Volumetric thermal expansion coefficient

Octahedral distortion prevents calculation of the volumetric thermal expansion coefficient of the phase stemming from the linear polyhedral thermal expansion values. However, volumetric thermal expansions may be obtained by least-squares regression of the observed PVT relations. Table 5.10 lists, for instance, the coefficients of a thermal expansion relation of the type... 

Often for solids, linear thermal expansion coefficients, a, = (1 /L) x (.9L/dT)P, are tabulated. For an isotropic substance (the same in all directions), we can relate aL to the volumetric thermal expansion coefficient, defined in Eq. (8), by considering a cube with V = L ... 

Volumetric thermal expansion coefficient of a "rubbery" polymer (T>Tg). 

Free-volume theory Molecular motion involves the availability of vacancies. The vacancy volmne is the free volume, Vp, of the liquid, approximately the difference in volume of the liquid, Vl, and crystalline, 14, forms. Vp is a function of temperature. D is a constant close to unity. The Williams-Landel-Ferry (WLF) equation uses a similar approach in which is the fraction of free volume at Tg, about 0.025, and Pl and Pc are the volumetric thermal expansion coefficients of the liquid and solid, respectively. 

The almost perfect orientation of p-aramid fibers is reflected in the anisotropic behavior of its thermal expansion coefficient. The linear expansion coefficient for these materials is negative (Table 13.2). Because the volumetric thermal expansion coefficient is not affected by orientation, the radial coefficient must increase as fiber orientation increases. The negative expansion coefficient of these materials has opened a whole field of applications in electronics (see section 13.8.4.2). 

Materials typically expand when heated. The expansion is characterized by the linear or volumetric thermal expansion coefficient. The three primary types of materials expand differently. Polymers expand more than metals, and metals expand more than ceramics. For many materials thermal expansion is related to the melting temperature of the material, also a relation between the thermal expansion coefficient of polymers and their elastic modulus is given. 

For most materials, when heated, the average amplitude of the atom s vibration within it increases, which increases the distance between the atoms thus causing the material to expand. Early developed thermomechanical microvalves, including bimetallic microvalves, were primarily based on this principle. In contrast to bimetallic microvalves, only one single-layered beam or membrane is usually involved in a thermal expansion microvalve. Generally, two coefficients are used to depict the thermal expansion in a piece of material the linear thermal expansion coefficient, a, and the volumetric thermal expansion coefficient, (i. These two coefficients relate the change in temperature to the change in the linear and volumetric dimensions of the material, respectively, and are defined as... 

TABLE 47 Ideal (X-ray) density (p), Gruneisen parameter (yy), isohoric heat capacity (Cy, Z = 4), entropy (S), bulk Kj) and shear (G ) moduli, and volumetric thermal expansion coefficient (ay) calculated for RGaOj (R = La-Gd) at 300 K using semi-classical approach (Senyshyn et al., 2005b]... 

FIGURE 83 Components of the thermal expansion tensor (A) and volumetric thermal expansion coefficient (B) versus rare earth cation radius in rare earth gallates. The dashed lines are guides for the eye. 

The volumetric thermal expansion coefficient can also be calculated from the free energy. The thermal expansion coefficient is defined as... 

When the sample is measured using a differential-type TMA, it is not necessary to make corrections [4]. The volumetric thermal expansion coefficient, a, cannot be measured by TMA. Generally, the expansion coefficient of solids is small rz = 3/j is currently used. However, for samples whose Poisson s constant is not 1.0, a... 

Since the first term of the right hand side of equation (11) can be considered a constant, the temperature coefficient of the refractive index An / AT is a measure of the difference between the electronic polarization coefficient and the volumetric thermal expansion coefficient p, where and p are defined as ... 

Density is one of the most basic thermophysical properties. As shown in Eq. (4.3), the temperature derivative of density corresponds to the volumetric thermal expansion coefficient P, which corresponds to the driving force for thermogravitational convection. 

As for the thermal expansion coefficient, the isotropy assumption was accomplished simply by using one-third of the volumetric thermal expansion coefficient, as suggested by Lipatov et al. (1991). It should be noted, however, that in special cases of conductive adhesive applications, directional conduction may be required, in which case the use of non-isotropic, directional material properties would be necessary. 

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