Modulus Of Single-walled Carbon Nanotubes Research Articles

Effect of defects and loading on prediction of Young’s modulus of SWCNTs

In this paper, the influence of various vacancy and Stone-Wales defects on the Young’s modulus of single-walled carbon nanotubes is investigated via a structural model. Dispersion in experimental results is the motivation for this work. Our results show that the type of method used (loading and boundary condition) for the prediction of the Young’s modulus of SWCNTs is very important for the results. The effect of different types of defects on the Young’s modulus is also studied for zigzag and armchair nanotubes with various aspect ratios (length/diameter). A comparison of our results with those of experimental methods indicates that for the exact prediction of the Young’s modulus of SWCNTs we need to apply the correct conditions.

Vibration Properties of Single Walled Carbon Nanotubes—A Comparison Between the Atomistic Simulations and Continuum Shell Modelling

Free vibration analysis of single-walled carbon nanotubes (SWCNTs) is carried out to study the natural frequencies and mode shapes. Appropriate continuum parameters are proposed for equivalent continuum modelling of SWCNTs. Atomistic simulations and continuum shell modelling are employed for the purpose. The formulation of the stiffness matrix computation of the atomistic model is brought out using a finite difference method to bypass the usage of Young’s modulus and wall thickness of SWCNTs. The importance of using equilibrium configuration is highlighted. The large scattered values of Young’s modulus and wall thickness of SWCNTs are reported based on the prior atomistic models. The appropriate set of continuum parameters are proposed based on the free vibration analysis of continuum shell model of SWCNTs. The computed natural frequencies and mode shapes given by the continuum shell model are close to that of atomistic computations, provided the wall thickness values (between 0.06 nm to 0.09 nm) and corresponding Young’s modulus of SWCNTs are chosen appropriately.

High accuracy curve fits for chirality, length and diameter dependent initial modulus of single walled carbon nanotubes

Many previous studies regarding the estimation of mechanical properties of single walled carbon nanotubes (SWCNTs) report that, the modulus of SWCNTs is chirality, length and diameter dependent. Here, this dependence is quantitatively described in terms of high accuracy curve fit equations. These equations allow us to estimate the modulus of long SWCNTs (lengths of about 100–120 nm) if the value at the prescribed low lengths (lengths of about 5–10 nm) is known. This is supposed to save huge computational time and expense. Also, based on the observed length dependent behavior of SWCNT initial modulus, we predict that, SWCNT mechanical properties such as Young's modulus, secant modulus, maximum tensile strength, failure strength, maximum tensile strain and failure strain might also exhibit the length dependent behavior along with chirality and length dependence.

Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration

Abstract Free vibrations of thirty-three armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) of aspect ratios (length/diameter) between ∼3 and ∼15 and having free ends have been studied using the MM3 potential. It is found that these tubes exhibit Rayleigh and Love inextensional modes of vibration. The lowest natural frequency of a mode of vibration corresponds to a circumferential wave number greater than one. Recall that a cylindrical shell of small aspect ratio and comprised of a linear elastic and isotropic material also exhibits the Rayleigh and the Love inextensional modes of vibration. In order to quantitatively compare frequencies of a shell with those of a SWCNT, we find geometric and material parameters for the shell in two ways. In the first approach, we require that the lowest Rayleigh and the radial breathing mode frequencies and the lowest frequencies of the axial and the torsional modes of vibration of a SWCNT match with the corresponding ones of the shell having length and mean diameter equal to those of the SWCNT. In the second technique, we account for the transverse inertia effects, and equate frequencies of the lowest Love, axial and torsional modes of vibration of a SWCNT to that of a shell. Each one of these two methods determines Young’s modulus and Poisson’s ratio of the material of the shell and its thickness, and enables us to explore similarities and differences between vibrations of a shell and of a SWCNT. It is found that the two techniques give very close results for the material and the geometric parameters of the shell, and hence of the SWCNT. The SWCNT thickness increases from ∼0.88 A to 1.37 A when the tube radius is increased from ∼3.6 A to 15 A and stays at 1.37 A for further increases in the tube radius. The wall thickness is essentially independent of the tube chirality. We use these results to provide an expression for the wall thickness in terms of the tube radius and the bond length in the initial relaxed configuration of a SWCNT. We also compare higher vibrational modes of shells and hollow cylinders with those of the corresponding SWCNTs. For a shell we use a first-order shear deformable shell theory (FSDST). Frequencies of a shell using the FSDST and of the hollow cylinder using the three-dimensional linear elasticity theory are computed with the finite element method. It is found that for low to moderate circumferential and axial wave numbers frequencies and mode shapes of the shell and of the hollow cylinder agree well with those of the corresponding SWCNT computed with the MM simulations.

Mechanical properties of nickel-coated single-walled carbon nanotubes and their embedded gold matrix composites

The effects of nickel coating on the mechanical behaviors of armchair single-walled carbon nanotubes (SWCNTs) and their embedded gold matrix composites under axial tension are investigated using molecular dynamics (MD) simulation method. The results show that the Young's moduli and tensile strength of SWCNTs obviously decrease after nickel coating. For armchair SWCNTs, the decreased ratio of the Young's moduli of SWCNTs with smaller radius is larger than that of SWCNTs with larger radius. A comparison is made between the response to Young's modulus of a composite with parallel embedded nanotube and the response of a composite with vertically embedded nanotube. The results show that the uncoated SWCNT can enhance the Young's modulus of composite under the condition of parallel embedment, but such improvement disappears under the condition of vertical embedment because the interaction between SWCNT and gold matrix is too weak for effective load transfer. However, the nickel-coated SWCNT can indeed significantly improve the composite behavior.

The effects of boron doping and boron grafts on the mechanical properties of single-walled carbon nanotubes

The effects of boron doping and boron grafts on the mechanical behaviour of armchair (6, 6) and zigzag (10, 0) single-walled carbon nanotubes (SWCNTs) under axial loading are investigated using the molecular dynamics (MD) simulation method. The results show that Young's moduli, the tension strength, the buckling loads and the buckling strains of SWCNTs decrease after functionalization. The influences of the distribution density of functionalization on Young's moduli of SWCNTs are also systematically studied. The results show that Young's moduli of SWCNTs gradually decrease with the increase in the distribution density. For three types of distribution density, i.e. 6.25%, 12.5% and 25.0%, the decreased magnitudes of Young's moduli of armchair (6, 6) SWCNT for boron doping (respectively boron grafts) are 10.4%, 16.9% and 31.2% (respectively 5.5%, 7.9% and 14.1%). The results further indicate that the degree of degradation of Young's moduli of SWCNT is more sensitive to boron doping than to boron grafts under the condition of the same distribution density. The general conclusions derived from this work may be of importance in devising high-performance Mg matrix composite materials.

Effect of ending surface on energy and Young's modulus of single-walled carbon nanotubes studied using linear scaling quantum mechanical method

Abstract By using a linear scaling self-consistent charge, density functional tight-binding (SCC-DFTB) method and an ab intio Dmol3 calculation, the energy and Young's modulus as a function of tube length for (10, 0) single-walled carbon nanotubes (SWCNTs) are investigated. It was found that with increasing the length of SWCNTs the Young's modulus increases rapidly, then, there is a slow increase, which ultimately approaches a constant value after the length is increased to ~20 nm, whereas a reversed variation tendency was found for the average energy of atoms in SWCNTs with a change of the tube length. We found that the characters of the length-dependent energy and Young's modulus stem from the changed Py-DOS of atoms in the ending region of the tube. Here one simple formula is proposed for quantitatively explaining the length-dependent energy and modulus.

High tensile modulus of carbon nanotube nano-fibers produced by dielectrophoresis

We report an experimental measurement of the high tensile modulus of single-walled carbon nanotube (SWNT) bundles and SWNT nano-fibers produced by dielectrophoresis. The average tensile modulus of the SWNT nano-fibers is 265 GPa, much higher than the carbon nanotube fibers spun by other techniques. The tensile modulus increases as the diameter of the fiber decreases due to changes in the dieletrophoretic forces that densify the packing of SWNT bundles. Tensile modulus of 643 ± 86 GPa and 443 ± 14 GPa were obtained, respectively, from a SWNT bundle of 10 nm diameter and a SWNT nano-fiber of diameter 55 nm with a packing density of 69%, respectively.

A comparative study of Young’s modulus of single-walled carbon nanotube by CPMD, MD and first principle simulations

Carbon nanotubes (CNTs), due to their exceptional magnetic, electrical and mechanical properties, are promising candidates for several technical applications ranging from nanoelectronic devices to composites. Young’s modulus holds the special status in material properties and micro/nano-electromechanical systems (MEMS/NEMS) design. The excellently regular structures of CNTs facilitate accurate simulation of CNTs’ behavior by applying a variety of theoretical methods. Here, three representative numerical methods, i.e., Car–Parrinello molecular dynamics (CPMD), density functional theory (DFT) and molecular dynamics (MD), were applied to calculate Young’s modulus of single-walled carbon nanotube (SWCNT) with chirality (3,3). The comparative studies showed that the most accurate result is offered by time consuming DFT simulation. MD simulation produced a less accurate result due to neglecting electronic motions. Compared to the two preceding methods the best performance, with a balance between efficiency and precision, was deduced by CPMD.

Wall thickness of single-walled carbon nanotubes and its Young's modulus

Young's modulus of single-walled carbon nanotubes (SWCNTs) was calculated by a modern pseudo-potential ab initio method. Using different models for wall thickness, we found that the Young's modulus of an SWCNT varies in a wide range from 0.94 to 5.81 TPa. Here, an SWCNT's wall thickness was obtained by electronic structure calculation. Using the wall thickness, we obtained that the Young's modulus of an SWCNT is 1.06–1.16 TPa, which is in agreement with the experimental result of 1.20–1.25 TPa.

Young’s moduli of functionalized single-wall carbon nanotubes under tensile loading

This paper quantitatively investigates the effect of chemical functionalization on the axial Young’s moduli of single-walled carbon nanotubes (SWCNTs) based on molecular mechanics (MM) simulation, in which the COMPASS force field is used to model the interatomic interactions in a nonfunctionalized nanotube or a functionalized nanotube grafted with vinyl groups. We obtain the axial Young’s moduli of both functionalized and nonfunctionalized SWCNTs. The influences of the number and distribution density of the sp3-hybridized carbon atoms and the radius and chirality of the SWCNTs on Young’s moduli are studied. The results indicate that Young’s moduli depend strongly on the chirality of the SWCNTs and the distribution density of the sp3-hybridized carbon atoms. A 37.50% content of sp3-hybridized carbon atoms may degrade Young’s modulus by up to 33.36%. In addition, MM simulations show that the functionalization of SWCNTs results in a decrease of Young’s moduli of the corresponding SWCNT/polyethylene composites.

Effect of Temperature on Elastic Properties of Single-Walled Carbon Nanotubes

In this article, effect of temperature on elastic properties of single-walled carbon nanotubes is investigated by means of a molecular structural mechanics model in which the primary bonds between two nearest-neighboring atoms are treated as a two-node dimensional Euler—Bernoulli beam. By considering the effect of environmental temperature on force constant values of bond stretching, bond angle bending, and torsional resistance, the corresponding basic parameters of these two-node dimensional Euler—Bernoulli beam elements are obtained under different environmental temperatures, respectively. Nano-scale finite element simulations of the elastic properties of single-walled carbon nanotubes under different environmental temperatures reveal that the elastic modulus of single-walled carbon nanotubes decreases significantly with the increase of temperature. It is noted that the Young's modulus of armchair nanotubes is more sensitive to environmental temperature due to the tube chirality. Finally, the relationship between elastic modulus of zigzag and armchair carbon nanotubes and environmental temperature is given by a simple formula.

Effect of Environmental Temperatures on Elastic Properties of Single-Walled Carbon Nanotube

Based on a method of molecular structural mechanics (MSM), the effect of environmental temperature on elastic properties of armchair and zigzag single-walled carbon nanotubes is investigated. Single-walled carbon nanotubes with different chiral vectors are considered as a molecular structural mechanics model, which is composed of the discrete molecular structures through the carbon-to-carbon bonds. By considering the effect of environmental temperature on force constant values of the bonds stretching, bonds angle bending and torsional resistance, the corresponding basic parameters of a truss of the single-walled carbon nanotubes are obtained in different environmental temperatures, respectively. Nanoscale structural mechanics simulation for the elastic properties of single-walled carbon nanotubes in different environmental temperatures reveals that the elastic modulus of single-walled carbon nanotubes decreases significantly with the increase of environmental temperature. It is noted that the Young's modulus of single-walled carbon nanotubes is more sensitive to environmental temperature than the shear modulus.

Deformation of isolated single-wall carbon nanotubes in electrospun polymernanofibres

Electrospinning has been used to prepare poly(vinyl alcohol) (PVA) nanofibres, with diameters rangingfrom 1 µm down to 20 nm, that contain dispersions of isolated, well-aligned, single-wall carbon nanotubes(SWNTs). The nanofibres were characterized by electron microscopy and Ramanspectroscopy. Single Raman radial breathing modes (RBMs) were found for the SWNTs inthe nanofibres which allowed the identification of particular nanotubes and indicateddebundling/separation of the original SWNT ropes. Moreover the results of polarized Ramanspectroscopy were consistent with the presence of isolated SWNTs, well-aligned along thenanofibre axes. The nanofibres were subjected to deformation and the position of the G andG′ bands was followed as a function of strain. It was found that large band shifts were obtained, indicatingthat there was good stress transfer from the PVA matrix to the nanotubes. A band shift of up to40 cm−1 for 1% strainwas found for the G′ band which is similar to that reported for the deformation of isolated nanotubes. Thisindicates that the Young’s modulus of SWNTs is in excess of 800 GPa.

Difference between bending and stretching moduli of single-walled carbon nanotubes that are modeled as an elastic tube

The authors show that an elastic tube model of a (5,5) carbon nanotube predicts stretching and bending moduli that differ by 19%. This is due to (1) differing energy storage mechanisms in each mode and (2) the inability of the tube model to capture these effects. Conventional tube models assume a common energy storage mechanism in stretching and bending. They show that energy is stored primarily through bond stretching/rotation and bond torsion/van der Waals interactions in stretching and bending, respectively. This knowledge underscores the need to use different moduli to predict stretching, bending, and combined bending and stretching when using the tube model.

The atomic-scale finite-element modelling of single-walled carbon nanotubes

In this paper, an atomic-scale finite-element (AFE) model is proposed for single-walled carbon nanotubes (SWCNTs), which are considered to behave like space-frame structures when subjected to loadings. To create the AFE models, three-dimensional beam elements are used to model the bonds between carbon atoms as loading-carrying elements, while the nodes are placed at the locations of carbon atoms to connect the loading-carrying elements. The material properties of beam elements can be determined by using a linkage between molecular and continuum mechanics. In order to evaluate the AFE model and its performance, the influence of tube wall thickness on Young's modulus of SWCNTs is investigated. It is found that the selection of wall thickness significantly affects the magnitude of the Young's modulus. For the values of wall thickness used in this study, the obtained values of Young's modulus agree well with the corresponding theoretical results. Furthermore, the results also illustrate that Young's modulus is inversely proportional to the wall thickness. The presented results demonstrate that the proposed AFE model can be used as a valuable tool for studying the mechanical behaviour of carbon nanotubes.

Finite element modeling of single-walled carbon nanotubes with introducing a new wall thickness

A three-dimensional finite element (FE) model for armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) is proposed. By considering the covalent bonds as connecting elements between carbon atoms, a nanotube is simulated as a space frame-like structure. Here, the carbon atoms act as joints of the connecting elements. To create the FE models, nodes are placed at the locations of carbon atoms and the bonds between them are modeled using three-dimensional elastic beam elements. Using Morse atomic potential, the elastic moduli of beam elements are obtained via considering a linkage between molecular and continuum mechanics. Also, a new wall thickness ( = bond diameter) equal to 0.1296 nm is introduced. In order to demonstrate the applicability of FE model and new wall thickness, the influence of tube wall thickness, diameter and chirality on the Young's modulus of SWCNTs is investigated. It is found that the choice of wall thickness significantly affects the calculation of Young's modulus. For the values of wall thickness used in the literature, the Young's moduli are estimated which agree very well with the corresponding theoretical results and experimental measurements. We also investigate the dependence of elastic moduli on diameter and chirality of the nanotube. The larger tube diameter, the higher Young's modulus of SWCNT. The Young's modulus of chiral SWCNTs is found to be generally larger than that of armchair and zigzag SWCNTs. The presented results demonstrate that the proposed FE model and wall thickness may provide a valuable tool for studying the mechanical behavior of carbon nanotubes and their application in nano-composites.

Dynamic elastic modulus of single-walled carbon nanotubes in different thermal environments

Abstract This Letter reports the result of investigation on the effect of loading rate (strain rate) on mechanical properties of armchair and zigzag nanotubes in different thermal environments, based on the molecular structural mechanics model in which the primary bonds between two nearest-neighboring carbon atoms are treaded as dimensional 2-node Euler–Bernoulli beam considering the effect of environmental temperature on force constant values of the bonds stretching, bonds angle bending and torsional resistance. Nanoscale finite element simulations of the dynamic Young's modulus of single-walled carbon nanotubes under different strain rates and environmental temperatures reveal that the dynamic Young's modulus of the single-walled carbon nanotubes increases with the increase of strain rate, and decreases significantly with the increase of environment temperature. It is significant that the dynamic Young's modulus of zigzag nanotubes is more sensitive to strain rate and environmental temperature due to the tube chirality.

The effect of impurities on the Young’s modulus of single-walled carbon nanotubes

The molecular dynamics method was used to investigate the effect of the impurities N,O,Si,P and S on the Young's moduli of armchair (5, 5) and zigzag (9, 0) single-walled carbon nanotubes. The results show that the Young's moduli of armchair (5, 5) and zigzag (9, 0) single-walled carbon nanotubes are 948 and 804 GPa, respectively. When the impurity concentration is less than 10%, the Young's moduli are approximately linearly decreasing with increasing of the impurity concentration, and the greatest decreasing ratio is induced by the impurity Si and the smallest by the impurity N. The decreasing rate of the Young's modulus increases with increase of the impurity atomic number when the impurity element is of the same period with the element C. The effect of the impurities on the Young's modulus of carbon nanotubes is the stronger when the period of the impurity element is different with the element C, and the decreasing rate of the Young's modulus decreases slightly with increasing of the impurity atomic number. The reasons are analyzed by the laws of the Young's potential energy variation of carbon nanotubes with impure and the electron cloud coupling between two atoms from the theory of the local density approximation based on the density functional theory.

Radial moduli of individual single-walled carbon nanotubes with and without electric current flow

Although the radial elastic moduli of single-walled carbon nanotubes (SWCNTs) were studied by simulation or calculation, no experimental results were presented. The authors have studied the radial deformability of different diameters of SWCNTs with and without current flow using nanoindentation with rigid silicon and diamond tips under atomic force microscopy. The radial elastic moduli of SWCNT deduced by the Hertz model were found to be strongly dependent on the diameter, i.e., SWCNTs with smaller diameter have larger moduli. The radial stiffness was also shown to be larger for a SWCNT with current than that without current.