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DOI: 10.1039/D0NA00217H
(Paper)
Nanoscale Adv., 2020, Advance Article

Brahmanandam Javvaji^{a},
Bohayra Mortazavi^{a},
Timon Rabczuk^{bc} and
Xiaoying Zhuang*^{de}
^{a}Chair of Computational Science and Simulation Technology, Department of Mathematics and Physics, Leibniz Universität Hannover, Applestr. 11, 30167 Hannover, Germany
^{b}Institute of Structural Mechanics, Bauhaus University Weimar, Marienstrasse 15, 99423 Weimar, Germany
^{c}College of Civil Engineering, Department of Geotechnical Engineering, Tongji University, Shanghai, China
^{d}Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
^{e}Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam. E-mail: xiaoying.zhuang@tdtu.edu.vn

Received
18th March 2020
, Accepted 16th May 2020

First published on 28th May 2020

Recent experimental advances [Liu et al., npj 2D Mater. Appl., 2019, 3, 23] propose the design of graphene nanoribbon springs (GNRSs) to substantially enhance the stretchability of pristine graphene. A GNRS is a periodic undulating graphene nanoribbon, where undulations are of sinus or half-circle or horseshoe shapes. Besides this, the GNRS geometry depends on design parameters, like the pitch's length and amplitude, thickness and joining angle. Because of the fact that parametric influence on the resulting physical properties is expensive and complicated to examine experimentally, we explore the mechanical, thermal and electromechanical properties of GNRSs using molecular dynamics simulations. Our results demonstrate that the horseshoe shape design GNRS (GNRH) can distinctly outperform the graphene kirigami design concerning the stretchability. The thermal conductivity of GNRSs was also examined by developing a multiscale modeling, which suggests that the thermal transport along these nanostructures can be effectively tuned. We found that however, the tensile stretching of the GNRS and GNRH does not yield any piezoelectric polarization. The bending induced hybridization change results in a flexoelectric polarization, where the corresponding flexoelectric coefficient is 25% higher than that of graphene. Our results provide a comprehensive vision of the critical physical properties of GNRSs and may help to employ the outstanding physics of graphene to design novel stretchable nanodevices.

Nonetheless, it has also been practically proven that graphene can show largely/finely tunable electronic, mechanical, thermal, optical and electromechanical properties, with accurately controlled mechanical straining,^{8–13} defect engineering^{14–17} or chemical doping.^{18–22} We remind that for many centuries, springs have been playing pivotal roles in the design and fabrication of various kinds of devices. The importance of springs originates from the fact that while the mechanical properties of a material is considered as its inherent properties and thus invariable, when it is shaped in the form of springs the subsequent structures can show superior stretchability and diverse mechanical responses. In particular, the design of spring like structures has been known as one of the most effective approaches to achieve highly stretchable and flexible moving components.

For the employment of graphene in flexible nanoelectronics, its ductile and highly rigid mechanical properties are undesirable. Therefore, engineering of the graphene design in order to improve its stretchability is a critical issue.^{23–27} To address this challenge, in a most recent exciting experimental advance by Liu et al.^{28} the old idea of spring design has been applied in the case of graphene to enhance its stretchability and flexibility via a nanowire lithography technology. This experimental study consequently raises questions concerning how the design of graphene springs can be improved to reach higher degrees of stretchability. In addition, the electronic and heat transport properties of these novel nanostructures should also be examined, in order to provide comprehensive visions for the design of nanodevices. As a common challenge in an electronic apparatus, the thermal conductivity of employed components should be high to effectively dissipate excessive heat. On the other hand, low thermal conductivity is a key requirement for the enhancement of the efficiency of thermoelectric energy conversion. As a common barrier, it is well known that the evaluation of the mechanical and transport properties of graphene springs by experimental tests is not only complicated but also time consuming and expensive as well. This study therefore aims to investigate the mechanical response and heat conduction properties of graphene springs via conducting extensive classical molecular dynamics simulations. Since graphene is the first member of 2D materials, commonly the experimental and theoretical methodologies that are applied for graphene can be extended to the other members of the 2D materials family. We are thus hopeful that the results obtained by this study may serve as valuable guides for future theoretical and experimental studies on the design of 2D spring material like structures.

(1) |

(2) |

Fig. 1 Unit cell representation and definition of various structural parameters for the (a) sinus shape and (b) horseshoe shape graphene nanosprings. |

The horseshoe shape design GNR (GNRH) was composed by connecting two circular arcs of the inner radius (h_{r}) with the intersecting angle (h_{θ}). When h_{θ} = 0°, the GNRH looks like a series of connected semi-circles and for h_{θ} > 0° and h_{θ} ≤ 45°, the horseshoe design is obtained. For h_{θ} more than 45°, the semi-circles merge with each other, which is not desirable. These parameters define the shape of a single horseshoe curve. Another curve with radius h_{r} + h_{t} creates a parallel curve, where h_{t} defines the thickness of the GNRH. We constructed the horseshoe-shaped region on a pristine graphene sheet and removed atoms in the other regions, which creates an initial atomic configuration for the GNRH system. After the initial preparation of the spring structures, we removed the carbon atoms bonded with a single carbon atom along the lateral edges since these atoms can lead to instability in simulations. It is worth noting that the carbon atoms belonging to the curved edges of the GNRS and GNRH system are not terminated with hydrogen atoms.

With the defined geometrical parameters and initial atomic configurations for the GNRS and GNRH, we consider the following parameter sets for estimating the mechanical and thermal properties. Parameter set s_{p} – s_{t} – s_{a}: the starting value for s_{p} is 9 nm (where the size effects are absent in pristine graphene^{13}), and s_{a} is 2.5 nm, where at least 10 carbon rings accompany in the lateral direction. The minimum possible value of s_{t} is 1.6 nm for this combination of s_{p} and s_{a} for having a reasonable thickness for these spring systems. Further, we increase the values of s_{p}, s_{t} and s_{a} by integral multiples from 1 to 5. Parameter h_{θ}: we consider h_{θ} to be 0°, 15°, 30° and 45° by keeping h_{r} at 2.5 nm and h_{t} as 1.6 nm. Using this choice of parameters, we explore the effect of h_{θ} on mechanical and thermal properties. Parameter h_{r} – h_{t}: in this set, h_{r} and h_{t} parameters are scaled by integral multiples from 1 to 5 starting from 2.5 and 1.6 nm at fixed h_{θ}. We fix the value of h_{θ} from the previous parameter set, which has shown exceptional mechanical properties. The spring structures from the different parameter sets are made by cutting from graphene sheets oriented along the zigzag direction. Our comparative results discussed in the ESI† document however confirm that the orientation of the original graphene sheet shows negligible effects on the estimated properties.

Fig. 2(b) shows the stress–strain response for parameter set h_{θ}, where h_{r} and h_{t} values are 2.5 and 1.6 nm, respectively. When h_{θ} = 0°, there is no stress rise up to a strain of 0.3. For the strain range 0.3 to 0.6, the deformation in atomic configuration raises the system stress followed by a failure. The strain range for the non-zero portion of the stress–strain curve shifts between 0.5 and 1.0 when h_{θ} is 15°. For h_{θ} equal to 30°, the non-zero portion of the stress–strain curve span the strain range of 0.94 to 1.5. For the 45° GNRH, this strain range increased to 2.4. Fig. 3(f)–(j) show the atomic configurations for a GNRH with a 45° connecting angle. The closeness between semi-circular segments in the GNRH develops strong repulsive interactions compared to those in the GNRS. Such repulsion largely deflects the atomic system. Further, mechanical stretching reduces atomic deflections by maintaining the stress levels via transforming the smooth circular GNRH segments to sharp peaks. Fig. 3(g) shows the atomic configuration with several peaks for the GNRH. For a strain greater than 1.5, Fig. 2(b) shows a linear stress–strain response due to the bond stretching. At a strain of 2, the GNRH system looks like a combination of thread and knots, as seen in Fig. 3(h), where stress concentrates near the thread portions. Further increase in the strain in the GNRH leads to bond failure. Fig. 3(i) and (j) at strain levels of 2.17 and 2.23 show the complete failure of the GNRH.

Interestingly, the GNRH with various h_{θ} values maintained the stress levels when increasing the strain range. We consider varying the width parameter h_{t} in the GNRH by keeping the other parameters constant to check its influence on the mechanical properties. The range of h_{t} is limited by the choice of the other two parameters. For example, consider that h_{r} is equal to 5 nm and h_{θ} is 45°. The maximum available value of h_{t} is 4 nm. When h_{t} is greater than 4 nm, the two circular cross-sections of the GNRH unit cell overlap with each other, which is not desirable. We consider h_{t} values to be 1.6, 2.4 and 3.2 and 4 nm and the corresponding RS values are 2.64, 2.44, 2.17 and 1.9, respectively. TS values are noted as 53.83, 38.06, 24.81 and 17.49 GPa (see the ESI†). When varying h_{θ}, the RS increased with a very low effect on TS (see circle markers with dotted and solid lines in Fig. 2(d)). However, variation of h_{t} has an effect on both RS and TS in the GNRH. The increase in thickness increases the separation between the stress centers and decreases the TS, which leads to early failure and decrease in RS. This finding implies that the change of h_{t} strongly influences both RS and TS of the GNRHs. The observations concerning the width effect on the mechanical response are in close agreement with those of an earlier report based on MD simulations.^{36}

When compared to the GNRS, the GNRH shows higher stress levels (dotted lines in Fig. 2(d)). From the structural point of view, the GNRH differs from GNRS in two factors, one is the smoothness of undulations and the second is the closeness between the undulations. The smoothness of circular arcs in the GNRH makes the stress distribute across all the boundary atoms. The increased number of atoms with higher per atom stress values increases the total stress in the atomic system. In the case of the GNRS, the lower number of atoms with higher per atom stress near the peaks of the sine curve makes the total stress lower. From Fig. 2(d), it can be seen that the TS values for the 2.5 – 1.6 – 0° GNRH and 9 – 1.6 – 2.5 GNRS are 39.72 and 31.93 GPa, respectively, which represents that the smoothness factor accommodates more number of atoms with high stress levels in the horseshoe shape design. The closeness between the undulations increases the deflections in the atomic system, which helps to avoid the bond stretching and stress rise. These deflections in GNRS and GNRH systems are measured using the standard deviation of the z-coordinates (),^{37} which is defined as , where z_{i} is the z-coordinate of the i^{th} atom and z_{0} is the averaged z-coordinate over N atoms. Fig. 2(c) shows the computed with respect to strain for the selected GNRS (9 – 1.6 – 2.5) and GNRH (parameter set h_{θ}) systems. initially increases with strain, which represents that the energy of given mechanical strain is used to increase the deflections in both GNRH and GNRS systems. After reaching a maximum deflection, the given tensile loading starts stretching the atomic system and decreasing the deflections which decrease . However, the magnitude of for the 0° GNRH configuration is high compared to that of the 9 – 1.6 – 2.5 GNRS, which supports that the repulsive interactions in the GNRH are heavier compared to those in the GNRS. increases with an increase in parameter h_{θ}. is highest for the 45° GNRH.

The very strong repulsions exist between the semi-circular rings due to the minimum spacing. The very high deflections and smoothness of circular cross-sections help to avoid the stress concentrations in the GNRH, which helps to enhance the mechanical properties. As a result, a very high value of RS is noted for the GNRH. With the increasing system size (arc length of the GNRH), RS tends to converge to a value of 2, which is about 17% higher than that of the graphene kirigami design,^{38,39} keeping the stress-levels identical.

As proposed in previous work,^{33} we use a microscale continuum model of the graphene springs to evaluate the effective thermal conductivity. This evaluation was carried out within the diffusion range, in which the phonon-boundary scattering vanishes. To this aim, a system is modeled by the finite element (FE) approach to establish connections between the effective thermal conductivity and nanoribbon's arc length. We apply inward and outward heat-fluxes on the two opposite sides of the GNRS as the boundary conditions. Using the measured temperature gradient along the heat transfer direction, the effective thermal conductivity was computed from Fourier's law. We then used a first order rational curve fitting to extrapolate the atomistic results (circular markers in Fig. 4(d) that correspond to the averaged κ/κ_{0} over several samples of s_{p} – s_{t} – s_{a} and the standard deviation among them) dominated by the phonon-boundary scattering to the diffusive transport by the FE simulations. As shown in Fig. 4(d), this approach could provide a very accurate estimation of thermal transport at different arc lengths, and reveals that the phonon-boundary scattering starts to vanish at large arc lengths.

For the GNRH systems κ/κ_{0} with varying h_{θ} is shown in Fig. 4(b). κ/κ_{0} for the 2.5 – 1.6 – 0° GNRH is about 0.0169, which is nearly the same as for the 9 – 1.6 – 2.5 GNRS. The constant thickness and similar scattering effects in these two spring systems produce the nearly equal thermal conductivity. Keeping thickness constant and increasing the joining angle h_{θ} to 15°, κ/κ_{0} decreased from 0.0169 to 0.0111. As h_{θ} increases, there is an increase in the radius of curvature of the junction region that connects the two semi-circular segments. The phonon transport through this increased curvature experiences significant scattering, which reduces the heat transfer and κ. The thermal conductivity for h_{θ} 30° and 45° is nearly the same.

We examine the thermal conductivity for the GNRH samples used in estimating the effect of width on mechanical properties in Section 3.1. The effective thermal conductivity for 2.5 – 1.6 – 45° and 5 – 1.6 – 45° are 0.0084 and 0.0055. This represents that increase in h_{r} increases the radius of curvature and produces more edge localized phonon modes. These modes do not contribute to thermal transport; as a result the thermal conductivity decreases for the 5 – 1.6 – 45° GNRH system (see the ESI†). However, the increase in h_{t} increases the number of phonon modes in the GNRH keeping the density of edge localized modes the same. This reduces the edge scattering and increases the phonon transport, thus increasing κ.^{44} Fig. 4(c) shows κ for increasing values of h_{r} and h_{t} keeping the joining angle h_{θ} as 45°. As the GNRS radius and thickness increase, the available region for heat transport increases, which helps to lower the scattering and increase κ/κ_{0} from 0.0084 to 0.0216. This increase of κ is small compared to that of the GNRS systems due to the large curvature induced scattering. We repeat the FE modeling for the GNRH with h_{θ} as 45° similar to the GNRS. The fitting between atomistic results and FE modeling is very good. However, GNRH fitting is converged at significantly larger cut lengths compared to GNRS fitting. This proves that curvature induced scattering reduces κ in the GNRH.

To calculate the polarization in the atomic system, we utilize the charge–dipole (CD) model along with the short-range bonded interactions (Tersoff potential). According to the CD model, each atom i is associated with charge q_{i} and dipole moment p_{i}.^{53,54} The mathematical CD formulation involves the various contributions from charge–charge, charge–dipole and dipole–dipole interactions to the total system short-range interaction energy. The minimization of the energy function gives the numerical values of q_{i} and p_{i}. The complete details about the CD model and estimation of charges and dipole moments can be found in ref. 17 and 52 and references therein.

For applying deformation, we have added left and right rectangular regions to the GNRS and GNRH systems by discarding the periodic boundary condition used in Section 2. These regions have equal s_{t} or h_{t} to those of the spring systems with 1 nm length along the spring longitudinal direction. The left and right regions help to hold the given displacement, particularly during the bending test, and relax the remaining system. We define a load cycle by prescribing the displacement of atoms to left and right regions for a 1 ps time period, followed by a relaxation for a 2 ps time period. Because of the nonperiodic boundaries, we perform simulations at different repetitions of sinus and horseshoe shapes in spring systems. These simulations help us to study the size effect on electromechanical properties. For every load cycle, we note the evolution of the atomic configuration, and the corresponding charge and dipole moments. The total polarization of the atomic system is the sum of all atomic dipole moments divided by the volume of the atomic system.

For tensile deformation, we apply a displacement in the longitudinal direction u_{x} = t_{load}l_{0} to the atomic system, where is the strain rate, equal to 1 × 10^{8} s^{−1} as used in Section 2, t_{load} is the loading time (1 ps) and l_{0} is the initial length of the atomic system in the longitudinal direction. The load cycles continued to reach a strain ε_{xx} of 0.4 for 9 – 1.6 – 2.5 and 2.5 – 1.6 – 0° systems. The strain limit 0.4 corresponds to the linear rise in the stress–strain response for these systems as shown in Fig. 2(a). At each load cycle, the polarization is measured and the variation with strain is plotted in Fig. 5(a). The variation in polarization with strain is nearly negligible for both GNRS and GNRH systems. The coefficient of variation for the polarization response is nearly equal to 1 for both the GNRS and GNRH, similar to non-piezoelectric pristine graphene.^{17} The cancellation of polarization at the sinus and horseshoe cut patterns makes these systems non-piezoelectric materials.

For bending deformation, we supply the following out-of-plane displacement field to the atomic system

(3) |

(4) |

(5) |

P_{z} = μ_{zxzx}K_{eff}.
| (6) |

From Fig. 5(b), it can be seen that the slope of the polarization to strain gradient curve decreases with the increase of h_{θ} in GNRH systems. The ratio of the change in the pyramidalization angles in GNRH and pristine graphene decreases to 1.27, 1.14 and 1.05 and 0.85 with h_{θ} at 0°, 15°, 30° and 45°, respectively. The decrease in θ_{σπ} decreases the dipole moment distribution, as seen from Fig. 6(c) to (d), which decreases the flexoelectric coefficient. For h_{θ} = 45°, the atoms on the lateral boundaries near the central line do not have dipole moments. The strong repulsions between these edges cancel the effect of pyramidalization, which decreases the flexoelectric coefficient.

In order to check the dependence of h_{t} on the polarization, we consider 2.5 – 2.4 – 0° and 2.5 – 3.2 – 0° GNRH configurations. For these configurations, the polarization variation and flexoelectric coefficient are nearly equal to those of the 2.5 – 1.6 – 0° GNRH. The increase in thickness does not affect the induced polarization and flexoelectric coefficient (see the ESI†). The increased thickness was unable to change the pyramidalization angle further, which makes the polarization comparable with that of smaller thickness systems.

Further, the result of the flexoelectric coefficient with length variation is given in Fig. 5(c) for GNRS and GNRH (h_{θ} = 0°). This result represents that there is a boundary effect when the arc length is less than 100 nm. For systems with an arc length about 30 nm, the local electric field is strongly affected by the left and right region of atoms. Here the imposed boundary condition constrains the natural motion of the interior atoms, which restricts the process of pyramidalization and controls the dipole moment. When increasing the arc length this effect slowly nullifies and the atomic configuration deforms to generate the dipole moments. For systems with lengths higher than 100 nm, the boundary effect is completely negligible and the flexoelectric coefficients turn into a stable value. Finally, the flexoelectric coefficient of the GNRS and GNRH-0° system is 0.25 times higher than that of pristine graphene.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0na00217h |

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