Bond stiffness

Theoretical analysis of certain features in the electromagnetic spectrum yields basic molecular parameters such as bond lengths and bond stiffness. We shall see presently that the mechanical spectra can be related to molecular parameters and not just modelistic characteristics as we have used until now. 

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. 

The stiffnesses of other bond types are calculated in a similar way (in general, the cumbersome sum described above is not needed because the interactions are of short range). The resulting hierarchy of bond stiffnesses is as shown in Table 6.1. 

At equilibrium, the heavy isotopes of an element will tend to be concentrated in substances where that element forms the stiffest bonds (i.e., bonds with high spring constants). The magnitude of the isotopic fractionation will be roughly proportional to the difference in bond stiffness between the equilibrated substances. Bond stiffness is greatest for short, strong chemical bonds these properties correlate with ... 

Physical properties of polymers, including solubility, are related to the strength of covalent bonds, stiffness of the segments in the polymer backbone, amount of crystallinity or amorphousness, and intermolecular forces between the polymer chains. The strength of the intermolecular forces is directly related to the CED, which is the molar energy of vaporization per unit volume. Since intermolecular attractions of solvent and solute must be overcome when a solute (here the polymer) dissolves, CED values may be used to predict solubility. 

Limited predictions, about vibration can be made based upon the mass of ffie Atoms involved and the stiffness of the bond between them. Atoms with greater mass resonate at lower frequencies stiffer bonds, such as double and triple bonds, resonate at higher frequencies. Pond strength and bond stiffness follow the same order sp >... 

The value of /is approximately 5 X 105 dyne/cm for single bonds and approximately two and three times this value for double and triple bonds, respectively (see Table 2.1). The force constant, /, can be thought of as a measure of bond stiffness. This force constant can be correlated with such properties as bond order and bond strength. Because the frequency is directly related to the square root of the force constant, we know that the frequency of bond vibrations should decrease as bonds decrease in strength. 

The CO overlap population is increased over the free molecule value on adsorption to Ni(I), Ni(II), and Ni(III), increasing in this order. Adsorption to Ni(0) decreases the overlap population. Thus, an increase in vibration frequency of CO is expected as the charge of nickel ion becomes more positive because overlap population is a measure of bond stiffness. This effect has been observed experimentally and illustrates a potentially valuable application of MO theory. 

The three optic types modes (vtz Aig and vixEg) the latter mode vtx Eg is doubly degenerate, only involve distortions to the O—O bonding interactions i.e. the sublattices move as rigid units. The frequencies of these modes are in effect the spring-constants of the O—0 interactions involved which can be related to the bond stiffness Xi /Xs. 

We have been able to compare the difference that full anti-ferroelectric ordering present in ice VIII has with the incomplete anti-ferroelectric ordering in two ice VII structures in terms of bond stiffness Xi As and charge movement The ability of the O—O... 

Figure 2 The variation with pressure for ice XI of the AIM quantities (a) ellipticity 8, and (b) bond stiffness X,i/A-3, of the hydrogen bonds (BCPs). The numbering scheme in the legend matches that in Figure 4 also the legend in Figure 2(b) is appropriate for Figure 2 (a).

Bond Type Bond Stiffness (N/m) Young s Modulus (GPa) Example... 

The true barriers to rotation, bond stiffness etc. are defined by the total electronic context of the molecule, including surrounding solvent molecules. In some cases, these barriers are not very sensitive to context and may be taken as... 

Polymer materials in fiber form possess superior mechanical properties compared to the same material in bulk polymer form. The reason for this is clear, the alignment and straightening of the polymer chains make better and better use of the bond stiffnesses and strengths that exist between the individual atoms from which a single polymer molecule is comprised. As the fibers become thinner, this effect is amplified until finally a theoretical maximum is reached. In order to assess the applicability and potential of various SPCs, it is useful to compare both the measured and theoretically achievable (theoretical calculation based on atomic forces) values of tensile strengths and stiffness for various polymer fibers. Table 19.1 presents a selection of thermoplastic polymers, summarizing their mechanical properties in terms of both measured and theoretically achievable fiber properties in comparison with its bulk state [24]. 

The ability of IR spectroscopy to probe the environment in which bonds vibrate allows it to be used to examine structural ordering of surface modifier molecules. If alkyl chains for example are in a liquid-like state of disorder the dispersion forces of attraction between chains are not maximised and vibration of the C-H bonds is influenced almost solely by the bond stiffness. If on the other hand the chains are in an ordered crystalline array, dispersion forces are maximised. This causes the C-H bonds to become very slightly longer (and therefore weaker) than when in the liquid state. This reduction in stiffness causes the asymmetric C-H stretching vibration frequency of a Cjg alkyl chain to reduce by 3-5 cm. Such frequency shifts have been used by Kellar and co-workers [48] to investigate ordering of oleic acid monolayers and by Liauw and co-workers [13, 49]. Vaia and co-workers [50] have also used this approach to examine the order of alkyl chains of quaternary alkyl ammonium intercalants in organo-clays for nanocomposite applications. 

This model was first suggested by Fraenkel and recently investigated by Fatkullin et al In the limit of high bond stiffness k —> oo this model becomes the freely jointed chain model. We note that unlike the Rouse model, this model is nonlinear, and the different Cartesian components are coupled. We will investigate the consequences of these complications in Section 1.06.3.3.2. In this chapter, we will use bond stiffness fe= 100. 

The exact position of a signal indicates subtle factors that affect bond stiffness, such as resonance. 

Using nominal nuclear masses gives k values of 958, 512,408, 311 Nm , respectively. Decreasing bond stiffness implies decreasing bond strengths. (Observed Do values are, in kJmol 564, 428, 363, 295, respectively.)... 

Alternative assessments of chain modulus have been derived from x-ray measurements of crystal lattice strain resulting from applied stress. Interestingly, the chain modulus was observed to be temperature sensitive. At —80, 20, and 120°C, x-ray-determined modulus values of 121, 97, and 65 GPa were reported [63]. This was explained on the basis of temperature-dependent interchain interactions, because temperature-dependent bond stiffness is not expected. The polymer can be viewed as an assembly of sinuous chains, packed together in a local direction, which are strengthened by their interaction. With increasing temperature, enhanced molecular mobility weakens these interchain interactions, and a lower tensile modulus results. 

Within the framework of this general model, other refinements are possible. For example, bond stiffness in the cation can be introduced via next nearest-neighbour interactions. More specific interactions can also be modelled. For example, some imidazoliums are believed to undergo n — n stacking interactions. 

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