QC & MD

An effective multiscale computing scheme based on QM/QC (quantum mechanics/quasicontinuum) is applied for simulation of Carbon nanotubes (CNTs) mechanics. First, quasicontinuum simulation of deformations of curved crystalline structures is conducted to examine the fully nonlocal behavior of CNTs with the aid of high-order interpolation functions and the ''cluster'' concept, which facilitates accurate energy approximation for crystals. Next, a multiscale computing approach based on QM/QC hybridization is devised, and applied for simulation of CNT mechanics. For studying electronic properties coupled with the mechanical deformation of CNTs, the change of the electrical properties from an initial semiconductor into metal under mechanical bending is investigated. Single-walled CNTs having various types of defects and subjected to uniaxial tension are considered for fracture. The theoretical strength of the CNTs in the presence of each defect is computed based on the QM/QC hybrid scheme, where in the defect neighborhood is modeled as a QM zone for a first-principle-based calculation using density functional theory (DFT), and the remaining area as a QC zone. This multiscale computing approach greatly improves the accuracy in the prediction of the failure strains of CNTs over a purely molecular mechanical or quasicontinuum method.

Recently, the electromechanical properties of CNTs have attracted much interest, largely based on their promise for application for ultra-sensitive electromechanical sensors. In a recent experiment (Tombler etal.,2000), it was demonstrated that the electrical conductance of a CNT changes when the middle part of a suspended nanotube is deformed by a sharp AFM tip. Using QM/QC hybridization, we investigate the change of electronic structure by bending a defect-free (10,0) CNT up to 90° degrees. The strain energies and calculated geometries versus the bending angles are shown in Figs. 1(a) and (b). For the bending simulation of a CNT, the chirality is (10,0) and the length is 12.9nm. One-third of the tube, which is the shaded middle zone in Fig. 1(b), is modeled with MM zone and with a QM subdomain located at the center of the MM zone, and other regions are coarse-grained with 160 cubic elements. The atomic structures optimized by the QM/QC hybrid model are similar to those from fully atomistic model or QC simulations, because the CNT is free from defects and there is no change in the bonding configurations between carbon atoms. The energy variation with respect to the bending angle in the hybrid simulations is almost the same as the results calculated using QC alone. A kink occurs at the center when the bending angle reaches approximately 52°, which is indicated by an abrupt protrusion on the energy curve in Fig. 1(a). The kink formation was observed to cause nonlinearity in the strain energy, and sp2–sp3 rehybridization. Upon bending, important changes in the local density of states of s and p electrons and increased s–p mixing are observed to develop. As the bending angle increases, the bandgap of the CNT decreases (see Fig. 1(c)), and the initially semiconducting CNT suddenly transforms to a metallic nanotube after the kink formation (see Fig. 1(d)). The electronic structure changes are expected to have important implications with respect to the low temperature electrical transport properties of CNTs and real applications as ultra-sensitive nanosensors.

The equilibrium configuration near the vacancy at each strain is obtained from the present QM/QC hybrid method for each of the semiconducting and the metallic tubes, and is presented in Figs. 2(a) and (b). In every pair of figures inside each box in Fig. 2, the upper one indicates the top view of the atomic configuration while the lower the side view. For the (10,0) CNT, an initial crack occurs, perpendicular to longitudinal tube axis, at the site of a bond close to the vacancy. In the metallic CNT, the failure propagates across the bonds with 30° from the longitudinal tube axis. Again, it is not possible to obtain this detailed configuration at the point of failure from the QC solution, as most of the force fields including TB-G2 and MTB-G2 are far from being accurate near failure point due to the bond breakage.

The movement of a 5|7 defect in the plastic deformations of a carbon nanotube (CNT) plays the role of dislocation glide in the plastic deformations of metals. This work is concerned with the atomic shift mechanism of the 5|7 defect and the energy barrier when the CNT is subjected to tensile loading. Action-derived molecular dynamics (ADMD) is applied to find the minimum energy path and the energy barrier. It is found that the tensile loads make it easy for the 5|7 defect to glide, and lower the energy barrier. The minimum load level that makes a 5|7 defect glide freely with no energy barrier in the presence of an adatom is obtained.

The initial state means the initial configuration, as given in Fig. 4(a), and the final state is the final configuration, as given in Fig. 4(c), which is are obtained after the shift of one 5|7 defect. Since the internal structure changes due to the shift of the 5|7 defect, the constitutive relation comes out different.

It is well known that the presence of adatoms substantially reduces the energy barrier for various processes that involve bond breakage and creation in CNTs and fullerenes. Indeed, it is reported that additional iodine atoms can assist the coalescence of C60 molecules inside single walled CNTs in the actual experimental environment. In this context, the change of energy barrier is explored for the 5|7 defect glide in the presence of a carbon additional atom.

Consider two armchair CNTs in the presence of an adatom under different boundary conditions. The first is subjected to no external forces, while the second is subjected to an external loading of the force constant . The potential energy profiles are shown in Figs. 7(a) and (b). Fig. 7(a) represents the overall potential energy profiles versus the image numbers or the step indices for the two CNTs with different force constants. The time from the first image to the last corresponds to the entire duration of the pathway from the time the adatom comes into the CNT to the time another atom leaves the CNT (see the green adatom and the yellow atom leaving the CNT as depicted in Figs. 8(a) and (c)). Each of the two atomic configurations that correspond to these energy profiles is shown in Figs. 8(a) and (c). The size of the time step is taken to be for this computation. One in Fig. 7(b) shows the zoomed-in energy profile between points A and B of Fig. 7(a), while the other in Fig. 7(b) shows the zoomed-in energy profile between points C and D in Fig. 7(a). This energy profile is obtained from another calculation after further refining the interval between A and B, and between C and D with the time step size of .

Fig. 7(a) shows that the energy barrier for the glide of the 5|7 defect is in the absence of the external loading, and zero when the CNT is subjected to the loading of . These are much lower than , which is the energy barriers in the absence of an adatom (see Fig. 8). This reveals that the adatom plays the role of a catalyst in the glide reaction by way of the SW transformation.

The novel computational scheme known as quasicontinuum (QC) has been widely utilized over the past decade for exploring extreme/multi-scale phenomena in the spatial domain, such as, mechanical behaviors of nanostructures or defect behaviors in crystalline materials. This work reports on the recent extension of the QC method to simulate mechanical behaviors or deformations of curved crystalline bodies such as carbon nanotubes (CNTs). In addition to QC implementation utilizing high-order triangular elements, this study presents a new QC approach based on what is known as “variable-node elements”. This proves to be extremely efficient when combined with a fully automatic adaptive refinement. Several numerical examples demonstrate the accuracy and effectiveness of the new method.

For the bending of a CNT, the chirality is (24,0) and the length and the diameter is 24.2nm and 1.9nm. The loading condition is given in terms of the prescribed displacement on both ends. The total number of atoms is 5,376, corresponding to 16,128 degrees of freedom. The number of degrees of freedom for the QC model is 7,680. The energy versus the bending angle is presented in Fig. 9, in which the comparison is made between the two solutions; the first is the solution from the QC with the present rectangular elements, and the second is from molecular mechanics. The good agreement between the two solutions is apparent. The undeformed mesh and several deformed configurations are shown in Fig. 9. In this bending simulation the atomic level stain is calculated as shown in Fig. 10.

The torsion of a CNT is simulated, with the same specification for the bending case. This model is described by the QC model with 2,916 representative atoms and 576 elements. This simulation is performed only 54.2% of the total degree of freedom. The undeformed mesh and several deformed configurations are shown in Fig. 11. The energy versus the torsion angle is presented in Fig. 11, in which the comparison is made between the two solutions; the first is the solution from the QC with the present rectangular elements, and the second is from MM. As shown in Fig. 11, the result of QC is in good agreement with the result of MM. However, in the torsion simulation, the deformation is less localized than the previous bending case. Therefore, the error of the torsion simulation is relatively larger than the bending case. In this twisting simulation the atomic level stain is calculated as shown in Fig. 12.

The example considered is the adaptive simulation for the bending of an armchair type CNT with chirality (40,40) under the loading shown in Fig. 13. The tube length and the diameter are 52.34 nm and 5.5 nm, respectively. The total number of the carbon atoms is 33,360, which corresponds to 100,080 degrees of freedom. The number of the nodes of the initial mesh is 984, corresponding to 2,952 degrees of freedom. The number of the total degrees of freedom increases to 36,510, including the inner displacement degrees of freedom, when the bending angle reaches 22˚ (see Fig. 13). In actuality, this example was chosen for adaptive refinement in the QC using the triangular element. The present solution from the QC by the 12-noded rectangular element is compared with each of the solutions from and from molecular mechanics in Fig. 13. The three solutions are shown to be in good agreement. The initial undeformed mesh and several deformed states are presented in Fig. 13. The QC by the rectangular and the variable-node element is much more efficient in terms of the solution time than the QC by the triangular elements. Furthermore, a mesh generation code is unnecessary for the rectangular element case, whereas the mesh generator TRIANGLE was used for the triangular element case.

Molecular dynamics simulation of nanoindentation and nanolithography by AFM are conducted to study homogeneous nucleation of various defects, and their subsequent development and interactions as well. In nanoindentation, dislocations found to be initiated on the slip plane off the loading axis and to form a loop, are emitted onto the bottom surface.

During nanolithography via AFM, dislocation loops are emitted along the top surface, and resourceful defect interactions such as, formation of void trail via the motion of a jog, and creation of extended nodes and stacking fault tetrahedrons are observed. The effects of the scratching depth, speed, and orientation on the resulting defects are discussed as well.

(Sukky Jun, Youngmin Lee, Sung Youb Kim and Seyoung Im, Large-scale molecular dynamics simulations of Al(111) nanoscratching, Nanotechnology, 15, 1169-1174, 2004)