The chemical bond strength describes the ability of a chemical bond holding two constituting atoms together. Many physical and mechanical properties of a material, such as melting point, activation energy of phase transition, tensile and shear strength, and hardness, are closely related to the bond strength (Kittle, 2004). However, a universal quantification of bond strength in crystal is hard to be performed due to the lack of effective microscopic parameters to characterize the bond strength. Usually, different characteristic parameters are chosen for different materials. For simple substances and covalent compounds, bond strength is correlated to cohesive energy. For ionic crystals, breaking a chemical bond means overcoming of the electrostatic interaction between anion and cation, which is defined as lattice energy and used to characterize bond strength. Correspondingly, several theoretical definitions of chemical bond strength have been proposed, such as Pauling’s definition for ionic crystals (Pauling, 1929), orbital scaling for covalent crystals (Hultgren, 1932), and two power-law expressions for a variety of materials (Brown & Shannon, 1973; Gibbs et al., 2003). These definitions of bond strength are only valid for some specific types of crystals, and a generalized model of bond strength has not been reached.
In a crystal, bond strength is an intrinsic property of chemical bond, and is regulated by the constitutional atoms as well as the crystal structure. From this viewpoint, bond strength is directly determined by the bond length and the shared bonding electrons. Obviously, greater bond strength would be expected with shorter bond length. The extent of electron sharing, related to the electronegativity difference of bond-forming atoms, is determined by the localized electron density in the binding region. It was found that the greater the localized electron density, the more the effective bonding electrons, and the stronger the bond strength (Gibbs et al., 2003). Most recently, we established a universal semi-empirical quantitative scale to describe the strength of chemical bond in crystals (Guo et al., 2009). The chemical bond strength is defined as the maximum force that a chemical bond can resist under the uniaxial tension along the bond direction which is called tensile unbinding force. We found that the bond strength only relies on two parameters, the bond length and effectively bonded valence electron number of a chemical bond.
In the following, the concept of effectively bonded valence electron number of chemical bond is introduced and the universal quantification model of chemical bond strength is established based on effectively bonded valence electron number and bond length exclusively. The correlation between ideal tensile strength and chemical bond strength is presented. This model allows a convenience determination of chemical bond strength for a variety of materials, ranging from covalent crystals to ionic crystals as well as low dimensional materials. Its application to low dimensional materials, such as graphene, h-BN sheet, and SWNT, are also presented.
As mentioned in the introduction, the shared bonding electrons in the binding region of two bonded atoms plays a vital role in determining the bond strength. To establish an effective quantification model of bond strength, we must find a practical way to estimate the population of these electrons.
Considering two atoms,
The EBVE numbers of diamond (0.707) and NaCl (0.163) are in good agreement with the Mulliken population. Some EBVE numbers of various covalent and ionic crystals are listed in our previous publication (Guo et al., 2009) as well as in the following text.
As soon as the tensile unbinding force of a bond is known, the ideal tensile strength of a crystal is easily accessible. For simple structural crystals shown in Figure 1, generally, the weakest tensile directions, such as of zincblende (ZB), of wurtzite (WZ), and of rock salt (RS) structures, are parallel to the axes of the bonds to be broken. Thus, the ideal tensile strength,, of a crystal along the weakest direction should be correlated with
where in the unit of m-2 is the number of the broken bonds per unit area on the plane, which has the lowest bond density.
Alternatively but more time-consumingly, the ideal tensile strength of a crystal can be determined from first-principles calculations (Roundy et al., 1999). We acquired the ideal tensile strength of a wide variety of covalent and ionic crystals with a single type of chemical bond with ZB, WZ, or RS structures from first-principles calculations, for which
Up to now, we are considering two bonding atoms,
3. Results and discussions
In this section, we will start with the calculation of bond strength in chosen types of materials to understand the relationship of bond strength and crystal structure as well as to trace the relations between the macroscopic properties and bond strength. We will end this section with the bond strength calculations for some low-dimensional materials, such as graphene, h-BN sheet, and SWNT, to demonstrate the effectiveness of our semi-empirical quantification model to these systems.
3.1. IV-A semiconductors
The IV-A semiconductors belong to the family of
Several points need to be mentioned for these complicated structures before we discuss the calculation results. Firstly, we give the hexagonal representation instead of the simple rhombohedral representation for r8 structure to show the structure more clearly. Secondly, Si atoms in Imma structure are eight-fold coordinated while Ge atoms in the same structure of are six-fold coordinated (Figure 2f). Thirdly, there are two types of coordination states in the most complicated structure Cmca, although these atoms are identical. We denote the ten-fold coordinated atoms located at 8d sites with white spheres, and the eleven-fold coordinated atoms at 8f sites with gray spheres, as shown in Figure 2g.
The lattice parameters and the calculated bond strengths are listed in Table I. Except the sh-Si and sh-Ge, the arithmetic average of the bond length are given in Table I for structures with different bond lengths. The bond strength of graphite, diamond, lonsdaleite, ZB-Si, WZ-Si, ZB-Ge, ZB-Sn, and ZB-SiC can be referred to our recent publication (Guo et al., 2009).
The bond strength of the IV-A materials as a function of bond length is presented in Figure 3. The tensile unbinding forces are unambiguously grouped by distinct
NB 8-Nionic crystals
Our simple model for chemical bond strength can easily be applied to ionic crystals, just like the above considered pure covalent and polar covalent ANB8-N materials (Guo et al., 2009). The elemental combinations of IA-VIIA, IB-VIIA, and IIA (except Be)-VIA tend to form ionic crystals. The typical structures of ionic ANB8-N materials are RS and CsCl. ZB and WZ structures are also founded for some IB-VIIA crystals (Shindo et al., 1965). These four structures have been presented in Figure 1 and 2. Other structures, such as (H) (Hull & Keen, 1994), P3m1 (Sakuma, 1988), P4/nmm (Liu, 1971), P21/m and Cmcm (Hull & Keen, 1999), can also be found in I-VIIA and IIA-VIA compounds and are shown in Figure 4. The concrete coordination state of each atom in these five structures is clearly shown. The distance between the two neighboring iodine atoms in P3m1-structured CuI is 4.353 Å. Consequently, this compound has a lamellar structure, significantly different from other dense structures.
In our previous work, the bond strength and uniaxial tensile strength of ten types of rocksalt structured compounds has been calculated (Guo et al., 2009). Crystal parameters and calculated bond strength of other ionic AB compounds are shown in Table 2. For the high pressure phases, the lattice parameters are given under compression. For the monoclinic structures of P21/m-structured AgCl, AgBr and AgI, angles are 98.4, 95.9, and 98.4, respectively. The unbinding tensile force versus bond length for I-VIIA and IIA-VIA compounds listed in Table 2 together with those given in previous work is shown in Figure 5. The bond strength of seventy six chemical bonds locates on seven parallel lines from top to the bottom with decreasing n
3.3. III-VI crystals
Next step is to treat complicate crystals of A
For B2O3 of P3
Next structure is C
The lattice parameters, bond lengths, and bond strength are listed in Table 3. Most of the chemical bonds are two-electron bond. The bond strength versus bond length for these A
An extensive analysis of the A
Generally, lower coordination number results in higher bonded valence electron number, especially when the valence electron numbers of bonded atoms is the same. However, this argument does not hold when the valence electron numbers are different. For example, the Si atoms forming Si-Si bond in hcp- and fcc-structured Si are twelve-fold coordinated, which is the highest among the chemical bonds discussed above. The bonded valence electron number n
3.4. Low dimensional systems
We now apply our semi-empirical model to evaluate the theoretical tensile strength of the low dimensional systems, such as graphene, h-BN sheet, SWNT. As we mentioned before, the highest effective bonded valence electron numbers of 0.943 and 0.856 occur for three-fold coordinated C-C bond in graphite and three-fold coordinated B-N bond in h-BN, respectively. This is consistent with the argument that the sp2 hybridized C-C bond in graphite is the strongest chemical bond (Coulson, 1950). Recently, graphene, a single atomic layer of graphite, has stirred enormous research interests owning to its exceptionally high crystallinity and electronic quality, as well as a fertile ground for applications (Geim & Novoselov, 2007). Experimentally, the mechanical properties of graphene have been identified with atomic force microscope (AFM) nanoindentation, giving a tensile strength of 130 GPa (Lee et al., 2008). Here we predict the theoretical tensile strength of graphene in <10> direction using the present model as follows
where R is the thickness of graphene taken as the interlayer separation 3.4 Å of graphite. F
Whilst graphene has a great application potential in microelectronics, hexagonal boron nitride (h-BN) sheets can find uses as an effective insulator in graphene based electronics. The mechanic properties of h-BN sheets have recently been investigated. With an elastic constant (E
Carbon nanotube (CNT), with the same covalent sp2 bonds formed between individual carbon atoms as graphen, is one of the strongest and stiffest materials. Direct tensile testing of individual tubes is challenging due to their small size (10 nm or less in diameter). There are several experimental efforts on the mechanical properties of CNT (Yu et al., 2000; Demczyk et al., 2002; Ding et al., 2007; Barber et al., 2005). However, the reported failure stress values display a large variance and are well below the theoretical predicted values in most cases (Ozaki et al., 2000; Mielke et al., 2004), which are attributed to the large number of defects presented on the nanotubes. Accurate measurements of tensile strength require high-quality CNT with well-defined sample parameters, as well as the elimination of measurement uncertainties. Notwithstanding, for zigzag single wall nanotube (SWNT), the theoretical tensile strength along the axial direction can be predicted with our simple model as,
Before we end this section, there is one last point need to be mentioned regarding to the dependence of the tensile strength on the direction of the applied tensile stress. A tensile stress tilted away from the axis of a chemical bond would generate a shear component with respect to the bond, and we cannot perform the tensile strength calculation with Eqn. 3 under this circumstance. However, if the shear unbinding strength can be expressed, the ideal strength along any specific direction of a crystal will be accessible. Further studies are therefore highly expected.
The bond strength of a variety of chemical bonds are analysized with our semi-empirical unbinding tensile force model. This model proves to be valid for a wide selection of crystals, as well as low-dimensional materials such as graphene and nanotubes. In this model, the chemical bond strength, defined as the tensile unbinding force F
This work was supported by NSFC (Grant Nos. 50821001 and 91022029), by NBRPC (Grant No. 2011CB808205).