Isothermal compressibility modulus

The modulus indicates that heat is absorbed (+), during die isodrermal expansion, but released (—) during die isothermal compression. In the adiabatic processes no heat is supplied or removed from die working gas, and so... 

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... 

Free Volume Versus Configurational Entropy Descriptions of Glass Formation Isothermal Compressibility, Specific Volume, Shear Modulus, and Jamming Influence of Side Group Size on Glass Formation Temperature Dependence of Structural Relaxation Times Influence of Pressure on Glass Formation... 

VII. ISOTHERMAL COMPRESSIBILITY, SPECIFIC VOLUME, SHEAR MODULUS, AND JAMMING ... 

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... 

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. 

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. 

Much of the work on the compressibility and bulk modulus of liquids reported in the literature was motivated by problems in mass hydraulic flow, such as raising a hydraulic fluid to a pressure in the range 68.9-137.8 MPa (10,000-20,000 Ib/in ) and circulating it through the hydraulic system. In this type of problem most of the emphasis is on the isothermal compressibility of the fluid. 

When considering the potential effect of pressure on a system, it is useful to recognize the magnitude of pressure required to significantly alter molecular and bulk properties. The isothermal compressibility, k (or its reciprocal K, the bulk modulus) (Eq. 2), gives an indication of the sensitivity of a system to pres-... 

The isothermal compressibility, written as or k, is the reciprocal of the bulk modulus. Beware of the... 

The bulk modulus K (= /H, the reciprocal of the bulk compliance) can be measured in compression with a very low height-to-thickness (h/i) ratio and unlubricatcd flat clamp surfaces. In pure compression with a high h/t ratio, and lubricated clamps, the compressive modulus (= /D, the reciprocal of the compressive compliance) will be measured. Any intermediate hjt ratios will measure part bulk and part compressive moduli. Hence it is vital for comparing samples to use the same dimensions in thermal scans and the same h/t ratio when accurately isotherming and controlling static and dynamic strains and frequencies. 

Both compressibilities are intensive measurable state functions, though K7 is proportional to a class I derivative, while Kg is proportional to one of class 111. Because volume decreases with increasing pressure, these definitions contain negative signs to make the compressibilities positive. Besides PvT experiments. Kg can also be obtained from measurements of the speed of sound. The reciprocal isothermal compressibility is called the hulk modulus. 

Throughout this chapter the values for a, w and Cp given by Gschneidner (1964) were used to convert isothermal compressibilities to adiabatic values. In all cases, however, the reciprocal of adiabatic compressibility ( s), i-e. the bulk modulus (K), is quoted in this chapter. From the experimental point of view, the most common method used to determine the isothermal compressibility ( t) involves the measurement of volume change (AV/Vo) associated with applied pressure (P) to obtain a relative volume change versus pressure relation. Several... 

K by the internal friction method as shown in fig. 8.47. The shear modulus decreased almost linearly from 293 to about 673 K where a sudden change in slope occurred the investigators attributed this to grain boundary relaxation effects. Bridgman (1954), Evdokimova and Genshaft (1965) and Frolov et al. (1969) have determined isothermal compressibilities. For the sake of comparison, these isothermal compressibilities have been converted to adiabatic bulk moduli (See table 8.11). 

Bulk modulus K is the ratio of change in pressure/volume compression (fractional change in volume), where K = V(APIAV). For solids, apphed stress is used instead of pressure, and A is a constant for a given material. When there is an adiabatic exchange of heat, we have bulk modulus at constant entropy [13]. When the bulk modulus is determined at constant temperature, we have the isothermal bulk modulus [13]. 

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. 

Because the relation between the bulk modulus and the appUed pressure involves a relative change in volume (AV/V) of the material, it is possible to introduce the isothermal compressibility of the material defined as ... 

Compressibility modulus of Langmuir monolayers can be determined from the slope of the 7t-A isotherms (Vollhardt D et al. (2006) Adv Colloid Interf Sci 127 83). The compressibility modulus Eo=r(dII/dr)x, where F=1/AN, and A, N, and n are molecular area, Avogadro s number, and surface pressme, respectively. 

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