Crystal Growth. Principles of Crystal Structure Determination. Tensor Nature of Physical Properties. Chapter 8 Mechanical Properties of Solids. Lattice Vibrations. Chapter 11 Dielectrics. Piezo Pyro and Ferroelectricity.

As the measurement of a single CNT and graphene has some difficulty especially for a certain type zigzag type or air-chairs type , it therefore causes our interest to present the exact forms of modeling at nanoscale. CNT has similar properties of fullerene molecules; its high strength has caused a widespread concern in various application fields.

Graphene is considered to be an atomic crystal with flat polycyclic aromatic hydrocarbons the structure is very stable.

It can be rolled into a barrel-shaped carbon nanotube [ 2 ], showing the similar properties of nanotube and fullerene molecule. However, the theoretical mechanical deformation analysis is so complicated that the nonlinear analytical method is needed where van der Waals force should be considered and the corresponding nonlinear transformation is needed.

### Introduction

Researches of CNT are mainly listed in Table 1. Thus, a new so-called atomic finite element method AFEM as a bridge between molecular dynamics and continuum mechanics [ 22 ] can be used to investigate elastic properties of covalent bond structures. In , the AFEM [ 22 ] is proposed, in which the interatomic bonds are modeled as nonlinear spring elements. In this chapter, carbon nanotubes CNT , hexagonal graphene, and portlandite CH are particularly investigated.

CNT, has its unique physical and chemical characteristics, has higher mechanical stability properties and impact resistance, thus can be used as a reinforcing material or composite material matrix. Generally, the force field is expressed in the form of steric potential energy. It depends solely on the positions of the nuclei constituting the molecule.

The total steric potential energy, omitting the electrostatic interaction, is a sum of energies due to valence or bonded interactions and nonbonded interactions [ 23 ]:.

Interatomic interactions in molecular mechanics by Li and Chou are shown in Figure 1. Li and Chou have proposed the molecular mechanics method MM [ 8 ], under the small deformation premise, to simplify the calculation, the simplest harmonic forms to merge the dihedral angle torsion and the improper torsion into a single equivalent term are adopted, which are as follows:.

Obviously, these formulas have established the equivalent relation between the macrobeam structure and microscopic molecular structure. Then, a mathematical modeling—AFEM modeling—of ionic crystals is proposed by means of interatomic potentials. Geometry of chemical bonds, nodal forces, and nodal displacements. Chemical bond element nodal forces and displacements are shown in Figure 2 b. The nodal force increment and the displacement increment in each loading step in the x -direction are:.

Continuum methods such as FEM use averaged description for modeling the material properties. As a result, they are not capable of modeling phenomena at the atomic scale. Localized nonlinear deformation, defects, nanoscale materials, and structures e. CNT types and characteristic parameters are classified in Table 2. Molecular mechanics method mainly includes two basic assumptions. The first assumption is the Born-Oppenheimer approximation, of which the electronic function is centered on the nuclear positions, with the electrons around them in an optimal distribution [ 25 ].

The second assumption is that each nucleus is seen as a classical particle. Electrons are not taken into account and atoms are simulated as spheres, with an assigned radius and charge, interacting by means of a collection of interatomic potentials [ 26 , 27 ]. The AFEM is proposed based on the development of nanotechnology across multiple length scales [ 22 ]. For a system composed of N atoms, the total energy of the system can be expressed as a sum of all the chemical energy stored:.

The minimum energy state is:. Each atom not only forms a direct bond with its three adjacent atoms, but also has an interaction with its six subadjacent atoms. Newton-Raphson iteration is as follows:.

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Each iteration needs to compute the total potential energy of the system and coordinates of the first and the second derivatives. K and P can be directly obtained by the interatomic interaction potential. The appropriate boundary conditions and finite elements are chosen to calculate the minimum energy state during iterative calculation, and the coordinates of each atom in the system can be obtained.

Figure 3 shows the interaction of van der Waals force of atoms A and atoms L among atoms. As is shown in Figure 3 , stiffness matrix for unit atom A is thus expressed as:. In summary, there are two types of interactions, including bonded and nonbonded: 1 for bonded type, it usually corresponds to strong covalent bonds, the number of neighbors is preset and is dictated by the lattice. Take the CNT for example, the carbon covalent bond often has carbon atoms 3—4 neighbors, and preset list of neighbors for each atom may be used throughout the calculations, used for crystalline solids with well-defined bonds.

Typical structure example is graphene; 2 for nonbonded type, it commonly corresponds to relatively weak forces, and the number of neighbors may be changing and is found based on the cutoff radius. AFEM modeling hypotheses of covalent bond and ionic bond. From Figure 4 , the ionic bond or metallic bond is taken into consideration, van der Waals interaction and electrostatic interaction Coulomb force can be separately calculated.

Besides, modeling of covalent bond is based on the theory of molecular mechanics method MM. Based on the AFEM modeling, the elastic stiffness tensors can be described and defined [ 28 ].

## Atomistic Theory of the Optical Properties (Optical Properties of Materials) Part 1

The expression of the total steric potential energy U is given out [ 23 ]. For ionic crystal, the Coulomb force should also be considered. So the ideal expression of U , considering the electrostatic interaction, is a sum of energies due to valence or bonded interactions and nonbonded interactions:.

The coupled field of van der Waals force and Coulomb force between atoms or charged ions may be obtained by using Lennard-Jones potential or other potentials. Nonlinear spring elements can represent Uvdw and Ucoul of ionic bond. Around the equilibrium position, van der Waals force is developed by the first-order Taylor:. For Uvdw , the nonlinear displacement-force characteristic of spring element as the first derivative of potentials must be defined by means of the program potentials [ 22 ], then by the coordination transformation.

Nonlinear spring element is used to represent interatomic force of Lennard-Jones potential. Generally, the expression of van der Waals force can be obtained by the Lennard-Jones potential [ 30 ]:. Schematic illustration of Lennard-Jones potential and van der Waals force is shown in Figure 5. Schematic illustration of Lennard-Jones potential and van der Waals force. As shown in Figure 5 a , repulsive force is more intense but the range is very short that prevents exchange of electrons from occupying the same region of space, since the forces exerted at larger distances are very small and can be neglected [ 23 , 31 ].

When the atomic distance is greater than 2.

## NSF Award Search: Award# - Atomistic and Continuum Models of Solids

As is shown in Figure 5 b , if the average distance r between the atoms increases, the attractive force of van der Waals force between pairs of atoms will resist the force caused by the increase of distance r. And if the average distance r between the atoms increases, the repulsive force between pairs of atoms will be able to resist the force caused by the decrease of distance r. Interactions of van der Waals consist of two terms: a repulsive term and an attractive term. The function of Lennard-Jones is known as a function of the potential of van der Waals type.

It is used to describe the intermolecular force between argon atoms i and j separated by a distance rij. Any two molecules attract each other at long separation distance and repel each other when they come closer [ 32 ]. When the interatomic spacing is greater than its unstressed value, the attractive forces between atoms must be greater than the repulsive forces the attractive forces balance both the repulsive forces and the external forces.

Repulsive forces are taken as positive and attractive as negative. This conveniently makes their potential energies positive and negative, respectively, in which the zero of potential energy is taken at large separation. As is shown in Figure 6 , covalent bonds e. Then, axial moduli of the structure, as a function of elastic constants, are derived and verified by cubic linear elements volume unit on ABAQUS. Nonlinear springs are introduced to describe the relationship between atoms of different initial distances.

When the van der Waals force is considered, the coordinate transformation needs to be done before carrying out the INP file. Input file usage:. Instead, we can define connectors that have spring-like elastic behavior. To obtain the homogenized moduli, the notion of homogenization has been introduced using the representative volume element RVE of the material [ 34 ]. When a solid model is under external load, microstresses and strains are induced; according to the statistical homogeneity assumption, an appropriate RVE can be defined and isolated.

On the RVE boundary, there exist definitive surface displacements and surface tractions.

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Through homogenization [ 35 ], mechanical behavior is described by a definitive constitutive law.