Journal of Nanomaterials

Volume 2015 (2015), Article ID 157423, 8 pages

http://dx.doi.org/10.1155/2015/157423

## Estimating Young’s Modulus of Single-Walled Zirconia Nanotubes Using Nonlinear Finite Element Modeling

^{1}Department of Mechanical Engineering, Universiti Teknologi PETRONAS, 31750 Tronoh, Perak, Malaysia^{2}Department of Chemical Engineering, Universiti Teknologi PETRONAS, 31750 Tronoh, Perak, Malaysia

Received 17 October 2014; Revised 13 December 2014; Accepted 15 December 2014

Academic Editor: Shiren Wang

Copyright © 2015 Ibrahim Dauda Muhammad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The single-walled zirconia nanotube is structurally modeled and its Young’s modulus is valued by using the finite element approach. The nanotube was assumed to be a frame-like structure with bonds between atoms regarded as beam elements. The properties of the beam required for input into the finite element analysis were computed by connecting energy equivalence between molecular and continuum mechanics. Simulation was conducted by applying axial tensile strain on one end of the nanotube while the other end was fixed and the corresponding reaction force recorded to compute Young’s modulus. It was found out that Young’s modulus of zirconia nanotubes is significantly affected by some geometrical parameters such as chirality, diameter, thickness, and length. The obtained values of Young’s modulus for a certain range of diameters are in agreement with what was obtained in the few experiments that have been conducted so far. This study was conducted on the cubic phase of zirconia having armchair and zigzag configuration. The optimal diameter and thickness were obtained, which will assist in designing and fabricating bulk nanostructured components containing zirconia nanotubes for various applications.

#### 1. Introduction

Zirconia (ZrO_{2}) is considered to be among the most important ceramic materials owing to its exceptional mechanical properties together with its stability at high temperatures [1]. It is used as a refractory in insulation, for metal coating, also as abrasives, enamels, and glazes, and as support material for catalysts [2], and due to its ion conductivity it is used in oxygen pumps for partial regulation [3], gas sensors [4], and high temperature fuel cells [5]. Also, ZrO_{2} is considered as one of the most radiation-resistant ceramics, thus having specific application in the nuclear industry [6].

At atmospheric pressure, ZrO_{2} has three phases or polymorphs. At high temperatures (above 2350°C), ZrO_{2} exists as cubic fluorite structure (*fm3m*), while at low temperatures (below 1150°C), monoclinic baddeleyite (*P211C*) structure dominates. A tetragonal phase exists at intermediate phase having* P42/nmc* symmetry [7]. Most applications of ZrO_{2} are based on its cubic polymorph which can be stabilized by doping with oxides such as CaO, MgO, CeO_{2}, and Y_{2}O_{3} [8]. Improved results are obtained by reducing the crystallite size to few nm, with an average of about 15 nm [9]; this results in ZrO_{2} nanomaterials in form of dots, slabs, sheets, and tubes at atomic scale level. For instance, ZrO_{2} is presently being tested as catalyst support for several reactions and displays a much higher activity than some other oxides [10]. Thus, for some years now, progress has been attained on the synthesis and study of various nanostructure materials containing zirconia. Significant consideration has been given to ZrO_{2} nanotube (ZNT) due to its existing and potential applications, such as components of oxygen sensors, host matrix for optical functional materials, and electrolytes in solid-oxide fuel cells [11].

Studies have shown that nanotubes have mechanical properties far superior when compared to that of bulk material [4]. These novel nanotubes (NTs) are projected to have high stiffness, wear resistance, strength, lower thermal conductivity, and high melting temperature [2] and are still retaining high plasticity because of their nearly infinite length-to-width ratio [12]. These higher material properties render such nanotubes suitable for a range of applications [11].

Progress in scanning probe methods, especially atomic force microscope (AFM), has assisted in providing novel settings to estimate some mechanical properties of nanotubes [12]. Calculations and interpretations of physical properties of ZNT and similar materials most often are conducted using AFM or electron microscopes: the scanning electron microscope (SEM) or the transmission electron microscope (TEM). Possibly the most common instrument in characterization of these materials is AFM [14], which applies very small forces (nN) and detects tiny displacements (nm). Several experimental investigations on the mechanical properties have been conducted for carbon nanotubes [15–17] and inorganic nanotubes [18] with Young’s modulus of ZNT detected experimentally to be between 30 and 52 GPa using nanoindentation setup [19].

However, several difficulties and challenges have been experienced during the mechanical characterization of nanotubes due to their tiny size and high cost of required equipment [14]. In addition, there are other problems related to specimen collection, handling, and setting and quantifying small forces and small deformations. Precise setting of the load application spot and analysis of the results further complicate the procedure [15].

Owing to some problems encountered during experimental analysis, theoretical modelling methods have been used recently to evaluate mechanical properties of NTs. Amongst the existing modelling techniques, the molecular dynamics (MD) method has been used widely [20] and is focused on the force field and total potential energy linked to the interatomic potentials of nanotubes. Based on this approach, the bonding and nonbonding potentials are represented in relation to the force constants and the distance change amongst the atomic bonds, and then elastic moduli are estimated by using different small-strain deformation modes [21]. But MD simulation is not effective for time-consuming or fixed problem(s); thus, it has limitation in the study of the mechanical properties of nanotubes. The other method is the continuum or finite element method [22], whereby nanotubes are made up of elements regarded as beams or shells that are subjected to bending, tension, or torsional loading. This facilitates the nanotube to be modelled as a shell-like or frame structure and the mechanical behaviour realized by finite element method or classical continuum mechanics [23].

The magnitude of axial Young’s modulus for carbon nanotubes (CNT) has been found out by simulation to be in the range of 1.0 TPa to 5.5 TPa [20, 24] with that of WS_{2}, MoS_{2}, and TiO_{2} nanotubes to be 143 GPa, 230 GPa, and 270 GPa, respectively [22, 23]. Much information is required on the mechanical behaviour of ZNT, as sufficient study has not been done experimentally and numerically [25]. Therefore, the purpose of this paper is to simulate the mechanical behaviour of ZNT when exposed to axial tension in order to estimate Young’s modulus using finite element (FE) approach.

#### 2. Modeling and Simulation

##### 2.1. Molecular Mechanics

Based on the concept of molecular mechanics, the total potential energy () is specified as sum of specific potential constituents as a result of interactions [21, 23, 24]:
where , , , and are energies related to bond stretching, bond-angle inversion, and bond torsion and inversion, respectively, and are based on bonding, while and are van der Waals and electrostatics interactions, respectively, and are not based on bonding. The energy terms in (1) can be expressed using different functions depending on the loading condition and type of material [26]. For the Zr–O bond in ZNT the ionic bonding dominates; thus, the significant parts of the potential energy are and and are represented by Buckingham and Coulomb expressions, respectively. For ZrO_{2}, the potential energy is expressed as a sum of two-body interactions of the form [21]
where , , and are constants describing the contributions of short-range interaction of each particular pair and is the distance between Zr and O_{2} atoms. Also and represent the charges on the pairs of ions and is the permittivity of free space.

The concept of energy equivalence can be used to link the force factors in molecular mechanics and the element stiffness in structural mechanics [15, 20, 22–24, 26] which will allow simulation of the mechanical behavior of ZNT. Based on Timoshenko’s theory of elasticity for beams, the relations between the beam strain energies and the harmonic potentials are expressed as [27] where and represent Young’s modulus and shear modulus, respectively, for Zr–O bond in form of beam with as diameter and as length. Also, and denote the stretching force and bending force constants, respectively.

The diameter, which is equivalent to thickness, and the Poisson ratio for the Zr–O structural bond element can be determined using analytical, mechanistic, or numerical models [26].

##### 2.2. Finite Element Modeling (FEM)

###### 2.2.1. Structure

The geometry of inorganic nanotubes is built on the similar models used for CNT where the tubes are supposed to be made by rolling up of nanosheets (NNS) to form a hollow cylinder and may be single- or multiwalled. For CNT, the basic structural unit is a single atomic layer, known as graphene [28]. The nanotube is defined by the translation vector and the chiral vector, , (, , , and are integers, and and are translation vectors of the 2D lattice) as shown in Figure 1. The nanotube of the chirality () is achieved by folding the layer in a manner that the chiral vector becomes the circumference of the nanotube. The orthogonality relations are used to define the NT chirality () harmonious with the initial 2D lattice periodicity [25]. By designation () is armchair, () is zigzag, and () is chiral [29].