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Confinement and controlling the effective compressive stiffness of carbyne
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 335709 (http://iopscience.iop.org/0957-4484/25/33/335709) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 13/06/2017 at 22:37 Please note that terms and conditions apply.
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Nanotechnology Nanotechnology 25 (2014) 335709 (9pp)
doi:10.1088/0957-4484/25/33/335709
Confinement and controlling the effective compressive stiffness of carbyne Ashley J Kocsis, Neta Aditya Reddy Yedama and Steven W Cranford Laboratory of Nanotechnology in Civil Engineering (NICE), Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA E-mail: [email protected] Received 16 April 2014, revised 21 May 2014 Accepted for publication 29 May 2014 Published 30 July 2014 Abstract
Carbyne is a one-dimensional chain of carbon atoms, consisting of repeating sp-hybridized groups, thereby representing a minimalist molecular rod or chain. While exhibiting exemplary mechanical properties in tension (a 1D modulus on the order of 313 nN and a strength on the order of 11 nN), its use as a structural component at the molecular scale is limited due to its relative weakness in compression and the immediate onset of buckling under load. To circumvent this effect, here, we probe the effect of confinement to enhance the mechanical behavior of carbyne chains in compression. Through full atomistic molecular dynamics, we characterize the mechanical properties of a free (unconfined chain) and explore the effect of confinement radius (R), free chain length (L) and temperature (T) on the effective compressive stiffness of carbyne chains and demonstrate that the stiffness can be tuned over an order of magnitude (from approximately 0.54 kcal mol−1 Å2 to 46 kcal mol−1 Å2) by geometric control. Confinement may inherently stabilize the chains, potentially providing a platform for the synthesis of extraordinarily long chains (tens of nanometers) with variable compressive response. Keywords: carbon, carbyne, confinement, compression, nanomechanics, nanorod (Some figures may appear in colour only in the online journal) 1. Introduction
between electronic or mechanical components at the atomistic scale. Due to its chemical stability, mechanical properties, and natural abundance, carbon’s applications could lead the way in nanotechnology applications [4]. This molecular ‘rod’ has potential as an effective wire or even structural element, and thus its mechanical behavior is of critical importance. It has been shown via first principles calculations that the axial stiffness, strength, and bending rigidity of carbyne exceeds even the exemplary properties of graphene and carbon nanotubes [5], primarily due to its single-atom crosssection. That being said, due to the onset of buckling (even due to thermal fluctuations), this material is not strong in compression (effectively behaving as a weak entropic spring). Its flexibility is between those of typical polymers and double-stranded DNA, wherein the mechanics must account for both bending rigidity and conformational entropy—i.e., under compression, carbyne can be viewed as a semi-flexible polymer. Thus, carbyne presents an interesting model material: relatively stiff with a persistence length on the order of
Carbon allotropes have the continuous interest of material scientists due to their molecular stability and promising electrical and thermal properties [1–3]. One of the newest potential carbon allotropes being investigated exhibits the bare minimum possible molecular structure—a monoatomistic chain of atoms known as carbyne [4–6]. Carbyne is a one-dimensional carbon allotrope of sp-hybridized carbon atoms [7]. Theoretically, it may take a cumulene ([=C=]) repeating double bond form, or a polyyne form with alternating single and triple bonds (–[C≡C]–; see figure 1(a)), which has been determined to be more energetically favorable [5]. Carbyne is known to have a relatively high axial tensile stiffness and, due to its monoatomistic ‘cross-section’, incredible specific strength and high surface area per given mass [5]. Carbyne’s practical applications include energy storage devices and nanoscale electronic devices [8]. It is also possible that carbyne could act as a structural connection 0957-4484/14/335709+09$33.00
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To such an end, herein we investigate the extent in which the effective compressive stiffness of carbyne can be controlled due to ideal cylindrical confinement. We note that— even while buckled—the entropic conformation of carbyne would provide resistance to axial deformation. As such, we are not concerned with the critical buckling load of the carbyne chains (which is meaningless due to thermal fluctuations), but rather the effective stiffness—i.e., the resistance to contraction along the axial direction. Specifically, supported by previous experimental observations, we consider a finite length of carbyne subject to compression confined to an effective carbon nanotube to see any variation in compressive stiffness using full atomistic molecular dynamics (MD) simulations. We subsequently explore the effect of: (1) molecular chain length, L; (2) temperature, T, and; (3) confinement radius, R.
2. Methods We first undertake an analysis of the baseline nanomechanical properties of carbyne chains, to enable comparison with chains subject to confinement, consistent with our model formulation. Specifically, we determine the axial stiffness, key stress/strain values, and bending rigidity by implementing mechanical test cases via MD with the first-principles–based ReaxFF force field [18]. While it has been demonstrated that the ReaxFF potential provides an accurate account of the chemical and mechanical behavior of carbon-based materials [4, 19–23], it is not as accurate as quantum-based ab initio methods [5]. However, due to the length and time scales required, first principles methods are not suitable. As such, characterization of the mechanical properties of carbyne also serves to validate the use of ReaxFF. ReaxFF is utilized to ensure the simulations are as accurate as possible and imitates the local chemical environment to allow dynamic alterations to the molecular and atomistic interactions [24]. Mechanical characterization is then followed by a suite of confined compression simulations. All simulations are implemented by the software package LAMMPS (http://lammps.sandia.gov/ ) [24].
Figure 1. (a) Chemical structure of carbyne, a linear monoatomistic
chain of carbon atoms with a repeating single–triple (polyyne) bond structure (also known as linear acetylene); (b) schematics of unconfined (free) compression where buckled shape is unconstrained compared to confined compression where lateral deformation is limited.
approximately 14 nm [5]—chain lengths less than this will tend to straighten to keep their bending energy low and thus act as a suitable structural element at the molecular scale; subject to compression, however, such rods will immediately lose structural integrity and buckle. At the macroscale, it has been shown that constrained slender rods can support higher compressive loads [9–12], akin to bracing the effective buckling length. Confinement may therefore provide a means to increase the effective strength and stiffness of carbyne chains subject to compression (figure 1(b)). Moreover, if variable, the amount of confinement may provide a mechanical means to enable the fine tuning of compressive stiffness, again similar to controlling the effective length (and critical buckling load) of a slender column through external bracing. One possible solution is the vacuum space inside carbon nanotubes, which offers an interesting medium for the inclusion, transportation and confinement of foreign molecules. Demonstrated by both first principles calculations [13–15] and experimental observation [14, 16, 17], carbyne chains are likely stable within the cavity of both single and multi-walled carbon nanotubes. As such, they may provide an ideal platform as axial, cylindrical confinement with a variable radius—e.g., an effective mechanical sheath or casing for a single carbyne chain.
2.1. Axial stiffness and stress response
To closely mimic first principles approaches [5], a temperature-free molecular mechanics approach is implemented on a representative unit cell of carbyne consisting of two carbon atoms (reflecting one instance of the repeating single-triple bond structure). Periodic boundaries are applied, effectively representing an infinite carbyne chain. The system is then subject to increments of strain along the chain axis, followed by energy minimization (conjugate gradient (CG) algorithm as implemented in LAMMPS with an energy tolerance of 10−6 kcal mol−1). During axial stretching, stress is calculated using a virial stress formulation. Virial stress is commonly used to relate to the macroscopic (continuum) stress in MD computations [25, 26]. A virial stress approach allows us to determine the components of the macroscopic stress tensor 2
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through the virial components, Sij , where: Sij =
⎡ ⎢ ⎣
⎤
N
∑ ∑ ri(ab) ⋅ F j′ (ab) ⎥, a∈Ω
⎦
b=1
(1)
which generates the six components of the symmetric stressvolume tensor, S, where N represents the number of individual atoms, b, interacting with atom a, through any multibody potential contribution, ‘I’ and ‘j’ the coordinate basis, r(ab) the distance between ‘a’ and ‘b’ along the ‘I’ direction, i and Fj'(ab) the scaled partial force on atom, a, due to all multibody contributions [27]. We note the velocity term is absent due to the minimization approach.
Figure 2. Model configuration of compressive deformation of confined carbyne chain. Dashed lines indicate ‘confinement radius’ of virtual nanotube (diameter = 2 R). Red atom indicates ‘dummy atom’ which drives the compressive deformation via incremental displacement, Δ.
energy state. As there is no data extracted when the dummy atom is under motion and allow equilibrium, we eliminate any rate dependence but maintain finite temperature effects. The process is repeated to determine the energy versus displacement response.
2.2. Bending rigidity
Due to the difficulty in dynamically bending carbyne chains, an energy-based molecular mechanics approach is also implemented to determine bending rigidity. A pre-formed chain ring with a varying number of carbon atoms is utilized. The ring structures maintain a constant radius of curvature without the need to impose artificial boundary conditions, and thus enables efficient energy minimization. A total of seven trials (seven rings with different radii) are run, each with a different number of carbon atoms (n = 8, 10, 12, 14, 16, 18 and 20). Each system is equilibrated (0.5 ns at 300 K) to ensure conformational stability, and minimized (CG algorithm with an energy tolerance of 10−6 kcal mol−1) to determine elastic bending strain energy.
3. Results and discussion 3.1. Mechanical characterization
First we discuss the baseline mechanical properties of carbyne, to both validate the use of the ReaxFF potential and to allow the theoretical treatment and comparison of the compressive behavior. Of key concern is accurately capturing the axial stiffness and bending rigidity, which are key properties for confined compression. 3.1.1. Theoretical axial stiffness and stress-strain response.
2.3. Confined compression
Each minimization per strain increment returns the total elastic strain energy of the carbyne system. Rather than extracting axial stiffness indirectly from a stress-strain or force-strain response, the tensile stiffness, C, of carbyne can be defined energetically as:
Compression simulations are carried out under a canonical (NVT) ensemble, time step of 0.5 × 10−15 s. Finite temperature is required to accurately capture the entropic contribution of the carbyne chain resistance. A 100 carbon atom, 13 nm carbyne chain is minimized and subsequently equilibrated at 100 K, 300 K, or 500 K. Length can be varied by fixing a portion of the chain. Following equilibration, a confining cylinder is introduced, with a harmonic repulsive potential, where: 2
Fconfine = K (r−R) ,
C=
1 ∂ 2E , a ∂ε 2
(3)
where a is the equilibrium unit cell length (here, a = 2.63 Å), E the strain energy per two carbon atoms (or, equivalently, per a single/triple bond pair) and ε the strain. We track the energy versus strain from the MD simulation, and fit the data to a fourth order polynomial (see figure 3(a)). Taking the derivative about ε = 0, we find a stiffness of 91.75 eV Å−1. This result is directly compared to ab initio results, utilizing published DFT data from Liu et al [5], and plotted on the same axis. While the two curves deviate at higher strains, the reported value of 95.56 eV Å−1 indicates a close match in terms of equilibrium stiffness, within 4% of the ab initio result. Beyond the axial stiffness, we also want to ensure an accurate force–displacement response of carbyne can be attained using ReaxFF. As such, we plot the virial stress of the simulated test with respect to strain. We note that that virial stress calculates terms in units of stress-volume, typically normalized by the atomistic volume of the system. Due to the 1D structure of carbyne, we normalize the virial
(2)
where K is the confinement stiffness (taken as 15 000 kcal mol−1 Å−1, representing a ‘stiff’ boundary), r the distance from the atom to the confining cylinder, and R the effective radius of confinement. By utilizing a variable confining radius, the cylinder can represent carbon nanotubes of different size. For the boundaries of carbyne, one end of the chain is fixed while the other is attached to a ‘dummy atom’ (figure 2). This atom acts as a virtual ‘force transducer’. To apply compression, the dummy atom is incrementally displaced by approximately 0.1 Å over a short timespan of 0.5 ps (an effective rate of 20 m s−1). After each displacement, the dummy atom is fixed, and the system is equilibrated at constant temperature (100 K, 300 K, or 500 K) for 0.5 ps to get an average potential energy (at the new, shortened/deformed length). Trial cases indicated that the displacement rate of the dummy atom is independent of the subsequent equilibrium 3
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Figure 4. (a) Minimized carbyne rings representing a set of systems
with constant curvatures; (b) EL versus κ plot used to determine the effective bending stiffness of carbyne. Note that the fitted line only appears linear due to the scaling of the plot—it is a quadratic, which provides a better statistical fit (R2 = 0.9997 compared to R2 = 0.998 for a linear fit).
Figure 3. Axial stretching response (a) potential energy versus strain; DFT data (blue circles) from previously published paper [5] and the current MD simulations (red continuous). Stiffness obtained from the quadratic term of the fitted polynomial; (b) virial stress versus strain response of carbyne indicating transition point at approximately 7 nN (green) and ultimate strength at approximately 11 nN (red). Linear trendline represents carbyne’s modulus of elasticity (Y = 313 nN).
the transition in stress response at approximately 10% strain —previous reports indicate a transition of bond length alternation (BLA) of carbyne under tension [5, 8, 30], with a clear transition at ε ≈ 0.1. The BLA affects the electronic density and resulting band gap of carbyne. This extreme sensitivity of band gap to mechanical tension is promising for many applications [31, 32], and is indirectly captured by the ReaxFF potential.
stress by the unit cell length only, without having to define an effective cross-sectional area (a similar assumption is used when characterizing graphene in two dimensions [21, 28]). As such, the virial stress is presented in units of force (figure 3(b)). First we can fit an effective modulus, Y, to the stress-strain response, via a linear fit at small deformation. A fit between ±5% strain results in a modulus of approximately Y ≅ 313 nN . We also observe an ultimate strength of 10.75 nN at 18% strain, which is an agreement with the ab initio prediction of 9.3 to 11.7 nN [5]. Of additional note is
3.1.2. Bending rigidity.
The minimized carbyne rings represented a set of systems with constant curvatures on the order of 0.24–0.60 Å−1 (n = 20 and n = 8 respectively; figure 4(a)). Combining the bending strain energy output of the simulations along with calculated values of perimeter for each ring results in normalized energy values (kcal mol−1 Å−1). Energy per length, EL, and the curvature (κ = 1/r) of each ring can be seen in table 1. The elastic
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Table 1. Bending stiffness summary; results of MD simulations involving curved polyyne carbyne chains involving 8–20 atoms.
Number of carbon atoms Parameter
Units −1
Bending energy, E Perimeter, L Energy per length, E/L Curvature, κ
kcal mol Å kcal mol−1 Å−1 Å−1
8
10
12
14
16
18
20
−1232.9 10.48 −117.6 0.5995
−1591.1 13.10 −121.5 0.4796
−1944.6 15.72 −123.7 0.3997
−2302.9 18.34 −125.6 0.3426
−2660.2 20.96 −126.9 0.2998
−3016.3 23.58 −127.9 0.2665
−3371.4 26.20 −128.7 0.2398
bending energy of a curved element can be calculated as: E=
∫
0
L
1 2 Dκ ds , 2
3.2.1. Unconfined compressive stiffness and theoretical bounds. An initial simulation is run using a confinement
radius of 100 Å making the carbyne chain virtually unconfined, thereby providing a baseline compressive resistance. We choose a constant temperature of 300 K to approximate room temperature conditions, and probe three chain lengths (65 Å, 98 Å, and 117 Å). To maintain consistency with the calculations for the confined compression simulations, equation (5) is used to calculate the energy displacement relationship, and the unconfined compressive stiffness results are easily approximated by a second order polynomial trendline. The unconfined compressive simulations for L = 65 Å, 98 Å, and 117 Å resulted in spring constants of k = 14.308 kcal mol−1 Å−2, 2.912 kcal mol−1 Å−2, 0.540 kcal mol−1 Å−2, respectively (see figure 6). The maximum stiffness can be considered from the modulus of pure axial compression, which we have calculated as Y ≅ 313 nN . To derive an equivalent spring stiffness, we can divide by the spring length, where:
(4a)
where D is the bending stiffness or bending rigidity of the element (D = EI for continuum beam elements, for example), as s is the local coordinates along the curve. Assuming constant curvature and bending stiffness: E=
1 2 Dκ L, 2
(4b)
1 2 Dκ . 2
(4c)
such that: EL =
We use the above quadratic form of bending strain energy due to curvature as it is both accepted in the small deformation regime and enables comparison with continuumscale theory (see [5] for example). By plotting EL versus κ (figure 4(b)) a fitted constant was obtained to determine the effective bending stiffness of carbyne using equation (4c). The outputted bending stiffness was 30.7 ± 0.6 (kcal mol−1) Å or 1.32 eV Å (R2 = 0.9997, indicating a statistically significant fit). Once again, this is in good agreement in magnitude with first principles results, which reported a larger bending rigidity of 82.8 (kcal mol−1) Å (3.56 eV Å) [5]. We note that carbon rings undergo so-called Jahn–Teller distortions [33–35], which is not captured by the ReaxFF potential, thereby partially explaining the deviation in derived stiffness. From this bending stiffness, the persistence length of carbyne (at 300 K) is on the order of 5 nm (where lp = D /kB T ).
k max =
Since harmonic spring like behavior is assumed in the simulations, a spring constant, k, is able to be calculated from the following: 1 2 kΔ , 2
(6)
For L = 65 Å, 98 Å, and 117 Å, kmax = 69.9 kcal mol−1 Å−2, 46.4 kcal mol−1 Å−2, and 38.8 kcal mol−1 Å−2, respectively. To achieve such a stiffness, the chain would remain straight (zero curvature) without buckling, and undergo homogenous contraction across its length—e.g., completely confined. Such a scenario is prevented by finite temperature effects that inevitably produce perturbations from a straight configuration. We also note there is a higher deviation from the maximum compressive stiffness as the length increases (e.g., 0.540 kcal mol−1 Å−2 to 38.8 kcal mol−1 Å−2 for L = 117 Å), corresponding to an increase in buckling modes. The minimum stiffness evokes the force-extension relation for a semi-flexible polymer, which associated an equivalent spring constant to a random wormlike chain with a stochastic end-to-end distance [36]. In the limit, the spring constant can be expressed as:
3.2. Confined compression simulations
U=
Y . L
(5) k min =
where U represents the total potential energy, k represents the spring constant, and Δ represents the total displacement of the spring. The energy-displacement relationship is plotted from the simulation, and the spring constants are calculated using the coefficient of the quadratic term of the second order polynomial trendline.
90D 2 , kB TL4
(7)
where D is the bending rigidity, kB Boltzmann constant, T the temperature, and L the chain length. Equation (7) serves as an asymptotic lower bound. For L = 65 Å, 98 Å, and 117 Å, kmin = 0.008 kcal mol−1 Å−2, 0.0015 kcal mol−1 Å−2, and 0.000 75 kcal mol−1 Å−2, respectively, at 300 K, which are 5
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Figure 5. Simulation snapshots of carbyne chain under deformation
(L = 117 Å, T = 300 K, R = 0.70 Å). (a) Initial configuration, (b) approximately 10% strain and (c) approximately 15% strain.
orders of magnitude less than the observed results. It is clear that the unconfined chain lies between the two extremes of upper and lower compressive stiffness—carbyne provides an intermediate case between a compression member (k max ) and a semi-flexible chain (k min ). 3.2.2. Variation in confinement radius, length and temperature. The MD simulations confined carbyne to
four different confinement radii: 0.05 Å, 0.35 Å, 0.70 Å, and 2.10 Å. The confinement radii 0.05 Å, 0.70 Å, and 2.10 Å represent (5, 5), (6, 6), and (8, 8) carbon nanotubes respectively, while the 0.35 Å value represents an intermediate case to better demonstrate the relationship between confinement, length, and temperature. It has been shown that the interaction energy of a concentric carbyne chain within a (10, 10) CNT is negligible [13], suggesting an upper limit to an idealized confinement radii (in terms of carbon nanotube encapsulation). The specific confinement radii are determined by accounting for the carbon–carbon atomistic interaction distance, (3.35 Å), between atoms and subtracting it from the radius of each specified nanotube. Each simulation with specific confinement radii is run at 100 K, 300 K, and 500 K. It has been shown that carbyne chains are stable within CNTs to a temperature of approximately 450 °C (approximately 725 K) [16, 17]. To confirm, we increased the temperature in our simulations by 50 K increments until the chain lost stability. It was determined that the carbyne chain ‘melts’ or has an upper limit of approximately 900 K. The stiffness of the carbyne chain is adjusted by altering the length of the chain from 130 Å to 117 Å, 98 Å, or 65 Å during different trials of the simulation. A total of 36 simulations were run to account for adjustments in confinement radius, temperature, and chain length. Example snapshots are depicted in figure 5. By adjusting the confinement radii, the amount of space that the carbyne chain has to deform is altered. In the simulations with the largest confinement radius—the virtual (8, 8) carbon nanotube—the carbyne chain has the most room to deform, resulting in the lowest compressive strength (e.g., for L = 117 Å at T = 300 K, k = 0.873 kcal mol−1 Å−2, approaching the unconfined value of 0.540 kcal mol−1 Å−2). When increasing the confinement (decreasing the radii), the stiffness increases, achieving a significant proportion of k max (see figure 6). For example, at R = 0.05 Å, L = 65 Å achieves
Figure 6. Energy–displacement behavior. Energy-displacement
results for five different virtual nanotube confinement radii: 0.05 Å, 0.35 Å, 0.70 Å, 2.10 Å, ∞ (unconfined) run at 300 K with chain length, L: (a) 117 Å (b) 98 Å (c) 65 Å. The plots also depict the upper and lower bounds, kmax = 69.9 kcal mol−1 Å2 and kmin = 0.000 75 kcal mol−1 Å2, respectively (black solid lines).
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Results also prove that with decreased chain length and confinement radius, carbyne can demonstrate a controlled variation in the effective modulus of elasticity, Y*. The effective modulus of elasticity can be calculated by rearranging the terms in equation (8a) to solve for Y: Y* =
at 100 K, 300 K and 500 K, continuous, dashed, and dash-dot lines respectively, for L = 65 Å, 98 Å and 117 Å, red, blue and green colors, respectively. The results indicate a slight but consistent increase in stiffness with temperature.
approximately 65% k max , while L = 117 Å achieves approximately 24% k max . Simulation results clearly show that with decreased chain length, the carbyne chain demonstrates an increase in compressive stiffness (figure 6). In terms of effective modulus, we can consider the carbyne chain as analogous to a truss element, where the the spring constant, k, is inversely proportional to the length of the carbyne chain, similar to equation (6), or: YA . L
(8b)
From equation (8b), it is evident that the modulus of elasticity is directly proportional to the spring constant, k, and hence, the overall stiffness. A radius of R = 0.386 Å is used to compute the modulus of elasticity, acquired from ab initio results from Liu et al [5]. Due to the relationship between compressive stiffness, confinement radius, temperature, and chain length amongst the 36 simulations, there is a two orders of magnitude difference between the highest spring constant and effective modulus (k = 46.130 kcal mol−1 Å−2; Y* ≈ 44 TPa) and the lowest spring constant and effective modulus (k = 0.662 kcal mol−1 Å−2; Y* ≈ 1 TPa). Correspondingly, the largest spring constant occurs at the smallest confinement radius (R = 0.05 Å), smallest chain length (L = 65 Å) and the highest temperature (T = 500 K), while the smallest spring constant occurs at the largest confinement radius (R = 2.10 Å), the largest chain length (L = 117 Å), and the lowest temperature (T = 100 K), (figure 7). For all of the confined carbyne simulations, the quadratic trendlines used to fit the energy displacement results produced a coefficient of determination of R2 > 0.944. Since this value is close to 1.00, where 1.00 statistically represents perfect regression, the computed trendlines demonstrate good fitted nonlinear regressions. Moreover, all k values determined from the energy displacement results for each of the carbyne chain lengths lie within the theoretical upper and lower boundaries of compressive stiffness demonstrating the validity of the range of simulation results.
Figure 7. Plot of the spring constant values versus confinement radii
k=
kL . A
(8a)
4. Conclusion
With this relationship, a decrease in the chain length should cause an increase in the spring constant value and hence, an increase in compressive stiffness. This result is most clear when removing the confinement of the virtual nanotube, keeping the temperature constant, and comparing only the length and the stiffness values of carbyne (see figure 6). With an increase in temperature, we observe that the effective stiffness marginally increases (figure 7). This is in opposition to semi-flexible polymer behavior, wherein higher temperature increases the compliance of a chain, thereby allowing easier coiling and bending (e.g., rigidity ∝T −1). Under confinement, however, the increased thermal energy can be thought of as increasing the pressure within the confined space, and this effectively increases the resistance to compressive deformation. In effect, more force is required to hold a confined coiled/deformed chain at high temperatures. The competition due to these competing factors is nontrivial, and beyond the scope of the current study. That being said, the observed temperature dependence is not significant.
A recent full-principles study by Liu et al has rigorously characterized the mechanical properties of carbyne, constituting a linear chain of repeating sp-hybridized carbon atoms, assessing axial stiffness, strength, and bending rigidity. While undoubtedly, first-principles methods are necessary to explore subatomistic features, we show that, under similar simulation conditions, ReaxFF agrees with more rigorous methods, resulting in similar values for stiffness (C = 91.75 eV Å−1 and Y = 313 nN), ultimate strength (10.75 nN at 18% strain), bending rigidity (D = 30.7 ± 0.6 (kcal mol−1) Å), and persistence length (approximately 5 nm). Relatively large carbyne systems can thus be efficiently explored using a MD framework, extending the time—and length-scales typically associated with first-principles methods, with little consequence on mechanical features. Taking into account the relative weakness of carbyne in compression, here we explored the potential to fine-tune compressive resistance by subjecting a chain to molecular 7
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confinement (similar to bracing of a column at the macroscale). By plotting the spring constants for the four different confinement radii and the three carbon chain lengths, it became clear that there is a strong relationship between stiffness and chain length, L. An increase in stiffness causes an increase in the spring constant value, modulus of elasticity, and hence, the propensity to resist deformation. The spring constants and modulus of elasticity are significantly larger for shorter chain lengths. There is a two orders of magnitude difference between the largest k and the smallest k each corresponding to the 0.05 Å confinement radius but with the largest value associated with the shortest chain length at 500 K, and the smallest value associated with the longest chain length at 100 K. These results confirm the hypothesis of controlling compressive stiffness, since the output demonstrates that a smaller confinement radius results in a higher stiffness value, k. Through adjustments to free length, confinement radii, and temperature, the effective spring stiffness can be varied by over an order of magnitude— an amazing result for the same system of atoms in a consistent molecular configuration. As an additional benefit, it has been reported that long carbyne chains (e.g., more than 20 carbon atoms) become energetically unstable [7, 37, 38], requiring large and bulky molecular end groups [6]. Confinement within carbon nanotubes may inherently stabilize the chains, providing a platform for the synthesis of extraordinarily long chains (tens of nanometers). As such, characterization of the behavior of confined carbyne may prove of interest beyond the tunability of compressive behavior. While carbyne is not known to be strong in compression, it is clear that by confining carbyne within a virtual nanotube, effective stiffness can be controlled. The strong relationship between carbyne’s compressive stiffness, confinement radius, and chain length demonstrates the possible opportunity to be able to control some of carbyne’s material properties including the ‘effective’ modulus of elasticity and/or spring constant. By effectively being able to design various behaviors from the same material—e.g., a finite length of carbyne —the future of carbyne as ‘nanowires’ and ‘nanorods’ in nanotechnology can be enhanced.
[2] Geim A K and Novoselov K S 2007 The rise of graphene Nat. Mater. 6 183–91 [3] Hirsch A 2010 The era of carbon allotropes Nat. Mater. 9 868–71 [4] Nair A K, Cranford S W and Buehler M J 2011 The minimal nanowire: mechanical properties of carbyne Europhys. Lett. 95 16002 [5] Liu M et al 2013 Carbyne from first principles: chain of C atoms, a nanorod or a nanorope ACS Nano 7 10075–82 [6] Chalifoux W A and Tykwinski R R 2010 Synthesis of polyynes to model the sp-carbon allotrope carbyne Nat. Chem. 2 967–71 [7] Jones R O and Seifert G 1997 Structure and bonding in carbon clusters C-14 to C-24: chains, rings, bowls, plates, and cages Phys. Rev. Lett. 79 443–6 [8] Hu Y H 2011 Bending effect of sp-hybridized carbon (carbyne) chains on their structures and properties J. Phys. Chem. C 115 1843–50 [9] Su T X et al 2013 Mechanism by which a frictionally confined rod loses stability under initial velocity and position perturbations Int. J. Solids Struct. 50 2468–76 [10] Wicks N, Wardle B L and Pafitis D 2008 Horizontal cylinderin-cylinder buckling under compression and torsion: review and application to composite drill pipe Int. J. Mech. Sci. 50 538–49 [11] Thompson J M T et al 2012 Helical post-buckling of a rod in a cylinder: with applications to drill-strings Proc. R. Soc. Lond. A 468 1591–614 [12] Domokos G, Holmes P and Royce B 1997 Constrained Euler buckling J. Nonlinear Sci 7 281–314 [13] Kuwahara R et al 2011 Encapsulation of carbon chain molecules; in single-walled carbon nanotubes J. Phys. Chem. A 115 5147–56 [14] Zhao X L et al 2003 Carbon nanowire made of a long linear carbon chain inserted inside a multiwalled carbon nanotube Phys. Rev. Lett. 90 187401 [15] Xu B et al 2009 Structural and electronic properties of finite carbon chains encapsulated into carbon nanotubes J. Phys. Chem. C 113 21314–8 [16] Nishide D et al 2007 Raman spectroscopy of size-selected linear polyyne molecules C2nH2 (n = 4–6) encapsulated in single-wall carbon nanotubes J. Phys. Chem. C 111 5178–83 [17] Nishide D et al 2006 Single-wall carbon nanotubes encaging linear chain C10H2 polyyne molecules inside Chem. Phys. Lett. 428 356–60 [18] Chenoweth K, van Duin A C T and Goddard W A 2008 ReaxFF reactive force field for molecular dynamics simulations of hydrocarbon oxidation J. Phys. Chem. A 112 1040–53 [19] Cranford S 2013 Thermal stability of idealized folded carbyne loops Nanoscale Res. Lett. 8 490 [20] Cranford S W and Buehler M J 2011 Mechanical properties of graphyne Carbon 49 4111–21 [21] Cranford S and Buehler M J 2011 Twisted and coiled ultralong multilayer graphene ribbons Modelling Simul. Mater. Sci. Eng. 19 054003 [22] Sen D et al 2010 Tearing graphene sheets from adhesive substrates produces tapered nanoribbons Small 6 1108–16 [23] Buehler M J 2006 Mesoscale modeling of mechanics of carbon nanotubes: self-assembly, self-folding, and fracture J. Mater. Res. 21 2855–69 [24] Cranford S W, Buehler M J and SpringerLink (Online service) 2012 Biomateriomics (Springer Series in Materials Science) (Dordrecht: Springer) [25] Tsai D H 1979 Virial theorem and stress calculation in molecular-dynamics J. Chem. Phys. 70 1375–82 [26] Zimmerman J A et al 2004 Calculation of stress in atomistic simulation Modelling Simul. Mater. Sci. Eng. 12 S319–32
Acknowledgements SWC acknowledge generous support from NEU’s CEE Department. NARY is thanked for his volunteer efforts and contribution to this study. 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 [39].
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[27] Thompson A P, Plimpton S J and Mattson W 2009 General formulation of pressure and stress tensor for arbitrary manybody interaction potentials under periodic boundary conditions J. Chem. Phys. 131 154107 [28] Huang Y, Wu J and Hwang K C 2006 Thickness of graphene and single-wall carbon nanotubes Phys. Rev. B 74 245413 [29] Cranford S and Buehler M J 2011 Twisted and coiled ultralong multilayer graphene ribbons Modelling Simul. Mater. Sci. Eng. 19 054003 [30] Cahangirov S, Topsakal M and Ciraci S 2010 Long-range interactions in carbon atomic chains Phys. Rev. B 82 195444 [31] Cretu O et al 2013 Electrical transport measured in atomic carbon chains Nano Lett. 13 3487–93 [32] Artyukhov V I, Liu M and Yakobson B I 2013 Mechanically induced metal–insulator transition in carbyne arXiv:1302.7250 [33] Bylaska E J, Weare J H and Kawai R 1998 Development of bond-length alternation in very large carbon rings: LDA pseudopotential results Phys. Rev. B 58 R7488–91
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Confinement and controlling the effective compressive stiffness of carbyne
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 335709 (http://iopscience.iop.org/0957-4484/25/33/335709) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 207.162.240.147 This content was downloaded on 13/06/2017 at 22:37 Please note that terms and conditions apply.
You may also be interested in: Tensile strength of carbyne chains in varied chemical environments and structural lengths Reza Mirzaeifar, Zhao Qin and Markus J Buehler Erratum: The minimal nanowire: Mechanical properties of carbyne A. K. Nair, S. W. Cranford and M. J. Buehler The minimal nanowire: Mechanical properties of carbyne A. K. Nair, S. W. Cranford and M. J. Buehler Twisted and coiled ultralong multilayer graphene ribbons Steven Cranford and Markus J Buehler Mechanical properties of single-walled carbon nanotubes: a comprehensive molecular dynamics study Hessam Yazdani, Kianoosh Hatami and Mehdi Eftekhari Graphene–polymer interfacial behavior Amnaya P Awasthi, Dimitris C Lagoudas and Daniel C Hammerand Abnormality in Fracture Strength of Polycrystalline Silicene Ning Liu, Jiawang Hong, Ramana Pidaparti et al. Tersoff potential with improved accuracy for simulating graphene in molecular dynamics environment G Rajasekaran, Rajesh Kumar and Avinash Parashar Mechanical properties of phosphorene nanotubes: a density functional tight-binding study V Sorkin and Y W Zhang
Nanotechnology Nanotechnology 25 (2014) 335709 (9pp)
doi:10.1088/0957-4484/25/33/335709
Confinement and controlling the effective compressive stiffness of carbyne Ashley J Kocsis, Neta Aditya Reddy Yedama and Steven W Cranford Laboratory of Nanotechnology in Civil Engineering (NICE), Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA E-mail: [email protected] Received 16 April 2014, revised 21 May 2014 Accepted for publication 29 May 2014 Published 30 July 2014 Abstract
Carbyne is a one-dimensional chain of carbon atoms, consisting of repeating sp-hybridized groups, thereby representing a minimalist molecular rod or chain. While exhibiting exemplary mechanical properties in tension (a 1D modulus on the order of 313 nN and a strength on the order of 11 nN), its use as a structural component at the molecular scale is limited due to its relative weakness in compression and the immediate onset of buckling under load. To circumvent this effect, here, we probe the effect of confinement to enhance the mechanical behavior of carbyne chains in compression. Through full atomistic molecular dynamics, we characterize the mechanical properties of a free (unconfined chain) and explore the effect of confinement radius (R), free chain length (L) and temperature (T) on the effective compressive stiffness of carbyne chains and demonstrate that the stiffness can be tuned over an order of magnitude (from approximately 0.54 kcal mol−1 Å2 to 46 kcal mol−1 Å2) by geometric control. Confinement may inherently stabilize the chains, potentially providing a platform for the synthesis of extraordinarily long chains (tens of nanometers) with variable compressive response. Keywords: carbon, carbyne, confinement, compression, nanomechanics, nanorod (Some figures may appear in colour only in the online journal) 1. Introduction
between electronic or mechanical components at the atomistic scale. Due to its chemical stability, mechanical properties, and natural abundance, carbon’s applications could lead the way in nanotechnology applications [4]. This molecular ‘rod’ has potential as an effective wire or even structural element, and thus its mechanical behavior is of critical importance. It has been shown via first principles calculations that the axial stiffness, strength, and bending rigidity of carbyne exceeds even the exemplary properties of graphene and carbon nanotubes [5], primarily due to its single-atom crosssection. That being said, due to the onset of buckling (even due to thermal fluctuations), this material is not strong in compression (effectively behaving as a weak entropic spring). Its flexibility is between those of typical polymers and double-stranded DNA, wherein the mechanics must account for both bending rigidity and conformational entropy—i.e., under compression, carbyne can be viewed as a semi-flexible polymer. Thus, carbyne presents an interesting model material: relatively stiff with a persistence length on the order of
Carbon allotropes have the continuous interest of material scientists due to their molecular stability and promising electrical and thermal properties [1–3]. One of the newest potential carbon allotropes being investigated exhibits the bare minimum possible molecular structure—a monoatomistic chain of atoms known as carbyne [4–6]. Carbyne is a one-dimensional carbon allotrope of sp-hybridized carbon atoms [7]. Theoretically, it may take a cumulene ([=C=]) repeating double bond form, or a polyyne form with alternating single and triple bonds (–[C≡C]–; see figure 1(a)), which has been determined to be more energetically favorable [5]. Carbyne is known to have a relatively high axial tensile stiffness and, due to its monoatomistic ‘cross-section’, incredible specific strength and high surface area per given mass [5]. Carbyne’s practical applications include energy storage devices and nanoscale electronic devices [8]. It is also possible that carbyne could act as a structural connection 0957-4484/14/335709+09$33.00
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To such an end, herein we investigate the extent in which the effective compressive stiffness of carbyne can be controlled due to ideal cylindrical confinement. We note that— even while buckled—the entropic conformation of carbyne would provide resistance to axial deformation. As such, we are not concerned with the critical buckling load of the carbyne chains (which is meaningless due to thermal fluctuations), but rather the effective stiffness—i.e., the resistance to contraction along the axial direction. Specifically, supported by previous experimental observations, we consider a finite length of carbyne subject to compression confined to an effective carbon nanotube to see any variation in compressive stiffness using full atomistic molecular dynamics (MD) simulations. We subsequently explore the effect of: (1) molecular chain length, L; (2) temperature, T, and; (3) confinement radius, R.
2. Methods We first undertake an analysis of the baseline nanomechanical properties of carbyne chains, to enable comparison with chains subject to confinement, consistent with our model formulation. Specifically, we determine the axial stiffness, key stress/strain values, and bending rigidity by implementing mechanical test cases via MD with the first-principles–based ReaxFF force field [18]. While it has been demonstrated that the ReaxFF potential provides an accurate account of the chemical and mechanical behavior of carbon-based materials [4, 19–23], it is not as accurate as quantum-based ab initio methods [5]. However, due to the length and time scales required, first principles methods are not suitable. As such, characterization of the mechanical properties of carbyne also serves to validate the use of ReaxFF. ReaxFF is utilized to ensure the simulations are as accurate as possible and imitates the local chemical environment to allow dynamic alterations to the molecular and atomistic interactions [24]. Mechanical characterization is then followed by a suite of confined compression simulations. All simulations are implemented by the software package LAMMPS (http://lammps.sandia.gov/ ) [24].
Figure 1. (a) Chemical structure of carbyne, a linear monoatomistic
chain of carbon atoms with a repeating single–triple (polyyne) bond structure (also known as linear acetylene); (b) schematics of unconfined (free) compression where buckled shape is unconstrained compared to confined compression where lateral deformation is limited.
approximately 14 nm [5]—chain lengths less than this will tend to straighten to keep their bending energy low and thus act as a suitable structural element at the molecular scale; subject to compression, however, such rods will immediately lose structural integrity and buckle. At the macroscale, it has been shown that constrained slender rods can support higher compressive loads [9–12], akin to bracing the effective buckling length. Confinement may therefore provide a means to increase the effective strength and stiffness of carbyne chains subject to compression (figure 1(b)). Moreover, if variable, the amount of confinement may provide a mechanical means to enable the fine tuning of compressive stiffness, again similar to controlling the effective length (and critical buckling load) of a slender column through external bracing. One possible solution is the vacuum space inside carbon nanotubes, which offers an interesting medium for the inclusion, transportation and confinement of foreign molecules. Demonstrated by both first principles calculations [13–15] and experimental observation [14, 16, 17], carbyne chains are likely stable within the cavity of both single and multi-walled carbon nanotubes. As such, they may provide an ideal platform as axial, cylindrical confinement with a variable radius—e.g., an effective mechanical sheath or casing for a single carbyne chain.
2.1. Axial stiffness and stress response
To closely mimic first principles approaches [5], a temperature-free molecular mechanics approach is implemented on a representative unit cell of carbyne consisting of two carbon atoms (reflecting one instance of the repeating single-triple bond structure). Periodic boundaries are applied, effectively representing an infinite carbyne chain. The system is then subject to increments of strain along the chain axis, followed by energy minimization (conjugate gradient (CG) algorithm as implemented in LAMMPS with an energy tolerance of 10−6 kcal mol−1). During axial stretching, stress is calculated using a virial stress formulation. Virial stress is commonly used to relate to the macroscopic (continuum) stress in MD computations [25, 26]. A virial stress approach allows us to determine the components of the macroscopic stress tensor 2
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through the virial components, Sij , where: Sij =
⎡ ⎢ ⎣
⎤
N
∑ ∑ ri(ab) ⋅ F j′ (ab) ⎥, a∈Ω
⎦
b=1
(1)
which generates the six components of the symmetric stressvolume tensor, S, where N represents the number of individual atoms, b, interacting with atom a, through any multibody potential contribution, ‘I’ and ‘j’ the coordinate basis, r(ab) the distance between ‘a’ and ‘b’ along the ‘I’ direction, i and Fj'(ab) the scaled partial force on atom, a, due to all multibody contributions [27]. We note the velocity term is absent due to the minimization approach.
Figure 2. Model configuration of compressive deformation of confined carbyne chain. Dashed lines indicate ‘confinement radius’ of virtual nanotube (diameter = 2 R). Red atom indicates ‘dummy atom’ which drives the compressive deformation via incremental displacement, Δ.
energy state. As there is no data extracted when the dummy atom is under motion and allow equilibrium, we eliminate any rate dependence but maintain finite temperature effects. The process is repeated to determine the energy versus displacement response.
2.2. Bending rigidity
Due to the difficulty in dynamically bending carbyne chains, an energy-based molecular mechanics approach is also implemented to determine bending rigidity. A pre-formed chain ring with a varying number of carbon atoms is utilized. The ring structures maintain a constant radius of curvature without the need to impose artificial boundary conditions, and thus enables efficient energy minimization. A total of seven trials (seven rings with different radii) are run, each with a different number of carbon atoms (n = 8, 10, 12, 14, 16, 18 and 20). Each system is equilibrated (0.5 ns at 300 K) to ensure conformational stability, and minimized (CG algorithm with an energy tolerance of 10−6 kcal mol−1) to determine elastic bending strain energy.
3. Results and discussion 3.1. Mechanical characterization
First we discuss the baseline mechanical properties of carbyne, to both validate the use of the ReaxFF potential and to allow the theoretical treatment and comparison of the compressive behavior. Of key concern is accurately capturing the axial stiffness and bending rigidity, which are key properties for confined compression. 3.1.1. Theoretical axial stiffness and stress-strain response.
2.3. Confined compression
Each minimization per strain increment returns the total elastic strain energy of the carbyne system. Rather than extracting axial stiffness indirectly from a stress-strain or force-strain response, the tensile stiffness, C, of carbyne can be defined energetically as:
Compression simulations are carried out under a canonical (NVT) ensemble, time step of 0.5 × 10−15 s. Finite temperature is required to accurately capture the entropic contribution of the carbyne chain resistance. A 100 carbon atom, 13 nm carbyne chain is minimized and subsequently equilibrated at 100 K, 300 K, or 500 K. Length can be varied by fixing a portion of the chain. Following equilibration, a confining cylinder is introduced, with a harmonic repulsive potential, where: 2
Fconfine = K (r−R) ,
C=
1 ∂ 2E , a ∂ε 2
(3)
where a is the equilibrium unit cell length (here, a = 2.63 Å), E the strain energy per two carbon atoms (or, equivalently, per a single/triple bond pair) and ε the strain. We track the energy versus strain from the MD simulation, and fit the data to a fourth order polynomial (see figure 3(a)). Taking the derivative about ε = 0, we find a stiffness of 91.75 eV Å−1. This result is directly compared to ab initio results, utilizing published DFT data from Liu et al [5], and plotted on the same axis. While the two curves deviate at higher strains, the reported value of 95.56 eV Å−1 indicates a close match in terms of equilibrium stiffness, within 4% of the ab initio result. Beyond the axial stiffness, we also want to ensure an accurate force–displacement response of carbyne can be attained using ReaxFF. As such, we plot the virial stress of the simulated test with respect to strain. We note that that virial stress calculates terms in units of stress-volume, typically normalized by the atomistic volume of the system. Due to the 1D structure of carbyne, we normalize the virial
(2)
where K is the confinement stiffness (taken as 15 000 kcal mol−1 Å−1, representing a ‘stiff’ boundary), r the distance from the atom to the confining cylinder, and R the effective radius of confinement. By utilizing a variable confining radius, the cylinder can represent carbon nanotubes of different size. For the boundaries of carbyne, one end of the chain is fixed while the other is attached to a ‘dummy atom’ (figure 2). This atom acts as a virtual ‘force transducer’. To apply compression, the dummy atom is incrementally displaced by approximately 0.1 Å over a short timespan of 0.5 ps (an effective rate of 20 m s−1). After each displacement, the dummy atom is fixed, and the system is equilibrated at constant temperature (100 K, 300 K, or 500 K) for 0.5 ps to get an average potential energy (at the new, shortened/deformed length). Trial cases indicated that the displacement rate of the dummy atom is independent of the subsequent equilibrium 3
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Figure 4. (a) Minimized carbyne rings representing a set of systems
with constant curvatures; (b) EL versus κ plot used to determine the effective bending stiffness of carbyne. Note that the fitted line only appears linear due to the scaling of the plot—it is a quadratic, which provides a better statistical fit (R2 = 0.9997 compared to R2 = 0.998 for a linear fit).
Figure 3. Axial stretching response (a) potential energy versus strain; DFT data (blue circles) from previously published paper [5] and the current MD simulations (red continuous). Stiffness obtained from the quadratic term of the fitted polynomial; (b) virial stress versus strain response of carbyne indicating transition point at approximately 7 nN (green) and ultimate strength at approximately 11 nN (red). Linear trendline represents carbyne’s modulus of elasticity (Y = 313 nN).
the transition in stress response at approximately 10% strain —previous reports indicate a transition of bond length alternation (BLA) of carbyne under tension [5, 8, 30], with a clear transition at ε ≈ 0.1. The BLA affects the electronic density and resulting band gap of carbyne. This extreme sensitivity of band gap to mechanical tension is promising for many applications [31, 32], and is indirectly captured by the ReaxFF potential.
stress by the unit cell length only, without having to define an effective cross-sectional area (a similar assumption is used when characterizing graphene in two dimensions [21, 28]). As such, the virial stress is presented in units of force (figure 3(b)). First we can fit an effective modulus, Y, to the stress-strain response, via a linear fit at small deformation. A fit between ±5% strain results in a modulus of approximately Y ≅ 313 nN . We also observe an ultimate strength of 10.75 nN at 18% strain, which is an agreement with the ab initio prediction of 9.3 to 11.7 nN [5]. Of additional note is
3.1.2. Bending rigidity.
The minimized carbyne rings represented a set of systems with constant curvatures on the order of 0.24–0.60 Å−1 (n = 20 and n = 8 respectively; figure 4(a)). Combining the bending strain energy output of the simulations along with calculated values of perimeter for each ring results in normalized energy values (kcal mol−1 Å−1). Energy per length, EL, and the curvature (κ = 1/r) of each ring can be seen in table 1. The elastic
4
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Table 1. Bending stiffness summary; results of MD simulations involving curved polyyne carbyne chains involving 8–20 atoms.
Number of carbon atoms Parameter
Units −1
Bending energy, E Perimeter, L Energy per length, E/L Curvature, κ
kcal mol Å kcal mol−1 Å−1 Å−1
8
10
12
14
16
18
20
−1232.9 10.48 −117.6 0.5995
−1591.1 13.10 −121.5 0.4796
−1944.6 15.72 −123.7 0.3997
−2302.9 18.34 −125.6 0.3426
−2660.2 20.96 −126.9 0.2998
−3016.3 23.58 −127.9 0.2665
−3371.4 26.20 −128.7 0.2398
bending energy of a curved element can be calculated as: E=
∫
0
L
1 2 Dκ ds , 2
3.2.1. Unconfined compressive stiffness and theoretical bounds. An initial simulation is run using a confinement
radius of 100 Å making the carbyne chain virtually unconfined, thereby providing a baseline compressive resistance. We choose a constant temperature of 300 K to approximate room temperature conditions, and probe three chain lengths (65 Å, 98 Å, and 117 Å). To maintain consistency with the calculations for the confined compression simulations, equation (5) is used to calculate the energy displacement relationship, and the unconfined compressive stiffness results are easily approximated by a second order polynomial trendline. The unconfined compressive simulations for L = 65 Å, 98 Å, and 117 Å resulted in spring constants of k = 14.308 kcal mol−1 Å−2, 2.912 kcal mol−1 Å−2, 0.540 kcal mol−1 Å−2, respectively (see figure 6). The maximum stiffness can be considered from the modulus of pure axial compression, which we have calculated as Y ≅ 313 nN . To derive an equivalent spring stiffness, we can divide by the spring length, where:
(4a)
where D is the bending stiffness or bending rigidity of the element (D = EI for continuum beam elements, for example), as s is the local coordinates along the curve. Assuming constant curvature and bending stiffness: E=
1 2 Dκ L, 2
(4b)
1 2 Dκ . 2
(4c)
such that: EL =
We use the above quadratic form of bending strain energy due to curvature as it is both accepted in the small deformation regime and enables comparison with continuumscale theory (see [5] for example). By plotting EL versus κ (figure 4(b)) a fitted constant was obtained to determine the effective bending stiffness of carbyne using equation (4c). The outputted bending stiffness was 30.7 ± 0.6 (kcal mol−1) Å or 1.32 eV Å (R2 = 0.9997, indicating a statistically significant fit). Once again, this is in good agreement in magnitude with first principles results, which reported a larger bending rigidity of 82.8 (kcal mol−1) Å (3.56 eV Å) [5]. We note that carbon rings undergo so-called Jahn–Teller distortions [33–35], which is not captured by the ReaxFF potential, thereby partially explaining the deviation in derived stiffness. From this bending stiffness, the persistence length of carbyne (at 300 K) is on the order of 5 nm (where lp = D /kB T ).
k max =
Since harmonic spring like behavior is assumed in the simulations, a spring constant, k, is able to be calculated from the following: 1 2 kΔ , 2
(6)
For L = 65 Å, 98 Å, and 117 Å, kmax = 69.9 kcal mol−1 Å−2, 46.4 kcal mol−1 Å−2, and 38.8 kcal mol−1 Å−2, respectively. To achieve such a stiffness, the chain would remain straight (zero curvature) without buckling, and undergo homogenous contraction across its length—e.g., completely confined. Such a scenario is prevented by finite temperature effects that inevitably produce perturbations from a straight configuration. We also note there is a higher deviation from the maximum compressive stiffness as the length increases (e.g., 0.540 kcal mol−1 Å−2 to 38.8 kcal mol−1 Å−2 for L = 117 Å), corresponding to an increase in buckling modes. The minimum stiffness evokes the force-extension relation for a semi-flexible polymer, which associated an equivalent spring constant to a random wormlike chain with a stochastic end-to-end distance [36]. In the limit, the spring constant can be expressed as:
3.2. Confined compression simulations
U=
Y . L
(5) k min =
where U represents the total potential energy, k represents the spring constant, and Δ represents the total displacement of the spring. The energy-displacement relationship is plotted from the simulation, and the spring constants are calculated using the coefficient of the quadratic term of the second order polynomial trendline.
90D 2 , kB TL4
(7)
where D is the bending rigidity, kB Boltzmann constant, T the temperature, and L the chain length. Equation (7) serves as an asymptotic lower bound. For L = 65 Å, 98 Å, and 117 Å, kmin = 0.008 kcal mol−1 Å−2, 0.0015 kcal mol−1 Å−2, and 0.000 75 kcal mol−1 Å−2, respectively, at 300 K, which are 5
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Figure 5. Simulation snapshots of carbyne chain under deformation
(L = 117 Å, T = 300 K, R = 0.70 Å). (a) Initial configuration, (b) approximately 10% strain and (c) approximately 15% strain.
orders of magnitude less than the observed results. It is clear that the unconfined chain lies between the two extremes of upper and lower compressive stiffness—carbyne provides an intermediate case between a compression member (k max ) and a semi-flexible chain (k min ). 3.2.2. Variation in confinement radius, length and temperature. The MD simulations confined carbyne to
four different confinement radii: 0.05 Å, 0.35 Å, 0.70 Å, and 2.10 Å. The confinement radii 0.05 Å, 0.70 Å, and 2.10 Å represent (5, 5), (6, 6), and (8, 8) carbon nanotubes respectively, while the 0.35 Å value represents an intermediate case to better demonstrate the relationship between confinement, length, and temperature. It has been shown that the interaction energy of a concentric carbyne chain within a (10, 10) CNT is negligible [13], suggesting an upper limit to an idealized confinement radii (in terms of carbon nanotube encapsulation). The specific confinement radii are determined by accounting for the carbon–carbon atomistic interaction distance, (3.35 Å), between atoms and subtracting it from the radius of each specified nanotube. Each simulation with specific confinement radii is run at 100 K, 300 K, and 500 K. It has been shown that carbyne chains are stable within CNTs to a temperature of approximately 450 °C (approximately 725 K) [16, 17]. To confirm, we increased the temperature in our simulations by 50 K increments until the chain lost stability. It was determined that the carbyne chain ‘melts’ or has an upper limit of approximately 900 K. The stiffness of the carbyne chain is adjusted by altering the length of the chain from 130 Å to 117 Å, 98 Å, or 65 Å during different trials of the simulation. A total of 36 simulations were run to account for adjustments in confinement radius, temperature, and chain length. Example snapshots are depicted in figure 5. By adjusting the confinement radii, the amount of space that the carbyne chain has to deform is altered. In the simulations with the largest confinement radius—the virtual (8, 8) carbon nanotube—the carbyne chain has the most room to deform, resulting in the lowest compressive strength (e.g., for L = 117 Å at T = 300 K, k = 0.873 kcal mol−1 Å−2, approaching the unconfined value of 0.540 kcal mol−1 Å−2). When increasing the confinement (decreasing the radii), the stiffness increases, achieving a significant proportion of k max (see figure 6). For example, at R = 0.05 Å, L = 65 Å achieves
Figure 6. Energy–displacement behavior. Energy-displacement
results for five different virtual nanotube confinement radii: 0.05 Å, 0.35 Å, 0.70 Å, 2.10 Å, ∞ (unconfined) run at 300 K with chain length, L: (a) 117 Å (b) 98 Å (c) 65 Å. The plots also depict the upper and lower bounds, kmax = 69.9 kcal mol−1 Å2 and kmin = 0.000 75 kcal mol−1 Å2, respectively (black solid lines).
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Results also prove that with decreased chain length and confinement radius, carbyne can demonstrate a controlled variation in the effective modulus of elasticity, Y*. The effective modulus of elasticity can be calculated by rearranging the terms in equation (8a) to solve for Y: Y* =
at 100 K, 300 K and 500 K, continuous, dashed, and dash-dot lines respectively, for L = 65 Å, 98 Å and 117 Å, red, blue and green colors, respectively. The results indicate a slight but consistent increase in stiffness with temperature.
approximately 65% k max , while L = 117 Å achieves approximately 24% k max . Simulation results clearly show that with decreased chain length, the carbyne chain demonstrates an increase in compressive stiffness (figure 6). In terms of effective modulus, we can consider the carbyne chain as analogous to a truss element, where the the spring constant, k, is inversely proportional to the length of the carbyne chain, similar to equation (6), or: YA . L
(8b)
From equation (8b), it is evident that the modulus of elasticity is directly proportional to the spring constant, k, and hence, the overall stiffness. A radius of R = 0.386 Å is used to compute the modulus of elasticity, acquired from ab initio results from Liu et al [5]. Due to the relationship between compressive stiffness, confinement radius, temperature, and chain length amongst the 36 simulations, there is a two orders of magnitude difference between the highest spring constant and effective modulus (k = 46.130 kcal mol−1 Å−2; Y* ≈ 44 TPa) and the lowest spring constant and effective modulus (k = 0.662 kcal mol−1 Å−2; Y* ≈ 1 TPa). Correspondingly, the largest spring constant occurs at the smallest confinement radius (R = 0.05 Å), smallest chain length (L = 65 Å) and the highest temperature (T = 500 K), while the smallest spring constant occurs at the largest confinement radius (R = 2.10 Å), the largest chain length (L = 117 Å), and the lowest temperature (T = 100 K), (figure 7). For all of the confined carbyne simulations, the quadratic trendlines used to fit the energy displacement results produced a coefficient of determination of R2 > 0.944. Since this value is close to 1.00, where 1.00 statistically represents perfect regression, the computed trendlines demonstrate good fitted nonlinear regressions. Moreover, all k values determined from the energy displacement results for each of the carbyne chain lengths lie within the theoretical upper and lower boundaries of compressive stiffness demonstrating the validity of the range of simulation results.
Figure 7. Plot of the spring constant values versus confinement radii
k=
kL . A
(8a)
4. Conclusion
With this relationship, a decrease in the chain length should cause an increase in the spring constant value and hence, an increase in compressive stiffness. This result is most clear when removing the confinement of the virtual nanotube, keeping the temperature constant, and comparing only the length and the stiffness values of carbyne (see figure 6). With an increase in temperature, we observe that the effective stiffness marginally increases (figure 7). This is in opposition to semi-flexible polymer behavior, wherein higher temperature increases the compliance of a chain, thereby allowing easier coiling and bending (e.g., rigidity ∝T −1). Under confinement, however, the increased thermal energy can be thought of as increasing the pressure within the confined space, and this effectively increases the resistance to compressive deformation. In effect, more force is required to hold a confined coiled/deformed chain at high temperatures. The competition due to these competing factors is nontrivial, and beyond the scope of the current study. That being said, the observed temperature dependence is not significant.
A recent full-principles study by Liu et al has rigorously characterized the mechanical properties of carbyne, constituting a linear chain of repeating sp-hybridized carbon atoms, assessing axial stiffness, strength, and bending rigidity. While undoubtedly, first-principles methods are necessary to explore subatomistic features, we show that, under similar simulation conditions, ReaxFF agrees with more rigorous methods, resulting in similar values for stiffness (C = 91.75 eV Å−1 and Y = 313 nN), ultimate strength (10.75 nN at 18% strain), bending rigidity (D = 30.7 ± 0.6 (kcal mol−1) Å), and persistence length (approximately 5 nm). Relatively large carbyne systems can thus be efficiently explored using a MD framework, extending the time—and length-scales typically associated with first-principles methods, with little consequence on mechanical features. Taking into account the relative weakness of carbyne in compression, here we explored the potential to fine-tune compressive resistance by subjecting a chain to molecular 7
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confinement (similar to bracing of a column at the macroscale). By plotting the spring constants for the four different confinement radii and the three carbon chain lengths, it became clear that there is a strong relationship between stiffness and chain length, L. An increase in stiffness causes an increase in the spring constant value, modulus of elasticity, and hence, the propensity to resist deformation. The spring constants and modulus of elasticity are significantly larger for shorter chain lengths. There is a two orders of magnitude difference between the largest k and the smallest k each corresponding to the 0.05 Å confinement radius but with the largest value associated with the shortest chain length at 500 K, and the smallest value associated with the longest chain length at 100 K. These results confirm the hypothesis of controlling compressive stiffness, since the output demonstrates that a smaller confinement radius results in a higher stiffness value, k. Through adjustments to free length, confinement radii, and temperature, the effective spring stiffness can be varied by over an order of magnitude— an amazing result for the same system of atoms in a consistent molecular configuration. As an additional benefit, it has been reported that long carbyne chains (e.g., more than 20 carbon atoms) become energetically unstable [7, 37, 38], requiring large and bulky molecular end groups [6]. Confinement within carbon nanotubes may inherently stabilize the chains, providing a platform for the synthesis of extraordinarily long chains (tens of nanometers). As such, characterization of the behavior of confined carbyne may prove of interest beyond the tunability of compressive behavior. While carbyne is not known to be strong in compression, it is clear that by confining carbyne within a virtual nanotube, effective stiffness can be controlled. The strong relationship between carbyne’s compressive stiffness, confinement radius, and chain length demonstrates the possible opportunity to be able to control some of carbyne’s material properties including the ‘effective’ modulus of elasticity and/or spring constant. By effectively being able to design various behaviors from the same material—e.g., a finite length of carbyne —the future of carbyne as ‘nanowires’ and ‘nanorods’ in nanotechnology can be enhanced.
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Acknowledgements SWC acknowledge generous support from NEU’s CEE Department. NARY is thanked for his volunteer efforts and contribution to this study. 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 [39].
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