Nonlinear scale

Hardness. The hardness (qv), or related property abrasiveness, is an important filler property. Hardness is determined by comparison to materials of known hardness on the Mohs scale. On this nonlinear scale, diamond is rated 10, quartz 7, calcite 3, and talc 1. The abrasiveness of a filler is also dependent on psd and the presence of impurities, eg, ka olin clay (Mohs hardness of 3) can be quite abrasive because of the presence of quartz impurities. 

Viscosity can also be determined from the rising rate of an air bubble through a Hquid. This simple technique is widely used for routine viscosity measurements of Newtonian fluids. A bubble tube viscometer consists of a glass tube of a certain size to which Hquid is added until a small air space remains at the top. The tube is then capped. When it is inverted, the air bubble rises through the Hquid. The rise time in seconds may be taken as a measure of viscosity, or an approximate viscosity in mm /s may be calculated from it. In an older method that is commonly used, the rate of rise is matched to that of a member of a series of standards, eg, with that of the Gardner-Holdt bubble tubes. Unfortunately, this technique employs a nonlinear scale of letter designations and may be difficult to interpret. 

Using the same reasoning as with the particle number distribution above, we observe that if the x- and y-axes are provided with the nonlinear scales, n and tf, defined by Eqs. (14.34) and (14.35), the mass distribution m x)/m t) can be described by a straight line... 

Eq. (10.19). These nonlinear scaling fields are a set of curvilinear coordinates in parameter space, following the flow lines of the nonlinear representation. 

Fig. 10.2. Schematic flow diagram on the critical surface close to a fixed point. Full lines show the flow of nonlinear scaling fields, where > A0 > 0. That relation is signalled by the broken lines which illustrate that the flow from a generic point on the critical surface approaches the fixed point from direction. The tangents of...

Equation (11,85) is the nonlinear scaling law, The linear scaling law results at the fixed point / = 1. As is easily derived with the help of Eq. (11.43) it takes the form discussed in Sect. 9.1 V becomes a function of s = CpRg and... 

Steinbrecher, T., Mobley, D.L., Case, D.A. Nonlinear scaling schemes for Lennard-Jones interactions in free energy calculations. J. Chem. Phys. 2007,127, 214108. 

Gaussian diffusion is by no means ubiquitous, despite the appeal of the central limit theorem. Indeed, many systems exhibit deviations from the linear time dependence of Eq. (1). Often, a nonlinear scaling of the form [14—16]... 

Quality of distance measurements can be improved by nonlinear scaling (Gauss-ian-type or exponential functions) of the distances S(a,b). 

The non-SI unit cm 1 (1 im 1 = 104cm J) is mostly used in IR, Raman and microwave spectroscopy. In this text, absorption spectra are shown on a wavenumber scale A(v), v = vjc = 1/2, [v] = im, mnning from right to left, so that the wavelengths, shown on top of the diagrams on a nonlinear scale, still increase to the right as is more customary to the chemist. To convert to a molecular energy scale we use the relation E = N. hc v to obtain the practical Equation 3.1. 

Fig. 5.4. The double-well radial potential for 4/ elements close to the point at which orbital collapse occurs. Note the highly nonlinear scales on both axes, designed to show as much detail as possible for both wells (after D.C. Griffin et al. [208]).

Figure 4 Scaling of the equilibrium position of a harmonic potential. The potential functions for initial and final state are given with minima at Xq and x respectively. The potential at the midpoint of the calculation, X = i, is shown using linear scaling of the potential (dotted) and nonlinear scaling (solid).

It is customary to characterize the modulus, stiffness, or hardness of rubbers by measuring their elastic indentation by a rigid die of prescribed size and shape under specified loading conditions. Various nonlinear scales are employed to derive a value of hardness from such measurements (Soden, 1952). Corresponding values of shear modulus G for two common hardness scales are given in Figure 1.18. 

---