Limits...
Dynamics of mechanical waves in periodic grapheme nanoribbon assemblies.

Scarpa F, Chowdhury R, Kam K, Adhikari S, Ruzzene M - Nanoscale Res Lett (2011)

Bottom Line: The acoustic wave dispersion characteristics of the GNRs are simulated using a Floquet-based wave technique used to predict the pass-stop bands of periodic mechanical structures.We show that the thickness and equilibrium lengths do depend on the specific vibration and dispersion mode considered, and that they are in general different from the classical constant values used in open literature (0.34 nm for thickness and 0.142 nm for equilibrium length).The thickness, average equilibrium length and edge type have to be taken into account when nanoribbons are used to design nano-oscillators and novel types of mass sensors based on periodic arrangements of nanostructures.PACS 62.23.Kn · 62.25.Fg · 62.25.Jk.

View Article: PubMed Central - HTML - PubMed

Affiliation: Advanced Composites Centre for Innovation and Science, University of Bristol, BS8 1TR Bristol, UK. f.scarpa@bristol.ac.uk.

ABSTRACT
We simulate the natural frequencies and the acoustic wave propagation characteristics of graphene nanoribbons (GNRs) of the type (8,0) and (0,8) using an equivalent atomistic-continuum FE model previously developed by some of the authors, where the C-C bonds thickness and average equilibrium lengths during the dynamic loading are identified through the minimisation of the system Hamiltonian. A molecular mechanics model based on the UFF potential is used to benchmark the hybrid FE models developed. The acoustic wave dispersion characteristics of the GNRs are simulated using a Floquet-based wave technique used to predict the pass-stop bands of periodic mechanical structures. We show that the thickness and equilibrium lengths do depend on the specific vibration and dispersion mode considered, and that they are in general different from the classical constant values used in open literature (0.34 nm for thickness and 0.142 nm for equilibrium length). We also show the dependence of the wave dispersion characteristics versus the aspect ratio and edge configurations of the nanoribbons, with widening band-gaps that depend on the chirality of the configurations. The thickness, average equilibrium length and edge type have to be taken into account when nanoribbons are used to design nano-oscillators and novel types of mass sensors based on periodic arrangements of nanostructures.PACS 62.23.Kn · 62.25.Fg · 62.25.Jk.

No MeSH data available.


Related in: MedlinePlus

Non-dimensional dispersion curves for zigzag (continuous lines) and armchair (dashed lines) (8,0) GNRs with different aspect ratios (AR). All the results are minimized for the thickness d only.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3211848&req=5

Figure 4: Non-dimensional dispersion curves for zigzag (continuous lines) and armchair (dashed lines) (8,0) GNRs with different aspect ratios (AR). All the results are minimized for the thickness d only.

Mentions: A more significant discrepancy between wave dispersion curves can be observed in Figure 2, when comparing the pass-stop band behaviour for the propagation along the zigzag and armchair directions. Only the first acoustic flexural wave dispersion characteristic is virtually unchanged, while for the other curves we observe a strong decrease in terms of magnitude, as well as mode inversion. The first stop band is significantly decreased by 25 GHz for kx = π - the armchair case gives a frequency drop of 39 GHz for the same propagation constant. Similar decreases in band gaps are observed for higher frequencies, while mode inversion (flexural to torsional) is observed for the armchair propagation around kx/π = 0.42, while for the armchair case the mode inversion is located around 0.8 kx/π. From the mechanical point of view, a possible explanation for this peculiar behaviour can be given considering the intrinsic anisotropy of the in-plane properties of finite size graphene sheets. Reddy et al. [17,41] have observed anisotropy ratios between 0.92 and 0.94 in almost square graphene sheets subjected to uni-axial loading, while similar orthotropic ratios have been identified also by Scarpa et al. [13]. The GNRs considered here have an aspect ratio close to 6, which induces the edges to provide a higher contribution to the homogenized mechanical properties due to Saint Venant effects [42]. A further confirmation of the effective in-plane mechanical anisotropy on the GNRs is apparent also from the non-dimensional dispersion curves shown in Figure 4. For that specific case, the GNRs have one side fixed (1.598 nm for the armchair, and 1.349 nm for the zigzag), with minimized thickness d equal to 0.074 and 0.077 nm and C-C bond equilibrium lengths of l = 0.142 nm for the armchair and zigzag cases, respectively. The dimensions of the nanoribbons are varied adjusting the aspect ratios (2.4 and 8), to obtain armchair and zigzag GNRs with similar dimensions. We have further nondimensionalised the dispersion curves using the values of the first dispersion relation (ω0) for the armchair configuration at kx = π/4. The GNR with an aspect ratio of 2.4 (Figure 4a) shows significant difference s in terms of dispersion characteristics between the armchair and the zigzag configurations, with a reduced band-gap of Δ(ω/ω0) equal to 3 for the armchair, against the value of 5 for the zigzag at the end of the first Brillouin zone (kx/π = 1). Between 4 <ω/ω0 <10, the wave dispersions appear to be composed by combinations of flexural plate-like modes with torsional components, with mode veering occurring between 0.45 < kx/π < 0.65. The zigzag-edged GNRs tend to show a narrowing of the nondimensional dispersion characteristics within the same ω/ω0 range considered. At higher non-dimensional wave dispersions, both armchair and zigzag nanoribbons tend to show beam-like dispersion characteristics [8,40,43]. The nanoribbons with higher aspect ratio (Figure 4b) show the pass-stop band behaviour typical of SS periodic structures made of Euler-Bernoulli beams [40]. However, while the first non-dimensional dispersion curve is identical, the following dispersion characteristics show a marked difference between zigzag and armchair configurations, with the zigzag GNRs having the highest ω/ω0 values. It is also worth of notice that while the zigzag configuration shows a dispersion curve provided by a torsional wave (straight line between 0 < kx/π < 0.62 at ω/ω0 = 37.4), the armchair GNR appears to be governed by flexural waves within the non-dimensional frequency interval considered. This type of behaviour suggests also that the specific morphology of the edges (combined with the small transversal dimensions of the GNRs) affect the acoustic wave propagation characteristics, both contributing to an overall mechanical anisotropy of the equivalent beams, as well as providing specific wave dispersion characteristics at higher frequencies. Moreover, the widening of the band gap observed in Figure 2 for the armchair configuration recalls some similarity to the variation of the energy gap of the electronic states noted in analogous armchair GNRs [3]. For a fixed width of 2.25 nm and and aspect ratio of 2.4 (i.e. 5.4 nm), the first pass-stop band at kx = 0 is located at 180 GHz. For the same fixed width but higher aspect ratio (8.0, corresponding to a transverse length of 18 nm), the same pass-stop band first frequency for kx = 0 is equal to 15 GHz, 12 times lower than the low aspect ratio case (Figure 4). Moreover, for the higher aspect ratio we observe aΔω = 18 GHz, while the lower aspect ratio provides a pass-stop band frequency interval Δω = 90 GHz, five times higher when compared for the armchair nanoribbons at AR = 2.4. Passing between lengths of 0.25 and 3 nm, Barone, Hod and Scuseria observe a decrease in energy gab by a factor of 3 for bare PBEs, and by 5 for bare HSEs [3]. When we consider the variation of the energy of the system proportional to the kinetic energy (and therefore approximately Δω2), ther ratio of the pass-stop bands for the armchair nanoribbons with different aspect ratios is compatible with the decrease of energy gap observed through DFT simulations [3].


Dynamics of mechanical waves in periodic grapheme nanoribbon assemblies.

Scarpa F, Chowdhury R, Kam K, Adhikari S, Ruzzene M - Nanoscale Res Lett (2011)

Non-dimensional dispersion curves for zigzag (continuous lines) and armchair (dashed lines) (8,0) GNRs with different aspect ratios (AR). All the results are minimized for the thickness d only.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3211848&req=5

Figure 4: Non-dimensional dispersion curves for zigzag (continuous lines) and armchair (dashed lines) (8,0) GNRs with different aspect ratios (AR). All the results are minimized for the thickness d only.
Mentions: A more significant discrepancy between wave dispersion curves can be observed in Figure 2, when comparing the pass-stop band behaviour for the propagation along the zigzag and armchair directions. Only the first acoustic flexural wave dispersion characteristic is virtually unchanged, while for the other curves we observe a strong decrease in terms of magnitude, as well as mode inversion. The first stop band is significantly decreased by 25 GHz for kx = π - the armchair case gives a frequency drop of 39 GHz for the same propagation constant. Similar decreases in band gaps are observed for higher frequencies, while mode inversion (flexural to torsional) is observed for the armchair propagation around kx/π = 0.42, while for the armchair case the mode inversion is located around 0.8 kx/π. From the mechanical point of view, a possible explanation for this peculiar behaviour can be given considering the intrinsic anisotropy of the in-plane properties of finite size graphene sheets. Reddy et al. [17,41] have observed anisotropy ratios between 0.92 and 0.94 in almost square graphene sheets subjected to uni-axial loading, while similar orthotropic ratios have been identified also by Scarpa et al. [13]. The GNRs considered here have an aspect ratio close to 6, which induces the edges to provide a higher contribution to the homogenized mechanical properties due to Saint Venant effects [42]. A further confirmation of the effective in-plane mechanical anisotropy on the GNRs is apparent also from the non-dimensional dispersion curves shown in Figure 4. For that specific case, the GNRs have one side fixed (1.598 nm for the armchair, and 1.349 nm for the zigzag), with minimized thickness d equal to 0.074 and 0.077 nm and C-C bond equilibrium lengths of l = 0.142 nm for the armchair and zigzag cases, respectively. The dimensions of the nanoribbons are varied adjusting the aspect ratios (2.4 and 8), to obtain armchair and zigzag GNRs with similar dimensions. We have further nondimensionalised the dispersion curves using the values of the first dispersion relation (ω0) for the armchair configuration at kx = π/4. The GNR with an aspect ratio of 2.4 (Figure 4a) shows significant difference s in terms of dispersion characteristics between the armchair and the zigzag configurations, with a reduced band-gap of Δ(ω/ω0) equal to 3 for the armchair, against the value of 5 for the zigzag at the end of the first Brillouin zone (kx/π = 1). Between 4 <ω/ω0 <10, the wave dispersions appear to be composed by combinations of flexural plate-like modes with torsional components, with mode veering occurring between 0.45 < kx/π < 0.65. The zigzag-edged GNRs tend to show a narrowing of the nondimensional dispersion characteristics within the same ω/ω0 range considered. At higher non-dimensional wave dispersions, both armchair and zigzag nanoribbons tend to show beam-like dispersion characteristics [8,40,43]. The nanoribbons with higher aspect ratio (Figure 4b) show the pass-stop band behaviour typical of SS periodic structures made of Euler-Bernoulli beams [40]. However, while the first non-dimensional dispersion curve is identical, the following dispersion characteristics show a marked difference between zigzag and armchair configurations, with the zigzag GNRs having the highest ω/ω0 values. It is also worth of notice that while the zigzag configuration shows a dispersion curve provided by a torsional wave (straight line between 0 < kx/π < 0.62 at ω/ω0 = 37.4), the armchair GNR appears to be governed by flexural waves within the non-dimensional frequency interval considered. This type of behaviour suggests also that the specific morphology of the edges (combined with the small transversal dimensions of the GNRs) affect the acoustic wave propagation characteristics, both contributing to an overall mechanical anisotropy of the equivalent beams, as well as providing specific wave dispersion characteristics at higher frequencies. Moreover, the widening of the band gap observed in Figure 2 for the armchair configuration recalls some similarity to the variation of the energy gap of the electronic states noted in analogous armchair GNRs [3]. For a fixed width of 2.25 nm and and aspect ratio of 2.4 (i.e. 5.4 nm), the first pass-stop band at kx = 0 is located at 180 GHz. For the same fixed width but higher aspect ratio (8.0, corresponding to a transverse length of 18 nm), the same pass-stop band first frequency for kx = 0 is equal to 15 GHz, 12 times lower than the low aspect ratio case (Figure 4). Moreover, for the higher aspect ratio we observe aΔω = 18 GHz, while the lower aspect ratio provides a pass-stop band frequency interval Δω = 90 GHz, five times higher when compared for the armchair nanoribbons at AR = 2.4. Passing between lengths of 0.25 and 3 nm, Barone, Hod and Scuseria observe a decrease in energy gab by a factor of 3 for bare PBEs, and by 5 for bare HSEs [3]. When we consider the variation of the energy of the system proportional to the kinetic energy (and therefore approximately Δω2), ther ratio of the pass-stop bands for the armchair nanoribbons with different aspect ratios is compatible with the decrease of energy gap observed through DFT simulations [3].

Bottom Line: The acoustic wave dispersion characteristics of the GNRs are simulated using a Floquet-based wave technique used to predict the pass-stop bands of periodic mechanical structures.We show that the thickness and equilibrium lengths do depend on the specific vibration and dispersion mode considered, and that they are in general different from the classical constant values used in open literature (0.34 nm for thickness and 0.142 nm for equilibrium length).The thickness, average equilibrium length and edge type have to be taken into account when nanoribbons are used to design nano-oscillators and novel types of mass sensors based on periodic arrangements of nanostructures.PACS 62.23.Kn · 62.25.Fg · 62.25.Jk.

View Article: PubMed Central - HTML - PubMed

Affiliation: Advanced Composites Centre for Innovation and Science, University of Bristol, BS8 1TR Bristol, UK. f.scarpa@bristol.ac.uk.

ABSTRACT
We simulate the natural frequencies and the acoustic wave propagation characteristics of graphene nanoribbons (GNRs) of the type (8,0) and (0,8) using an equivalent atomistic-continuum FE model previously developed by some of the authors, where the C-C bonds thickness and average equilibrium lengths during the dynamic loading are identified through the minimisation of the system Hamiltonian. A molecular mechanics model based on the UFF potential is used to benchmark the hybrid FE models developed. The acoustic wave dispersion characteristics of the GNRs are simulated using a Floquet-based wave technique used to predict the pass-stop bands of periodic mechanical structures. We show that the thickness and equilibrium lengths do depend on the specific vibration and dispersion mode considered, and that they are in general different from the classical constant values used in open literature (0.34 nm for thickness and 0.142 nm for equilibrium length). We also show the dependence of the wave dispersion characteristics versus the aspect ratio and edge configurations of the nanoribbons, with widening band-gaps that depend on the chirality of the configurations. The thickness, average equilibrium length and edge type have to be taken into account when nanoribbons are used to design nano-oscillators and novel types of mass sensors based on periodic arrangements of nanostructures.PACS 62.23.Kn · 62.25.Fg · 62.25.Jk.

No MeSH data available.


Related in: MedlinePlus