
==== Front
Commun Chem
Commun Chem
Communications Chemistry
2399-3669
Nature Publishing Group UK London

1261
10.1038/s42004-024-01261-9
Article
Molecular geometry specific Monte Carlo simulation of the efficacy of diamond crystal formation from diamondoids
Tang Ta 1
Park Sulgiye 2
http://orcid.org/0000-0001-8072-9237
Devereaux Thomas Peter 34
http://orcid.org/0000-0001-5174-9546
Lin Yu lyforest@slac.stanford.edu

4
Jia Chunjing chunjing@phys.ufl.edu

5
1 https://ror.org/00f54p054 grid.168010.e 0000 0004 1936 8956 Department of Applied Physics, Stanford University, 348 Via Pueblo, Stanford, 94305 CA USA
2 https://ror.org/00f54p054 grid.168010.e 0000 0004 1936 8956 Department of Earth and Planetary Sciences, Stanford University, 367 Panama Mall, Stanford, 94305 CA USA
3 https://ror.org/00f54p054 grid.168010.e 0000 0004 1936 8956 Department of Materials Science and Engineering, Stanford University, 496 Lomita Mall, Stanford, 94305 CA USA
4 https://ror.org/05gzmn429 grid.445003.6 0000 0001 0725 7771 Stanford Institute for Materials and Energy Sciences (SIMES), SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, 94025 CA USA
5 https://ror.org/02y3ad647 grid.15276.37 0000 0004 1936 8091 Department of Physics and Quantum Theory Project, University of Florida, 2001 Museum Road, Gainesville, 32611 FL USA
1 9 2024
1 9 2024
2024
7 1948 11 2023
29 7 2024
© The Author(s) 2024
2024
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Diamondoids are a class of organic molecules with the carbon skeletons isostructural to nano-diamond, and have been shown to be promising precursors for diamond formation. In this work, the formation of diamond crystals from various diamondoid molecule building blocks was studied using our developed molecular geometry specific Monte Carlo method. We maintained the internal carbon skeletons of the diamondoid molecules, and investigated how the carbon-carbon bonds form between diamondoid molecules and how efficient the process is to form diamond crystals. The simulations show that higher diamondoid molecules can produce structures closer to a diamond crystal compared with lower diamondoid molecules. Specifically, using higher diamondoid molecules, larger bulk diamond crystals are formed with fewer vacancies. The higher propensity of certain diamondoids to form diamond crystals reveals insights into the microscopic processes of diamond formation under high-pressure high-temperature conditions.

Diamondoids are a series of hydrogen-terminated nanometer-sized hydrocarbons that can be used to synthesize high-quality diamond crystals. Here, the authors use Monte Carlo simulations to study the potentials of different diamondoids in constructing diamond crystals with the assumption that the carbon skeletons keep intact, and find that higher diamondoid molecules are most suitable.

Subject terms

Atomistic models
Molecular self-assembly
Solid-state chemistry
Chemical synthesis
100006151 DOE | SC | Basic Energy Sciences (BES) No. DE-AC02-76SF00515 100006151 DOE | SC | Basic Energy Sciences (BES) No. DE-SC0022216 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Diamondoids are a series of hydrogen-terminated nanometer-sized hydrocarbons where the carbon skeletons represent the nano-diamond structures1. Various diamondoid molecules, including the lower diamondoids (adamantane C10H16, diamantane C14H20, and triamantane C18H24) and higher diamondoids (C22H28 and higher polymantanes), have been utilized to synthesize diamond crystals using high pressure high temperature (HPHT)2–6 or chemical vapor deposition (CVD)5,7–9 methods, in the quest of finding better materials to make higher quality diamond crystals at a reduced cost.

In the HPHT experiments, diamond from lower diamondoid molecules forms at much lower temperatures compared with conventional (hydro)carbon allotropes at a comparable pressure, suggesting a reduced transformation energy barrier, owing to the structural similarities between diamondoids and the bulk diamond2. Formation of diamond crystals from diamondoids at HPHT usually involves hydrogen dissociation2. The hydrogen-stripped carbon skeletons then reorient to form carbon-carbon bonding that builds into larger diamond crystals. The diamondoids-to-diamond formation can be seen as a three-dimensional origami process for the carbon skeletons. Since different diamondoid molecules have different geometries of the carbon skeletons, specific types of diamondoids can offer the advantage of constructing diamond crystals without supplemental breaking of internal carbon-carbon bonds; while other types of diamondoid molecules form lower quality diamond with increased carbon vacancies due to increased necessity to reorient the carbon skeletons.

In order to aid in the experimental search for the optimal candidates and conditions for diamond synthesis, we performed a Monte Carlo simulation to study the potentials of different diamondoids in constructing diamond crystals. Six diamondoid molecules are studied: adamantane C10H16, diamantane C14H20, triamantane C18H24, [121]tetramantane C22H28, [1(2,3)4]pentamantane C26H32, and [1212]pentamantane C26H32. Our Monte Carlo results show that higher diamondoids have a higher efficacy to form diamond crystals compared with lower diamondoids, given the carbon skeleton for each diamondoid molecule remains intact without breaking internal C-C bonds during materials synthesis.

Results

Monte Carlo simulations were conducted on six diamondoid molecules, including adamantane, diamantane, triamantane, [121]tetramantane, [1(2,3)4]pentamantane, and [1212]pentamantane, with the final structure of diamond formation using [1(2,3)4]pentamantane shown in Fig. 1. For all the simulations, we used an NPT-ensemble. Specifically, we set the initial temperature Ti = 0.12Eb ≈ 5000 K, final temperature Tf = 0.024Eb ≈ 1000K, and pressure P = 0.012Eb ≈ 10GPa. We set the C-C bond formation energy Eb = 3.6eV. In each set of the simulation for different diamondoid molecules, we kept the total number of carbon atoms approximately the same (see Table 1).Fig. 1 The final structure of diamond formation using the [1(2,3)4]pentamantane molecule in our simulation, visualized in three different perspectives.

The blue atoms highlight one [1(2,3)4]pentamantane molecule.

Table 1 Number of diamondoid molecules used at each Monte Carlo simulation with a fixed total number of carbon atoms set to 1000, 5000, and 10000, respectively

	Number of molecules M	
#C	#Ada	#Dia	#Tria	#Tetra	#Penta	
1000	100	71	56	45	38	
5000	500	357	278	227	192	
10000	1000	714	556	455	385	
#C stands for the number of carbon atoms, #Ada stands for the number of adamantane molecules, #Dia stands for the number of diamantane molecules, etc.

For each set of parameters, we performed ten independent simulations until the newly formed C-C bond numbers are converged. Figure 2a, b visualize an representative structural update for increasing bond numbers before and after one Monte Carlo step. Figure 2c shows the C-C bond number as a function of the Monte Carlo step for adamantane of total 5000 carbon atoms on ten independent Monte Carlo simulations. The C-C bond number for each Monte Carlo run follows a trend, in which the bond number increases quickly at the initial stage during heating, after which it slows down and saturates, suggesting the convergence of structural optimizations. Figures 1 and 3 show a few representative examples of the final diamond structures formed from diamondoid molecules. The C-C bond number convergence for each diamondoid molecule and different total numbers of carbon atoms are shown in Fig. S1.Fig. 2 Schematics of the Monte Carlo processes and the bond number convergence in the simulations.

Schematics of the (a) old and (b) new configurations after one Monte Carlo update. The picked molecule M0 (blue) form bonds with the other carbon atoms in the cluster (grey), with one more bond being formed in the new configuration. c Convergence of the structural optimization characterized by the saturation of the C-C bond number. The simulations for adamantane with a total number of 5000 carbon atoms have been shown. The ten lines in the plot show the number of C-C bonds formed among adamantane molecules during ten independent Monte Carlo simulations. Each Monte carlo simulation consists of 2 million Monte Carlo steps of heating at Ti = 5000K, 2 million Monte Carlo steps of annealing with a linear temperature change from Ti = 5000 K to Tf = 1000 K, and 10 million Monte Carlo steps of cooling at Tf = 1000 K.

Fig. 3 The initial and final structures with a total number of carbon atoms set to 1000.

The first row are the structures of the diamondoid molecules. The second row are the randomly initialized structures, and the third row are the final structures after Monte Carlo simulations. Column (a–f) corresponds to the six diamondoid molecules we studied.

Once the structures have converged, we characterized the final structure’s crystalline quality by first counting the number of vacancies (see Fig. 4a). Here, vacancies are defined as empty carbon sites inside the final bulk diamond-like structure. We find that for #C = 10,000, the number of vacancies decreases as we move from lower diamondoids to higher diamondoids; and [1(2,3)4]pentamantane forms fewer vacancies compared with [1212]pentamantane.Fig. 4 Efficacy of diamond crystal formation from various diamondoid molecules as revealed by our Monte Carlo simulations.

a Number of vacancies normalized by the total number of carbon atoms for increasing number of carbon atoms 1000(green), 5000(blue) and 10000(red). The error bar shows the standard deviation of ten independent Monte Carlo simulations. Lower number of vacancies means higher crystal quality. b Ratio of surface sites that are compatible with diamond crystal structure for increasing number of carbon atoms 1000(green), 5000(blue) and 10000(red). The error bar shows the standard deviation of ten independent Monte Carlo simulations. A higher ratio means a faster growth rate.

To further assess the diamondoid molecules’ ability to grow into larger diamond crystals, we calculated the ratio of compatible surface sites. We defined the compatibility as the ability for a diamondoid molecule to attach to a surface site in a manner that is compatible with the diamond crystal structure. The compatible ratio is obtained by dividing the number of compatible sites to the total number of surface sites. In Fig. 4b, we see that such a ratio is higher for higher diamondoids, and [1(2,3)4]pentamantane is better than [1212]pentamantane. A higher ratio means a higher probability to add diamondoids onto the surface sites, thus a faster growth rate. We note that the ratio decreases for all diamondoids as we increase the total number of carbon atoms, and when the ratio reaches zero, the diamond structure cannot grow any further. These results suggest that higher diamondoids can grow faster and form larger diamonds with fewer defects in the bulk.

Discussion

Our Monte Carlo simulations show that for a fixed number of total carbon atoms, higher diamondoid molecules form a diamond crystal with fewer vacancies inside as-formed structures, and more compatible sites on the surface to potentially form a larger diamond crystal, compared to lower diamondoid molecules. Both findings suggest that higher diamondoids have a higher efficacy in forming a diamond crystal than lower diamondoids.

Our theoretical trends align consistently with the experimental results of diamond crystal formation from lower diamondoids using the HPHT method2, where diamond could be more easily formed from triamantane than from adamantane and diamantane, with a lower critical temperature at a given pressure. Such consistency may be attributed to the fact that the internal C-C bonds of lower diamondoids have a lower likelihood of dissociating at an applied high pressure and high temperature. In diamond synthesis using the HPHT method with adamantane, ab initio molecular dynamics simulations have indicated that part of the C-C skeleton can remain intact during the transformation to diamond2. However, whether the C-C skeleton will predominantly remain intact for larger diamondoids during diamond formation using the HPHT method has not been investigated. To further address this issue, theories utilizing large-scale molecular dynamics simulations10 may be required, where the internal carbon-carbon bond breaking is properly integrated. The different results of diamond growth using diamondoids under HPHT and CVD7 conditions further emphasize that crystallization during these two methods intrinsically follows very different microscopic mechanisms; thus finding an optimal crystal growth method can be critical to materials synthesis and advanced materials design. Fundamental understanding of these dynamical processes associated with the CVD and HPHT diamond synthesis from diamondoids can further guide the bottom-up synthesis of diamond with desirable defect centers from doped molecular precursors11–15. The numerical approach we present here can potentailly be applied to a broad variety of systems for materials synthesis, such as self-assembly, HPHT and CVD processes, and more.

Methods

When discussing algorithms used for modeling crystal growth, kinetic Monte Carlo emerges as a commonly employed method, such as its early application in atomic modeling of surface growth of GaAs using molecular beam epitaxy16. Additionally, Monte Carlo simulations of ‘units of growth’, which involve space-filling tiles or Voronoi polyhedra, present a coarse-grained alternative. This approach has proven applicable to a wide range of crystal types, encompassing porous crystalline materials, metal-organic frameworks, and ionic crystals17. On the other hand, in contexts such as high-pressure materials synthesis, where the focus lies solely on the final structure rather than the dynamical process of materials formation, commonly utilized approaches include genetic algorithms for molecular crystals18 and evolutionary algorithms for a broad range of materials systems19,20.

In our study, we implement Monte Carlo simulations to examine the optimal structures formed from various diamondoid molecules. Constraints were imposed on the internal C-C bonds within each molecule, while those C-C bonds formed between the two molecules ‘were set to’ have the same bond angle and length as those within the bulk diamond lattice. Ab initio simulations predicted that the C-H bond dissociation energy was lower than that of the C-C bonds, so that dehydrogenation occurred prior to the C-C bond dissociation—an assumption shown reasonable for the formation of diamond using adamantane molecules2. Therefore, in our Monte Carlo simulations, only the carbon atoms and the C-C bond formation between diamondoid molecules are considered. Our algorithm can be regarded as a specific implementation of kinetic Monte Carlo, by taking advantage of the geometry of the diamondoids molecules and the isostructural carbon skeleton of diamondoids with nano-diamond. Here, the ‘unit of growth’ aligns with the carbon skeleton of the diamondoid molecules. Simultaneously, the problem is approached atomically, with each carbon occupying a site in a diamond lattice and the resulting C-C bonds being treated at the atomic level. Our algorithms leverage both the concept of the ‘unit of growth’ and atomic modeling methods, enabling efficient handling of large systems.

The simulations were conducted within a closed box with dimensions of L × L × L (L = 40) in unit of diamond unit cell. To initiate the simulations, M molecules were randomly placed within the box. In order to expedite the initialization process, the i-th molecule was initialized to form at least one bond with the existing structures created by the preceding i-1 molecules. This initialization process ensured that the M molecules formed a connected structure.

Algorithm 1

Each Monte Carlo step in our molecular geometry specific simulation

V ← (xmax − xmin) × (ymax − ymin) × (zmax − zmin), where, for example, xmax is the maximum x coordinate of all the carbon atoms in the cluster.

S0← surface of the cluster. Surface consists of carbon atoms, which form <4 bonds with other carbon atoms.

Randomly pick a molecule M0 such that M0∩S0≠∅ and remove it.

n0 ← number of bonds broken when removing M0

Enn← empty nearest neighbor sites of the cluster after M0 is removed, i.e., for any atom that remains in the cluster, all its empty nearest neighbor sites belong to Enn

Randomly pick an empty carbon atom site C1∈Enn. This guarantees the carbon atom with more surrounding empty sites has a higher probability to be selected to potentially form a C-C bond with atoms in M0.

counter  ← 0

succeed  ← False

(We will attempt “MAX_TRY = 10” times to put the molecule at the new position. “counter” records how many times we have tried. At each try, a random atom in M0 is chosen to be put at C1 and a random orientation of M0 around C1 is chosen. The possible orientations are restricted to those such that the C-C bonds of M0 align with those of the diamond lattice of the cluster).

repeat

Randomly pick an atom C0 ∈ M0.

Try to put C0 at the position of C1 and form bonds between C0 and the existing atoms in the cluster

if Bonds can form s.t. M0 does not overlap with existing atoms in the cluster then

succeed  ← True

end if

counter  ← counter  + 1

until succeed ∣∣ counter ≥ MAX_TRY

if succeed then

n1 ← number of bonds formed after putting M0 to the new position

ΔNb ← n1 − n0

Calculate the new volume V and volume difference ΔV

p ← random number between 0 and 1

if p<exp(−β(−EbΔNb+PΔV)) then

M0 jump to the new position

end if

end if

if M0 does not jump to a new position then

put M0 back to the original position

end if

In each Monte Carlo step, we randomly select a diamondoid molecule to possibly update its position. The update of this diamondoid molecule position, or we call it a ‘movement’, breaks old C-C bonds and forms new C-C bonds with the remaining carbon atoms. (See Algorithm 1 for details) We calculated the enthalpy difference ΔH = -EbΔNb + PΔV before and after the movement of this diamondoid molecule, where ΔNb is the number of net carbon-carbon bonds formed between diamondoid molecules (if fewer C-C bonds formed than the C-C bonds breaking, ΔNb would be negative), Eb is the carbon-carbon bond formation energy, P represents the pressure, and V is the volume of the minimal cubic box that contains all the diamondoid molecules. To optimize the sampling efficiency of potential newly formed C-C bonds, we employ a weighted sampling approach for atoms in diamondoid structures as well as the subsequently formed connections. This molecular geometry specific approach ensures that C atoms surrounded by more empty sites are assigned higher weights during the sampling process. (See Algorithm 1 for details). The probability for which we accept the movement is exp(−βΔH), where β = 1/kBT. When exp(−βΔH)≥1, we accept this movement; when 0<exp(−βΔH)<1, we accept this movement with the probability exp(−βΔH). The initial structure was further perturbed by setting a high initial temperature Ti, compared to the C-C bond dissociation energy. The temperature was then gradually reduced to a final low temperature Tf, at which we performed enough Monte Carlo simulations until the structure was converged. Specifically, 2 million Monte Carlo steps are performed at Ti, then temperature is linearly lowered to Tf in 2 million Monte Carlo steps. Finally, 10 million Monte Carlo steps are performed at Tf.

Supplementary information

Peer Review File

Supplementary Information

Supplementary information

The online version contains supplementary material available at 10.1038/s42004-024-01261-9.

Acknowledgements

We would like to thank Ting Ge and Shuo Li for insightful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract No. DE-AC02-76SF00515. Parts of the computing for this project were performed on the National Energy Research Scientific Computing Center (NERSC), a US Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.

Author contributions

C. Jia, Y. Lin and T. Tang designed the project. T. Tang developed the codes and ran the simulations. C. Jia, Y. Lin, T. Tang and S. Park made the figures. T. Tang, S. Park, T. P. Devereaux, C. Jia, and Y. Lin contributed to the writing of the manuscript.

Peer review

Peer review information

Communications Chemistry thanks Qiang Zhu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

Data supporting the findings of the current study are available within the article and from the authors upon request.

Code availability

The source code for our Monte Carlo simulations for diamondoids is accessible at github repository: https://github.com/taaatang/Diamondoid.git.

Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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