[Paper Review] Defining graphenic crystallites in disordered carbon: moving beyond the platelet model

Kate J. Putman, Nigel A. Marks+4 more|arXiv (Cornell University)|Dec 13, 2022

Graphene research and applications4 citations

TL;DR

This study redefines graphenic crystallites in disordered carbon using large-scale atomistic simulations (1 million atoms) that reveal a continuous, curved 3D network of hexagonal carbon rings, moving beyond the traditional platelet model. It shows that XRD-derived $L_{\mathrm{a}}$ is skewed toward larger sizes, with over 98% of crystallites smaller than $L_{\mathrm{a}}$, and more than 75% smaller than half of $L_{\mathrm{a}}$, challenging long-standing interpretations of XRD data.

ABSTRACT

We develop a picture of graphenic crystallites within disordered carbons that goes beyond the traditional model of graphitic platelets at random orientation. Using large atomistic models containing one million atoms, we redefine the meaning of the quantity La extracted from X-ray diffraction (XRD) patterns. Two complementary approaches are used to measure the size of graphenic crystallites, which are defined as regions of regularly arranged hexagons. Firstly, we calculate the X-ray diffraction pattern directly from the atomistic coordinates of the structure and analyse them following a typical experimental process. Second, the graphenic crystallites are identified from a direct geometrical approach. By mapping the structure directly, we replace the idealised picture of the crystallite with a more realistic representation of the material and provide a well-defined interpretation for $L_a$ measurements of disordered carbon. A key insight is that the size distribution is skewed heavily towards small fragments, with more than 75% of crystallites smaller than half of $L_a$.

Motivation & Objective

• To redefine the structural meaning of $L_{\mathrm{a}}$ extracted from XRD patterns in disordered carbon, moving beyond the outdated platelet model.
• To resolve the long-standing ambiguity in interpreting $L_{\mathrm{a}}$ by linking it to a realistic, geometry-based definition of graphenic crystallites.
• To clarify the relationship between XRD measurements and actual crystallite size distributions in disordered carbon.
• To demonstrate that graphenic crystallites are not isolated platelets but part of a continuous, curved 3D network with in-plane order decoupled from stacking.

Proposed method

• Generation of large-scale atomistic models (up to 1 million atoms) of disordered carbon using annealed molecular dynamics to self-assemble a 3D interconnected network of curved graphene-like sheets.
• Direct calculation of X-ray diffraction (XRD) patterns from atomistic coordinates using the full Debye scattering equation, following standard experimental analysis procedures.
• Geometric identification of graphenic crystallites via a direct search for regularly arranged hexagonal rings in the 3D structure, defining them as regions of in-plane order.
• Comparison of XRD-derived $L_{\mathrm{a}}$ with the actual size distribution of crystallites identified geometrically to assess the validity and interpretation of $L_{\mathrm{a}}$.
• Analysis of crystallite shape, size distribution, and eccentricity to characterize the morphology of graphenic regions.
• Systematic variation of density (0.5–2.0 g/cc) to test the robustness of findings across different disordered carbon structures.

Experimental results

Research questions

• RQ1 How does the XRD-derived $L_{\mathrm{a}}$ relate to the true size distribution of graphenic crystallites in disordered carbon?
• RQ2 To what extent do traditional platelet models misrepresent the actual morphology and connectivity of graphenic regions in disordered carbon?
• RQ3 Are graphenic crystallites confined to stacked regions, or can they exist independently in non-stacked, curved layers?
• RQ4 What is the actual shape and size distribution of graphenic crystallites when defined geometrically from atomistic models?
• RQ5 How does curvature in the carbon network influence the interpretation of $L_{\mathrm{a}}$ from XRD data?

Key findings

• XRD-derived $L_{\mathrm{a}}(10)$ is significantly larger than the arithmetic mean of the crystallite size distribution, with the mean crystallite size estimated at ~7 Å compared to $L_{\mathrm{a}}(10)$ of 19–22 Å.
• More than 98% of graphenic crystallites are smaller than the $L_{\mathrm{a}}(10)$ value extracted from XRD, indicating that $L_{\mathrm{a}}$ is not representative of the majority of crystallites.
• Over 75% of crystallites are smaller than half of $L_{\mathrm{a}}(10)$, highlighting a strong skew in the size distribution toward small fragments.
• Graphenic crystallites are predominantly circular, with ~73% having a pseudo-eccentricity $\varepsilon < 0.3$, indicating near-circular morphology.
• The in-plane ordering (graphenic crystallites) is not confined to stacked regions; crystallites exist outside stacking zones, and stacking can occur in non-graphenic layers.
• The study confirms that $L_{\mathrm{a}}$ is dominated by intensity weighting from larger crystallites, explaining why it overestimates the typical crystallite size in disordered carbon.