Transition bulk modulus

Material properties can be further classified into fundamental properties and derived properties. Fundamental properties are a direct consequence of the molecular structure, such as van der Waals volume, cohesive energy, and heat capacity. Derived properties are not readily identified with a certain aspect of molecular structure. Glass transition temperature, density, solubility, and bulk modulus would be considered derived properties. The way in which fundamental properties are obtained from a simulation is often readily apparent. The way in which derived properties are computed is often an empirically determined combination of fundamental properties. Such empirical methods can give more erratic results, reliable for one class of compounds but not for another. 

The trends in several ground state properties of transition metals have been shown in Figs. 2, 3 and 15 of Chap. A and Fig. 7 of Chap. C. The parabolic trend in the atomic volume for the 3-6 periods of the periodic table plus the actinides is shown in Fig. 3 of Chap. A. We note that the trend for the actinides is regular only as far as plutonium and that it is also broken by several 3 d metals, all of which are magnetic. Similar anomalies for the actinides would probably be found in Fig. 15 of Chap. A - the bulk modulus - and Fig. 7 of Chap. C - the cohesive energy if more measurements had been made for the heavy actinides. 

Spiering et al. (1982) have developed a model where the high-spin and low-spin states of the complex are treated as hard spheres of volume and respectively and the crystal is taken as an isotropic elastic medium characterized by bulk modulus and Poisson constant. The complex is regarded as an inelastic inclusion embedded in spherical volume V. The decrease in the elastic self-energy of the incompressible sphere in an expanding crystal leads to a deviation of the high-spin fraction from the Boltzmann population. Pressure and temperature effects on spin-state transitions in Fe(II) complexes have been explained based on such models (Usha et al., 1985). 

The cohesive energy, equilibrium atomic volume, and bulk modulus across a transition metal series may now be evaluated by choosing the following simple exponential forms for ( and h(R), namely... 

Fig. 7.12 The theoretical ( ) and experimental (x) values of the equilibrium band width, Wigner-Seitz radius, cohesive energy, and bulk modulus of the 4d transition metals. (From Pettifbr (1987).)...

The transition from the expanded state to the collapsed one and vice versa is controlled by diffusion of the solvent in the gel [56, 57], It was found [56] that the kinetics of swelling and deswelling of the gel is determined by local motions controlled by the diffusion equation in which the diffusion coefficient is given by the ratio of the bulk modulus to the frictional factor (between network and liquid). Whereas in our samples with a volume 1 cm3, the transition from one to another equilibrium state takes several days, for submicron spheres this time... 

Neutral NIPA gel is the most extensively studied among known gels from the standpoint of phase transition, and thus, various physical properties around the transition have been reported. These include the shear and bulk modulus [20, 24], the diffusion constant of the network [25], spinodal decomposition [26], specific heat [21], critical properties of gels in mixed solvents [8] and the effect of uniaxial [27] and hydrostatic [28] pressures on the transition, and so... 

The bulk modulus (K) is unaffected by the (1 transition it is thus not surprising to find that it is independent of the frequency/ strain rate. 

The bulk modulus depends practically only on cohesion. It does not exhibit viscoelastic effects and depends only slightly on temperature, except at the glass transition where it varies by a factor of about 2. 

The tensile and shear modulus differ from the bulk modulus in fact that they are influenced by the molecular mobility, and therefore display an almost discontinuous decrease at each secondary transition. Their relative variation at Tg is 10-100 times higher than that of the bulk modulus. The Poisson s ratio varies in the opposite sense in the region T < Tg. It tends toward 0.5 when the temperature increases beyond Tg. 

Bulk moduli and pressure derivatives. Results for the bulk modulus and its pressure derivative for all three HMX polymorphs obtained from fitting simulation-predicted isotherms to the equations of state discussed above are summarized in Table 7. For all data sets, we include fits to the Us-Up form (Eq. 18) and both weighting schemes for the third-order Birch-Mumaghan equation of state (Eqs. 20 and 21). In the case of the experimental data for /THMX, values for the moduli based on Eqs. 18 and 20 were taken from the re-analysis of Menikoff and Sewell. Two sets of results are included in the case of Yoo and Cynn, since they reported on the basis of shifts in the Raman spectra a phase transition with zero volume change at 12 GPa. Simulation data of the /T HMX isotherm due to Sorescu et al. were extracted by hand from Fig. 3b of their work. 

Hartmann already pointed out that the reducing parameter B0 is equal to the compression modulus or bulk modulus K, extrapolated to zero temperature and pressure and that T0 is related to the glass transition temperature. 

To make this point quantitatively requires a knowledge of the radial interactions other than tho.se from the d states (Eq. 20-13). Assuming such terms are the same across a transition. series, and fitting them to the bulk modulus for one metal, does, in combination with Eq. (20-13), predict a minimum in Iq at the center of the... 

From Eq.21 it follows that the imaginary party of the effective bulk modulus K is much larger than the imaginary part of K. Therefore, sound waves are strongly attenuated in this type of material. The viscoelastic polymer-air microbubbles composites are particularly useful in the design of broadband, transition type anechoic coatings for underwater... 

As an example, Fig. 8 shows the fracture toughness for PMMA and Fig. 9 the fringe pattern transition at the critical temperature, whereas Fig. 10 shows the lateral face of the sample with the crack-tip above and below the critical temperature. It has also been shown that neither the bulk modulus nor the craze stress varies near the critical temperature (Fig. 11 and 12). It seems that the local material property varying near that particular temperature is the craze stiffness, as shown in Fig. 13. 

Fig. 11. Bulk modulus of PMMA near the critical temperature T there is no transition at T, from Ref courtesy of Chapman and Hall, Ltd.

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