10.2 Interaction Between Atoms The internal energy of a material is a consequence of the atomic arrangement and depends on the forces that keep together the atoms. There are forces that bind together the elementary particles that form an atom [7]. In the current theory of constitution of matter these forces are the weak forces, the strong forces and the electromagnetic forces. The terms weak or strong have nothing to do with the actual magnitude of the forces but it results from a comparison of the product of the force between two particles, called Fm, times the square of the distance between the two particles r, Fmr 2 compared to the product (h/2π)c, where h is the Plank constant and c is the speed of light. The electromagnetic forces are the forces that bind together or repel charged particles. The positively charged protons are bound to electrons to form atoms, this binding effect results in the neutral charge of atoms. Atoms are neutral, then: what forces make atoms to stick together to form a crystal? The forces that bind together atoms are called in the literature of physics residual electromagnetic forces and result from sharing electrons that is, electrons of one atom interact with the protons of the other atom. These forces are weak forces in the sense that has been previously defined and the carriers of these forces are photons. The main types of bonding between atoms of interest to mechanics of materials are classified as hetero-polar, mono-polar, Van der Waals, and metallic bonding [8]. In the hetero-polar (ionic) bonding, one atom gives up one electron to become positive and the other atom gains an electron to become negatively charged. In the homopolar (covalent) bonding, both atoms remain neutral and share an electron pair. Van der Waals forces include attraction and repulsion forces between atoms, molecules, resulting from changes in the polarization of the interacting particles, for example repulsion forces caused by the Pauli Exclusion Principle. Metallic bonding is a very important category for the mechanics of materials because of the importance of metals in technical applications. Metals are characterized by delocalization of electronic shells, that is, electrons are shared by all the atoms in a cluster, the Fermi layer. The bonding can be characterized by the attractive force of the non localized electrons and positively charged metallic ions. From the point of view of mechanics of materials it is interesting to point out relative magnitude of different types of bonds with distances between particles. The covalent bonds change as e r, the ionic bond as 1/r and the Van der Waals force as 1/r6. The ionic forces that bind metals are very strong and reflect in the high temperature of melting of metals and high modulus of elasticity. 10.3 Models of Chemical Bonds In Sect. 10.2 a summary of the principal types of chemical bonds between atoms has been presented and the dependence of these bonds with the distance between nuclei indicated. The actual force fields between atoms require Quantum Mechanics computations. In classical Mechanics, in the process of molecular modeling, a force field is represented by a selected function that includes parameters that can be experimentally determined. The function provides the potential energy of a system of particles, atoms or molecules. An example is given in Fig. 10.1, the Lennard-Jones potential is given by [9, 10], ΦlJ ¼4εLJ σLJ r h i 12 σLJ r h i 6 ð10:1Þ The meanings of the parameters εLJ andσLJ included in Eq. (10.1) are given in Fig. 10.1. εLJ is a bonding energy, the order of magnitude is 10 1 (nN)nm; σ LJ is a distance, in the equilibrium position it is approximately σLJ ¼ro ¼0:3 nm. Following the conventional symbols utilized in the literature, the subscripts LJ have been added to avoid confusion with the same symbols used in Continuum Mechanics to indicate strain and stress as well. The atoms adopt a position that minimizes their potential energy for a given temperature. As indicated in Fig. 10.1, if r <σLJ, the atoms experience repulsive forces that become extremely high, that is in 3-D the atom becomes an almost rigid sphere. On the contrary, the separation of the atoms reduces the binding forces that asymptotically become zero. In Fig. 10.1, it is also represented the harmonic potential used as an approximation for the analysis of dynamical problems. If we compute the derivative of energy function Eg (see Fig. 10.2a), the inter-atomic force function is obtained. It can be seen that when the energy has a minimum, the mutual forces are zero, condition of equilibrium at a given temperature. Assuming that the force is equal to a constant E (modulus of elasticity, see Fig. 10.2b) times the displacement Δr, F ¼EΔr, leads to 82 C.A. Sciammarella et al.
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