Chemistry-mediated Ostwald ripening in carbon-rich C/O systems at extreme conditions

There is significant interest in establishing a capability for tailored synthesis of next-generation carbon-based nanomaterials due to their broad range of applications and high degree of tunability. High pressure (e.g., shockwave-driven) synthesis holds promise as an effective discovery method, but experimental challenges preclude elucidating the processes governing nanocarbon production from carbon-rich precursors that could otherwise guide efforts through the prohibitively expansive design space. Here we report findings from large scale atomistically-resolved simulations of carbon condensation from C/O mixtures subjected to extreme pressures and temperatures, made possible by machine-learned reactive interatomic potentials. We find that liquid nanocarbon formation follows classical growth kinetics driven by Ostwald ripening (i.e., growth of large clusters at the expense of shrinking small ones) and obeys dynamical scaling in a process mediated by carbon chemistry in the surrounding reactive fluid. The results provide direct insight into carbon condensation in a representative system and pave the way for its exploration in higher complexity organic materials. They also suggest that simulations using machine-learned interatomic potentials could eventually be employed as in-silico design tools for new nanomaterials.


Supplementary Discussion
Here we provide an overview of the chemical evolution occurring in the reactive fluid during the appearance and growth of the carbon clusters. As shown in panel a of Supplementary Figure 1, the fluid surrounding the clusters is comprised of a variety of small species, though only CO, CO 2 , and O are present in notable quantities. Figure 1 panel b indicates that these small species account for approximately 20% of all carbon in the system, at any given time. The remaining carbon is distributed between clusters, i.e. any set of carbon atoms containing at least 10 members separated by no more than r CC = 1.9 Å and "poly" material, i.e. species that are neither nominal clusters nor the small molecules mentioned above. We note that at early times this "poly" material encompasses the polymeric fragments which are precursors to the first carbon clusters, while at late times it contains the remnants of clusters that have been consumed via Ostwald ripening.    As noted in the main manuscript, an important part of the cluster evolution is the elimination of interior oxygen; Supplementary Figure 2 provides an overview of this carbon cluster purification kinetics. Panel a shows a spatially-resolved view of the oxygen content averaged over all the clusters, indicating that initially, oxygen is fairly abundant and largely uniformly distributed. As time progresses, oxygen is found to migrate to cluster exteriors (i.e. larger r/ R G values). The apparent concomitant oxygen localization at cluster interiors shown in panel a of Supplementary Figure 2 before 10 ps likely arises from small clusters merging via direct contact (aggregation).
We quantified the elimination kinetics by plotting the amount of oxygen in all clusters larger than the average r/ R G (i.e. clusters growing via Ostwald ripening), divided by the total number of atoms in these clusters, as a function of time (Supplemen-tary Figure 2, panel b). The resulting data are well described by a simple exponential relaxation model (x = a 0 + a 1 e −a2t , where a 0 = 0.0078, a 1 = 0.18, and a 2 = 0.22); this indicates that the growing clusters likely contain a small fraction of impurity oxygen (i.e. <1%), suggesting non-zero solubility of oxygen in liquid carbon at these conditions. Supplementary Figure 3, shows that each system with more than ≈ 1.25 × 10 3 atoms follows the classical cluster growth trend discussed in the main text (i.e. linear increase in R g 3 with time). In the three larger systems, we observe two distinct regimes, distinguished by the inflection point at t ≈ 5 ps separating initial appearance and growth stages, whereas for the smallest cluster-containing system (≈ 1.25 × 10 3 atoms), three regimes emerge. The final flat linear portion of the latter data set is attributed to system size effects, where formation of a single cluster and ensuing growth saturation is observed. Linear fits to the data indicate that growth kinetics are consistent for the larger systems.
We note that the classical growth kinetics observed for the present system enables prediction of R g as a function of time, and comparisons with carbon recovery experiments. For example, our recent UF laser-drive shock experiments on CO 1 yielded amorphous carbon clusters of r = 2.5-15 nm, 1 suggesting onset quenching times of ≈ 1-2 ns, in agreement with the experimental time scales.
We provide the average cluster reduced radial density profiles, ρ(r/R g ) and radii of gyration, R g in Supplementary Figure 4 panels a through d, for each system with at least 1.25 × 10 3 atoms. We note that, for clarity, the present analysis considers only carbon atoms within a cluster; since oxygen is located exclusively at the exterior in mature clusters, this assumption has minimal influence on later-time results. In order to generate a given reduced radial density profile, a radial density profile was computed for each relevant cluster. A ρ(r/R g ) histogram was then constructed by binning each density profile according to r/R g . For clarity, the results presented herein correspond to hyperbolic tangent fits (see main text) to each histogrammed ρ(r/R g ).
For all system sizes, average cluster profiles -Supplementary Figure 4 -transition from extended, lowdensity curves at early time to curves indicating an interior density of approximately 3 g/cm 3 , with a steep cluster interface. We also find that in all but the 1.25 × 10 3 atom system, which forms a single cluster, average cluster radius of gyration, R g and minimum radius of gyration, R g,min values are consistent, with the former value increasing from approximately 0.4 to 1.0 nm in the first 0.25 ps; R g,max values increase with system size.
Here we investigate the effects of system size on emergent system kinetics through time-resolved analysis of i. percentage of system carbon participating in cluster formation, %C sys,clu (Supplementary Figure 5 panel a), ii. total cluster volume fraction, f v,clu (Supplementary Figure 5 panel b), and iii. total number of clusters present in the system, n clu (Supplementary Figure 5 panel c). Note that f v,clu is computed based on cluster R g values. The results indicate that the evolution trends for each quantity are similar for different system sizes and consistent with Ostwald ripening evolution, and that the number of clusters is most severely affected by finite size effects. The late time values for the percentage of participating carbon and total volume fraction of clusters are in agreement for all systems, and appear to converge as the system size is increased.
Atomic composition plays a key role in determining carbon clustering kinetics. For example, energetic materials with different oxygen balance generally exhibit detonation properties differences that can be attributed in part to distinct carbon condensation effects. 2 As noted in the main text, introducing a third (and possibly forth) atom type in our current modeling paradigm should make such simulation studies possible. As a stepping stone to these future efforts we investigate the effects of decreasing the C:O ratio (from 1:1) on the carbon precipitation kinetics. We note that astrophysical studies have suggested that the C:O ratio of both rocky planets and their stars (i.e. under extreme temperature and pressure) is critical for determining overall mineralogy. 3 The present studies were conducted for systems of ≈ 1.25 × 10 4 atoms containing 5, 10, 15, 30, 40, 50, and 75% CO 2 , and the balance CO. (CO 2 % and CO as utilized here strictly refer to initial concen- trations.) Each of these simulations were conducted at 6500 K and the density determined to yield the same initial pressure as the 0% CO 2 simulations discussed in the main text. Although the system size is relatively small, finite size effects are generally less significant in systems with greater oxygen concentration (less excess carbon). We note that no cluster formation was observed over 0.25 ns for systems with CO 2 ≥ 50%, suggesting that at these thermodynamic conditions they are in a single phase region. Supplementary Figure 6 shows radial composition in the 5:6 C:O system. Cluster chemical evolution proceeds in close similarity with the 1:1 C:O case described in the main text, with early time clusters exhibiting approximately 20% oxygen-containing interiors, and purifying to nearly neat carbon interiors at late time. The 5:6 system also exhibits formation of an oxygen enrichment surface layer, which narrows (relative to the cluster radius) as the cluster matures. The primary difference compared with the 1:1 system is the C/O fractional composition at large r/R g . Supplementary Figure 7 shows that maximum values of %C sys,clu (panel a) and f v,clu (panel b) are found to decrease as initial CO 2 content is increased from 0 to 40% (C:O ratio decreased from 1:1 to 1:1.4 ), leveling off at approximately 57% (9%) and 45%(5%), respectively. Systems containing less carbon were found to produce fewer clusters, and to undergo slower kinetics (see Supplementary Figure 7 panel c and Supplementary Figure 8 panels a through c) during the first ≈ 50 ps, likely due to nucleation effects. At later times, minimal differences in the evolution of average, minimum, and maximum R G are observed, as expected for Ostwald ripening kinetics, with the exception of the CO 2 = 40 % case, where a single cluster is formed.  Figure. 7: From top to bottom: average percentage of system carbon participating in cluster formation, %C sys,clu , average cluster volume fraction, f v,clu , and natural log of the average number of clusters, ln n clu for systems containing ≈ 1.25×10 4 atoms, with 0 to 40% of the initial composition CO 2 (as indicated by line color) and the balance, CO.  Figure. 8: From top to bottom: Average ( Rg ), average maximum Rg,max ), and average minimum R g,min ) cluster radius of gyration for systems of varied CO:CO 2 ratio. Line color indicates CO 2 %.