Construction of Bituminous Coal Structure via a Molecular Dynamics-Coupled Hybrid Reverse Monte Carlo Approach

Abstract

Bituminous coal is structurally disordered and chemically heterogeneous, which complicates atomistic construction when models are assembled from representative structural fragments or constrained by limited measurements. Using demineralized, vacuum-dried Shenmu bituminous coal, we developed a molecular dynamics-coupled hybrid reverse Monte Carlo (MD–HRMC) workflow under ReaxFF to construct and validate a three-dimensional periodic atomistic CHONS model containing 10,001 atoms in a 45.6 Å cubic cell (atomic fractions: C 51.27%, H 41.38%, O 6.62%, N 0.53%, and S 0.21%). High-energy synchrotron X-ray total scattering provided an experimental pair distribution function (PDF) as the primary structural constraint, while a ReaxFF-based energetic term penalized chemically implausible local environments. A staged MD–HRMC annealing protocol yielded concurrent stabilization of the PDF misfit and the ReaxFF potential energy, and the resulting model reproduced the experimental PDF over 0–10 Å. An independent comparison with quantitative solid-state ¹³C NMR and XPS supported the constructed chemical structure, including an aromatic carbon fraction of f _a’ = 0.612 compared with the NMR-derived value of 0.57, as well as the predominance of nonoxygenated carbon environments. The constructed structure further revealed medium-range organization characterized by small- to intermediate-sized aromatic domains, locally confined but poorly connected void space, and a dominant covalent skeleton coexisting with a disconnected fragment population. Mean-squared displacement analysis indicated clear dynamical heterogeneity within the model. The resulting model provides an experimentally constrained and chemically validated atomistic representation of Shenmu bituminous coal and may serve as a structurally grounded basis for future refinement of atomistic coal models and for exploring coal structure–behavior relationships at the molecular level.

1. Introduction

Coal continues to play a major role in the global energy mix, and its clean and efficient utilization remains a longstanding focus in energy science and engineering. , A molecular-level understanding of coal pyrolysis, gasification, and related conversion processes ultimately requires atomistic models that faithfully represent the disordered and heterogeneous structure of coal. , Such models are indispensable for revealing the microscopic chemical reaction mechanisms governing coal conversion and for enhancing overall utilization efficiency.

Driven by advances in characterization techniques such as solid-state ¹³C NMR, X-ray photoelectron spectroscopy (XPS) and high-resolution transmission electron microscopy (HRTEM)multimodal spectroscopic constraints have emerged as the foundational paradigm for coal molecular models. − In a classic review, Mathews and Chaffee systematically summarized more than one hundred coal molecular models proposed between 1942 and 2010, and pointed out that coal structure representation has evolved from conceptual “hand-drawn 2D” sketches to “computable 3D” models, with increasing emphasis on large-scale, multicomponent and porous structures. Building on this progress, Castro-Marcano et al. developed a multicomponent atomistic representation aimed at capturing the structural continuum and broad molecular-weight distribution of coal. Expanding upon this concept, pioneering works by Zhang et al. and Yoo et al. have successfully assembled massive multimolecular systemsoften comprising tens of thousands of atomsto intrinsically capture diverse molecular-weight distributions and enable large-scale reactive simulations. Collectively, these studies have driven coal molecular modeling from “average structural units” toward truly “multi-molecular three-dimensional systems”.

Although such developments substantially advanced coal molecular representations, the explicit use of pair distribution function information in coal studies has remained relatively limited. Early work by Grigoriew , examined the interpretation of pair functions and diffraction patterns for coal-like laminar amorphous materials, while Castro-Marcano et al. later compared experimental PDF data with molecular models of the Argonne Premium coals. More recently, Li et al. used atomic-scale structural analysis informed by PDF-related characterization to examine anthracite structure. Taken together, these studies show that PDF-based analysis is relevant to coal structure, but in most cases it has been used for structural interpretation or post hoc model evaluation rather than as a direct optimization target during atomistic model generation.

Nevertheless, spectroscopy-based bottom-up assembly strategies, while highly successful in reproducing elemental compositions and functional-group distributions, they often struggle to represent the true amorphous continuum of the coal matrix. This approach inherently neglects whether artificially packing these massive fragment combinations into a constrained volume is topologically and energetically reasonable. Such artificial packing frequently induces severe steric hindrance, which tends to disrupt the short- to intermediate-range pair correlations natively measured by the experimental total-scattering pair distribution function (PDF). Consequently, subsequent brief molecular-dynamics (MD) relaxations often trap these assembled systems in deep local energy minima rather than achieving true energetic self-consistency under the applied force field. , As emphasized by Mathews and co-workers, this typically leaves artificial packing artifacts and residual strain, critically limiting the ability to represent the seamless 3D connectivity of coal organic matter. ,

In the fields of disordered carbon and amorphous materials, reverse Monte Carlo (RMC) and its hybrid variants have been developed precisely to reconcile the tension between scattering constraints and chemical validity. , Pikunic et al. used experimental structure factors in conjunction with an adapted RMC algorithm to reconstruct the microstructure of amorphous carbon, thereby demonstrating the feasibility of “experimental scattering + statistical inversion” for carbonaceous solids. However, conventional RMC lacks explicit chemical-bond constraints and readily generates nonphysical features such as unrealistic bond lengths and an excess of small rings. To overcome these issues, Jain et al. and Nguyen et al. introduced energy penalty terms into the RMC acceptance criterion, giving rise to the hybrid reverse Monte Carlo (HRMC) method, which has been successfully applied to the atomistic modeling of porous carbons and activated carbons. Farmahini et al. subsequently extended HRMC to silicon-carbide-derived nanoporous carbons and amorphous carbons, using multistage, multiconstraint modeling protocols to compare competing structural models generated under different schemes. , Building on this foundation, Bousige et al. proposed the MD-HRMC approach, in which molecular-dynamics annealing is embedded within HRMC so that dynamical evolution assists in crossing energy barriers, achieving simultaneous minimization of scattering misfit and potential energy and thereby reconciling structural disorder with chemical plausibility. Recent studies further indicate that combining scattering constraints with dynamics or energetic penalties has become an important direction for modeling atomistic structures of amorphous carbon; however, applications of such methods to coal, a multicomponent organic solid, remain relatively scarce.

More generally, PDF-guided construction of disordered carbonaceous materials does not imply a unique single structure, and the resulting configuration may depend on the imposed constraint set, the chosen force-field description, and the efficiency of phase-space sampling. To address these methodological gaps, we present a ReaxFF-driven MD–HRMC construction workflow to model Shenmu bituminous coal. To capture the essential short- to medium-range packing order (up to 10 Å) while maintaining computational tractability, the model scale is set at 10,001 atoms within a ∼45.6 Å periodic cell. Elemental composition and helium pycnometry density are used exclusively to parametrize the initial cell stoichiometry and volume. In the present work, these generic limitations are mitigated by combining the experimental PDF with a coal-optimized ReaxFF energy constraint and strict geometric screening within a staged MD–HRMC annealing scheme. The experimental PDF provides the primary structural constraint during refinement, whereas the ReaxFF term penalizes sterically strained or chemically implausible local environments through a reactive bonding description. Quantitative solid-state ¹³C NMR and XPS are employed as independent postgeneration validations of chemical plausibility rather than as fitting targets. On this basis, the constructed model is further examined in terms of intermediate-range aromatic organization, local free-volume characteristics, covalent-connectivity partitioning, and connectivity-resolved mobility, thereby providing an experimentally constrained atomistic framework for analyzing the structural heterogeneity of Shenmu bituminous coal across multiple length and dynamical scales.

2. Experimental Section

2.1. Sample Preparation

A sample of Shenmu bituminous coal was ground to −200 mesh. Demineralization was performed using a HCl/HF/HCl acid treatment procedure detailed elsewhere to minimize the mineral derived contributions during subsequent analyses. The residue was thoroughly washed with excess distilled water until the filtrate reached neutral pH, and then dried in a vacuum oven at 80 °C to constant weight.

2.2. Elemental Analysis and True Density

The ultimate analysis for the demineralized, vacuum-dried sample (C, H, N, S, and O) was performed using an elemental analyzer (Vario EL III, Elementar, Germany). The true density of the demineralized sample was obtained by means of a helium pycnometer (AccuPyc II 1345, Micromeritics, USA). These measurements were used as the compositional and density inputs for subsequent model construction.

2.3. X-ray Photoelectron Spectroscopy (XPS)

The surface elemental composition and chemical states were analyzed by XPS using a Kratos Axis Supra spectrometer equipped with a monochromatic Al Kα X-ray source (1486.7 eV, 150 W). A wide survey spectrum was first recorded, followed by high-resolution narrow scans for C, O N and S. The binding-energy scale was calibrated by setting the C 1s peak of adventitious carbon to 284.8 eV, and peak fitting of the high-resolution spectra was performed with the Avantage software package.

2.4. Solid-State ¹³C NMR Spectroscopy

To obtain a quantitative description of the carbon skeletal structure, solid-state ¹³C CP/MAS NMR experiments were performed on a Bruker AVANCE III 600 spectrometer equipped with a 4 mm MAS probe and operating at a ¹³C resonance frequency of 150.9 MHz. The demineralized coal powder was packed into zirconia rotors and spun at a MAS frequency of 12 kHz. The relative fractions of the different carbon types were quantified by peak deconvolution of the ¹³C CP/MAS spectra following the parametrization procedure of Solum et al. ,

2.5. Synchrotron X-ray Diffraction

High-quality total-scattering data for atomistic structural modeling were collected by high-energy synchrotron X-ray diffraction (SR-XRD) at beamline ID31 of the European Synchrotron Radiation Facility (ESRF). The incident X-ray energy was 75.051 keV, corresponding to a wavelength of 0.16520 Å. The demineralized coal powder was loaded into a cylindrical sample holder between Kapton windows, and two-dimensional diffraction images were recorded in transmission mode using a Pilatus CdTe 2 M area detector. To ensure data accuracy, background scattering from the empty Kapton cell was measured and subtracted, and the instrument geometry was calibrated using the NIST standard reference material SRM 660b (LaB₆).

The two-dimensional diffraction patterns were azimuthally integrated with the pyFAI software package, including flat-field, geometric, solid-angle and polarization corrections. The resulting one-dimensional intensity data were then processed with PDFgetX3 to obtain the experimental total-scattering pair distribution function (PDF) for subsequent structural modeling.

2.6. MD–HRMC Construction Framework

To construct a bituminous-coal model that simultaneously matches experimental structural data and remains energetically stable, we developed a hybrid construction framework that couples reactive molecular dynamics (ReaxFF–MD) with hybrid reverse Monte Carlo (HRMC) based on the open-source fullrmc platform. The objective function combines a PDF-fitting term, a ReaxFF potential-energy term, and hard geometric constraints, enabling the model to satisfy experimental scattering statistics while maintaining chemically reasonable local environments. The overall workflow follows a simulated-annealing schedule coupled to an MD–HRMC composite sampling cycle (Scheme ).

1. Schematic Workflow of the MD–HRMC Construction Scheme.

2.6.1. Construction of the Initial Model

The initial model was constructed directly from the experimentally determined composition and density of the demineralized Shenmu coal sample. After demineralization, the ash yield decreased to 0.53 wt %, indicating that most inorganic mineral matter had been removed. For model construction, the elemental composition was taken on a dry, mineral-matter-free basis, giving 79.2 wt % C, 5.36 wt % H, 13.62 wt % O, 0.96 wt % N, and 0.87 wt % S. These values correspond to atomic fractions of 51.27 C, 41.38 H, 6.62 O, 0.53 N, and 0.21 S, respectively. The true density was determined by helium pycnometry to be 1.36 g cm^–3. By fixing the total number of atoms at 10,001 and enforcing consistency with the experimental density and elemental composition, the edge length of the cubic box was calculated to be 45.6 Å. The amorphous starting configuration was generated using the pdbparser package, which populates a periodic simulation cell according to the prescribed integer atom counts, target density, and box size, thereby producing a motif-free initial structure while avoiding unphysical atomic overlaps. This procedure provides a chemically unbiased starting point for subsequent MD-HRMC construction. To assess the sensitivity of the construction to the initial atomic arrangement, the as-generated amorphous structure was further subjected to short MD perturbations of 1000 and 5000 steps, and the resulting snapshots were used as alternative starting configurations. Together with the unperturbed configuration, these two perturbed structures served as the starting points for three independent constructions, with all other simulation parameters and constraints kept strictly identical.

2.6.2. Experimental PDF Constraint and X-ray Scattering Factors

The PDF data used for structural fitting were obtained from high-energy synchrotron X-ray total scattering. After subtraction of the empty-background signal and application of geometry-related corrections, the two-dimensional diffraction images were integrated into a one-dimensional scattering profile and processed with PDFgetX3 for data normalization and real-space transformation. Following the PDFgetX3 workflow, the experimental data were first normalized to the total scattering structure factor S(Q), from which the reduced structure function was constructed as

$$F(Q)=Q[S(Q)-1]$$ 1

Within the program, the measured experimental function is corrected at the level of F(Q) through an empirical treatment of the background contribution, yielding a corrected function with the proper high-Q asymptotic behavior. The experimental reduced pair distribution function is then obtained by sine Fourier transformation,

$$G_{\exp }(r)=\frac{2}{\pi }{\int }_{Q_{\min }}^{Q_{\max }}F(Q)\sin (Qr)\mathrm{d}Q$$ 2

Accordingly, G _exp(r) represents the experimentally derived reduced real-space pair-correlation information after total-scattering normalization and background correction.

Following the PDFFIT convention summarized by Keen, the reduced pair distribution function G(r) and the pair distribution function g(r) are related by

$$g_{\exp }(r)=1+\frac{G_{\exp }(r)}{4\pi {\rho }_{0}r}$$ 3

where ρ₀ is the average atomic number density of the system. The transformed g _exp(r) was used as the target function in the subsequent HRMC fitting, so that the experimental and simulated quantities were compared within the same normalized real-space framework.

For each trial configuration, the simulated total pair distribution function g _sim(r) was constructed directly from the histogram of all interatomic distances under periodic boundary conditions. For each element pair i, j, the pair counts were normalized by the shell volume and the corresponding reference average pair density, and the total simulated function was then obtained by summing over all element pairs with the appropriate weights,

$$g_{\mathrm{sim}}(r)=\sum _{i,j}{w}_{ij}\frac{n_{ij}(r)/v(r)}{N_{ij}/V}$$ 4

where n _ij (r) is the number of i-j atom pairs whose distances fall within the shell volume v(r), N _ij is the total number of such pairs in the system, and V is the simulation cell volume. To ensure consistency with the elemental sensitivity of X-ray total scattering, the contributions from different atomic pairs were weighted using effective pair weights w _ij constructed over the selected Q range from atomic form factors parametrized according to Waasmaier and Kirfel. As a result, g _sim(r) represents an X-ray–weighted total pair distribution function.

The structural discrepancy between experiment and simulation was defined at the level of g(r) and expressed as the normalized squared residual,

$${\chi }^2=\frac{\sum _i{[g_{\mathrm{sim}}(r_i)-g_{\exp }(r_i)]}^2}{\sum _i{[g_{\exp }(r_i)]}^2}$$ 5

The fitting range was set to r = 0–10 Å with a step size of Δr = 0.01 Å. This range captures the principal short-range bonding information and the key medium-range packing features of the coal structure, while reducing the influence of noise, finite-size effects, and Fourier termination ripples at larger r. The choice of g(r) as the fitting quantity is appropriate for two reasons. It corresponds directly to the histogram-based quantity produced on the simulation side, namely g _sim(r), and it approaches unity in the large-r limit, providing a normalized measure of pair correlations relative to a spatially uniform reference state. In this representation, the experimental constraint and the simulated result can therefore be compared point by point within the same normalized real-space framework.

2.6.3. ReaxFF Energy Constraint and Acceptance Criterion

In addition to the experimental PDF constraint, an energy penalty based on ReaxFF was introduced into the HRMC acceptance criterion to suppress non–physical configurations such as unrealistic bond lengths and an excessive number of small rings. Energies were evaluated in LAMMPS using the coal-specific ReaxFF parameter set Coal-FF and called from fullrmc through the Python interface with GPU acceleration enabled.

After each HRMC trial move, the total potential-energy change was computed as ΔE = E _new – E _old. To bring the energy term to a numerical scale comparable to that of the PDF misfit in the Metropolis criterion, we express it in a dimensionless per-atom form. First, the per-atom energy increment is defined as

$$\mathrm{\Delta }\stackrel{-}{u}=\frac{\mathrm{\Delta }E}{N_{\mathrm{tot}}}$$ 6

where N _tot is the total number of atoms in the system. An energy-scale factor ω is then introduced, and the dimensionless energy term is written as

$$\mathrm{\Delta }U=\frac{\mathrm{\Delta }\stackrel{-}{u}}{\omega }$$ 7

The factor ω is chosen to be on the order of a typical C–C bond energy and is further adjusted to account for the relatively high hydrogen content of the coal sample. Specifically, ω is taken to be proportional to (N _tot/N _nonH)E _C, where N _nonH is the number of non-hydrogen atoms and E _C denotes a representative C–C bond energy E _C = 7 eV (=161.4235 kcal mol^–1).

During HRMC sampling, the acceptance probability P _acc for a trial configuration is determined by the Metropolis criterion. Different forms are used for H and non-H atoms (C, O, N, S). For moves involving non-H atoms, both the structural misfit Δχ² and the energy change ΔU are taken into account:

$$P_{\mathrm{acc}}^{\mathrm{non}‐\mathrm{H}}=\min \left[1,\exp \left(-\frac{\mathrm{\Delta }{\chi }^2+\mathrm{\Delta }U}{T_{\chi }}\right)\right]$$ 8

For moves involving H atoms, which do not contribute to the X-ray PDF, the acceptance probability depends only on the energy change:

$$P_{\mathrm{acc}}^{\mathrm{H}}=\min \left[1,\exp \left(-\frac{\mathrm{\Delta }U}{T_{\chi }}\right)\right]$$ 9

where T _χ is the “effective temperature” used in the Metropolis criterion and is gradually reduced according to the simulated-annealing schedule.

2.6.4. Simulated Annealing and MD–HRMC Sampling Protocol

Once the acceptance criteria are defined, it is necessary to specify how the configurations evolve in the high-dimensional configuration space. In this work, we adopt a simulated-annealing strategy combined with composite MD–HRMC sampling: in an outer annealing loop the “effective temperature” is gradually lowered, and at each temperature plateau ReaxFF–MD global relaxation and HRMC local sampling are performed in an alternating fashion.

To realize a smooth transition from high-temperature global exploration to low-temperature local refinement, a geometric cooling schedule is applied to the nominal physical temperature T _phys. Given the initial and final temperatures T _initial = 3000 K and T _final = 300 K, and the total number of outer annealing cycles N _cyc = 100, the cooling factor α is defined as

$$\alpha ={\left(\frac{T_{\mathrm{final}}}{T_{\mathrm{initial}}}\right)}^{1/N_{\mathrm{cycle}}-1}$$ 10

The physical temperature in the k-th annealing cycle (k = 0,..., N _cyc – 1) is then

$$T_{\mathrm{phys}}^{(k)}=T_{\mathrm{initial}}{\alpha }^k$$ 11

To give the Metropolis exponent a clear physical meaning, we introduce a dimensionless “effective temperature” T _χ ^(k) proportional to T _phys ^(k):

$$T_{\chi }^{(k)}=\frac{k_{\mathrm{B}}{T}_{\mathrm{phys}}^{(k)}}{\omega }=T_{\chi ,0}{\alpha }^k,T_{\chi ,0}=\frac{k_{\mathrm{B}}{T}_{\mathrm{initial}}}{\omega }$$ 12

where k _B = 0.001987 kcal mol^–1 K^–1 is the Boltzmann constant and ω is the energy-scale factor introduced in Section 2.3.3.

Parallel to the temperature annealing protocol, the maximum displacement amplitude for HRMC trial moves, Δr _max, is gradually reduced from 0.5 to 0.05 Å over the course of the simulation. This reduction facilitates a smooth transition from coarse-grained global rearrangements to fine-scale local adjustments.

Within each annealing cycle, the system first underwent an NVT ReaxFF–MD relaxation in LAMMPS using Coal-FF at a fixed thermostat temperature T _MD = 500 K. Equations of motion were integrated for 5,000 steps with a time step of 0.25 fs using KOKKOS-GPU acceleration. The MD stage serves to remove highly unfavorable overlaps and relax accumulated local stresses, thereby repairing the global topology before stochastic refinement.

After MD relaxation, HRMC refinement was performed at the current annealing stage by attempting 10⁵ single-atom trial moves. In each trial move, a single atom was randomly selected. Trial configurations are first screened by hard geometric constraints to eliminate nonphysical overlaps, and any move that produces an interatomic distance below 0.9 r _{0, ij} (with r _{0, ij} denoting the corresponding Coal-FF equilibrium bond length) is rejected outright. Configurations passing the geometric check are accepted or rejected using the Metropolis criteria in Sections and with the current T _χ ^(k). Convergence was assessed by tracking the cycle-wise evolution of the PDF misfit χ², total potential energy, and acceptance ratio. Regarding computational cost, while a standard geometric RMC completes in ∼4 h but yields severely defective structures, and a conventional HRMC fails to converge even after 480 h due to extremely low acceptance rates, this GPU-accelerated MD-HRMC workflow successfully achieves a chemically and topologically realistic construction in approximately 150 h.

2.7. ReaxFF–MD Relaxation and Connectivity-Resolved Transport Analysis

To further relax the MD–HRMC–constructed coal configuration under a reactive force-field description and to generate a dynamical trajectory for transport statistics, reactive molecular dynamics (ReaxFF–MD) simulations were performed in LAMMPS under the NVT ensemble. Unwrapped-coordinate trajectories were saved for postprocessing. The subsequent analysis follows a connectivity-defined operational partitioning scheme: the system is first decomposed into a dominant covalently connected backbone and a set of disconnected finite fragments based on covalent connectivity; mean squared displacement (MSD) is then computed for each population in a common drift-corrected reference frame, and effective diffusion coefficients (D) are extracted from the linear regime of MSD–time relations.

2.7.1. ReaxFF–MD Trajectory Generation and Coordinate Output

All ReaxFF–MD simulations were conducted under periodic boundary conditions using units real and atom style charge in LAMMPS. Charge equilibration was performed at every time step using fix qeq/Reaxff. Production dynamics were integrated in the NVT ensemble at 300 K with a time step of 0.25 fs. The resulting trajectory was used for subsequent connectivity-resolved analysis. The subsequent connectivity-resolved analysis was performed in Python using MDAnalysis trajectory I/O, NumPy for array-based MSD evaluation, and SciPy for linear fitting of effective diffusion coefficients.

2.7.2. Connectivity-Defined Partitioning into Skeleton and Disconnected Fragment Components

To enable connectivity-resolved dynamical diagnostics, the constructed structure was exported to a molecular file format and parsed as a molecular graph, where nodes represent atoms and edges represent covalent bonds. Covalent bonds for graph construction were assigned by bond perception using Open Babel. The system was decomposed into connected components based on covalent connectivity. The largest connected component (by atom count) was defined as the skeleton, while all remaining components not covalently connected to the skeleton were merged into the disconnected fragment population.

Within the disconnected fragment population, fragments were further divided into two operational subgroups according to the number of carbon atoms n _C in each fragment: fragments with n _C > 0 were labeled as carbon-containing disconnected fragments, whereas fragments with n _C = 0 were labeled as carbon-free disconnected fragments. This labeling is used solely for separate reporting of MSD and the corresponding effective diffusion coefficient (D) for these two subgroups.

2.7.3. MSD Calculation and Diffusion Coefficients

To evaluate whether the connectivity-defined partition between the skeleton component and the disconnected fragment population also corresponds to a meaningful difference in motion, MSD was computed from the unwrapped-coordinate trajectory after removing rigid-body drift using a skeleton-based reference frame. MSD curves were evaluated using time-origin averaging: for each lag time τ, squared displacements were averaged over all available time origins t ₀ to reduce noise and to minimize dependence on the selection of the initial frame. To avoid statistical instability at long lag times (where the number of available time origins decreases sharply), the maximum lag time was truncated to half of the total number of frames.

Effective diffusion coefficients D were extracted from the linear regime of MSD–time relations using the three-dimensional Einstein relation:

$$\mathrm{MSD}(t)=6Dt$$ 13

The linear fitting window was selected using an automatic window optimization within an admissible range.

From the 300 K ReaxFF–MD trajectory, MSD and D were computed for: (i) skeleton atoms; (ii) atoms in the disconnected fragment population; and (iii) fragment center-of-mass (COM) trajectories, which provide a complementary descriptor for fragment-scale translational motion and allow comparison of mobility differences between carbon-containing and carbon-free disconnected fragments.

3. Results and Discussion

3.1. Convergence of the Simulation and Energy Evolution

The coupled MD–HRMC simulated-annealing trajectory is summarized in terms of annealing cycles. As described in Section , each cycle consists of an NVT ReaxFF–MD relaxation step followed by an HRMC refinement batch at the same temperature plateau, after which the annealing temperature is updated according to the prescribed cooling schedule.

Figure a shows the scheduled decrease in temperature together with the HRMC trial-move acceptance rate. The temperature follows a smooth cooling profile, while the acceptance rate drops from about 60–70% in the early cycles to approximately 20–30% in the late cycles. At high temperature, a large fraction of trial moves is accepted, indicating broad accessibility of configurational space under the combined structural and energetic constraints. As the temperature is reduced, the acceptance rate becomes lower, consistent with a shift toward more selective, local refinement.

1.

Convergence and dynamic evolution during the MD–HRMC simulated annealing process. (a) The HRMC trial move acceptance rate and the effective system temperature as a function of annealing cycles. (b) Variation of HRMC total energy and PDF Error (χ²) as a function of annealing cycles.

Figure b tracks the total potential energy and the PDF Error, χ², as a function of cycle. Both quantities decrease rapidly during the early stage, suggesting that MD relaxation and HRMC sampling jointly remove highly unfavorable configurations while improving agreement with the experimental PDF. The curves then approach a plateau, where only small fluctuations are observed and no systematic drift is evident over the remaining cycles. Together with the low acceptance rate at late cycles, these trends indicate that the construction has reached a stable regime within the present sampling protocol. All subsequent structural analyses are based on configurations sampled in this plateau regime.

Three independent constructions were performed with all simulation parameters and constraints kept strictly identical to assess sensitivity to the initial random atomic arrangement. The main structural statistics were consistent across the three runs, and the full comparison is provided in Section S1 of the Supporting Information; the following discussion is based on one representative construction.

3.2. Fit of the Model to the Experimental PDF

Figure compares the simulated g(r) and G(r) profiles of the constructed model with the experimental data. The simulated pair distribution function, g(r), was computed directly from the final MD-HRMC configuration utilizing the fullrmc framework.

2.

Comparison of the experimental and simulated pair distribution functions for the constructed coal model. (a) Comparison of experimental and simulated g(r) over 0–10 Å; (b) Comparison of experimental and simulated G(r).

In g(r), the simulated curve reproduces the experimental features well over 0–10 Å. In the short-range region (r < 2.5 Å), the simulated and experimental curves agree closely. The first major peak centered at roughly 1.4–1.5 Å corresponds directly to the nearest-neighbor C–C covalent bonds, representing a structural average of aromatic (∼1.40 Å) and aliphatic (∼1.54 Å) environments. Furthermore, the matched position of the second coordination shell at approximately 2.4–2.5 Å corresponds to next-nearest-neighbor C–C distances (e.g., meta-carbons within aromatic rings). The accurate reproduction of these peak shapes and positions suggests that the constructed structure captures a chemically realistic distribution of local bonding geometries without obvious distortion. Extending into the medium-range region (2.5–8 Å), the model continues to reproduce the main features of the experimental profile. The second peak near 2.5 Å and the broader feature around 3.5–4.0 Å are both matched in position and relative prominence, and the intervening minima are also captured. The agreement across these features indicates that the model reflects not only isolated peak positions but also the overall short- to medium-range packing correlations represented by g(r). Across 0–10 Å, the residual curve (Δg) fluctuates around zero without a persistent positive or negative bias over an extended r interval. The resulting normalized misfit is χ² = 0.00758 for r = 0–10 Å. This level of agreement supports the use of the constructed configuration as a structural basis for subsequent chemical validation and connectivity-resolved transport analyses.

The G(r) comparison in Figure b provides a more stringent test of the fit at larger pair distances because the additional r-weighting enhances the contribution of correlations in the intermediate-range regime. In this representation, the constructed model reproduces the main peak–trough sequence and the relative amplitude variation of the experimental curve over approximately 1–5 Å. This agreement is significant because the G(r) signal in this range reflects not only local bonding geometry but also higher-order pair correlations associated with the organization of the structural framework, including short/intermediate-range packing motifs. At larger r, where the experimental signal progressively decays and becomes noisier, the simulated curve still follows the overall attenuation trend without obvious systematic deviation.

Taken together, the agreement in g(r) and G(r) indicates that the constructed structure captures the principal pair-statistical features of the experimental total-scattering data at both local and intermediate length scales. This does not imply that the intermediate-range topology is uniquely determined by the PDF fit alone; rather, it shows that the model is statistically consistent with the experimentally constrained short- and intermediate-range structure. The chemical plausibility and structural interpretation of the model are therefore examined further below using independent spectroscopic validation and topology- and connectivity-based analyses.

3.3. Validation of the Chemical Structure

With the structural fit to the PDF and the energy evolution established, we next evaluate whether the constructed configuration reproduces chemically meaningful local environments. The MD–HRMC–ReaxFF model is assessed against quantitative solid-state ¹³C NMR and high-resolution XPS (C 1s, N 1s, and S 2p). The comparison is organized at three levels: (i) the carbon-skeleton composition, (ii) oxygen-containing functional groups, and (iii) the occurrence modes of low-abundance heteroatoms. This structure allows the spectroscopic constraints to serve as independent checks on the chemical plausibility of the model.

3.3.1. Carbon Skeleton Structure

Quantitative solid-state ¹³C NMR provides a direct constraint on the distribution of carbon types in coal. Using a Solum et al. type peak deconvolution and parametrization scheme, we obtained the NMR-derived parameters summarized as follows. In this framework, f _a denotes the total sp²-hybridized carbon fraction, f _a ^C denotes the carbonyl or carboxyl carbon fraction, f _a’ denotes the aromatic carbon fraction after correction for oxygen-bound nonaromatic sp² carbons, and f _al denotes the sp³-hybridized (aliphatic) carbon fraction. The NMR-derived values are f _a = 0.650, f _a ^C = 0.08, f _a’ = 0.57 and f _al = 0.350.

To validate the constructed carbon skeleton, we compared the aromatic carbon (f _a’) partition inferred from ¹³C NMR with that computed from the final MD–HRMC configuration. Carbon environments in the model were assigned using bond perception in Open Babel. After excluding the carbonyl and carboxyl carbons (f _a ^C) from the total sp²-hybridized carbon count, the model-derived f _a’ is 0.612. These values are close to the NMR-derived fractions, with an absolute deviation of 4.2 percentage points. This result indicates that the constructed model reproduces the main carbon-skeleton characteristics of the sample with relatively small deviation in the aromatic/aliphatic carbon partitioning.

Notably, f _a’ and f _al were not enforced explicitly in the MD–HRMC objective function. Instead, the carbon skeleton features emerge from the combined enforcement of PDF consistency and ReaxFF energetics during simulated annealing. The agreement in Figure therefore serves as an independent spectroscopic validation rather than a fitted target.

3.

Comparison of the carbon aromatic carbon fraction (f _a’) and aliphatic carbon fraction (f _al) derived from quantitative solid-state ¹³C NMR and from the final MD–HRMC configuration.

To visualize the constructed framework and provide an additional internal consistency check, Figure shows the three-dimensional configuration of the model and representative local motifs extracted from the covalent connectivity network. The global view (Figure a) illustrates a highly cross-linked framework within the periodic simulation cell, while the extracted motifs (Figure b,c) highlight finite aromatic domains and representative linkages between neighboring aromatic units. These motifs are qualitatively consistent with the NMR-derived picture of an aromatic-rich backbone with a smaller aliphatic fraction that primarily appears as short bridges and terminal substituents.

4.

Three-dimensional morphology and representative local motifs of the constructed coal model. (a) Global view of the periodic simulation cell. (b) Representative aromatic-rich motif extracted from the covalent connectivity network. (c) Representative cross-linking motif showing aliphatic and ether linkages between neighboring aromatic units.

We further examined local bonding geometry using bond-angle and coordination-number statistics derived from the covalent connectivity network. The C–C–C angle distribution Figure a is concentrated between the ideal sp² (120°) and sp³ (109.5°) geometries, with a dominant population near 120° and a secondary contribution toward 109.5°. The coordination-number statistics (Figure b) show that three-coordinated carbon is predominant, whereas four-coordinated carbon forms a smaller but non-negligible population, consistent with the presence of aliphatic bridges and saturated sites. Only minor fractions exhibit atypical coordination in the perceived bond network. Overall, the motif-level visualization and local-geometry statistics complement the ¹³C NMR comparison and indicate that the constructed structure maintains chemically reasonable carbon environments at the atomic scale.

5.

Local-geometry statistics of the constructed structure based on covalent bond perception (Open Babel). (a) C–C–C bond-angle distributions for carbon atoms assigned as sp² and sp³. (b) Distribution of carbon coordination numbers in the covalent network.

On this validated backbone, we next examine whether oxygen-containing functional groups and low-abundance heteroatom environments in the model are consistent with the C 1s, N 1s, and S 2p components resolved from XPS.

3.3.2. Oxygen-Containing Functional Groups

Figure shows the XPS survey spectrum together with the high-resolution C 1s, N 1s, and S 2p regions. Peak assignments and relative areas are summarized in Table . Following common used C 1s deconvolution scheme for coal, the envelope was resolved into four components: nonoxygenated carbon (C–C/C–H, 284.8 eV), C–O (286.1 eV), CO (287.5 eV), and O–CO (289.2 eV). The surface chemistry is dominated by C–C/C–H (76.49%), with C–O as the second-largest contribution (13.97%). More highly oxidized motifs are less abundant, with CO accounting for 2.79% and O–CO for 6.75%. Overall, the C 1s analysis indicates that the surface carbon chemistry of Shenmu bituminous coal is dominated by nonoxygenated carbon environments, while oxygen occurs mainly in singly bonded C–O functionalities, with smaller contributions from CO and O–CO-type carbon.

6.

XPS spectra of the coal sample. (a) Survey spectrum. High-resolution spectra and peak deconvolutions for (b) C 1s, (c) N 1s, and (d) S 2p.

1. XPS Peak Assignments and Relative Areas for the C 1s, N 1s, and S 2p Regions of the Coal Sample.

element	assignment	binding energy (eV)	area (%)
C 1s	C–C/C-H	284.8	76.49
	C–O	286.1	13.97
	CO	287.5	2.79
	O–CO	289.2	6.75
N 1s	pyrrolic N	400.1	44.96
	pyridinic N	398.6	26.91
	quaternary N	401.2	28.13
S 2p	thiophenic S	163.9, 165.1	20.5
	sulfoxide S	164.7, 165.9	34.1
	sulfide S	163.3, 164.5	21.4
	sulfone S	168.0, 169.3	24.0

Figure compares these C 1s-derived fractions with bond-type statistics extracted from the final MD–HRMC–ReaxFF configuration. The model reproduces the qualitative hierarchy observed in XPS, with C–C/C–H as the dominant environment followed by C–O, and with CO and O–CO remaining minor. Quantitatively, the model yields a higher C–C/C–H fraction (88.1% vs 76.49%) and a substantially lower O–CO fraction (0.12% vs 6.75%) than the XPS result; C–O is slightly lower (11.1% vs 13.97%), and CO is also reduced (0.76% vs 2.79%).

7.

Comparison of C 1s component fractions from XPS with bond-type statistics extracted from the final MD–HRMC–ReaxFF configuration.

Several factors can contribute to these differences. XPS preferentially probes near-surface environments, whereas the present MD–HRMC–ReaxFF structure represents a periodic bulk configuration. In addition, component assignment in XPS deconvolution is not identical to topology-based functional-group recognition in the model. In this context, we interpret the C 1s comparison primarily as support for the dominant bonding environments and their ordering, while noting that strongly oxidized motifsespecially O–COare likely underrepresented in the bulk model relative to the surface-sensitive XPS measurement. This carbon–oxygen baseline provides context for interpreting the N 1s and S 2p speciation discussed below.

3.3.3. Nitrogen and Sulfur Heteroatom Speciation

To assess whether the constructed structure captures heteroatom environments, nitrogen and sulfur statistics extracted from the final MD–HRMC–ReaxFF configuration were compared with the XPS-derived speciation (Table ). In our N 1s deconvolution, pyrrolic-N is the dominant component, while pyridinic-N and quaternary/graphitic-N contribute as secondary components. For S 2p, the deconvolution reported in Table follows a constrained spin–orbit doublet model (fixed 2p_3/2:2p_1/2 area ratio, fixed splitting, and equal line widths within each doublet). Because N and S occur at low abundance, their impact on the PDF constraint is limited; their local environments are therefore governed mainly by the energetic regularization introduced through the ReaxFF term. For this reason, the comparison is used to evaluate dominant trends and qualitative consistency rather than to require one-to-one quantitative agreement.

2. Comparison between XPS-Derived N 1s/S 2p Speciation and Model-Derived Nitrogen/Sulfur Classifications (Open Babel-Based Bond Perception).

element	experimental XPS	fraction/%	model statistics	fraction/%
N 1s	pyridinic N	26.91	2-coordinated N	46.8
	pyrrolic N	44.96	3-coordinated N	46.8
	quaternary N	28.13	4-coordinated N	6.4
S 2p	sulfide S	21.4	thioether/thiol S	76.2
	thiophenic S	20.5	thiophenic S	9.5
	sulfoxide	34.1	oxidized S	14.3
	sulfone	24.0		not identified

In the model, nitrogen environments were characterized using a coordination-based descriptor derived from bond perception. This descriptor provides an operational way to separate N sites (two-, three-, and four-coordinated; N2/N3/N4), but it does not map directly onto XPS-assigned chemical states (pyridinic, pyrrolic, and quaternary). With this limitation in mind, the model nonetheless reflects the coexistence of multiple nitrogen environments, consistent with the mixed speciation implied by the N 1s spectrum.

For sulfur, the S 2p fit indicates that oxidized sulfur species contribute substantially in the XPS measurement (sulfoxide and sulfone; Table ), by contrast, the model is dominated by reduced sulfur environments (thioether/thiol and thiophenic sulfur) with a smaller oxidized fraction, and no sulfone category is identified under the present topology-based classification (Table ). Given the surface sensitivity of XPS, this difference is consistent with a scenario in which oxidized sulfur is enriched near the surface, while the periodic bulk model reflects the interior chemical framework. Taken together, the XPS comparisons support the chemical plausibility of the constructed structure in terms of the dominant carbon environments and the presence of diverse N/S occurrence modes.

3.4. Intermediate-Range Topology and Spatial Organization

The macroscopic physicochemical properties of an amorphous coal matrix are governed not only by local atomic configurations, but also by the spatial organization of aromatic units and the topological distribution of three-dimensional free volume. With this in mind, the constructed model was further characterized, following the spectroscopic validation, from two complementary perspectives: intermediate-range aromatic topology and spatial organization.

3.4.1. Distribution of Aromatic Cluster Size and Statistics of Intermediate-Range Aromatic Domains

Aromatic domains in the constructed model were identified and quantified using an Open Babel–based postprocessing workflow, in which six-membered carbon rings were first extracted from the constructed bonding network and then grouped into aromatic domains according to coplanarity and fused-topology relationships. The Figure results show that the aromatic carbon in the cross-linked network does not remain as isolated local sp² islands, but instead develops into a large number of recognizable fused-ring clusters during the construction. In terms of the overall size distribution, the aromatic population is dominated by small domains containing one or two six-membered rings, whereas larger fused aromatic domains composed of three or more rings are already present but remain comparatively limited in abundance.

8.

Carbon-number-weighted distribution of aromatic-domain size in the constructed coal model, together with representative three-dimensional topological motifs. Aromatic domains were classified according to the number of coplanarly fused six-membered rings as small (1–2 rings), medium (3–4 rings), and large (≥5 rings). The inset structures are representative polycyclic aromatic fragments extracted directly from the constructed covalent network and illustrate the short- to intermediate-range aromatic aggregation features captured by the model.

This topological description provides a physically complementary perspective to the overall aromaticity derived from spectroscopic analysis. Under the combined influence of the experimental PDF constraint and local ReaxFF energetics, the construction appears to preferentially resolve local densification and short-range aromatic organization. The emergence of polycyclic aromatic units suggests that the resolved structure has developed a measurable degree of intermediate-range aromatic ordering under the experimental constraints. At the same time, the continued growth of more extended aromatic sheets remains relatively restrained, which likely reflects a combination of the stochastic nature of the construction pathway and the intrinsic densification tendency of the amorphous network. Therefore, a distribution dominated by small polycyclic aromatic clusters can be reasonably interpreted as being consistent with a highly fragmented and localized mode of aromatic organization within the coal matrix.

3.4.2. Free-Volume and Geometric Accessibility Analysis Based on Zeo++

To provide complementary geometric information on local cavities, confined space, and potential migration pathways in the constructed coal structure, Zeo++ analysis was performed using a He probe with a radius of 1.2 Å. The analysis was used to quantify the maximum included sphere diameter (D _i), the maximum free sphere diameter (D _f), the probe-occupiable accessible volume (POAV), the probe-occupiable nonaccessible volume (PONAV), the number of connected channels, and the number of isolated pockets.

The Zeo++ results indicate that the present model contains local cavities of measurable size (D _i = 3.6 Å), showing that the constructed framework is not a perfectly dense and pore-free homogeneous solid. By contrast, the maximum free sphere diameter (D _f = 1.2 Å), which governs long-range geometrical accessibility, is substantially smaller than the effective kinetic size of typical probe molecules. Using a helium probe with a radius of 1.2 Å, zero probe-occupiable accessible volume (POAV) and identifies no continuous percolation channels. Instead, the available free volume is strictly partitioned into 142 isolated pockets, corresponding to a minimal probe-occupiable nonaccessible volume (PONAV) of 0.0022 cm³ g^–1. These results indicate that, although the constructed framework retains local cavities of finite size, these cavities are not connected through bottlenecks wide enough to support long-range accessibility at the He-probe scale.

These geometric features are spatially consistent with the limited extension of the aromatic domains discussed earlier. In the absence of large, continuous aromatic sheets, the three-dimensional frameworkgenerated by small aromatic clusters interwoven with complex aliphatic cross-linkstends to develop a compact and highly partitioned organization, giving rise to confined pockets rather than open transport channels. Because the simulation cell was constrained by the experimental helium pycnometry density, the constructed structure was expected to evolve toward a relatively dense skeletal framework, rather than a full representation of the hierarchical pore system of bulk coal. Within this scope, the present model provides an atomistic description of the cross-linked solid matrix and its confined local free volume. Further extension toward a multiscale porous representation may require explicit pore-generation strategies, such as virtual pore doping, to incorporate larger-scale void space beyond the dense skeletal phase.

3.5. Connectivity-Resolved Dynamics

3.5.1. Network–Fragment Topology from Covalent Connectivity Decomposition

Covalent connectivity analysis indicates that the constructed configuration is not a single fully connected macromolecule. Instead, it forms a composite topology comprising a dominant skeleton (the largest connected covalent network) and a disconnected population of finite fragments (Table ). The skeleton contains 7,221 atoms (72.2%) and defines the primary covalent backbone, whereas the remaining 2,780 atoms (27.8%) are distributed over 681 disconnected fragments (Table ).

3. Geometric Free-Volume Metrics of the Constructed Coal Model Obtained from Zeo++ Analysis, Including the Maximum Included Sphere Diameter (D _i), Maximum Free Sphere Diameter (D _f), Probe-Occupiable Accessible Volume (POAV), and Probe-Occupiable Nonaccessible Volume (PONAV).

parameter/definition	value	unit
D i (maximum included sphere diameter)	3.6	Å
D f (maximum free sphere diameter)	1.2	Å
POAV (probe-occupiable accessible volume)	0	cm3·g–1
PONAV (probe-occupiable nonaccessible volume)	0.0022	cm3·g–1

4. Connectivity-Based Decomposition of the Constructed Coal Model and Operational Grouping of the Disconnected Population.

component label	operational definition	number of atoms	atom fraction (%)	number of fragments
skeleton (host)	largest connected covalent network	7,221	72.2	1
disconnected fragment	all disconnected finite clusters	2,780	27.8	681
carbon-containing	disconnected clusters with carbon	2,003	20.0	305
carbon-free	disconnected clusters without carbon	777	7.8	376

3.5.2. Connectivity-Resolved MSD and Effective Diffusivity

A connectivity label is only useful if it maps onto a physically meaningful difference. To validate this distinction, we computed drift-corrected MSDs from the ReaxFF–NVT trajectory following the procedure in Section , using the skeleton center-of-mass as the reference to remove global translation.

Figure shows that the skeleton exhibits a nearly flat MSD over the sampled time window, whereas the disconnected population displays a substantially larger and steadily increasing MSD. When the disconnected population is separated by n _C, the fragment COM MSDs further reveal internal heterogeneity: the C-free subset shows the largest displacements, and the organic subset remains intermediate. The resulting observation is consistent with a low-mobility covalent backbone coexisting with a higher-mobility fragment population.

9.

Drift-corrected mean squared displacement (MSD) at 300 K from the 500 ps ReaxFF–NVT trajectory. MSD is reported for atoms in the skeleton (largest covalent network) and for fragment center-of-mass (COM) trajectories of the disconnected population, separated into carbon-containing disconnected fragments (n _C > 0) and carbon-free disconnected fragments (n _C = 0).

To quantify this hierarchy and check its robustness, effective diffusion coefficients were extracted from the linear regime of the MSD–time relation. Figure summarizes the results across four representative fitting intervals. While the absolute values vary with the fitting range, the mobility ranking is stable: the skeleton remains the least mobile, and the disconnected population remains higher by roughly one to 2 orders of magnitude under the same extraction protocol. These D values are used here as effective, trajectory-based descriptors for comparing populations within the constructed model, rather than as macroscopic transport coefficients for bulk coal.

10.

Effective diffusion coefficients extracted from linear fits to MSD­(t) = 6Dt under four representative fitting intervals. Results are reported for the skeleton, the full disconnected population, and the carbon-containing disconnected fragments subset to assess the robustness of the mobility ranking to the window-selection procedure.

3.5.3. Composition and Plausibility of the Mobile Small-Molecule Population

To further clarify the chemical meaning of the disconnected fragment population identified by the connectivity analysis, all fragments not belonging to the largest covalently connected skeleton, as defined in Section , were extracted and further processed using an Open Babel–based postanalysis workflow. On this basis, the disconnected fragments were subjected to fragment-level compositional classification, molecular-formula counting, and plausibility screening.

Further classification of the 305 carbon-containing disconnected fragments shows a clear predominance of small, weakly polymerized species. The median fragment size is 5 atoms, with an interquartile range of 4–7 atoms, and the largest carbon-containing disconnected fragment contains 53 atoms. In compositional terms, hydrocarbon fragments and oxygen-containing aliphatic fragments dominate, accounting for 154 and 139 fragments, respectively, or 50.5 and 45.6% of the classified carbon-containing disconnected fragments population. On an atom-weighted basis, these two classes contribute 39.7 and 55.4% of the carbon-containing disconnected fragments atoms, respectively. Nitrogen-containing fragments are much less abundant (9 fragments, 3.0%), and only one oxygen-containing aromatic fragment was identified. Overall, the carbon-containing disconnected population is dominated by small aliphatic fragments and low-carbon oxygen-containing species, whereas larger aromatic species remain rare (Table ).

5. Compositional Stratification and Quality-Control Summary of the Connectivity-Defined Disconnected Fragment Population in the Constructed Coal Model, Including Fragment Counts, Atom Counts, Size Statistics, and Dominant Compositional Classes.

metric	value
total mobile fragments	681
total mobile atoms	2780
chemically interpretable carbon-containing disconnected fragments	305
proportion of carbon-containing disconnected fragments (%)	44.8
atoms in carbon-containing disconnected fragments	2003
proportion of atoms in carbon-containing disconnected fragments (%)	72.1
maximum size of a carbon-containing disconnected fragment	53

Beyond the carbon-containing disconnected fragments, the constructed model also contains a small population of nonorganic or ultralight disconnected species. Some of these, particularly H₂O and certain small oxygen-bearing species, indicate that a minor fraction of atoms in the present structure remains only weakly integrated rather than being fully incorporated into the main covalent skeleton. The compositional statistics further show that the carbon-containing disconnected population is dominated by small aliphatic and oxygenated aliphatic fragments, with only minor nitrogen-containing species and trace oxygenated aromatic fragments. Overall, the disconnected population constitutes a connectivity-defined partition within the present constructed structure, and its carbon-containing subset shows the closest qualitative correspondence to the weakly integrated molecular material commonly discussed in coal. The occurrence of ultralight disconnected species nevertheless indicates a remaining limitation of the present construction; future refinement of the construction strategy may therefore benefit from additional topological or fragment-size constraints to promote a more realistic distribution of atoms within the coal framework.

3.5.4. Relationship to Network–Mobile Fraction Descriptions of Coal

Coal structure has long been discussed in terms of a cross-linked macromolecular network that coexists with a more weakly bound molecular or mobile fraction, often described as a “mobile or molecular phase” or as associated molecular units within the network framework. ,

Within this context, the present construction provides a connectivity-based analogue of the same descriptive framework. The covalently connected skeleton corresponds to the dominant network backbone, whereas the disconnected population represents a weakly integrated component of the present constructed structure. Importantly, the dynamics analysis provides a self-consistency test: the population identified as disconnected by topology is also the population that exhibits the larger MSD growth and higher effective diffusivity. Among these disconnected species, it is the carbon-containing subset that shows the closest qualitative correspondence to the weakly integrated molecular material commonly discussed in coal.

This positioning is intended to facilitate comparison with prior coal-structure representations that invoke a network together with a more weakly integrated component. It does not require assuming thermodynamic two-phase equilibrium; rather, it links a connectivity-based structural partition to a trajectory-based mobility hierarchy within a single experimentally constrained three-dimensional atomistic model. In this sense, the present construction establishes a structurally and chemically validated atomistic model of Shenmu bituminous coal, while further revealing a topology–mobility hierarchy that is consistent with classical network–mobile descriptions of coal structure.

4. Conclusion

We developed an MD-coupled HRMC construction workflow under ReaxFF to construct and validate a three-dimensional periodic atomistic model of Shenmu bituminous coal, constrained primarily by experimental X-ray PDF data together with energetic regularization. The protocol yields concurrent stabilization of the PDF misfit and the ReaxFF potential energy and reproduces the experimental PDF over 0–10 Å, consistent with realistic short- and medium-range pair correlations. Independent comparison with quantitative solid-state ¹³C NMR and XPS supports the constructed carbon speciation (model f _a’ = 0.612 vs NMR f _a’ = 0.57) and the predominance of nonoxygenated carbon environments, with expected deviations from surface-sensitive XPS responses. The constructed structure further reveals medium-range organization characterized by small- to intermediate-sized aromatic domains, locally confined but poorly connected void space, and a dominant covalent skeleton coexisting with a disconnected fragment population. MSD analyses show that these disconnected components exhibit substantially greater mobility than the main skeleton, indicating clear dynamical heterogeneity within the model. Together, these results support the structural and chemical validity of the constructed atomistic model of Shenmu bituminous coal and provide a structurally grounded basis for further refinement of atomistic coal representations.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c00765.

• Three-dimensional atomic coordinates of the representative constructed bituminous coal model used for the main analyses in this study (PDB)

• Detailed analysis of the reproducibility of three independent constructions, including the evolution of normalized PDF misfit and potential energy, comparisons of experimental and simulated g(r) and G(r) profiles, and tabulated comparisons of chemistry- and connectivity-related structural descriptors (PDF)

L.K.: Conceptualization, methodology, software, validation, formal analysis, investigation, data curation, writingoriginal draft, and visualization. X.Y.: Software, formal analysis, and validation. J.D.: Data curation, investigation, and visualization. K.Y.: Methodology, formal analysis, writingreview and editing, and supervision. Z.Z.: Conceptualization, resources, writingreview and editing, supervision, project administration, and funding acquisition.

The authors declare no competing financial interest.