Isothermal bulk modulus

Kejrwords equation of state, free volume, bulk modulus, isothermal compressibility, isobaric expansivity, surface tension, pressure-volume-temperature relationship (P-V-T), pol)mner miscibility, injection molding. 

Liquid Temp T (°C) Density p( kg/nr) Specific gravity S Absolute viscosity m(N s/itt) Kinematic viscosity (m2/s) Surface tension [Pg.489]

Equations of state for solids are often cast in terms of the bulk modulus, Kp, which is the inverse of the isothermal compressibility, Kp, and thus defined as... 

Figure 2.15 Pressure-volume data for diamond, SiC>2-stishovite, MgSiC>3 and 8102-quartz based on third order Birch-Murnaghan equation of state descriptions. The isothermal bulk modulus at 1 bar and 298 K are given in the figure.

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 ... 

Table 5.35 Molar volnme (cm /mole), isobaric thermal expansion (K" ), and isothermal bulk modulus (Mbar) of pyroxene end-members according to Saxena (1989)...

Table 6,11 Isothermal bulk modulus and first derivative for silicate melt components (after Bottinga, 1985).

R. ZwanzigandR. D. Mountain, J. Chem. Phys. 43,4464 (1965) show that the modulus Goo and the isothermal compressibility are determined by similar integrals containing the pair correlation function and the interparticle potential for simple Lennard-Jones fluids. The adiabatic (zero frequency) bulk modulus Ko equals —y(0P/0P) j, which clearly is a kind... 

From Eq. (2.5), some thermodynamic relations readily follow. The isothermal bulk modulus K is given by... 

Here, V denotes specific volume, K denotes bulk modulus, subscripts P,V, S and T denote isobaric, isochoric, isentopic and isothermal conditions, respectively s is the second-rank strain tensor, and C is the fourth-rank elastic tensor. 

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. 

These are the 2D analogs of the bulk isothermal compressibility and bulk modulus, respectively. The scaling concepts introduced above clearly make sense in terms of the compressibility/elasticity as well. For polymers where the interface is a good solvent, the lateral modulus, sometimes called static di-lational elasticity, is small whereas it becomes larger as the interface becomes poorer. 

Straightforward measurements of elastic properties of materials can be made via high-pressure static compression experiments, in which X-ray diffraction (XRD) is used to measure the molar volume (V), or equivalently the density (p), of a material as a function of pressure (P). The pressure dependence of volume is expressed by the incompressibility or isothermal bulk modulus (Kt), where Kp = —V(bP/bV)p. 

This isothermal bulk modulus (Kj) measured by static compression differs slightly from the aforementioned adiabatic bulk modulus (X5) defining seismic velocities in that the former (Kj) describes resistance to compression at constant temperature, such as is the case in a laboratory device in which a sample is slowly compressed in contact with a large thermal reservoir such as the atmosphere. The latter (X5), alternatively describes resistance to compression under adiabatic conditions, such as those pertaining when passage of a seismic wave causes compression (and relaxation) on a time-scale that is short compared to that of thermal conduction. Thus, the adiabatic bulk modulus generally exceeds the isothermal value (usually by a few percent), because it is more difihcult to compress a material whose temperature rises upon compression than one which is allowed to conduct away any such excess heat, as described by a simple multiplicative factor Kg = Kp(l + Tay), where a is the volumetric coefficient of thermal expansion and y is the thermodynamic Griineisen parameter. 

Raman spectra of cubic BN to 21 GPa, in the temperature range 300-723 K, were used to derive values for the isothermal bulk modulus at ambient and high temperature.56 IR data (vBN of c-BN) were used to follow the formation of BN films by mass-selected B and N ion deposition.57 Raman data were used to characterise BN single-walled nanotubes (SWNT), formed by substitution from SWCNT by B203/N2 treatment.58 Assignments for such species were made using the results of DFT and ab initio calculations.59 61... 

The bulk modulus K is defined as the reciprocal of the isothermal compressibility, and Young s modulus E is defined as the ratio of longitudinal tensile stress and longitudinal tensile strain ... 

Diffraction experiments at high pressures provide information concerning the compression-induced changes of lattice parameters and, thus, sample volume. In pure phases of constant chemical composition and in the absence of external fields, the thermodynamic parameters volume V, temperature T and pressure P are related by equations of state, i.e. each value of a state variable can be defined as a function of the other two parameters. Some macroscopic quantities are partial differentials of these equations of state, e.g. the frequently used isothermal bulk modulus Bq of a phase at a defined temperature and zero pressure 5q = — Fq (9P/9F) for T= constant and P = 0, with the reciprocal of Bq V) being the isothermal compressibility k. Equations of state can also be formulated as derivatives of thermodynamic functions like the internal energy U or the Helmholtz free-energy F. However, for practical use the macroscopic properties of solids are often described by means of semi-empirical equations, some of which will be discussed in more detail. 

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