Mechanical Properties and Defect Sensitivity of Diamond

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Mechanical Properties and Defect Sensitivity of Diamond Nanothreads Ruth Roman, Kenny Kwan, and Steven Cranford Nano Lett., Just Accepted Manuscript • DOI: 10.1021/nl5041012 • Publication Date (Web): 18 Feb 2015 Downloaded from http://pubs.acs.org on February 21, 2015

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Nano Letters

Mechanical Properties and Defect Sensitivity of Diamond Nanothreads Ruth E. Roman1, Kenny Kwan1, and Steven W. Cranford1* 1

Laboratory of Nanotechnology In Civil Engineering, Department of Civil and Environmental Engineering, Northeastern University, Boston, MA 02115 *Corresponding author: [email protected]

Abstract: One of the newest carbon allotropes synthesized are diamond nanothreads. Using molecular dynamics, we determine stiffness (850 GPa), strength (26.4 nN), extension (14.9%), and bending rigidity (. × N-m2). The 1D nature of the nanothread results in a tenacity of . × N-m/kg, exceeding nanotubes and graphene. As the thread consists of repeating Stone-Wales defects, through steered molecular dynamics (SMD), we explore the effect of defect density on the strength, stiffness and extension of the system.

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Keywords: carbon allotropes; diamond nanothreads; mechanical properties; Stone-Wales

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defects; molecular dynamics

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The continuing focus of improvement and development of nanotechnology has unequivocally

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made carbon allotropes one of the key subjects of material science research. Their inherent

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molecular stability, in combination with promising electrical and thermal properties, spanning

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different forms of allotropes,1-10 have enticed the material science community. Recently, a new

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structural form of carbon was successfully synthesized from benzene precursors into a so-called

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diamond nanothread.11 The core of the nanothread is a long, thin strand of carbon atoms arranged

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in a diamond-like tetrahedral motif (a close-packed sp3-bonded carbon structure), the first

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member of a new class of sp3 materials created under high-pressure solid-state reactions.

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Uniquely, the presence of multiple covalent bonds across their cross-section removes them from

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the realm of traditional polymers while the lack of a central hollow disqualifies the term

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nanotube. Such threads represent a hybrid molecular structure, resembling a 3D molecular truss.

ACS Paragon Plus Environment

Nano Letters Mech. Prop. of Dia. Nanothreads

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The diamond thread is the result of controlled high-pressure conditions that force the reactions of

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benzene “chains” into an semi-ordered one-dimensional (1D) saturated nanomaterial that is

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formed by linked diamondoid-like structures (see Fig.1A; additional description in Supporting

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Information). Based on the matching between experimental and modeling work,11 the nanothread

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can be considered variation of a hydrogenated (3,0) nanotube12 with distributed Stone-Wales

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transformation defects (C-C dimers rotated by 90ο). Note that, due to the 1:1 C/H stoichiometric

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ratio, the thread can be thought of not as diamond (which is pure carbon), but rather a rolled

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sheet of hydrogenated graphene (or graphene), which has pure sp3 bonding and the necessary

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hydrogens. Due to the sp3 structure, the mechanical properties of such threads may be promising,

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but are currently unknown. Understanding of the mechanical behavior including limit states such

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as fracture stress and maximum strain is critical to explore the usability of such nanothreads as a

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nanomaterial, and to extend its use within nanobundles and/or nanofabrics. To this end, here the

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mechanical properties of the nanothread are explored considering an atomistically large model,

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using full atomistic molecular dynamics (MD).

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We apply a suite of in silico testing methods where full atomistic calculations of mechanical test

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cases is implemented to derive a set of parameters to characterize the diamond nanothread,

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similar to methods characterizing previous molecular systems.3-5, 13 We first confirm our model

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is structurally similar to the synthesized nanothreads. The axial stiffness as well as the

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strength/strain relationship are then investigated (via both quasi-static and dynamic steered

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molecular dynamic (SMD) approaches), bending rigidity is defined (via energy minimization

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molecular mechanics), and the mechanical effect of the Stone-Wales defects are scrutinized (via


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SMD). The atomistic simulations are performed using ReaxFF potential to describe interactions -

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an empirical bond-order-dependent potential that allows for fully reactive atomistic scale

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molecular dynamics simulations of chemical reactions, including bond rupture, in which

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dissociation and reaction curves are obtained from fitting to quantum chemistry calculations

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15

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of many carbon systems. 16-37

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First, to assess the stability of the nanothread and the ReaxFF potential, we compare the energy

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of the system to the potential energy of a representative sheet of graphane (e.g., hydrogenated

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graphene). Comparative analysis of the potential energies indicate a minimal difference in the

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potential energy (less than 1%) per carbon atom, on the order of the energy change from rolling

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graphene to a CNT and suggesting a stable allotrope (see Supporting Information). To confirm

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our model is representative of previous synthetic efforts, we first assess the molecular structure

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by calculating the radial distribution function (RDF), g(r), of a short equilibrium simulation (1

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ns) at finite temperature (300K). We use a model thread with a length of approximately 8 nm.

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The RDF is plotted in Fig. 1B, and matches previously published results.11 The observed nearest-

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neighbor carbon–carbon distance of approx. 1.52 Å is characteristic of dominant sp3 bonding,

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while the dominant peak at approx. 1.1 Å is consistent with the proposed 1:1 C/H stoichiometry.

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Successive peaks are due to second nearest C-C and C-H correlations, as well as H-H pairs. The

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peaks are more distinct in this model compared to experimental results11 due to the analysis of a

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single thread – interthread pairwise correlations can result in slight variations and broadening

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due to thread endcaps, cross-links, etc. The inclusion of a finite number of Stone-Wales defects

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is necessary to produce the thread structure and representative RDF. To further explore the local

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and global effects of the Stone-Wales defects, we consider four different structurally

14,

. Previous studies have successfully implemented ReaxFF for the behavior and characterization


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representative models containing 1 to 4 defects, resulting in only nominal variations in the

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characterization. Thus, each of the models can be considered structurally representative.

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However, energetic analysis indicates that adding a defect increases the potential energy of the

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thread by approximately 12 kcal/mol.

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We next apply uniaxial tensile strain along the thread axis to determine both stiffness and

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maximum strength/strain. A thread with two Stone-Wales defects is chosen as the representative

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model system (we note that two defects is necessary to produce a “unit segment” thread

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structure; see Supporting Information). Using a quasi-static approach, cycles of incrementally

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applied strain and minimization are undertaken (see Supporting Information). Subject to

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strain, stress is calculated using a virial stress formulation, and can be plotted (see Fig. 2A). The

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traditional definition of stress is difficult on quasi-one-dimensional materials due to the

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ambiguity of cross-section definition. To alleviate this ambiguity, we also report an equivalent

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1D stress (units of force), which are independent of cross-section, and thus easier to compare

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across other 1D systems (see Supporting Information). The process is repeated until failure is

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incurred, resulting in the maximum stress (134.3 GPa or 26.4 nN) and strain (14.9%). Once the

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stress and strain are determined, the axial stiffness can then be simply calculated by:

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= ∆⁄∆

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To maintain the assumption of linear elasticity, the stiffness is determined using a linear fit of the

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stress v. strain data up to = 4%. This range was selected to fit the initial linear regime of stress-

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strain, as there is a slight stiffening behavior beyond approximately 5% strain. We also note that

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due to the ambiguity of the cross-section of the nanothread (e.g., a 0.5 nm diameter,

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approximated by the van der Waal radii of opposite hydrogen atoms), the label of ‘‘modulus’’ is

(1)


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only intended as a placeholder for the tensile stiffness. The continuum interpretation of modulus,

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E, is not reflective of such threads, and is merely used here as a convenient convention for

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comparison. Using this approach, the stiffness of the nanothread is approximately 850 GPa (167

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nN). In comparison to CNTs, we note that the stiffness of the nanothreads is ≈85% that of the

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theoretical value of 1 TPa.38 As an alternate calculation of stiffness, we also plot the potential

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energy density, U, versus strain (Fig. 2B). This enables a direct future comparison to ab initio

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energy methods, such as DFT. The stiffness can then be expressed as:

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= ⁄

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Fitting the MD data with all terms of a fourth-order polynomial (accounting for nonlinearities)

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yields a tensile stiffness of E = 774 GPa (152 nN) at = 2% (e.g., midpoint of the previous fit),

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which is within 10% of the stiffness attained by the virial stress-strain approach, indicating

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consistency between the two methods. Discrepancy can be attributed to the fact that only axial

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stress was accounted for in the virial approach, neglecting higher order effects from shear

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contributions, for example. A fourth-order fit was chosen by increasing the polynomial order

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until sufficient fit (R2 > 0.98) was achieved.

(2)

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As depicted, the ultimate stress of the thread is approximately 134.3 GPa or 26.4 nN. This is over

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twice the strength of carbyne (single carbon chains) recently touted as one of the strongest new

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materials39. We can also determine the specific strength, or tenacity, of the nanothread, which is

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independent of the presumed cross-section and only dependent on the length of the thread, where

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= ⁄, e.g., the ultimate stress normalized by density. This results in = 4.13 × 10 N-

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m/kg, potentially making these diamond nanothreads the strongest fabricated material by density

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to date.


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While stiff in tension, we next assess the flexibility of the threads – do they behave more akin to

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rods or cables? Through the equilibrium simulations, it is observed that the diamond nanothreads

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are not an extremely rigid structure, and thus cannot be subjected to bending loads like a beam to

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assess flexural response. Nor do the threads behave like flexible polymers, and thus

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conformational methods (e.g., Kuhn length) cannot be applied to approximate bending rigidity.

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As such, we determine out-of-plane bending stiffness/rigidity by introducing a virtual sticky

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surface to deform along with the nanothread into curved configurations of various radii (Fig. 3A;

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see Supporting Information). The bending stiffness is obtained by curve fitting with energy

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versus curvature data points (Fig. 3B), using the following expression for elastic bending energy:

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= "# $

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where U is the system strain energy, D is the bending stiffness, and κ is the prescribed curvature.

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The resulting bending stiffness is calculated to be approximately 770 (kcal/mol)-Å, or 5.35 ×

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10& N-m2. This makes the nanothread a somewhat rigid yet flexible molecule (a (5,5) CNT,

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for example, has a rigidity on the order of 100,000 (kcal/mol)-Å, while carbyne has a rigidity of

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30 to 80 (kcal/mol)-Å). In terms of persistence, where $' = "/)* , kB the Boltzmann constant,

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T the temperature, and thus kBT the thermal energy, the diamond nanothread has a persistence

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length of approximately 130 nm at 300K. Compared to polymers ( $' ≈ 1 nm) or double-

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stranded DNA ($' ≈ 50 nm), diamond nanothreads are relatively rigid, while compared to CNTs

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($' = 10 − 100 µm), they are relatively flexible. Thus, they behave more like a flexible (albeit

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strong) cable rather than a rod or beam.

!

(3)

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One deviation between the structure of nanothreads and related nanotubes is the introduction of

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Stone-Wales defects. These defects break the axial symmetry of the structure, and eliminate the

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central hollow of the tube-like geometry. Observation of simulated tensile tests also indicated

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that failure consistently occurred at the defect, suggesting a localization mechanism. To probe

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the effect of such defects, we investigate the local strain under tensile loading, variation in local

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stiffness, and predict overall thread behavior based on defect density.

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First, the nanothread was divided into segments, to isolate the local strains associated with the

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bonds adjacent the Stone-Wales defects. We plot the global stress versus local strain (Fig. 4A).

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There is a clear difference between the “pristine” segments and the defective regions. The

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pristine segments depict consistent stress-strain response, with an effective stiffness of

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approximately 1089 GPa (214 nN). Note that these regions are effectively nanotubes, and it is no

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surprise that this stiffness agrees with previous calculated values of such tubes.40 The strain at

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failure is also on the order of 12%. In contrast, the defect regions are significantly more

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compliant – fitting a modulus to the stress-strain data results in stiffness of 306 GPa (60 nN) and

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347 GPa (68 nN) for the two defects, respectively. Note that that scatter in the plot is due to the

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relatively small length and resolution of the strain measure (e.g., 1% strain is on the order of

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hundredths of Ångstrom). Regardless, there is a clear trend – the strain is localized and much

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greater at the defects.

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To predict the mechanical response as a function of defect density, we consider the nanothread as

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a system of serial springs (Fig. 4B). This has two significant implications: (1) the ultimate

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strength is unaffected by number of defects and (2) the stiffness will decrease with number of


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defects. If the defect density is known, e.g., - = ./$/ , where n is the number of defects and L0

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is the total length of the thread, theoretically we can predict the stiffness of the thread via a

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simple rule of mixtures, where:

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0.1 = 2

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where Ld is the effective length of the defect region, and Ed and E0 the local stiffness of each

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section. Assuming Ld ≈ 4.8 Å (measuring the approximate axial length across the defect area),

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then 021 = 833 to 867 GPa (164 to 170 nN) depending on the value of - , which agrees well

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with the molecular mechanics results for n = 2 (850 GPa or 167 nN).

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+

047 345 1 ! 67 47

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(4)

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To confirm the model prediction, we implement steered molecular dynamics (SMD) to

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investigate the stiffness and strength of the nanothread subject to applied loads at a low but finite

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temperature (100K), rather than incremental affine strain (see Supporting Information). This

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will facilitate any dynamic strain localization during applied loading and ultimate failure event,

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and more accurately represent experimental strength measures. Rate dependence was checked by

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varying loading velocity (see Supporting Information). Threads with 2, 3, and 4 defects were ruptured

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via SMD, and the stress-strain response extracted from the force-displacement results (see Fig.

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4C).

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We find that the threads all break at the same load, approximately 128 GPa (25.1 nN). This is

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within 5% of the measured molecular mechanics strength of 134.3 GPa (26.4 nN) indicating a

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slight weakening due to temperature. From a statistical perspective, one may expect a weakening

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with the introduction of more defects, as failure may occur at a greater number of locations,

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assuming a distribution of stresses and local variance. This would also result in a length


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dependence on strength. Due to the limited number of defects (n≤4) and constant length, this

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behavior was not observed. However, this could also be due to the limited temperature of the

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SMD simulations, which limits stress variation across the thread. Such statistical measures are

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beyond the scope of the current first-order characterization. Moreover, while the SMD load is

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applied in the axial direction, the thread is able to move out of alignment, which may also cause

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additional weakening. In consideration of these effects, the 5% deviation is deemed acceptable.

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In terms of stiffness, we see a consistent decrease with an increase in defect density, with

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modulii of 808 GPa (158 nN) 763 GPa (150 nN) and 665 GPa (131 nN) for n = 2, 3, and 4

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respectively (measured between strains of 3%-8%). Note again, there is a slight increase in

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compliance attributed to temperature effects (808 GPa compared to 850 GPa; a 4.9% difference).

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From the theoretical prediction, 031 = 746 to 786 GPa while 041 = 675 to 720 GPa,

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consistent with the observed reduction in magnitudes (see Fig 5). With the introduction of the

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different numbers of defects, the stiffness present a degradation as a function of the defect

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density. We also see an increase in ultimate strain (strain at failure), between 19.6% and 22.7%

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(note that the increase in ultimate strain is due to initial straightening of the thread due to the

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curvature imposed by temperature effects; the initial gauge length was measured relative to the

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slightly curved equilibrated structure, not a straight thread).

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Of interest, we also track the bond-order (BO) evolution of the carbon-carbon bonds within the

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nanothread during extension (see Supporting Information). From BO inspection, we can make

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two conclusions: First, the system maintains its sp3 character throughout the tensile loading until

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failure, which could be key in exploiting electrical or optical properties of the nanothreads.

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Second, only the bonds primarily aligned with the axis of extension (e.g., along the thread


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length) undergo a reduction in BO, implying tension, whereas the bonds transverse to the axis

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undergo a nominal increase in BO, implying slight compression. Thus, the thread acts marginally

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like a nanotruss, wherein the transverse bonds serve as stiffening members rather than a majority

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of the bonds under tension. This can further explicate (a) the localization of failure on the defect

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region (which is unreinforced by transverse bonds), and (b) the deviation between virial

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approach and energy approach in stiffness.

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In summary, we investigated the mechanical properties of diamond-like nanothreads of carbon,

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to assess the potential high strength and stiffness, which allows identifying the imminent benefits

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and mechanical limits of its application in different areas. It was predicted that, due to the

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diamond-like structure, these nanothreads may have exhibited extraordinary properties such as

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strength and stiffness higher than that carbon nanotubes or conventional high-strength polymers.

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We have shown an axial stiffness on the order of nanotubes (approximately 665 to 850 GPa or

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131 to 167 nN) depending on defect density and characterization method. A specific strength

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( 3.94 × 10 to 4.13 × 10 N-m/kg) exceeding all known fabricated materials. This specific

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property of the diamond nanothread is particularly encouraging, making it a material that can be

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featured as the solution to the ever-challenging space elevator problem. Moreover, the strength is

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independent of the number of Stone-Wales defects, perhaps facilitating the synthesis of large-

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scale thread systems (e.g., bundles or fabrics) without significant loss of capacity (unlike carbon

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nanotubes). In 2D and 3D crystalline materials, it is well-known that the density of defects

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affects the plasticity and brittle-ductile transitions. Here, the nanothread is different because (a) it

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is inherently 1D and (b) the defects are not an error in an otherwise ordered crystalline structure,

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but rather necessary to define the nanothread (otherwise, it would be a nanotube), and the


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molecular structure is essentially a Stone-Wales defect in geometry alone. Finally, we show the

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thread can be represented by a simple springs-in-series model and, while the strength remains the

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same, the stiffness can be predicted by the linear defect density (ρd=n/L0). Due to the linear

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spring-like arrangement, one defect must carry the load just as 2, 3, or 4 defects would, thus the

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“brittleness” in terms of sudden failure does not change. Indeed, the strength drop due to defects

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here is not the same as a strength drop due to stress concentrations, as usually encountered with a

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defected system – due to the linear arrangement, all sections (with defect or defect-free) must

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carry the same load. Thus, additional defects do not weaken the system, but define the upper

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bound of strength. In terms of plasticity, we have also shown that the ultimate strain increases

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when the number of defects increases.

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These nanothreads may be the first member of a new class of diamond-like nanomaterials based

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on a strong tetrahedral core, and determining the mechanical limits of such systems is necessary

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to aid application development. While strong in tension, the relatively low bending rigidity may

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facilitate the design of thread bundles or weaved systems, lacking the mechanical persistence of

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CNTs. The lack of crystallinity may enable allowances of defects with suffering mechanical

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degradation. Functionalized nanothreads offer the potential for improved load transfer through

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covalent bonding, efficiently transferring their mechanical strength to a surrounding matrix and

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thus allowing for technological exploitation in fibres or fabrics. Once again, carbon has provided

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yet another exemplary allotrope to explore and engineer.

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Acknowledgements R.R., K.K., and S.W.C. acknowledge generous support from NEU’s CEE Department. The calculations and the analysis were carried out using a parallel LINUX cluster at NEU’s Laboratory for Nanotechnology In Civil Engineering (NICE). Visualization has been carried out using the VMD visualization package.41


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Figure 1: Diamond nanothread structure. A. Simulation snapshot of diamond nanothread model; approximately 8 nm in length with two Stone-Wales defects (see inset) located approximately 2 nm from each edge. From a modeling perspective it can be considered as a variation of a hydrogenated (3,0) nanotube with Stone-Wales defects. B. Radial distribution function (RDF), g(r), of a short equilibrium simulation (1 ns) of diamond nanothread at finite temperature (300K). The RDF and matches previously published results11 with key peaks representing the 1:1 carbon-hydrogen stoichiometry and thread structure. Multiple lines with deviation represent multiple runs with variation in number of defects (n = 1,2,3 and 4), indicating negligible structural change with defect density.


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Figure 2: Molecular mechanics tensile test. A. Virial stress versus strain for quasi-static incremental strain loading (strain increments of 0.0001 followed by energy minimization). Fitted stiffness (ε

Mechanical Properties and Defect Sensitivity of Diamond

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