Bulk modulus calculation

Studies of the thermodynamic properties obtained from solutions of the SSOZ equation have been much more extensive but the success has been mixed. The first such calculations were those of Lowden and Chandler who obtained the pressure of hard diatomic fluids from the RISM (SSOZ-PY) equation. They used two routes to the equation of state a compressibility equation of state in which they integrated the bulk modulus calculated from the site-site correlation functions via... 

The self-consistent (SC) solutions are an approximate approach, in which a particle of one phase of given shape (e.g., spheres, needles or discs) is surrounded by the composite material. This is illustrated in Fig. 3.12, in which a spherical particle is surrounded by an effective medium that represents the elastic properties of the composite. For the spherically shaped particles, one can again use Eq. (3.14) but now H=4pJ i is used for the bulk-modulus calculation and... 

Fig. 101. (a) Volume change and (b) bulk modulus of TmSedjiTeoji, vs. pressure at 300 K. The dashed curve in (a) is the expected volume change for divalent Tm ions. The dashed curve in (b) is the bulk modulus calculated irom the independently measured Cn and c,2. (After Boppart and Wachter 1984b.)... 

Fig. 20.15. The bulk modulus of Smi-xY S compounds. In the critical concentration region the bulk modulus undergoes an abrupt decrease indicating extreme softening of the lattice. The bulk modulus calculated from elastic constants obtained from sound velocity measurements on single crystals (from Penney et al., 1975).

The bulk modulus calculated with two-body increments only is 0.132 Mbar, considerably lower than the experimental value, and similar to the result of LDA (in the rhombohedral lattice), 0.187 Mbar. In the hep lattice the LDA bulk modulus does not change much, with a value of 0.190 Mbar. This strong underestimation of the bulk modulus with LDA is in contrast to what would normally be expected for an overbound structure. The two-body increments with only -correlation are still repulsive, as mentioned above. When only the -correlation of the three-body increments is included, the bulk modulus increases to a value of 0.383 Mbar. The final result of the method of increments, with the inclusion of -correlation for... 

The help of W. Sievers with bulk modulus calculations and with part of the drawings is gratefully acknowledged. 

There has not been as much progress computing the properties of crystals as for molecular calculations. One property that is often computed is the bulk modulus. It is an indication of the hardness of the material. 

For crystalline polymers, the bulk modulus can be obtained from band-structure calculations. Molecular mechanics calculations can also be used, provided that the crystal structure was optimized with the same method. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

The bulk modulus of an ideal SWNT crystal in the plane perpendicular to the axis of the tubes can also be calculated as shown by Tersoff and Ruoff and is proportional to for tubes of less than 1.0 nm diameter[17]. For larger diameters, where tube deformation is important, the bulk modulus becomes independent of D and is quite low. Since modulus is independent of D, close-packed large D tubes will provide a very low density material without change of the bulk modulus. However, since the modulus is highly nonlinear, the modulus rapidly increases with increasing pressure. These quantities need to be measured in the near future. 

The detailed first principles study of the three stable polymorphs has been performed recently using the LCAO technique The main drawback of that work is that no cell optimization was performed for anatase or brookite. The energy-volume curves that were used to calculate the bulk modulus, B, and its pressure derivative, B, have been produced by varying the volume with the c/a ratio and fractional atomic coor nates being fixed at experimental values which makes results unreliable. 

Table 1. Calculated properties of ratile, anatase, brookite, and columbite phases. Relative deviation from experimental values is shown in brackets. Structural experimental data are from 19,20,21,9 respectively. Bulk modulus of ratile extrapolated to 0 K is from 2.

Table 2 gives our calculated results for the equilibrium volume Vq, bulk modulus Bq, and enthalpy of formation AH. Theoretical results refer to T=0, uncorrected for zero point motion, whereas experimental values refer to room temperature. Note that the extensive quantities AH and Vq arc reported per atom in the present paper, i.e., divided by the total number of atoms. As well known the LDA underestimates the volume. Comparing the bulk modulus for T3 and D8s we see that the addition of Si to pure Ti has a large (26 %) effect on the bulk modulus, indicating that p electrons of Si have a strong effect on the bonding in this system. 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

Table4.6 Lattice constants a, volume V, cohesive energy and bulk modulus 6 for fee gold from nonrelativistic and relativistic pseudopotential DFT calculations (from Ref [402]).

The behavior of cristobalite PON has been studied as a function of pressure. No in situ evidence for pressure-induced amorphization was noticed. Whereas cristobalite Si02 displays four crystalline phases up to 50 GPa (195), PON remains in a cristobalite phase (193, 196). By using Raman spectroscopy and synchrotron X-ray diffraction, Kingma et al. (193, 197) observe a displacive transformation below 20 GPa to a high-pressure cristobalite-related structure, which then remains stable to at least 70 GPa. The high value of the calculated bulk modulus (71 GPa) (196) is indicative of the remarkable stiffness of the phase. 

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

This is the isothermal bulk modulus. Thus we can use our simulation data in Figure 5.1 and calculate a modulus for a hard sphere system. Equations (5.14) to (5.16) form an interesting hierarchy of equations ... 

Figure 5.3 The calculated hard sphere bulk modulus (line) compared with the data gathered by Vrij et al.7 (shown as the points)...

Table 5.22 Bulk modulus and thermal expansion for various compositional terms of the garnet phase. The experimental range for bulk moduli is relative to a common value of K = 4. The calculated values are from Ottonello et al. (1996). References for thermal expansion are as follows (1) Ottonello et al. (1996) (2) Skinner (1956) (3) Suzuki and Anderson (1983) (4) Isaak et al. (1992) (5) Armbruster and Geiger (1993). ...

While the results and conclusions are consistent with the asperity contact model discussed earlier, the data does not unambiguously demonstrate the connection to asperity deformation. One of the complicating assumptions in Ref. [14] was that the shear modulus used in the comparison was a composite modulus calculated from the bulk material properties of each component in a two-pad stack. If asperity deformation is a dominant factor, a more appropriate value is the shear modulus of the contacting member. 

Knowledge of the sample pressure is essential in all high-pressure experiments. It is vital for determinations of equations of state, for comparisons with other experimental studies and for comparisons with theoretical calculations. Unfortunately, one cannot determine the sample pressure directly from the applied force on the anvils and their cross-sectional area, as losses due to friction and elastic deformation cannot be accurately accounted for. While an absolute pressure scale can be obtained from the volume and compressibility, by integration of the bulk modulus [109], the most commonly-employed methods to determine pressures in crystallographic experiments are to use a luminescent pressure sensor, or the known equation of state of a calibrant placed into the sample chamber with the sample. W.B. Holzapfel has recently reviewed both fluorescence and calibrant data with the aim of realising a practical pressure scale to 300 GPa [138]. 

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