Thermal expansion coefficients isotropic

For materials that are isotropic, that is, have the same properties in all directions, it can be shown that ay = 3ai. A material that has different properties in different directions is said to be anisotropic. Thus, a linear expansion coefficient, if no direction of measurement is explicitly stated, implies an isotropic material. Conversely, a volume thermal expansion coefficient implies an anisotropic material, and one should exercise caution when deriving linear thermal expansion coefficients from volume-based measurements. 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

THERMAL EXPANSION COEFFICIENT. The change in volume per unit volume per degree change in temperature (cubical coefficient). For isotropic solids the expansion is equal in all directions, and the cubical coefficient is about three times the linear coefficient of expansion. These coefficients vary with temperature, but for gases at constant pressure the coefficient of volume expansion is nearly constant and equals 0.00367 for each degree Celsius at any temperature. 

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 ... 

Fig. 4. Linear thermal expansion coefficients as a junction of temperature ( ) isotropically cured at C (o) and ( a ) oriented polymer filament (200 pm diameter) measured in the axial and lateral direction, resp.

In this relation, a and a, are the thermal expansion coefficients of the substrate and film. These depend on the temperature T. If the film is homogeneous and elastically isotropic, the in plane thermoelastic stress is expressed by ... 

Most micromechanical theories treat composites where the thermoelastic properties of the matrix and of each filler particle are assumed to be homogeneous and isotropic within each phase domain. Under this simplifying assumption, the elastic properties of the matrix phase and of the filler particles are each described by two independent quantities, usually the Young s modulus E and Poisson s ratio v. The thermal expansion behavior of each constituent of the composite is described by its linear thermal expansion coefficient (3. It is far more complicated to treat composites where the properties of some of the individual components (such as high-modulus aromatic polyamide fibers) are themselves inhomogeneous and/or anisotropic within the individual phase domains, at a level of theory that accounts for the internal inhomogeneities and/or anisotropies of these phase domains. Consequently, there are very few analytical models that can treat such very complicated but not uncommon systems truly adequately. 

Although not explicitly stated, the discussion so far is only strictly true for isotropic, e.g., cubic, polycrystalline materials. Crystals that are noncubic and consequently are anisotropic in their thermal expansion coefficients behave quite differently. In some cases, a crystal can actually shrink in one direction as it expands in another. When a polycrystal is made up of such crystals, the average thermal expansion can be very small, indeed. Cordierite and lithium-aluminosilicate (LAS) (see Fig. 4.4) are good examples of this class of materials. As discussed in greater detail in Chap. 13, this anisotropy in thermal expansion, which has been exploited to fabricate very low-a materials, can result in the buildup of large thermal residual stresses that can be quite detrimental to the strength and integrity of ceramic parts. 

The thermal expansion coefficients of cubic materials are isotropic and hence do not exhibit this phenomenon. 

Although the thermal expansion coefficient is actually defined in terms of the volume of the substance, this value is somewhat difficult to measure. As a result, the expansion coefficient for glasses is usually only determined in one direction, i.e., the measured value is the linear thermal expansion coefficient, Kl. The true and average linear thermal expansion coefficients are given by Eqs. 7.3 and 7.4, respectively, where V is replaced by L in each equation. Since glasses are usually isotropic materials with relatively small thermal expansion coefficients, Ov = 3ml can be used to approximate with very little error in calculation. 

If a component is at a different temperature than its surroimding attachments then stresses will develop. For example, a rod attached to a rigid constraint will be placed under a thermal stress if it is at a different temperature than the constraint. If the constraint is at temperature and the rod at temperature T, a strain a(TQ—r,) will develop in the rod, where a is the thermal expansion coefficient of the rod. If the rod is linearly elastic, the thermal stress ctj. will be given by aj=Ea Tf —T, where Eis Young s modulus. Clearly, the situation is more complex if the rod can creep, as these stresses may relax over time. For this chapter, it will be assumed that the ceramics are linearly elastic and isotropic, in order to set out the basic principles. 

Glass is isotropic, but most other fibres have orientated structures and they possess properties (electrical conductivity, thermal expansion coefficient and degree of solvent swelling) which differ according to the direction of measurement. 

A different approach to the prediction of thermal expansion coefficients was taken by Schapery (1968), who calculated upper and lower bounds for both isotropic and anisotropic composites. The method is applicable to systems containing an arbitrary number of constituents and an arbitrary phase geometry. In some cases, the bounds coincide, and exact solutions may be found in other cases, approximations only may be derived. In a simple two-component system, Schapery obtained the following expression for the volumetric expansion coefficient of the composite ... 

CeCu2ln show isotropic behavior in the [100] and [110] directions as expected for a cubic material (Oomi et al., 1990). The thermal expansion coefficient of CeCu2ln below 100 K, however, is larger compared with that of LaCu2ln, which has no 4/ electrons. 

As will be seen below, we obtained similar and sometimes even better results with carbon layers. It is well known that so-called isotropic carbon is especially compatible with blood /7/. It is deposited as pyrolytic carbon in a fluidized bed of a hydrocarbon-noble gas mixture at relatively low pyrolysis temperatures between 1200 and 1500 C (LT = low temperature). Since this process can on ly be used with a heat-stable substrate material with low thermal expansion coefficients, we (and others /8/) used vacuum-coating processes for the deposition of similar coatings onto polymers. In our work we used two processes with one, the results of which will be reported in sect. 5, the carbon is separated out of a hydrocarbon-noble gas mixture in the set-up shown in Fig.2. However, not heat, as in pyrolysis, but rather a glow discharge is used for the decomposition. 

In the same way as for thermal conductivity, the coefficient of linear thermal expansion of a composite material is a complex function of the thermal expansion coefficients of the matrix (a ) and that of the reinforcement (aj). In the particular case of orthotropic composite materials, the thermal expansion coefficient of each component (i.e., matrix, reinforcement) is a tensor quantity [a ] with only three components a,j, and along the major axis, that is, one in the axial direction (a ) and two in the transversal directions (oc and a j). In the particular case of transversely isotropic materials such as for instance fiber reinforced composites, the axial coefficient of thermal expansion of the material, expressed in W.m K can be approximated by the rule-of-mixtures by means of the Young s moduli of the matrix and of the fiber ... 

Vf= fiber volume fraction v = Poisson s ratio of the matrix Ep E = tensile moduli of fiber and matrix, respectively The equivalent thermal expansion coefficients of an isotropic matrix is modeled as,... 

For determining the coefficient of linear expansion, it is a common practice to generate AL/Lq curves with and without the sample. The two cimves are subsequently subtracted to compensate for the expansion of the sample holder. For convenience, it is hest to use a standard reference material, eg, an isotropic metal such as alumimun or platinum (52), as the sample holder. In Figure 24, we see that, compared to other isotropic materials, quartz (fused silica) has a very low thermal expansion coefficient (0.6 x 10 /°C). It is also easily formed. If the upper... 

The thermal expansion behaviour of ultra high modulus polyethylene is very anisotropic. Transverse to the draw direction the thermal expansion coefficient is positive and comparable to that for isotropic polymer. In the draw direction the coefficient is negative and very small ( v 10" / ). For low molecular weight polymers the value... 

Values of linear thermal expansion coefficients of commonly used non-metallic thin film, interlayer or substrate materials are given in Table 2.1 over a broad range of temperatures of practical interest. Table 2.2 provides corresponding values of linear thermal expansion coefficients for polycrys-talline metals. Table 2.3 lists the room temperature values of elastic modulus and Poisson s ratio for a wide variety of polycrystalline and amorphous materials with isotropic elastic properties, which are commonly used as thin films, interlayers or substrates. The anisotropic elastic properties of cubic and hexagonal single crystals are given in Tables 3.1 and 3.2, respectively, in the next chapter. 

A well-known example of this is that cubic crystals are optically isotropic, which means that the dielectric permittivity has spherical symmetry in a cubic crystal. Another example is that the thermal expansion coefficient of a cubic crystal is independent of direction. In fact, if it were not, the crystal would lose its cubic symmetry if it were heated. Thus, as far as thermal expansion is concerned, a cubic crystal looks isotropic just as it does optically. Since, according to Neumann s principle, the physical properties of a crystal may be of higher symmetry than the crystal, we will generally find that they range from the symmetry of the crystal to the symmetry of an isotropic body. A more general example of higher symmetry in properties is that such physical properties characterized by polar second rank tensors must be centrosymmetric, whether the crystal has a center of symmetry or not, cf. Fig. 27. For, if a second rank tensor T connects the two vectors p and q according to... 

It is often said that group 432 is too symmetric to allow piezoelectricity, in spite of the fact that it lacks a center of inversion. It is instructive to see how this comes about. In 1934 Neumann s principle was complemented by a powerful theorem proven by Hermann (1898-1961), an outstanding theoretical physicist with a passionate interest for symmetry, whose name is today mostly connected with the Hermann-Mau-guin crystallographic notation, internationally adopted since 1930. In the special issue on liquid crystals by ZeitschriftfUr Kristal-lographie in 1931 he also derived the 18 symmetrically different possible states for liquid crystals, which could exist between three-dimensional crystals and isotropic liquids [100]. His theorem from 1934 states [101] that if there is a rotation axis C (of order n), then every tensor of rank rcubic crystals, this means that second rank tensors like the thermal expansion coefficient a, the electrical conductivity Gjj, or the dielectric constant e,y, will be isotropic perpendicular to all four space diagonals that have threefold symme-... 

---