Atomic stiffness

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

To understand the origin of the modulus, why it has the values it does, why polymers are much less stiff than metals, and what we can do about it, we have to examine the structure of materials, and the nature of the forces holding the atoms together. In the next two chapters we will examine these, and then return to the modulus, and to our bar-chart, with new understanding. 

To conclude, the concept of bond stiffness, based on the energy/distance curves for the various bond types, goes a long way towards explaining the origin of the elastic modulus. But we need to find out how individual atom bonds build up to form whole pieces of material before we can fully explain experimental data for the modulus. The... 

In the previous chapter, as a first step in understanding the stiffness of solids, we examined the stiffnesses of the bonds holding atoms together. But bond stiffness alone does not fully explain the stiffness of solids the way in which the atoms are packed together is equally important. In this chapter we examine how atoms are arranged in some typical engineering solids. 

As we showed in Chapter 4, atoms in crystals are held together by bonds which behave like little springs. We defined the stiffness of one of these bonds as... 

A comparison of these predicted values of E with the measured values plotted in the bar-chart of Fig. 3.5 shows that, for metals and ceramics, the values of E we calculate are about right the bond-stretching idea explains the stiffness of these solids. We can be happy that we can explain the moduli of these classes of solid. But a paradox remains there exists a whole range of polymers and rubbers which have moduli which are lower - by up to a factor of 100- than the lowest we have calculated. Why is this What determines the moduli of these floppy polymers if it is not the springs between the atoms We shall explain this under our next heading. 

This is because rubber, like many polymers, is composed of long spaghetti-like chains of carbon atoms, all tangled together as we showed in Chapter 5. In the case of rubber, the chains are also lightly cross-linked, as shown in Fig. 5.10. There are covalent bonds along the carbon chain, and where there are occasional cross-links. These are very stiff, but they contribute very little to the overall modulus because when you load the structure it is the flabby Van der Waals bonds between the chains which stretch, and it is these which determine the modulus. 

The moduli of metals, ceramics and glassy polymers below Tq reflect the stiffness of the bonds which link the atoms. Glasses and glassy polymers above are leathers, rubbers or viscous liquids, and have much lower moduli. Composites have moduli which are a weighted average of those of their components. 

In the last chapter we examined data for the yield strengths exhibited by materials. But what would we expect From our understanding of the structure of solids and the stiffness of the bonds between the atoms, can we estimate what the yield strength should be A simple calculation (given in the next section) overestimates it grossly. This is because real crystals contain defects, dislocations, which move easily. When they move, the crystal deforms the stress needed to move them is the yield strength. Dislocations are the carriers of deformation, much as electrons are the carriers of charge. 

When other elements dissolve in a metal to form a solid solution they make the metal harder. The solute atoms differ in size, stiffness and charge from the solvent atoms. Because of this the randomly distributed solute atoms interact with dislocations and make it harder for them to move. The theory of solution hardening is rather complicated, but it predicts the following result for the yield strength... 

The first patent on the chlorination of polyethylene was taken out by ICI in 1938. In the 1940s scientists of that company carried out extensive studies on the chlorination process. The introduction of chlorine atoms onto the polyethylene backbone reduces the ability of the polymer to crystallise and the material becomes rubbery at a chlorine level of about 20%, providing the distribution of the chlorine is random. An increase in the chlorine level beyond this point, and indeed from zero chlorination, causes an increase in the Tg so that at a chlorine level of about 45% the polymer becomes stiff at room temperature. With a further increase still, the polymer becomes brittle. 

Bar, G., Delineau, L., Hafele, A. and Whangbo, M.H., Investigation of the stiffness change in, the indentation force and the hydrophobic recovery of plasma-oxidized polydimethyl-siloxane surfaces by tapping mode atomic force microscopy. Polymer, 42(8), 3627-3632 (2001). 

The quantity p (mu) is called the effective mass of the molecular vibration (some people call it the reduced mass ). As we anticipated, the frequency is higher for stiff bonds (large k) and low atomic masses (low p). We see that, by measuring... 

According to the distance from probe to the sample, three operation modes can be classified for the AFM. The first and foremost mode of operation is referred to as contact mode or repulsive mode. The instrument lightly touches the sample with the tip at the end of the cantilever and the detected laser deflection measures the weak repulsion forces between the tip and the surface. Because the tip is in hard contact with the surface, the stiffness of the lever needs to be less than the effective spring constant holding atoms together, which is on the order of 1 — 10 nN/nm. Most contact mode levers have a spring constant of <1 N/m. The defection of the lever can be measured to within 0.02 nm, so for a typical lever force constant at 1 N/m, a force as low as 0.02 nN could be detected [50]. 

Since the idea that all matters are composed of atoms and molecules is widely accepted, it has been a long intention to understand friction in terms of atomic or molecular interactions. One of the models proposed by Tomlinson in 1929 [12], known as the independent oscillator model, is shown in Fig. 13, in which a spring-oscillator system translates over a corrugating potential. Each oscillator, standing for a surface atom, is connected to the solid substrate via a spring of stiffness k, and the amplitude of the potential corrugation is. ... 

The assumption of independent oscillators allows us to study a simplified system containing only one atom, as illustrated in Fig. 14 where x and Xq denote, respectively, the coordinates of the atom and the support block (substrate). The dynamic analysis for the system in tangential sliding is similar to that of adhesion, as described in the previous section. For a given potential V and spring stiffness k, the total energy of the system is again written as... 

Stability of the atomic system depends on the spring stiffness and the potential corrugation, or more specifically, depends on the ratio of k/. The system would become more stable if the stiffness increases or the potential corrugation decreases, which means less energy loss and lower friction. 

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