that this objective has been reached thanks to the workflow implemented in this study, a simulation of the aggregation in heptane of the mixture T2 in the presence of a nonylphenol resin asphaltene inhibitor, at a concentration of 1 wt%, has been performed. Due to intellectual property restrictions, the exact form of the inhibitor cannot be published, but we show in Figure 6 its general structure, which is sufficient for the purpose of this showcase.

Figure 6.

2D representation of the molecular structure of the inhibitor.

The aggregation behavior of the mixture T2 in the presence of the inhibitor (in black) is compared in Figure 7 to the case without (in green) already discussed in the previous section. To facilitate the comparison and avoid short-term fluctuations, the 20 ns moving averages are presented for the aggregation number. Moreover, while in the aggregation number only the asphaltene molecules are considered, the inhibitor molecules are included in the calculation of the radius of gyration of the cluster they belong to, in order to avoid drawing erroneous conclusions from artificially low aggregation numbers resulting from the limitations of this metric. An example of this type of artifact would be the following case: if two clusters of asphaltene were connected by one or two inhibitors in between while not interacting directly with each other, the aggregation number would be low, whereas there would actually be a very large cluster of asphaltene and inhibitors. On the other hand, the inclusion of the inhibitors in the calculation of the radius of gyration would yield large values, thus revealing the limitation of the inhibitor performance. For future investigations focusing on the inhibition of the aggregation process, the definition of additional observables, e.g., a modified version of the aggregation number, should facilitate the identification of the situations in which inhibitors are failing to prevent aggregation and end up embedded in the aggregates. Here, the nonylphenol resin inhibitor limits the aggregation of the T2 mixture to g_n = 23.0 ± 1.9, which is less than half the aggregation number of the case without inhibitor, as summarized in Table 4. The average radius of gyration is also reduced, which confirms the very good performance of this inhibitor. Considering that the concentration of inhibitor is only 1 wt%, we can conclude that the nonylphenol resin inhibitor is qualitatively (a different definition of the aggregation number was used) a better-performing asphaltene inhibitor than n-octylphenol,³⁰ which needed 7 wt% to limit the aggregation of less-aggregating asphaltenic systems. Nevertheless, it is worth noting that such a concentration, namely 1 wt%, is still 2 orders of magnitude larger than usual operating conditions. However, further decreasing the concentration of the inhibitor would require performing the simulations on much larger systems, yielding a computational cost far beyond the reach of any systematic study.

Figure 7.

Aggregation number g_n (top left) from simulations of mixture T2 in heptane with (in black) and without (in green) inhibitor molecules. 20 ns moving averages are used to filter short-term fluctuations of the aggregation dynamics. The distributions of the properties of the asphaltene clusters from these simulations are also represented (using the same color code): namely the radius of gyration (top right), the estimated density (bottom left,) and the relative shape anisotropy (bottom right), accumulated over the last 40 ns of the simulations.

Table 4. Summary of the Simulations of T2 with and without Inhibitor^a.

mixture	asph model	inhibitor	solvent	gn240	av Rg	av density	av κ2
T2	A3 A29 A85	no	heptane	52.6 ± 1.3	27.2 ± 3.7	0.3 ± 0.0	0.3 ± 0.1
T2	A3 A29 A85	1 wt %	heptane	23.0 ± 1.9	18.5 ± 2.4	0.4 ± 0.0	0.2 ± 0.1

^aFor the aggregation number, g_n²⁴⁰, we report the 20 ns average window at the end of the simulation. For the radius of gyration, R_g (in Å), the estimated density (in g/mol/ų) and the relative shape anisotropy, κ², we report the average value over the last 40 ns of the simulations.

To gain further insights into the action of the asphaltene inhibitor, Figure 8 compares the percentage of asphaltene molecules participating in aggregates (top) and the composition of the asphaltene aggregates at 220 ns (bottom) between system T2 (left) and T2 in the presence of the inhibitor (right). The composition of the asphaltene aggregates for T2 + inhibitor along the trajectory can be found in Figure S46. Interestingly, when the inhibitor is present in the system (5 molecules in total), a larger percentage of asphaltene participates in aggregates, namely 100% already at 40 ns, but overall, at 220 ns there is a much larger number of aggregates (5 aggregates with inhibtor in the system instead of 2 aggregates without inhibitor), and the largest aggregate is much smaller (66 molecules with inhibitor in the system versus 87 molecules without inhibitor). Figure S46 shows that the composition of the aggregates in the presence of inhibitor is very similar from 100 to 220 ns. Therefore, the interactions between molecules of inhibitor and asphaltene aggregates of medium size successfully permit preventing their merger into aggregates of larger size.

Figure 8.

Percentage of asphaltene molecules participating in aggregates (top) and composition of the asphaltene aggregates at 220 ns (bottom) between system T2 (left) and T2 in the presence of the inhibitor (right).

3. Conclusion

In this study, the combination of unsupervised machine learning and molecular dynamics simulation has permitted a thorough investigation of the role of asphaltene molecular polydispersity on the aggregation process. Indeed, we first performed an upfront selection, via unsupervised machine learning, of a series of asphaltene model molecules representative of a broad and diverse catalogue specifically designed for the purpose of molecular dynamics simulations. Then, we studied the aggregation of these molecules via single asphaltene model simulations in toluene and heptane solvents. With the exception of the most strongly aggregating molecule, namely A29, aggregation in toluene has been weak. In heptane, single asphaltene model simulations—in agreement with recent simulation works—have shown that, even though π–π interactions can be a very strong driver for asphaltene aggregation, other molecular features such as polar aliphatic chains and heteroatoms contribute significantly to this process. Furthermore, the different aggregation behaviors of the representative asphaltene model molecules in heptane have confirmed their ability to capture the diversity of asphaltene. Afterward, the effect of asphaltene molecular polydispersity on the aggregation process has been investigated by simulations of ternary and quaternary mixtures of these molecules. Overall, the effect of asphaltene molecular polydispersity is complex, difficult to disentangle, and diverse: depending on the composition of the mixture antagonistic, synergistic, and triggering effects have been observed. These findings illustrate again the necessity to account for molecular polydispersity when studying the asphaltene aggregation process and its inhibition. Finally, this work has also permitted deploying a robust simulation protocol for the in silico evaluation of the performance of asphaltene inhibitors, as demonstrated by the case study presented with the nonylphenol resin inhibitor. In future works we intend to build on the developments presented here to investigate and compare the behavior of a series of asphaltene inhibitors.

4. Computational Methods

4.1. Unsupervised Machine Learning

The first step in the selection of a series of representative asphaltene model molecules from the catalogue of Law et al.⁴¹ was to generate digital molecular structures of the model molecules released in the form of 2D and 3D molecular representations. To accelerate the digitalization, we have used an open-source optical chemical structure recognition (OCSR) Java-based tool called MolVec.⁵⁰ Despite such a tool, this is still a cumbersome process, but it is worth pointing out that OCSR tools have been reviewed recently⁵¹ and that many developments are currently in process in this field. Additional comments on the information of the original catalogue of 100 molecules are presented in the Supporting Information. SMILES codes (simplified molecular input line entry system)⁵² of the model molecules have been obtained via MolVec, and their 3D molecular structures have been generated using RDKit.⁵³ The 3D molecular structures have been used as inputs for the calculation of 3D molecular descriptors by the free software Mordred⁵⁴ and later on to perform MD simulations when deemed relevant.

At this point, each of the 100 molecules from the catalogue of Law et al.⁴¹ is described by 1826 3D molecular descriptors. Then, this highly multidimensional representation of the chemical space of asphaltene is reduced to three dimensions via a principal component analysis (PCA),⁵⁵ a linear dimensionality reduction method, using the implementation of the Scikit-learn library.^56,57 In order to identify groups of similar molecules, an unsupervised cluster analysis has been performed, using the standard Kmeans algorithm of Scikit-learn. Figure 9 displays the 3D PCA representation of the catalogue of 100 molecules using one color per cluster. It is worth mentioning that the first 3 PCA components account for 37.5%, 21.9%, and 8.8% of the explained variance ratio, respectively, hence a total of 68.2%. In this case the Kmeans algorithm had been set up to identify 7 clusters, and it is interesting to point out that the 11 resin molecules have correctly been assigned to a cluster of their own, namely cluster #3 in Figure 1, leaving 6 clusters of asphaltene. More details, such as the effect of the dimensionality reduction method (either using PCA or the uniform manifold approximation and projection method, UMAP, which is an nonlinear method⁵⁸), the difference between performing the dimensionality reduction before or after the cluster analysis, and the effect of the parameters of the clustering analysis (choice of number of clusters and choice of clustering algorithm) are reported in Figures S1–S4 and Tables S1 and S2 in the Supporting Information. However, it is important to mention that we have verified that such details only affected the cluster assignation of a few molecules at the frontier between clusters. When large numbers of molecules are considered in cluster analysis, the closest molecule to each cluster centroid is often chosen as a representative for the cluster. In our specific case, the number of molecules is moderate, and therefore we have looked at all the assignations and have chosen the molecules represented in Figure 1 as representative of their clusters, ensuring that none of these were at the frontier between clusters and affected by the setup of the clustering analysis. These molecules have subsequently been used in the MD simulations. The approach reported here is general and can be extended to include future developments of asphaltene models. The detailed composition of each cluster is given in the Supporting Information.

Figure 9.

Representation of the catalogue of 100 molecules from Law et al.⁴¹ after 3D principal component analysis and Kmeans clustering analysis. Each point represents a molecule, and colors reflect cluster assignation. The labels of the molecules are consistent with the original work.⁴¹

4.2. Simulation Details

While a variety of force fields (FF), all-atom, united-atom, and coarse-grained, have been used for the simulation of asphaltene aggregation,³⁰ overall any modern and well-validated FF can be considered a reasonable choice, as pointed out by Headen et al.³⁴ Indeed, by way of comparison, there are still uncertainties about larger issues, such as asphaltene structures and the exact composition of their systems. In this work, we have used the GAFF force field⁵⁹ in combination with AM1-BCC atomic partial charges, as validated by Glova et al.⁴⁷ Besides, the GAFF force field had already been used in asphaltene simulations.⁶⁰⁻⁶² The topologies of the simulated molecules (available in additional files in the Supporting Information) were generated using the ACPYPE tool,⁶³ which builds on Antechamber.⁶⁴ GPU-accelerated MD simulations were performed with the GROMACS simulation code (version 2020.4),⁶⁵⁻⁶⁸ which also served to construct the simulation boxes via random insertion of molecules (for both position and orientation). Single asphaltene model simulations with 40, 100, and 200 asphaltene molecules served as benchmarks, before settling on 100 molecules as the best compromise between system size and simulation cost. The number of solvent molecules (toluene and heptane) in each cubic simulation box was defined so as to ensure an asphaltene concentration of 7 wt% for each system. Therefore, the largest simulated system, namely 200 molecules of asphaltene model A3 in heptane, contained 621,082 atoms. When the aggregation inhibitor, a nonylphenol resin, was included in the simulations, its number of molecules was obtained from the 7:1 target ratio between asphaltenes and inhibitor. Then the number of solvent molecules was tuned to adjust the concentration of the inhibitor at 1 wt%.

The adopted simulation protocol can be summarized as follows. After construction of a cubic periodic simulation system, a steepest decent energy minimization is performed until all forces decreased below 100 kJ/mol/nm. Then, a 3 ns MD equilibration simulation in the isobaric–isothermal (NPT) ensemble was run using GROMACS’ velocity rescaling thermostat⁶⁹ and a Berendsen barostat.⁷⁰ Afterward, production MD simulations were carried out for 240 ns with a Nosé–Hoover thermostat^71,72 and Parrinello–Rahman barostat.^73,74 All simulations were performed at a temperature of 300 K and a pressure of 1 bar. Equations of motion were integrated using the leapfrog algorithm⁷⁵ with a time step of 2 ps while keeping hydrogen bonds rigid via the LINCS algorithm.⁷⁶ To account for long-range electrostatic interactions the particle-mesh Ewald (PME) algorithm⁷⁷ was employed, whereas a plain cutoff (PME could not be used with GPU yet) with a standard correction for energy and pressure was adopted for long-range dispersion interactions, in both cases with a cutoff value of 1.25 nm.

4.3. Aggregation Observables

The aggregation number captures the aggregation state of asphaltenic systems by counting the number of asphaltene molecules constituting an aggregate (also called a cluster). In this study, two molecules are considered to belong to the same aggregate if the shortest distance between atoms of the two molecules is inferior to a threshold value of 3.5 Å. This definition follows the findings of Headen et al.,³⁴ who showed that, when asphaltene molecules are clustered, their shortest distance clearly decreases below 3.5 Å. Moreover, Ghamartale et al.,³⁰ who studied the same asphaltene molecules, argued that such a threshold is applicable because the range of hydrogen bond lengths is between 2.70 and 3.30 Å. Furthermore, they provided an interesting discussion about the different possibilities for defining such a criterion, with a focus on the effect of using distances calculated between the center of mass between molecules (instead of interatomic distances) which can be less suited to properly account for the irregular packing of aggregates. Both the number-average aggregation number, g_n, and z-average aggregation number, g_z, have been used in asphaltene publications. Even though some prefer using g_z(18,30) over g_n,³⁴ no compelling argument was found in the papers or in the original ref (78) that actually contains a third definition, namely, the weight-average aggregation number g_w. These three observables can be obtained from different experimental techniques: g_n via membrane osmometry, g_w via static light scattering, and g_z via intrinsic viscosity measurements. As this study relies on MD simulations, we have followed—as for other observables—the definitions from Headen et al.³⁴ and used g_n as the aggregation number, which is strictly the average number of asphaltene molecules per aggregate

1

with n_i being the number of aggregates of g_i molecules. It is important to point out that the sums of eq 1 start from 2; thus, the monomers are excluded. To quantify the size of polymers or macromolecules in solution,⁷⁹ the radius of gyration (R_g) is defined as

2

with r_i being the position vector of atom i and r_cm being the position vector of the center of mass of the aggregate. Furthermore, information relative to the shape of the aggregate can be extracted from the gyration tensor (S)⁸⁰

3

in which the sums run over all atoms i of the aggregate and cm again refers to the center of mass. The diagonalization of the gyration tensor, S = diag(λ₁, λ₂, λ₃), permits obtaining the eigenvalues (principal moments) ordered as λ₁ ≥ λ₂ ≥ λ₃. Alternatively to eq 2, the radius of gyration can directly be obtained from the sum of the eigenvalues: R_g² = λ₁ + λ₂ + λ₃. From these eigenvalues, an estimation of the dimensionality and the symmetry of the aggregates can be provided by κ², the relative shape anisotropy:

4

κ² values span between 0, for a perfectly spherical cluster, and 1, for a linear chain. Still from the eigenvalues, it is possible to estimate the density of the asphaltene aggregates. The volume of each aggregate is approximated by the volume of an hypothetical effective ellipsoid having the same principal moments as the gyration tensor. Hence the axes a, b, and c of such an ellipsoid would be equal to and the volume encompassing the aggregate is

5

The mass of the aggregate is calculated as the sum of the mass of each asphaltene molecule populating the cluster. Therefore, the estimated density can be written as

6

where m_i is the mass of the ith molecule of the aggregate. In this work, the atomic positions have been written out each 10 ps (5000 time steps), and thus all observables of the aggregation state of the system, namely the aggregation number, radius of gyration, relative shape anisotropy, and density estimate, have been computed for trajectory frames each 10 ps. The aggregation number is represented with both 5 and 20 ns moving averages to better guide the eye. Additionally, in order to describe the equilibrium state of the asphaltenic systems, the average of the other observables in each recorded trajectory frame and their distributions are calculated during the last 40 ns of each run (from 200 to 240 ns of MD simulation).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c07120.

• Additional figures and tables as described in the text and additional details and results from the ML procedure and MD simulations (PDF)

• Topologies (ZIP)

The authors declare no competing financial interest.