Hertz solution

We will confine ourselves to axisymmetric indentors, noting of course that fairly explicit results can also be given for an ellipsoidal indentor, by virtue of the classical Hertz solution. For a spherical indentor we may approximately take... 

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

Often, Hertz s work [27] is presented in a very simple form as the solution to the problem of a compliant spherical indentor against a rigid planar substrate. The assumption of the modeling make it clear that this solution is the same as the model of a rigid sphere pressed against a compliant planar substrate. In these cases, the contact radius a is related to the radius of the indentor R, the modulus E, and the Poisson s ratio v of the non-rigid material, and the compressive load P by... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

Some recent papers permit an exciting outlook on the degree of sophistication of experimental techniques and on the kind of data which may be available soon. In the field of NMR spectroscopy, a publication by Hertz and Raedle 172> deals with the hydration shell of the fluoride ion. From nuclear magnetic relaxation rates of 19F in 1M aqueous solutions of KF at room temperature, the authors were able to show that the orientation of the water molecules in the vicinity of fluoride ions is such that the two protons are non-equivalent. A geometry is proposed for the water coordination in the inner solvent shell of F corresponding to an almost linear H-bond and to an OF distance of approximately 2.76 A, at least under the conditions chosen. 

Hertz, H.G., Chap VII.2 of Structure of Water and Aqueous Solutions (ed. W.A.P. Luck) Verlag Chemie Physik Verlag (1974). 

Similarly with the raising of the b.p. in violet or reddish-violet soln. of iodine in benzophenone, carbon disulphide, ethyl chloride, chloroform, carbon tetrachloride, ethylene chloride or benzene or in brown soln. of ethyl alcohol, methyl alcohol, thymol, ethyl ether, methylal, or acetone. The values for the last three solvents were rather low, presumably because of the chemical action of solute on solvent. High values with benzene are attributed to the formation of a solid soln. of solvent and solid. Confirmatory results were found by J. Hertz with naphthalene, and by E. Beckmann and P. Wantig with pyridine. The results by I. von Ostromisslensky (o-nitrotoluene), by G. Kriiss and E. Thiele (glacial acetic acid), and by H. Gautier and G. Charpy indicate polymerization, but they are not considered to be reliable. 

Henry s law The partial pressure of a gas in equilibrium with gas dissolved in a solution is proportional to the concentration of dissolved gas P = Hdissolved gas]. The constant k is called the Henry s law constant. It is a function of the gas, the liquid, and the temperature, hertz. Hz Unit of frequency, s-1. heterogeneous Not uniform throughout. 

NMR spectroscopists express coupling constants (J values) in hertz ESR spectroscopists usually express coupling constants of organic radicals in solution in units of gauss (G). Multiplication of A in (8.115) by h/gefie converts it from hertz to gauss we have... 

Heinrich Hertz long ago solved the problem of a thermal wave in front of a heated surface which is moving with constant velocity. This solution, first applied to a flame by Michelsohn, has the form... 

In essence, the chemical shift of a nucleus such as proton ( I I) is its resonance frequency. It is usually expressed in parts per million (ppm) relative to a standard. The most common standard is tetramethylsilane [(CH3)4Si, TMS] which defines 0 on the delta (8) scale and 10 on the older, less used t scale. A small amount of TMS is typically added to the NMR solution to be examined. The presence of an internal standard minimizes experimental variations. This is particularly important because the chemical shift is typically a change of only a few hertz per megahertz, hence the part per million (ppm) scale. The separation of peaks will be greater in hertz at higher field but spectra obtained at different field strengths are comparable on the ppm scale. Common reference standards are listed in Table 6.32. 

Fig. 8. Classification of ESR spectra (—296 K) of solutions of alkali metals in a variety of nonaqueous solvents rM is the electron-cation encounter lifetime, and A is the metal hyperfine coupling constant, in hertz THF = tetrahydrofuran, EA = ethylamine, DG = diglyme, MA = methylamine (—220 K), 1,2PDA = 1,2-propanediamine, EDA = ethylenediamine, AM = ammonia (—240 K).

The polarizations attributable to biopolymers in solution are of course both coupled to the solvent medium and superposed on polarizations attributable to the solvent. The usual pattern of dispersion, as discussed by Schwan in the paper following this one, is indicated schematically in Fig. 3(a) the B relaxation or relaxations of biopolymers are found at frequencies from a few hertz to a few tens of megahertz, while the y relaxation is for small solvent molecules at gigahertz frequencies, e.g., for aqueous solutions not far from the 18 GHz relaxation frequency (8 picosecond relaxation time) of pure water. In the few cases for which the intermediate frequency range has been studied in... 

By comparison, a saturated methine carbon (C-H) has a CSA of only 25 ppm because the mobility of electrons around the carbon nucleus is much less in an sp3-hybridized carbon and depends much less on the orientation of the C-H bond with respect to B0. In solution-state NMR we only see the isotropic chemical shift, < iso, and the fixed-position chemical shifts and the CSA value are obtained from solid-state NMR measurements. Although CSA does not affect chemical shifts in solution, it does contribute to NMR relaxation and can be exploited to sharpen peaks of large molecules such as proteins in solution. For large molecules, such as proteins, nucleic acids, and polymers, or in viscous solutions, molecular tumbling is slow and CSA broadens NMR lines due to incomplete averaging of the three principle chemical shift values on the NMR timescale. Like isotropic chemical shifts, CSA in parts per million is independent of magnetic field strength B0 but is proportional to B0 when expressed in hertz. Because linewidths are measured in... 

An experimental basis has been suggested by Hepler (1969) for the classification of solutes for structure formers, (32F /9r2)>0 and for structure breakers <0. More recently, Ben-Naim (1975) has demonstrated how the difference between solubilities of a solute in D20 and H20 can also be used in a similar fashion. Ben-Naim (1972a, 1973a) has also shown from theoretical arguments how it is possible for a solute to stimulate H-bond formation, i.e. structure-formation, between water molecules. However, the effect of a structure-former on water is not straightforward, the induced structure being apparently different from that of pure water at a lower temperature (Hertz, 1970). 

Nevertheless, we must guard against too ready an acceptance of static structural models for aqueous solutions. Just as in the case of water itself, a complete picture of these solutions requires information concerning the dynamic parameters describing the motions of solute and solvent molecules (Goldammer and Hertz, 1970 Hertz, 1964 1970). Hertz in particular has emphasized the importance of dynamic models (Franks, 1973b). 

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