Acoustic energy dissipation and thermalization in carbon nanotubes: Atomistic modeling and mesoscopic description

The exchange of energy between low-frequency mechanical oscillations and high-frequency vibrational modes in carbon nanotubes (CNTs) is a process that plays an important role in a range of dynamic phenomena involving the dissipation of mechanical energy in both individual CNTs and CNT-based materials. The rates and channels through which acoustic energy deposited instantaneously in individual CNTs is dissipated are investigated in a series of atomistic molecular dynamics simulations. Several distinct regimes of energy dissipation, dependent on the initial stretching or bending deformations, are established. The onset of axial or bending buckling marks the transition from a regime of slow thermalization to a regime in which the energy associated with longitudinal and bending oscillations is rapidly damped. In the case of stretching vibrations, an intermediate regime is revealed in which dynamic coupling between longitudinal vibrational modes and the radial “squash” mode causes delayed axial buckling followed by a rapid transfer of energy to high-frequency vibrations. The results of the atomistic simulations are used in the design and parameterization of a “heat bath” description of thermal energy in a mesoscopic model, which is capable of simulating systems consisting of thousands of interacting CNTs. Two complementary methods for the description of mechanical energy dissipation in the mesoscopic model are developed. The relatively slow dissipation of acoustic vibrations in the absence of buckling is described by adding a damping force to the equations of motion of the dynamic elements of the mesoscopic model. The sharp increase in the energy dissipation rate at the onset of buckling is reproduced by incorporating a hysteresis loop into the strain energy that accounts for localized thermalization in the vicinity of buckling kinks. The ability of the mesoscopic model to reproduce the complex multistep processes of acoustic energy dissipation predicted by the atomistic simulations is demonstrated in mesoscopic simulations of free stretching and bending vibrations of individual CNTs.

Figure 1

Snapshots from a mesoscopic simulation of the high-velocity impact of a spherical projectile with a diameter of 100 nm, a density of 2.8 g/cm${}^3$ and an initial velocity of 1000 m/s on a free-standing 20-nm-thick CNT film. The snapshots are shown for 55, 125, and 255 ps after the onset of the impact. The film has a density of 0.2 g/cm${}^3$, and the CNTs in the film are arranged in a continuous network of bundles. The nanotubes are colored by their local kinetic energy, and the projectile is not shown in the snapshots.

Figure 2

Schematic illustration of the mapping of (a) the atomistic model of a (10,10) CNT to a chain of coarse-grained particles and (b) a unit cell ring of the (10,10) CNT to a closed chain of 10 point masses. The coarse-grained representations are used in the analysis of the partitioning of the vibrational energy between the longitudinal, bending, and radial modes of the CNT.

Figure 3

The evolution of the energies of the LA, BA, Rad, and HB modes in simulations performed for CNTs equilibrated at a temperature of 568 K and homogeneously stretched at time $t=0$ with initial axial strains of 3$\%$ (a), 4$\%$ (b), and 6$\%$ (c). In the right panels, the values of energy are converted to temperature units for each group of modes by dividing the energy values by the product of the corresponding number of modes present in each group and the Boltzmann constant. These three simulations represent the three distinct regimes of stretching energy dissipation identified in the simulations.

Figure 4

The evolution of the energy (converted to temperature units as explained in the text) of the LA modes in simulations performed for initial temperatures of 41 and 1054 K with the same axial strain of 2$\%$ (a) and for initial strains of 1$\%$ and 3$\%$ at the same initial temperature of 568 K (b).

Figure 5

The rate of energy dissipation from the LA modes in the first stretching regime (initial strain does not exceed 3$\%$) as a function of the excess LA energy. The data points are obtained by averaging over ten simulations performed for each combination of initial strain and initial temperature. Isothermal contours of the fitted two-dimensional function given by Eq. (12) are drawn for several temperatures ranging from 200 to 1200 K.

Figure 6

The evolution of the energies of the LA, BA, Rad, and HB modes as well as the radial breathing and squash modes in simulations performed for a CNT equilibrated at a temperature of 41 K and homogeneously stretched at time $t=0$ with initial axial strains of 3$\%$ (a), 4$\%$ (b), and 6$\%$ (c). The horizontal dashed lines mark the critical value of the second radial mode energy required for the onset of axial buckling.

Figure 7

Snapshots of atomic configurations from a simulation performed with an initial temperature of 1054 K and an initial axial strain of 3.6$\%$ are shown in (a). As the length of the (10,10) CNT oscillates, the energy in the second radial mode increases (as can be seen from the ripples in the cross section) until an axial buckling kink is formed at about 9 ps (denoted by arrows). Atoms are colored according to their instantaneous kinetic energies, with red indicating higher energy. Enlarged views of two orthogonal projections of the segment of the CNT undergoing axial buckling are shown in (b). Schematic representations of CNT cross sections with atomic displacements corresponding to the second radial mode are shown in (c).

Figure 8

The evolution of the energy of the LA modes (top) and the second radial mode (bottom) in two simulations performed for the same initial temperature of 41 K and initial axial strain of 3.2$\%$. The only difference between the two simulations is the instantaneous distribution of the atomic positions and velocities at the time of the stretching deformation. An axial buckling kink is formed at $t$ $=$ 30 ps in simulation $a$ and is not observed in simulation $b$. The horizontal dashed line marks the critical value of the second radial mode energy required for the onset of axial buckling.

Figure 9

The probability of the formation of an axial buckling kink and observing the immediate dissipation of the LA energy (a) and the mean waiting time until this event occurs (b). Shown with the probabilities are the best fits of logistic regressions for each initial temperature. The transition to the third regime is defined where the mean waiting time drops to less than one period of stretching oscillation (less than 3 ps); this transition occurs between 4.2$\%$ and 5.8$\%$ initial strain for initial temperatures 1054 and 41 K, respectively. Error bars are not shown in (b) for strains at which the formation of a kink was observed in only one simulation.

Figure 10

Snapshots of atomic configurations from a simulation of a CNT with an initial temperature of 568 K and an initial homogeneous axial strain of 6$\%$ are shown in (a). Atoms are colored according to their instantaneous kinetic energies, with red indicating higher energy. An enlarged view of a segment of the CNT undergoing axial buckling at a time of 2 ps is shown in (b).

Figure 11

Snapshots of atomic configurations from a simulation of a CNT with an initial temperature of 1054 K and an initial homogeneous axial strain of 10$\%$. Atoms are colored according to their instantaneous kinetic energies, with red indicating higher energy.

Figure 12

The evolution of the energies of the LA, BA, Rad, and HB modes in simulations performed for CNTs equilibrated at a temperature of 568 K and bent into an arc with a constant radius of curvature equal to 30 nm (a) and 15 nm (b) at time $t=0$. In the right panels, the values of energy are converted to temperature units for each group of modes by dividing the energy values by the product of the corresponding number of modes present in each group and the Boltzmann constant. The periodic formation of a bending buckling kink in (b) is reflected in the large oscillations in the stretching and radial energies observed until approximately 50 ps as well as in the faster decay of the bending energy during this time.

Figure 13

Snapshots of atomic configurations in a simulation of free bending vibrations with an initial temperature of 1054 K and an initial radius of curvature of 15 nm (bending regime II). The snapshots are shown for the first half period of the first bending oscillation after the initial deformation. Atoms are colored according to their instantaneous kinetic energies, with red indicating higher energy.

Figure 14

The rate of energy dissipation from the excited BA modes as a function of the excess BA energy for all initial radii of curvature (a) and for initial radii of curvature greater than the critical value for dynamically forming a bending buckling kink, approximately 27 nm (b). The vertical dashed line in (a) marks the critical value of the excess bending energy $E_{\mathrm{BA}}^{*}$ corresponding to the onset of bending buckling and the transition from the first to the second regime of bending energy dissipation. The data points are obtained by averaging over ten simulations performed for each combination of initial bending strain and initial temperature. Isothermal contours of the fitted function given by Eq. (13) are shown in (b) for temperatures ranging from 100 to 1200 K. The temperature-independent fit to the buckling decay rates is shown by a dashed line in (a).

Figure 15

An illustration of the hysteresis approach to the localized energy dissipation upon buckling in the mesoscopic model. The compressive part of the axial strain energy is shown schematically in (a), and an enlarged view of the hysteresis region is shown in (b). The blue arrows in (b) represent the path followed when a buckling kink is created. The red segments of the plot are the irreversible parts of the strain energy hysteresis. $\Delta E_{\mathrm{str}}^{\mathrm{bcl}}$ is the amount of energy transferred to the heat bath of the CNT in each buckling cycle.

Figure 16

The evolution of the energies of the LA, BA, and HB modes in mesoscopic simulations performed for CNTs equilibrated at a temperature of 568 K and homogeneously stretched at time $t=0$ with initial axial strains of 3$\%$ (a) and 6$\%$ (b). The analogous plots for the atomistic simulations are shown in Figs. 3a and 3(c). In contrast to the atomistic simulations, the Rad modes are included within the HB modes in the mesoscopic simulations.

Figure 17

The evolution of the energies of the LA modes in simulations performed for the same conditions as in Fig. 16b with two values of $|{\varepsilon }_{\mathrm{str}}^{\mathrm{bcl}-\max }|$, 0.45 and 0.1. The value of ${\varepsilon }_{\mathrm{str}}^{\mathrm{bcl}-\max }$ defines the size of the buckling region, which is approximately 5 nm for $|{\varepsilon }_{\mathrm{str}}^{\mathrm{bcl}-\max }|=0.45$ and approximately 16 nm for $|{\varepsilon }_{\mathrm{str}}^{\mathrm{bcl}-\max }|=0.1$.

Figure 18

The evolution of the energies of the LA, BA, and HB modes in mesoscopic simulations performed for CNTs equilibrated at a temperature of 568 K and bent into an arc with a constant radius of curvature equal to 30 nm (a) and 15 nm (b) at time $t=0$. The analogous plots for the atomistic simulations are shown in Figs. 12a and 12b. In contrast to the atomistic simulations, the Rad modes are included within the HB modes in the mesoscopic simulations.