Helical carbon nanotube arrays: thermal expansion
Introduction
The single walled carbon nanotube (SWCN) exhibits anisotropic thermal expansion properties at the nanoscale that may find utility at the macroscale. The development of compositions and material forms that begin at the nanoscale and provide a pathway to the prediction of thermal expansion properties at the macroscale is the challenge addressed in the present work. The image of a nanotube array appeared in Science [2] (Fig. 1) showing what appears to be a hexagonal packing of collimated carbon nanotubes. The assembly of large numbers of carbon nanotubes of finite length into collimated arrays or “yarns” could produce a fiber of several microns in diameter. An element of helicity is introduced as it is in textile fiber yarns to enhance load transfer between the discontinuous nanotubes. Twists in multi-walled carbon nanotube ribbons have been observed [3]. Gommans et al. [4] have reported the use of electrophoresis processes to draw carbon nanotube assemblies from solution and Vigolo et al. [5] report the use of flow alignment processes to produce single walled carbon nanotube “fibers”. The need, therefore, to develop models that predict the effective thermal expansion properties of nanotube assemblies and their composites is obvious. Primary application for such a structural form is as fiber reinforcement in polymer composites, but there are likely to be many other applications where thermal expansion control is desired.
This paper is a companion to reference [1] and much of the descriptive material and justifications for choices of models and properties are discussed in more detail there. It is true that in their synthesis SWCN are produced in discontinuous arrays, not individual and continuous nanotubes. The introduction of twist into the array has also been observed in the literature, but the concept of a twisted array of SWCN that mimics the staple fiber yarn has not yet been observed. Therefore, the present study is an effort to estimate the properties of a geometry of SWCN array that is yet to be realized experimentally, but which offers the potential for the production of continuous fiber consisting of discontinuous SWCN.
Section snippets
Approach
The approach to development of an understanding of the thermoelastic properties of carbon nanotube reinforcements, consisting of arrays of collimated carbon nanotubes suspended in polymeric matrices, utilizes an analogy to a concentric, layered cylinder wherein micromechanics methods have been utilized to develop effective layer properties. This approach is consistent with that employed in composite mechanics. A large number of discontinuous nanotubes are assembled into a cylindrical geometry
Micro/nanomechanics
The elastic properties of each layer of the multi-layered cylinder are determined by the theory of Cox described in [6] wherein the nanotubes are treated as the discontinuous and collimated fiber phase suspended in a polymeric matrix. Kumar et al. [7] recently pointed out a patent application by Duprie and Michel [8] who reported results for the Young's modulus of polypropylene fibers containing 2.35% volume fraction carbon nanotubes. These fibers were produced by a melt spinning process with a
Concentric cylinder analogy
Consider the concentric cylinder nomenclature shown in Fig. 3. The radial displacement, u, axial displacement, w and tangential displacement, v are shown along with the stress components in cylindrical coordinates. The displacement field for this configuration was developed by Pagano et al. [12]. For axisymmetric deformation axial extension and shearing twist, the displacements are shown to be functions of the radial and axial coordinates:The corresponding mechanical strain
Determination of effective coefficients of thermal expansion
The multi-layer analysis described in reference [1] provides for the determination of the effective coefficients of thermal expansion of the array: αz, αθ and αzθ. First the terms w0 and v0 are set equal to zero and the effective thermal force and torque are determined for a given change in temperature, ΔT:Next, the corresponding values of u0, v0 and w0 are determined using the influence coefficients A, B, C and D:
The influence coefficients are be
Results and conclusions
Fig. 5 shows the influence of helical angle on the effective Young's modulus of the array as reported in Ref. [1]. Results presented in Fig. 6 show that the axial coefficient of thermal expansion of the SWCN dominates the effective axial coefficient of thermal expansion of the array for small angles of twist, while the thermal coefficient of the polymer matrix strongly influences the axial coefficient at large angles of twist. The effective circumferential coefficient of thermal expansion of
Acknowledgements
The authors wish to acknowledge the support of the NASA Langley research Center under Cooperative agreement NCC-1-02002 and collaboration of colleagues in the materials modeling and nanotechnology programs.
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