You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We use [*ab initio*]{} density functional calculations to study hydrogen-induced disintegration of single- and multi-wall carbon fullerenes and nanotubes. Our results indicate that hydrogen atoms preferentially chemisorb along lines in sp$^2$ bonded carbon nanostructures, locally weakening the carbon bonds and releasing stress. For particular structural arrangements, hydrogen helps to relieve the accumulated stress by inducing step-wise local cleavage leading to disintegration of the outermost wall.'
author:
- Savas Berber
- David Tománek
title: |
    Hydrogen-induced disintegration of fullerenes and nanotubes:\
    An *ab initio* study 
---

Exposure to hydrogen is known to cause embrittlement in metals[@DT067], a major concern in materials science. It is an intriguing question, whether local decohesion due to chemisorbed hydrogen may cause similar damage in sp$^2$ bonded carbon nanostructures, such as fullerenes and nanotubes[@Dresselhaus96], and whether it may have helped to efficiently cleave graphene[@Dairibbon08]. So far, past research has focussed on beneficial side-effects of the interaction of hydrogen with nanocarbons, including easier separation/debundling and the prospect of reversible hydrogen storage. Since both fullerenes and nanotubes are under internal stress due to the local curvature, hydrogen-induced local decohesion may facilitate fracture and cause irreversible damage in nanocarbons as it does in metals[@DT067].

Here we present an [*ab initio*]{} density functional study of bonding and structural changes in sp$^2$ carbon nanostructures exposed to hydrogen. Our results indicate that hydrogen atoms preferentially chemisorb along lines on the outermost wall of single- and multi-wall carbon fullerenes and nanotubes, locally weakening the carbon bonds and releasing stress. For particular arrangements of adsorbed hydrogen atoms, we identify the microscopic process of step-wise local cleavage leading to the disintegration of the outermost wall as a way to release the accumulated stress.

To gain fundamental insight into the effect of hydrogen on the structural integrity of sp$^2$ bonded carbon nanostructures, we studied the total energy and optimum structure of pristine and hydrogenated fullerenes and nanotubes. Our results are based on the density functional theory (DFT) within the local density approximation (LDA) and first-principles pseudopotentials. We used primarily the SIESTA[@soler2002] code with a double-$\zeta$ basis set with polarization orbitals, the Perdew-Zunger[@Perdew81] parametrization of the exchange-correlation functional, and norm-conserving [*ab initio*]{} pseudopotentials[@Troullier91] in their fully separable form[@Kleinman82]. The accuracy of our results was verified using the PWSCF plane-wave code with ultrasoft pseudopotentials. We used $10^{-2}$ eV/[Å]{} as a strict gradient convergence criterion when determining optimum geometries. Constrained structure optimization has been used to investigate energy barriers, transition states, and generally to identify the optimum reaction pathway.

Published theoretical results include studies of the bonding character of isolated H atoms on graphite flakes, nanotubes and fullerenes[@jeloaica99; @DT193; @DT198] and the preferential arrangement of hydrogen pairs on a graphene layer. Other investigations address the relative stability of hydrogen pairs for different adsorption geometries [@Crespi03] and coverages[@allouche06b] of hydrogen on graphene. Theoretical studies of hydrogen-covered nanotubes focussed on adsorbate-related electronic structure changes[@GulserenPRB02] and the relative stability of different adsorption arrangements. More recent studies considered the possibility of axial cleavage in nanotubes in the presence of exohedrally adsorbed H atoms. No results have been reported so far about the origin of hydrogen-induced structural damage in fullerene-like structures and its relation to stress-induced cleavage of nanotubes.

Same as in graphene, each carbon atom in fullerenes and nanotubes has three neighbors to form very stable sp$^2$ bonds. Unlike in planar graphene, the most stable sp$^2$ carbon allotrope, stress is associated with the nonzero local curvature in fullerenes and nanotubes. The small fullerenes C$_{20}$ and C$_{60}$ are spherical, with all atoms equivalent and subject to the same stress. With increasing size, however, free-standing fullerenes start resembling icosahedra, with twenty near-planar triangular facets spanning the surface. In these structures, stress associated with higher local curvature concentrates near the thirty edges connecting adjacent pentagons. In carbon nanotubes, on the other hand, all atoms are equivalent and the stress associated with local curvature is distributed uniformly throughout the structure.

![(Color online) Equilibrium structure and bonding character in H-covered fullerenes and nanotubes. (a) Hydrogen chemisorption on the C$_{960}$ fullerene along strained lines with larger curvature. (b) Lines of chemisorbed hydrogen, shown in end-on and side view, yield the most stable adsorption on the (6,6) carbon nanotube. (c) Relaxed atomic structure near a line of chemisorbed hydrogen atoms. (d) Difference charge density $\Delta{\rho}$ indicating hydrogen-induced sp$^2\!\!{\rightarrow}$sp$^3$ local bonding change in graphitic nanocarbons. Contours of $\Delta{\rho}$ in the plane indicated in (c) are superimposed with the atomic lattice.\[Fig1\]](Fig1){width="0.9\columnwidth"}

Even though fullerenes with more than 100 atoms have not been observed, large icosahedral shells are believed to form the outermost shell of multi-wall fullerenes, called bucky onions[@Ugarte-onion-Nature92]. Focussing on the faceted C$_{960}$ hollow shell structure in Fig. \[Fig1\](a), increased stress makes edge atoms more reactive and prone to bind hydrogen. Molecular hydrogen does not dissociate on planar graphene, but does so with an energy gain on fullerenes, with the dissociated hydrogen pair preferentially binding on top of adjacent carbon atoms. In the following, we will define adsorption energies per hydrogen pair, related to the free H$_2$ molecule. As expected, the energy gain depends on the local stress in the substrate, with values ranging from $1.20$ eV at the twelve pentagonal corners to $1.35$ eV at the edge sites and $0.34-0.67$ eV on the facets of the C$_{960}$ fullerene.

Calculations at higher coverages suggest that after the first two atoms are adsorbed, additional hydrogens should preferentially adsorb along straight lines on the substrate[@DT193]. We believe that formation of this energetically preferred adsorption pattern can be achieved by surface diffusion of hydrogen atoms at temperatures found during many hydrogenation reactions, since the activation barrier for sigmatropic rearrangement of chemisorbed hydrogen atoms is only ${\approx}1$ eV[@DT193]. As suggested earlier and illustrated in Fig. \[Fig1\](c), carbon atoms at icosahedral edges in large fullerene cages are more reactive than the facet sites. Since the dissociative chemisorption energy of hydrogen along the edges exceeds that on fullerene facets by ${\approx}0.7$ eV per H$_2$ molecule, we expect hydrogen atoms to adsorb preferentially along lines connecting pentagon corners.

Even though there are no preferential adsorption sites on a carbon nanotube, atomic hydrogen is believed to adsorb preferentially along lines parallel to the tube axis to release stress. Possible initial adsorption geometries for hydrogen on the $(6,6)$ carbon nanotube are shown in Fig. \[Fig1\](b). In the left panel, we depict the optimum nanotube geometry following hydrogen adsorption along three zigzag lines parallel to the axis. The calculated binding energy of 0.54 eV per hydrogen pair reflects also the energy associated with the tube relaxation to a triangular cross-section. Presence of hydrogen apparently locally reduces the flexural rigidity of the nanotube wall, converting its circular cross-section into a triangle. Other favorable local adsorption patterns include exo- or endohedrally adsorbed hydrogens forming double-lines along the nanotube axis. The initial geometry of such a combined exo/endohedral adsorption arrangement is shown in the middle panel, and an exohedral double-line arrangement of hydrogens is shown in the right panel of Fig. \[Fig1\](b). As we discuss later on, hydrogen-induced local decohesion causes significant structural rearrangements in both cases. The energy gain associated with hydrogen chemisorption was found to be $0.87$ eV per hydrogen pair in this case.

To obtain fundamental insight into the origin of hydrogen-induced local decohesion that is general and independent of a particular graphitic structure, we studied the effect of a line of chemisorbed hydrogen atoms on the structure and bonding in a graphene layer, as shown in Fig. \[Fig1\](c). Hydrogen atoms bind preferentially on top of carbon atoms, causing a local buckling. A line of chemisorbed hydrogen atoms is expected to form a crease in the initially planar graphene[@Dairibbon08]. In Fig. \[Fig1\](c) we constrained the crease angle to equal that between adjacent facets on the icosahedron in Fig. \[Fig1\](a). To get insight into the nature of the adsorption bond, we inspected the charge density difference ${\Delta}{\rho}=\rho{\rm
(H/graphene)}-\rho{\rm (H)}-\rho{\rm (graphene)}$, representing the charge redistribution in the system upon hydrogen adsorption. The spatial distribution of ${\Delta}{\rho}({\bf r})$ is shown in Fig. \[Fig1\](d) in the plane schematically indicated in Fig. \[Fig1\](c). Hydrogen adsorption causes only local charge rearrangement leading to a charge accumulation in the H-C $\sigma$ bonds, depleting the charge in the local C-C bond region, as expected in case of sp$^3$ bonding. Both the charge rearrangement and the local buckling are strong indicators for a local sp$^2{\rightarrow}$sp$^3$ transition. As suggested above, hydrogen pairs bind preferentially to neighboring carbon atoms, thereby changing an initially strong sp$^2$ bond to a weaker, strained sp$^3$ bond. This local decohesion resembles to some degree the microscopic mechanism of hydrogen embrittlement in some metals[@DT067].

Besides decohesion, local change from sp$^2$ to sp$^3$ bonding is also associated with a net increase in the C-C bond length by typically 8%. In partially hydrogen-covered systems, carbon atoms connected to lines of hydrogen are under stress. The consequences can be best illustrated in the fullerene structure shown in Fig. \[Fig1\](a), where hydrogen only attaches to the edges. The hydrogen-free planar facets can be thought as incompressible, same as small graphene flakes. Since the corners of the triangular facets determine the edge length, sp$^3$ bonded carbon atoms along the edges are prevented from increasing their interatomic distance, subjecting the edges to stress. The energy associated with accumulated stress increases linearly with the edge length $L$. For a critical value $L_{crit}$, this energy will exceed the activation barrier ${\Delta}E$ to break a C-C bond as a first step to initiating a cleavage that eventually causes the disintegration of the structure. As we show later on, we find that ${\Delta}E{\approx}1.7$ eV. This amount of stress energy is accumulated in a hydrogenated fullerene edge of critical length $L_{crit}{\approx}15$ nm. We thus expect hydrogen-induced cleavage to occur only in fullerenes with $L>15$ nm, with a diameter exceeding ${\approx}28$ nm and containing more than $10^5$ C atoms. Such large fullerenes never occur as single-wall structures, but may form the outermost wall of a multi-wall bucky onion. In large bucky onions, the spherical walls inside exert radial pressure on the outermost wall, thus enhancing the hydrogen-induced stress and reducing the critical diameter for disintegration of the outermost wall. We expect that multi-wall fullerenes exposed to hydrogen should peel layer-by-layer from the outside in a process appearing as exfoliation.

![Zipper mechanism of hydrogen-induced disintegration of strained sp$^2$ carbon nanostructures. (a) Periodically warped graphene as model strained system. (b) Optimum structure and energetics for breaking a bond pair. (c) Sequence of bond cleavages resulting in a crack. \[Fig2\]](Fig2){width="0.75\columnwidth"}

In the following, we study the microscopic mechanism leading to cleavage at the minimum energy cost. To obtain a general understanding of the fracture process that does not depend on a particular system, we created generic creases by placing a periodic array of hydrogen lines along the armchair direction on alternating sides of an infinite graphene monolayer, as seen in Fig. \[Fig2\](a). The energetics and atomic arrangement during the initial cleavage are depicted in Figs. \[Fig2\](b) and \[Fig2\](c). Cleavage at the crease is caused by locally weakening the affected bonds. We found that this is best achieved by adsorbing additional hydrogen atom pairs on both sides of the initial hydrogen line, as shown in the left panel of Fig. \[Fig2\](c). Such a local arrangement of adsorbed hydrogens is a favorable prerequisite to cleave an edge segment. We found that the energy needed to break one bond is larger than the energy to break two adjacent bonds, shown in a dark color in the left panel of Fig. \[Fig2\](c). Continuing this concerted bond breaking mechanism should cleave the edge, causing the structure to disintegrate.

The energetics of cleavage in Fig. \[Fig2\](b) shows the total energy change with respect to the initial structure as a function of the bond length $d$ across the crack, which we use as the reaction coordinate. We consider the full range of $d$ values occurring during the cleavage reaction and, for each value of $d$, we globally optimize the structure. Consequently, the calculated transition paths for the disintegration of hydrogenated fullerenes and nanotubes represent the optimum cleavage path in unconstrained systems. Following an initial energy investment of $1.7$ eV, the bonds break abruptly, leading to the final structure depicted in the right panel of Fig. \[Fig2\](c). There is a net energy gain of ${\approx}2$ eV associated with the bond breaking, caused by releasing the accumulated stress. The initially stressed sp$^3$ bonded crease segment converts into a pair of more stable sp$^2$ bonded, hydrogen terminated graphene edges.

We next calculated the energy barriers associated with breaking specific bonds in search of the optimum pathway to complete the crease cleavage after it was initiated. The optimum sequence of bonds to break, along with the intermediate structures and the energetics of the process, is depicted in Fig. \[Fig2\](d). Interestingly, the smallest energy investment to propagate the fracture following the first step in Fig. \[Fig2\](b) does not involve the neighboring pair of C-C bonds on the opposite side of the ridge, but rather the more distant pair of C-C bonds on the same side of the ridge. Our results indicate that the energy barrier to propagate the fracture decreases at successive steps. As a matter of fact, it may be reasonable to assume that the energy released following the initial step may activate the zipper-like cleavage at the crease without further energy investment. Once cleavage is initiated, the unzipping process transforms the stressed crease into two overlapping graphene edges in an exothermic reaction.

![Hydrogen-assisted disintegration of (a) the C$_{240}$ fullerene and (b) the (6,6) carbon nanotube. Structural snap shots along the optimum reaction pathway are depicted in the left panels. The structural transformations of the nanotube in (b) are shown in end-on view. Carbon atoms are shown as grey, hydrogen atoms as larger black spheres. The right panels show the total energy change in the system, with the data points corresponding to the structures in the left panels. \[Fig3\]](Fig3){width="0.9\columnwidth"}

Having studied the microscopic cleavage process in the warped graphene model, we next turn to hydrogen-induced disintegration of specific fullerene and nanotube structures. As we discussed above, we expect hydrogen-induced disintegration to proceed spontaneously only in very large fullerenes, which are typically multi-wall structures. In the smaller C$_{240}$ fullerene, we find this reaction to be energetically activated. The energetics of the C$_{240}$ disintegration process, induced by hydrogen adsorption along the icosahedral edges, is shown in Fig. \[Fig3\](a) along with snap-shots of intermediate structures. Unlike in the model system addressed in Fig. \[Fig2\], the hydrogen coverage has been gradually increased as the cleavage progressed, as an alternative scenario to initiating unzipping. Both in the warped graphene model and in the C$_{240}$ fullerene, we find the disintegration process to be exothermic, following an initial energy investment of ${\approx}2$ eV.

Whereas large fullerenes are more susceptible to hydrogen-induced fracture due to the stress build-up in the hydrogenated edges, we find nanotubes that are narrow to be more susceptible to fracture due to the stress caused by their high uniform curvature. Both in fullerenes and nanotubes with multiple walls, hydrogen will first adsorb on the outermost wall, causing local decohesion leading to bond breaking and eventual disintegration of the outermost wall. Continuing exposure to hydrogen will cause a layer-by-layer peeling that will appear as exfoliation.

As suggested by our results in Fig. \[Fig1\](b), hydrogen adsorption concentrates the deformation along lines parallel to the tube axis, reducing the flexural stress in the hydrogen-free region. The right panels of Fig. \[Fig1\](b), depicting hydrogen covered nanotubes, illustrate how hydrogen-induced decohesion along adsorption lines may cause large structural changes. The detailed cleavage process of an infinitely long $(6,6)$ carbon nanotube, covered by a complete chain of hydrogens adsorbed exohedrally and a neighboring chain adsorbed endohedrally, is described in Fig. \[Fig3\](b). The sequence of snap shots, presented in an end-on view, depicts intermediate structures, including a scroll[@DT138], encountered during a conjugate gradient optimization from an unrelaxed initial structure. This process is only spontaneous if all hydrogens adsorb in the optimum way to form completely filled lines of adsorbed hydrogen, and is activated otherwise. The final structure with a horseshoe-like cross-section, stabilized by the attraction of the hydrogenated zigzag edges, may furthermore undergo an activated conversion to a planar graphene strip with hydrogen-terminated edges.

In conclusion, we used [*ab initio*]{} density functional calculations to study hydrogen-induced disintegration of single- and multi-wall carbon fullerenes and nanotubes. Our results indicate that hydrogen atoms preferentially chemisorb along lines in sp$^2$ bonded carbon nanostructures, locally weakening the carbon bonds and releasing stress. Contrary to common belief that chemisorbed hydrogen causes no harm to carbon nanostructures, we show that irreversible structural changes may be caused by chemisorbed hydrogen that can initiate local cleavage leading to the disintegration of the outermost wall. In experimental observations of multi-wall structures, this could be interpreted as hydrogen-induced exfoliation.

We thank Glen P. Miller for inspiring this study by his experimental observations in bucky onions. This work was funded by the National Science Foundation under NSF-NSEC Grant No. 425826 and NSF-NIRT Grant No. ECS-0506309. Computational resources have been provided by the Michigan State University High Performance Computing Center.

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