A structural mechanics approach for predicting the mechanical properties of carbon nanotubes

Based on molecular mechanics, a structural mechanics model of carbon nanotubes (CNTs) was developed with special consideration given to the bending stiffness of the graphite layer. The potentials associated with the atomic interactions within a CNT were evaluated by the strain energies of beam elements which serve as structural substitutions of covalent bonds in a CNT. In contrast to the original model developed by Li and Chou (Int. J. Solids Struct. 40(10):2487–2499, 2003), in the current model the out-of-plane deformation (inversion) of the bond was distinguished from the in-plane deformation by considering a rectangular cross-section for the beam element. Consequently, the model is able to study problems where the effect of local bending of the graphite layer in a carbon nanotube is significant. A closed-form solution of the sectional properties of the beam element was derived analytically. The model was verified through the analysis of rolling a graphite sheet into a carbon nanotube. Using the present model, the buckling behavior of nanotubes under bending is simulated. The predicted critical bending angle agrees well with molecular dynamics simulations.     

An elastic model for bioinspired design of carbon nanotube bundles

Collagen fibers provide a good example of making strong micro-or mesoscale fibers from nanoscale tropocollagen molecules through a staggered and crosslinked organization in a bottom-up manner.Mimicking the architectural features of collagen fibers has been shown to be a promising approach to develop carbon nanotube(CNT)fibers of high performance.In the present work,an elastic model is developed to describe the load transfer and failure propagation within the bioinspired CNT bundles,and to establish the relations of the mechanical properties of the bundles with a number of geometrical and physical parameters such as the CNT aspect ratio and longitudinal gap,interface cross-link density,and the functionalizationinduced degradation in CNTs,etc.With the model,the stress distributions along the CNT-CNT interface as well as in every individual CNT are well captured,and the failure propagation along the interface and its effects on the mechanical properties of the CNT bundles are predicted.The work may provide useful guidelines for the design of novel CNT fibers in practice.     

Detailed investigation on single water molecule entering carbon nanotubes

The behavior of a water molecule entering carbon nanotubes (CNTs) is studied. The Lennard-Jones potential function together with the continuum approximation is used to obtain the van der Waals interaction between a single-walled CNT (SWCNT) and a single water molecule. Three orientations are chosen for the water molecule as the center of mass is on the axis of nanotube. Extensive studies on the variations of force, energy, and velocity distributions are performed by varying the nanotube radius and the orientations of the water molecule. The force and energy distributions are validated by those obtained from molecular dynamics (MD) simulations. The acceptance radius of the nanotube for sucking the water molecule inside is derived, in which the limit of the radius is specified so that the nanotube is favorable to absorb the water molecule. The velocities of a single water molecule entering CNTs are calculated and the maximum entrance and the interior velocity for different orientations are assigned and compared.     

Analysis of the vibrational behavior of the composite cylinders reinforced with non-uniform distributed carbon nanotubes using micro-mechanical approach

Nanocomposites are a promising new class of structural materials for the aerospace structural components. This paper presents a detailed theoretical investigation of dynamic characteristics of cylinders made of carbon nanotube-reinforced composites. The cylinders are modeled as a cylindrical shell consisting of an isotropic matrix reinforced with transversely isotropic carbon nanotubes. Two different types of carbon nanotube reinforcements are considered: single-walled carbon nanotubes (SWNTs) and multi-walled carbon nanotubes (MWNTs). The effects of carbon nanotube aspect ratio, dispersion, alignment and volume fraction on the elastic modulus are analyzed using the Eshelby–Mori–Tanaka theory. Mass and stiffness matrices are obtained via Ritz method and natural frequencies of the structure are derived through solving the eigenvalue problem. Finally, the effects of the CNT distribution on mode shapes and natural frequencies are discussed.     

Effect of dispersion method on thermal conductivity and stability of nanofluid

Preparing a stable nanofluid with high thermal conductivity is of a great concern. In order to find an optimum dispersion method to achieve a better performance, five different carbon nanotube (CNT) structures, namely SWNTs (single wall CNT), DWNTs (double wall CNT), FWNTs (few wall CNT) and two different MWNTs (multiwall nanotubes) were synthesized to prepare nanofluids with three different dispersion methods namely functionalization, SDS/ultrasonic probe and SDS/ultrasonic bath. The experiments reveal that the best stability and thermal conductivity are associated with the functionalized nanofluids. Specifically, for the times after 50 h, the functionalized profiles begin to level off due to their higher stability, while the other two paths continue their declining trend.     

An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes

A finite-deformation shell theory is developed for single-wall carbon nanotubes (CNTs) based on the interatomic potential. The modified Born rule for Bravais multi-lattice is used to link the continuum strain energy density to the interatomic potential. The theory incorporates the effect of bending moment and curvature for a curved surface, and accurately accounts for the nonlinear, multi-body atomistic interactions as well as the CNT chirality. It avoids the amibiguous definition of nanotube thickness, and provides the constitutive relations among stress, moment, strain and curvature in terms of the interatomic potential.     

Dynamic response of slacked single-walled carbon nanotube resonators

This paper presents an investigation of the dynamics of electrically actuated single-walled carbon nanotube (CNT) resonators including the effect of their initial curvature due to fabrication (slack). A nonlinear shallow arch model is utilized. A perturbation method, the method of multiple scales, is used to obtain analytically the forced vibration response due to DC and small AC loads for various slacked CNTs of higher and lower aspect ratio. Results of the perturbation method are verified with those obtained by numerically integrating the equations of a multi-mode reduced-order model based on the Galerkin procedure. The effective nonlinearity of the CNT is calculated as a function of the slack level and the DC load. To handle computational problems associated with CNTs of small radiuses, results based on a nonlinear cable model are also demonstrated. The results have indicated that the quadratic nonlinearity due to slack has dominant effect on the dynamic behavior of the CNT.     

Cooperative deformation of carboxyl groups in functionalized carbon nanotubes

Functionalized carbon nanotubes have tremendous potential for nanotechnology applications such as in the fabrication of polymeric carbon fibers. However, approaches to design carbon nanotube structures by using functional groups as glue and carbon nanotubes as stiff building blocks to reach superior mechanical strength and toughness at the fiber level with limited amount of materials remains poorly understood. Inspired by the outstanding mechanical properties of spider silk, here we present a bio-inspired structural model of carbon nanotube based fibers connected by weak hydrogen bonds (H-bonds) formed between functional carboxyl groups as the molecular interface. By applying shear loading, we study how the deformation of H-bonds in functional groups is affected by the structural organization of the carboxyl groups, as well as by the geometry of constituting carbon nanotubes. The analysis of H-bond deformation fields is used to compute the extent of significant deformation of inter-CNT bonds, defining a region of cooperativity. We utilize an exponential function (exp (?x/ξ)) to fit the deformation of H-bonds, with the cooperative region defined by the parameter ξ, and where a higher value of ξ represents a weaker exponential decay of displacements of carboxyl groups from the point where the load is applied. Hence, the parameter ξ characterizes the number of carboxyl groups that participate in the deformation of CNTs under shear loading. The cooperativity of deformation is used as a measure for the utilization of the chemical bonds facilitated by the functional groups. We find that for ultra-small diameter CNTs below 1 nm the external force deforms H-bonds significantly only within a relatively small region on the order of a few nanometers. We find that the mechanical properties of carbon nanotube fibers are affected by the organization of H-bonds in functional carboxyl groups. Both, the grouping of functional groups into clusters, and a specific variation of the clustering of functional groups along the CNT axis are shown to be potential strategies to improve the cooperativity of deformation. This allows for a more effective utilization of functional groups and hence, larger overlap lengths between CNTs in fibers. The effect of structural organization of functional groups is not only significant in very small diameter CNTs, but also in larger diameter CNTs as they are most commonly used for engineering applications. Notably larger-diameter CNTs naturally show a larger cooperative deformation range. Our model can be applied to other functional groups attached to CNTs, and could in principle also include strong bonds such as covalent or ionic bonds, or other weak bonds such van der Waals forces or dipole–dipole interactions.     

Stiffness threshold of randomly distributed carbon nanotube networks

For carbon nanotube (CNT) networks, with increasing network density, there may be sudden changes in the properties, such as the sudden change in electrical conductivity at the electrical percolation threshold. In this paper, the change in stiffness of the CNT networks is studied and especially the existence of stiffness threshold is revealed. Two critical network densities are found to divide the stiffness behavior into three stages: zero stiffness, bending dominated and stretching dominated stages. The first critical network density is a criterion to judge whether or not the network is capable of carrying load, defined as the stiffness threshold. The second critical network density is a criterion to measure whether or not most of the CNTs in network are utilized effectively to carry load, defined as bending–stretching transitional threshold. Based on the geometric probability analysis, a theoretical methodology is set up to predict the two thresholds and explain their underlying mechanisms. The stiffness threshold is revealed to be determined by the statical determinacy of CNTs in the network, and can be estimated quantitatively by the stabilization fraction of network, a newly proposed parameter in this paper. The other threshold, bending–stretching transitional threshold, which signs the conversion of dominant deformation mode, is verified to be well evaluated by the proposed defect fraction of network. According to the theoretical analysis as well as the numerical simulation, the average intersection number on each CNT is revealed as the only dominant factor for the electrical percolation and the stiffness thresholds, it is approximately 3.7 for electrical percolation threshold, and 5.2 for the stiffness threshold of 2D networks. For 3D networks, they are 1.4 and 4.4. And it also affects the bending–stretching transitional threshold, together with the CNT aspect ratio. The average intersection number divided by the fourth root of CNT aspect ratio is found to be an invariant at the bending–stretching transitional threshold, which is 6.7 and 6.3 for 2D and 3D networks, respectively. Based on this study, a simple piecewise expression is summarized to describe the relative stiffness of CNT networks, in which the relative stiffness of networks depends on the relative network density as well as the CNT aspect ratio. This formula provides a solid theoretical foundation for the design optimization and property prediction of CNT networks.     

Nonlinear analysis of a SWCNT over a bundle of nanotubes

The deformation of a single wall carbon nanotube (SWCNT) interacting with a curved bundle of nanotubes is analyzed. The SWCNT is modeled as a straight elastic inextensible beam based on small deformation. The bundle of nanotubes is assumed rigid and the interaction is due to the van der Waals forces. An analytical solution is obtained using a bilinear approximation to the van der Waals forces. The analytical results are in good agreement with the results of two numerical methods. The results indicate that the SWCNT remains near the curved bundle provided that its curvature is below a critical value. For curvatures above this critical value the SWCNT breaks contact with the curved bundle and nearly returns to its straight position. A parameter study shows that the critical curvature depends on the stiffness of the SWCNT and the absolute minimum energy associated with the van der Waals forces but it is independent of the SWCNT's length in general. An analytical estimate of the critical curvature is developed. The results of this study may be applicable to composites of nanotubes where separation phenomena are suspected to occur.     

碳纳米管复合膜的制备及其摩擦学性能

通过混酸对碳纳米管(CNTs)纯化,然后应用稀土溶液对纯化CNTs进行功能化,采用分子自组装技术在羟基化的玻璃基片表面制备了碳纳米管复合膜.运用原子力显微镜(AFM)及扫描电子显微镜(SEM)观察了薄膜的表面形貌,使用X射线光电子能谱仪(XPS)分析了薄膜表面典型元素的化学状态,并采用UMT-2MT摩擦试验机评价了薄膜的摩擦磨损性能.研究结果表明:通过硅烷偶联剂3-巯丙基三甲氧基硅烷(MPTS)的磺酸基化学吸附功能,稀土改性后的碳纳米管可以成功组装到氧化后的硅烷化表面.当组装碳纳米管复合膜后,基片表面的摩擦系数由无膜时的0.85降到了0.10,表明复合膜可以降低基片的摩擦系数,并且在较低载荷下具有较好的耐磨性能,显示了其在微机构表面改性方面良好的应用前景.     

Analysis of the entanglements in carbon nanotube fibers using a self-folded nanotube model

Carbon nanotube (CNT) fibers have shown superb mechanical properties, and have high potential to be used as reinforcements in multifunctional composites. CNT entanglements always exist in CNT fibers and play a crucial role in affecting their mechanical properties. In this study, the CNT entanglement is modeled as two connecting self-folded CNTs (SFCNTs). At large aspect ratios, a CNT is energetically favorable to be self-folded due to the van der Waals interactions between different parts of the CNT. The geometrical characteristics of the SFCNTs, such as the critical length for self-folding as well as the critical effective width and length, are investigated by using both an exact theoretical model and an approximate theoretical model. The tensile properties of the SFCNTs have been examined by using both the approximate theoretical model and atomistic simulations. Good agreements are achieved in the results of these two approaches.     

The influence of mechanical deformation on the electrical properties of single wall carbon nanotubes

Recent experimental studies and atomistic simulations have shown that carbon nanotubes (CNTs) display strong interplay between the mechanical deformation and electrical properties. We have developed a simple and accurate method to determine atom positions in a uniformly deformed CNT via a continuum analysis based on the interatomic potential. A shift vector is introduced to ensure the equilibrium of atoms. Such an approach, involving only three variables for the entire CNT, agrees very well with the molecular mechanics calculations. We then study the effect of mechanical deformation on the band gap change of single wall CNTs under tension, torsion, and combined tension/torsion via the k-space tight-binding method. Prior studies without this shift vector lead to significant overestimation of the band gap change. It is established that the conducting CNTs may easily become semi-conducting ones subject to mechanical deformation, but the semi-conducting CNTs never become conducting ones upon deformation.     

Mechanics of hydrogen storage in carbon nanotubes

A continuum mechanics model is established for hydrogen storage in single- and multi-wall carbon nanotubes (CNTs) and the bundle of single-wall CNTs. The model accounts for the deformation of CNTs, and van der Waals interactions among hydrogen molecules and between hydrogen and carbon atoms. The analytical expressions of hydrogen storage (number of hydrogen molecules per unit volume) in CNTs are obtained, and are validated by atomistic simulations. CNTs are categorized as tiny, small, medium and large CNTs; tiny CNTs cannot achieve the goals of hydrogen storage (62 kg/m³ and 6.5 wt% of hydrogen set by the US Department of Energy) without fracture; small CNTs are strained during hydrogen storage; medium CNTs can achieve the above goals without the strain and do not self collapse; and large CNTs may self collapse upon the release of hydrogen.     

MOLECULAR DYNAMICS SIMULATION OF POLYMER/CARBON NANOTUBE COMPOSITES

In this paper, we present classical molecular dynamics(MD) simulations of model polymer/CNT composites constructed by embedding a single wall(10,10) CNT into two different amorphous polymer matrices: poly(methyl methacrylate) and poly{(m-phenylene-vinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]}, respectively, with different volume fractions. The simulation results support the idea that it is possible to use CNTs to mechanically reinforce an appropriate polymer matrix, especially in the longitudinal direction of the nanotube. The comparison of the simulation results with the macroscopic rule-of-mixtures for composite systems showed that for strong interfacial interactions, there can be large deviations of the results from the rule-ofmixtures. In order to verify this study, results obtained have been compared with those given by Elliott and Han(2007).     

Three-dimensional theory of stability of a carbon nanotube in a matrix

The three-dimensional theory of stability of a carbon nanotube (CNT) in a polymer matrix is presented. The results are obtained on the basis of the three-dimensional linearized theory of stability of deformable bodies. Flexural and helical (torsional) buckling modes are considered. It is proved that the helical (torsional) buckling modes occur in a single CNT (the interaction of neighboring CNTs is neglected) and do not occur in nanocomposites (the interaction of neighboring CNTs is taken into account) __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 23–37, January 2006.     

Dispersion of multi-walled carbon nanotubes in PDMS/PB blend

In this study, the sequential dispersion of multi-walled carbon nanotubes (CNTs) in PDMS/PB (polydimethylsiloxane/polybutene) blends and the change of blend morphology by the dispersion of CNTs were investigated by rheological and morphological observations. The dispersion of CNTs into PDMS/PB blend was accomplished by the dilution of the CNT master (2?wt.% CNT in PDMS) in PDMS/PB blend using an extensional mixer. The morphological study shows that under the extensional flow, CNTs in the dispersed CNT master phase are mainly broken up by tip-streaming and the continuous pinching-off of PDMS drops during morphology evolution enhances the dispersion of CNT. It has been shown that CNTs can be disentangled as in the case of dispersing CNTs in a Boger fluid. Rheological data and TEM observations show that it is not simply a mixing of two phases and the CNTs in the master phase can be dispersed in the single CNT level.     

Coarse-grained potentials of single-walled carbon nanotubes

We develop the coarse-grained (CG) potentials of single-walled carbon nanotubes (SWCNTs) in CNT bundles and buckypaper for the study of the static and dynamic behaviors. The explicit expressions of the CG stretching, bending and torsion potentials for the nanotubes are obtained by the stick-spiral and the beam models, respectively. The non-bonded CG potentials between two different CG beads are derived from analytical results based on the cohesive energy between two parallel and crossing SWCNTs from the van der Waals interactions. We show that the CG model is applicable to large deformations of complex CNT systems by combining the bonded potentials with non-bonded potentials. Checking against full atom molecular dynamics calculations and our analytical results shows that the present CG potentials have high accuracy. The established CG potentials are used to study the mechanical properties of the CNT bundles and buckypaper efficiently at minor computational cost, which shows great potential for the design of micro- and nanomechanical devices and systems.     

Mechanics of deformation of single- and multi-wall carbon nanotubes

An effective continuum/finite element (FE) approach for modeling the structure and the deformation of single- and multi-wall carbon nanotubes (CNTs) is presented. Individual tubes are modeled using shell elements, where a specific pairing of elastic properties and mechanical thickness of the tube wall is identified to enable successful modeling with shell theory. The incorporation and role of an initial internal distributed stress through the thickness of the wall, due to the cylindrical nature of the tube, are discussed. The effects of van der Waals forces, crucial in multi-wall nanotubes and in tube/tube or tube/substrate interactions, are simulated by the construction of special interaction elements.The success of this new CNT modeling approach is verified by first comparing simulations of deformation of single-wall nanotubes with molecular dynamics results available in the literature. Simulations of final deformed configurations, as well strain energy histories, are in excellent agreement with the atomistic models for various deformations. The approach was then applied to the bending of multi-wall carbon nanotubes (MWNTs), and the deformed configurations were compared to corresponding high-resolution images from experiments. The proposed approach successfully predicts the experimentally observed wavelengths and shapes of the wrinkles that develop in bent MWNTs, a complex phenomenon dominated by inter-layer interactions. Presented results demonstrate that the proposed FE technique could provide a valuable tool for studying the mechanical behavior of MWNTs as single entities, as well as their effectiveness as load-bearing entities in nanocomposite materials.