Coupled quantum mechanical/molecular mechanical modeling of the fracture of defective carbon nanotubes and graphene sheets

Abstract

Coupled quantum mechanical/molecular mechanical ͑QM/MM͒ calculations were used to study the effects of large defects and cracks on the mechanical properties of carbon nanotubes and graphene sheets. The semi-empirical method PM3 was used to treat the QM subdomains and a Tersoff-Brenner potential was used for the molecular mechanics; some of the QM calculations were also done using density functional theory ͑DFT͒. Scaling of the Tersoff-Brenner potential so that the modulus and overall stress-strain behavior of the QM and MM models matched quite closely was essential for obtaining meaningful coupled calculations of the mechanical properties. The numerical results show that at the nanoscale, the weakening effects of holes, slits, and cracks vary only moderately with the shape of the defect, and instead depend primarily on the cross section of the defect perpendicular to the loading direction and the structure near the fracture initiation point. The fracture stresses for defective graphene sheets are in surprisingly good agreement with the Griffith formula for defects as small as 10 Å, which calls into question the notion of nanoscale flaw tolerance. The energy release rate at the point of crack extension in graphene was calculated by the J-integral method and exceeds twice the surface energy density by 10% for the QM͑DFT͒/MM results, which indicates a modest lattice trapping effect.

Using theory and computation to model nanoscale properties

“…Fig. 4a shows the types of theory that we have used (61,63) to study nanotube mechanical properties. Here we show a vacancy defect in what is otherwise a perfect nanotube.…”

“…To circumvent the limitations of calculations that use QM methods for all atoms, we have implemented QM/MM methods in which QM calculations are done for a patch of the nanotube close to the defect, and then a molecular mechanics (MM) force field is used to represent the region around the patch (63). In addition, we have also studied MM/CM methods in which the molecular mechanics calculations (which are still atomistic) are interfaced with continuum mechanics (CM) to extend the nanotube stress field to large distances from the initial defect (61).…”

Applicability of Continuum Fracture Mechanics in Atomistic Systems

“…However, the validity of LEFM and especially fracture toughness as a constant from nanoscale to macroscale has not been carefully examined. Even though some researchers [83][84][85][86][87] tried to compare the atomistic simulation results with the classic Griffith theory of fracture, their studies do not demonstrate the existence of inverse square root singular stress field at the atomistic level.…”

“…If we focus on engineering materials, one of the attractive findings is the carbon nanotube due to its extremely high stiffness and strength (~ 1 TPa for Young's modulus and ~ 100 GPa for tensile strength). Since Iijima's discovery in 1991 [6], a number of publications have been devoted to carbon nanotubes' material properties [7][8][9][10][11][12][13], strength (failure toughness) [14][15][16] and further application on composite materials [17][18][19][20][21]. Meanwhile, many researchers also focused on the nanocomposites composed of nanoparticles, e.g., silica (SiO 2 ), alumina (Al 2 O 3 ), titanium dioxide (TiO 2 ), etc., due to the greater reactive surface area per unit volume compared with large particles [22][23][24][25].…”

Response to “Comment on ‘On the tensile strength distribution of multiwalled carbon nanotubes’ [Appl. Phys. Lett. 92, 206101 (2008)]”

“…This was proposed and discussed 9,10 but never experimentally proved in any convincing way. As a matter of fact, recent work 11,12 demonstrates that continuum fracture mechanics ͑the Griffith formulation͒ is applicable to cracklike defects as small as 10 Å, without any nanoscale modification of the theory of fracture. …”