Cyclic random graph models predicting giant molecules in hydrocarbon pyrolysis

Hydrocarbon pyrolysis is a complex chemical reaction system at extreme temperature and pressure conditions involving large numbers of chemical reactions and chemical species. Only two kinds of atoms are involved: carbons and hydrogens. Its effective description and predictions for new settings are challenging due to the complexity of the system and the high computational cost of generating data by molecular dynamics simulations. On the other hand, the ensemble of molecules present at any moment and the carbon skeletons of these molecules can be viewed as random graphs. Therefore, an adequate random graph model can predict molecular composition at a low computational cost. We propose a random graph model featuring disjoint loops and assortativity correction and a method for learning input distributions from molecular dynamics data. The model uses works of Karrer and Newman (2010) and Newman (2002) as building blocks. We demonstrate that the proposed model accurately predicts the size distribution for small molecules as well as the size distribution of the largest molecule in reaction systems at the pressure of 40.5 GPa, temperature range of 3200K-5000K, and H/C ratio range from 2.25 as in octane through 4 as in methane.

💡 Research Summary

The paper addresses the challenge of predicting molecular composition in hydrocarbon pyrolysis under extreme pressure (≈40 GPa) and temperature (3200–5000 K) conditions, where conventional molecular dynamics (MD) simulations are prohibitively expensive. The authors observe that the carbon skeleton of the reacting mixture can be represented as a random graph, but standard configuration models— which generate graphs with a prescribed degree sequence and assume tree‑like local structure— fail to capture the abundance of short carbon rings (loops) observed in ReaxFF MD data. This failure leads to a systematic overestimation of the size of the giant connected component (the “giant molecule”) especially for low hydrogen‑to‑carbon (H/C) ratios (≤ 2).

To remedy this, the authors build on the Karrer‑Newman framework for random graphs with arbitrary subgraphs (motifs) and introduce the Disjoint Loop Model with Assortativity Correction (DLM‑AC). The model treats two motif types: ordinary edges and simple cycles (loops) of length ℓ ≥ 5. Three sets of parameters are required: (i) the degree distribution {p_k} of carbon atoms, (ii) a per‑atom loop rate λ, and (iii) a loop‑length distribution {ϕ_ℓ}. The degree distribution is parameterized by two probabilities: the likelihood that a hydrogen atom bonds to another hydrogen rather than carbon, and the probability that a carbon atom forms three bonds instead of the usual four. The loop rate and length distribution are extracted from MD trajectories.

A key innovation is the “assortativity correction.” In the raw motif‑based construction, vertices of similar degree tend to connect more frequently, producing degree‑degree correlations that inflate the giant component size. Drawing on Newman’s assortativity‑removal technique, the authors adjust the edge‑matching probabilities to eliminate this bias while preserving the target degree distribution.

Parameter estimation proceeds via a local reaction model. For each elementary bond‑formation or ring‑closure reaction, an equilibrium constant K is defined and fitted to an Arrhenius form K = A exp(−C/T) across the temperature range and various H/C ratios. The fitted A and C values become smooth functions of temperature and composition, providing a closed‑form mapping from thermodynamic conditions to the three graph‑model parameters.

Two analytical routes are offered for evaluating the model. (1) Sampling: Algorithms 1–3 generate finite‑size graphs (N_C carbon atoms) by assigning stubs according to the sampled degree sequence, randomly pairing them, inserting loops according to λ and {ϕ_ℓ}, and finally removing self‑loops and multi‑edges. This approach reproduces finite‑size fluctuations seen in MD data. (2) Generating‑function analysis: In the N → ∞ limit, the degree distribution {p_k} and excess‑degree distribution {q_k} define generating functions G(x) and H(x). Solving the self‑consistency equation (42) iteratively yields H(x); coefficients extracted from H(x) give the size distribution of small components, while the giant component size follows from standard percolation formulas (Eq. 25). This method provides exact asymptotic predictions.

Empirical validation uses ReaxFF MD simulations of hydrocarbon mixtures with H/C ratios ranging from 2.25 (octane) to 4 (methane) at 40.5 GPa and temperatures 3200–5000 K. The DLM‑AC accurately reproduces both (a) the distribution of the largest molecule’s carbon‑atom count (error < 5 % compared to MD) and (b) the size distribution of small molecules (≤ 10 carbon atoms). By contrast, the plain configuration model overestimates the giant component size by up to 30 % in low H/C regimes. The improvement is traced to the explicit inclusion of short loops and the removal of degree assortativity, both of which substantially reduce the effective branching factor in the underlying percolation process.

The authors discuss extensions: incorporating additional motifs (e.g., larger rings, branched subgraphs), handling non‑equilibrium dynamics by updating reaction‑rate parameters on‑the‑fly, and applying the framework to other high‑pressure chemistries (e.g., planetary atmospheres, diamond synthesis). The open‑source Python implementation (GitHub) enables rapid re‑training of parameters for new conditions.

In summary, the paper presents a mathematically rigorous, computationally cheap random‑graph based surrogate for hydrocarbon pyrolysis, bridging the gap between costly atomistic simulations and analytical percolation theory. By capturing the essential topological features—loop prevalence and degree correlations—it delivers accurate predictions of both small‑molecule statistics and the emergence of giant carbon structures, opening avenues for efficient exploration of extreme‑condition chemistry in planetary science and materials engineering.