Thermal Expansions

Thermal expansion depends on variations in the interatomic forces with temperature. These forces are strong for covalent bonds and weak for van der Waals forces. For example, for quartz, all atoms are three-dimensionally fixed 

The thermal expansion can be derived from the equation of state it is equal to the change in volume which the body undergoes without external pressure. Putting P = 0 in (5.83), we obtain for the thermal expansion e = (V - Vq)/Vq 

Now assume that = Y5 independent of the mode s in this case, y = y sind we obtain 

The relation (5.90) is known as the GrUneisen relation it predicts that the volume expansion coefficient has the same temperature dependence as the specific heat. From (5.89,90) we see that the GrLineisen parameter characterizes the thermal expansion just as the Debye temperature characterizes the specific heat. The GrLineisen relation is approximately satisfied for many compounds with values of y between 1 and 2. 

An example of a model which gives a mode-independent GrLineisen parameter is the Debye approximation where 03 = (9/9p)wp. Using (5.72) and ho3p = kgOp (oj Debye frequency, Debye temperature), we obtain 

We note that for LiF, the mean GrLineisen parameter y(T) is nearly temperature independent indicating that for this compound, the simple GrLineisen relation (5.90) is well satisfied. This is shown in Fig.5.6 which illustrates 

Differential thermal expansion between various components of calandrias, especially shell-and-tube types, has an important effect on the mechanical design of the equipment. Some types of tubular exchangers incorporate into the basic design and fabrication means to provide for thermal expansion. Other types, specifically fixed tubesheet units, must often be provided with expansion joints in the shell to meet specified differential thermal expansion between the tubes and shell. Another critical area affected by thermal expansion in fixed tubesheet units is the shell-to-tubesheet juncture. Gasketed joints may also be affected by thermal expansion therefore both the flange design and the gasket selection in 

a is the volume thermal expansion coefficient, Vq is the molar volume, and P is the compressibility. The formulae for a and P are given by Equations 16.4 and 16.5. 

For condensed systems such as solids and liquids, the difference between Cp and Cv is negligibly small at ordinary temperatures. 

Thermal expansion is measured in two ways volume and linear thermal expansion coefficients. The formula for the volume thermal expansion coefficient has already been given in Equation 16.4. The formula for the linear thermal expansion coefficient is given by Equation 16.6. 

Particularly the thermal expansion of / -eucryptite has been investigated by many authors. It appears that the expansion characteristics depend somewhat on the Al/Si order and possibly on other influences of thermal history, but the data given in Table 2.1 can be considered as typical. It should also be pointed out that coefficients of thermal expansion measured on polycrystalline aggregates by dilatometry can give different values due to internal stress and microcracking if the expansion of the crystals is highly anisotropic [2.27]. For / -eucryptite, several models have been proposed to explain the thermal expansion characteristics. 

The ideas of Cillery and Bush were taken up again in 1974 by Moya et al. [2.28]. They showed that the negative thermal expansion of /3-eucr q)tite would require its isothermal linear compressibility (X) parallel to the c axis to be negative if the thermal expansion was dominated by elastic effects. The same group Hortal et al. [2.29]) then measured compressibilities on /3-eucryptite single crystals and found X = —(1.13 1) x 10 cm /dyne and Xy = +(22.4 6) x 10 cm /dyne. 

So there is indeed a large anisotropy in the elastic properties, and the compressibility parallel to the c axis is probably negative, which led Hortal et al. to the conclusion that elastic effects govern the expansion behavior of /3-eucryptite. 

Upon extended heat treatment, lattice constants and thermal expansion of / -eucryptite change slightly from those at room temperature. If these changes are assumed to be caused by increased disorder and a concomitant partial occupation of the octahedral sites even at room temperature, then the model by Schulz predicts the observed changes correctly. It is also in agreement with the fact that the Mg alumino-silicates have positive thermal expansion (because the Mg ions already occupy the octahedral sites at room temperature). 

More recently, the crystal structure of / -eucryptite has been studied by single-crystal neutron diffraction, which gives the most reliable information on Li location and site occupancies. Guth made his analysis at room temperature and at 530°C [2.6] Steinmann worked at 767°C [2.7]. Their results confirm that the differences in occupation of the Li sites with tetrahedral coordination become smaller with increasing temperature. Significant site occupation for the octahedral position could not be detected even at 767°C. This casts some doubt on the validity of Schulz s model, but does not strictly disprove it because the occupancy of 10% required by the model at 767 °C may be too small to be detected even by neutron diffraction. 

The coefficient of thermal expansion is defined as the fractional change in length or volume of a material for a unit change in temperature. The coefficient of thermal expansion of plastics is 

Reproduced with permission from Handbook of Polyolefins Synthesis and Properties, 1st Edition, Eds., C. Vasile and R.B. Seymour, Marcel Dekker, New York, NY, USA, 1993. Copyright Marcel Dekker, 1993. 

As shown in Fig. 4.4,[ 1 thermal expansion increases with increasing temperature but this increase is not linear and is slightly more rapid at high temperature. 

It is well known that solids expand upon heating. The extent of the expansion is characterized by a coefficient of linear expansion a, defined as the fractional change in length with change in temperature at constant pressure, or 

The origin of thermal expansion can be traced to the anharmonicity or asymmetry of the energy distance curve described in Chap. 2 and reproduced in Fig. 4.3. The asymmetry of the curve expresses the fact that it is easier to pull two atoms apart than to push them together. At 0 K, the total energy of the atoms is potential, and the atoms are sitting at the bottom of the well 

In general, the asymmetry of the energy well increases with decreasing bond strength, and consequently the thermal expansion of a solid scales inversely with its bond strength or melting point. For example, the thermal expansion coefficient of solid Ar is on the order of 10 whereas for 

Perusal of Table 4.3, in which the mean thermal expansion coefficients of a number of ceramics are listed, makes it clear that a for most ceramics lies between 3 and 10 x 10 °C . The functional dependence of the fractional increase in length on temperature for a number of ceramics and metals is shown in Fig. 4.4. Given that the slope of these lines is a, one can make the following generalizations  

Ceramics in general have lower a values than metals. 

We recall that the coefficient of thermal expansion is expressible in terms of cross fluctuations between the (macroscopic) volume and the enthalpy in the P, P, N ensemble (Section 1.4). Here again, we are concerned with ffuctuations in molecular quantities only. 

Using the expression (5.46) for the volume, assuming that has been 

Note that here we encounter only fluctuations in the volume of the VP of singles and pairs of particles. The compressibility relation derived in Section 3.9 is admittedly simpler than (5.69), yet, as in (5.69), it also employed MDF s of order two. 

In concluding this section, we recall that at present, the most reliable source of information on the ordinary MDF s are the direct computational procedures using either the Monte Carlo or the molecular dynamic method. It is expected that these methods will also provide the appropriate information on the GMDF s. Computation of the latter should not pose any additional difficulties to those already encountered in the computation of ordinary MDF s. Once we get such information on the singlet and pair GMDF s, all of the quantities discussed in this section can be computed easily using one- and two-dimensional integrals. 

To minimise stresses during cell fabrication and cell operation, thermal expansion of the cathode should be matched with other SOFC component materials, especially electrolyte and interconnect. The thermal expansion coefficient (TEC) of undoped LaMnOa is 11.2 0.3 x 10 in the temperature range 35-1000 C [28]. Table 5.1 summarises the TECs of undoped and doped LaMnOa [28,29]. 

In the slightly A-site-deficient LaMnOa (Lao.ggMnOa), the TEC values are lower than in the stoichiometric composition. This is due to a crystal structure change caused by the A-site deficiency. With Sr doping in Lao.ggMnOa, the TEC values increase with increase in the concentration of Sr. 

Recently, Mori et al. [29] have observed a thermal expansion behaviour which exhibits some anomalous dependence on dopant concentration that is, there is a minimum in TEC around a dopant concentration of 0.1-0.2. The reported minimum TEC values are about 10 x 10 and 11 x 10 for Lao.8Cao.2MnOa and Lao.gSro.iMnOa, respectively. They also observed a 

LaCoOa-based cathodes show TEC values of about 20 x 10 which are too high compared to that of YSZ. Attempts have been made to reduce TEC by doping with Sr and other alkaline earth ions. 

Although a detailed description of cathode reactions and polarisations is given in Chapter 9, a brief discussion of the cathode reaction mechanisms is included here to elucidate several aspects of materials issues in cathode development. As discussed in the previous section, the cathode reduces oxygen molecules to 

A number of different but related notations are used to describe the thermal expansion of matter  

Experimental data available suggest that the thermal expansivity of a glass is of the same order as that of the crystal and that no great error is made by putting  

It is common practice to report the expansivities immediately below (eg) and above (t ) the transition. 

Empirically it has been found that often q (= -ap = -ep2) is practically independent of temperature over a wide range (see Bondi, 1968c). 

The expansivity of the rubbery or liquid polymer is always larger than that of the glassy or crystalline polymer. 

A Kelvin foam model with planar cell faces was used (a. 17) to predict the thermal expansion coefficient of LDPE foams as a function of density. The expansion of the heated gas is resisted by biaxial elastic stresses in the cell faces. However SEM shows that the cell faces are slightly wrinkled or buckled as a result of processing. This decreases the bulk modulus of the 

DMTA graph of storage Young s modulus and damping (tan 5) versus temperature, for EVA foam of density 

Compute the cross-sectional area of each layer A  

Since the layers are loaded in series, the same force Fj acts on each layer. Hence the stress on layer / is  

Polyolefins have a spectmm of mechanical properties, ranging from almost-rigid EPP mouldings for helmets and factory containers to mbbery high density EPDM foams for mouse mats and grips on hand tools. Rather than list all these areas, five specific areas are described in detail. 

Most materials increase in volume as the temperature is increased, a feature called thermal expansion. There are, though, an increasing number of solids known that contract as the temperature increases. 

Thermal expansion is a most important property in practice. It is used in most everyday thermometers. The shattering of ordinary glass on being cooled rapidly is due to the thermal contraction of the outer layers, and is prevented in special glasses such as Pyrex glass or fused silica, which have low thermal expansion. The thermal expansion of components in electronic devices is important, and the difference in thermal expansion of materials in a construction can lead to grave difficulties. The coincidence of the thermal expansion of steel and concrete at normal temperatures allows the use of steel-reinforced concrete in buildings. 

The mean coefficient of linear thermal expansion, mean of material is the increase in length per 

In the case of a solid with different mean thermal expansion coefficients along x, y and and z axes ax, ay and respectively)  

The linear expansivity of a solid, a, is defined as the increase in length per unit length at a given temperature  

In nanoparticle-polymer composites, thermal stability is one of the most important property enhancements. Recently, some theoretical efforts have been made to predict the thermal stability of such composites. For example, FEM and the theory of Chow have been used to predict the thermal expansion of clay-polymer nanocomposites. The results indicate that it is possible to considerably reduce and eventually match the thermal expansion of metal and polymer parts by dispersing a small amount of exfoliated muscovite mica platelets into a polymer matrix. Moreover, reduction is controlled by the product of aspect ratio and volume fraction of the platelets. 

Lee et al. [33] predicted the coefficient of thermal expansion of composites with aligned isotropic fiber- and disk-like fillers and its dependence on the aspect ratio of fillers. They found that the longitudinal coefficients of thermal expansion decrease and approach that of fillers. However, the transverse coeflBcients may increase or decrease with the aspect ratios. They also developed a new model, based on 

There is a very large anisotropy in the linear thermal expansivities of these oriented polymers. Early studies showed that in polyethylene the expansivity parallel to the orientation direction a, was negative and apparently very close to the value (—12 X 10 K ) for the c-axis expansion of the crystalline regions obtained from X-ray measurements. This result was attributed to the high degree of crystal continuity, and did not appear to be controversial. More recent work however, has 

It can be shown that this model gives the general result 

The cubic symmetry of CeSe (NaCl type), see p. 10, is preserved in the magnetically ordered state (Tn = 5K), Ott et al. [6], Hulliger et al. [6]. The weak anomaly in the linear thermal expansion coefficient along [100] around the magnetic phase transition, observed in dilato-metric measurements, is interpreted as a phase transition that is purely magnetic in origin [5]. 

Dark yellow single crystals have the lattice parameter a = 5.9923(6) A, a = 5.9924(4) A [5] at room temperature. Values for powdered samples range between a = 5.982 A, Guittard, Benacerraf [7], and a = 6.00 A, see, for example, Smolenskii et al. [8] a = 6.06 A for CeSe prepared from 002803 plus sodium. Banks et al. [4]. The temperature dependence of the lattice constant down to 4 K is shown in Fig. 32 a is almost temperature independent below 50 K [6]. The ionicity of CeSe is almost 40%, Gudat et al. [9]. 

The bulk modulus K = 0.472 x 10 N/m (= 472 kbar) and the pressure derivative dK/dp = 3.03 were calculated from interatomic potentials and cohesive energies, Jain, Shanker [11]. 

The linear thermal expansion coefficient of CeSe single crystals along [100] as studied by dilatometric measurements in the range 1.6 to 13 K has a weak but distinct maximum (a 10 K ) at the Neel temperature Tn = 5.6 K, Hulliger et al. [5]. 

Tray temperature often varies from one set of operating conditions to another (e.g., it is higher when a distillation column is pressured than when it is depressured). Often, the tray temperature is not the same as the shell temperature. Tray design should allow for thermal expansion of tray sections. Failure to do so may result in tray buckling or beam warping. 

To avoid thermal-expansion problems, one must (1) provide adequate fastener spacing or slotted holes (2) establish satisfactory clearances between the trays and the shell, IV2 in per 10 ft has been recommended (86)] and (3) refrain from welding chordal members such as downcomer panels, beams, or tray sections at both ends to the column shell. 

The a(t) depends nonhnearly on temperature in the low-temperature range, and the a(t) decreases with the feature sizes of nanostructures [8,9]. EXAFS investigations [10] revealed that in small gold particles, the temperature dependence of the first neighbor distance is different from that of the macrocrystalline counterpart. In the largest size samples, a reduction in the thermal expansion happens, whereas in the smallest ones, a crossover from an initial thermal expansion to a thermal contraction presents. 

According to Cardona [11], the lattice thermal expansion of a cubic crystal could be expressed in terms of the Griineisen mode parameter, y, and lattice vibration in the frequency of 

From the phonon nonlinerities, Griineisen [12] derived the expression for the volume TEC that is proportional to the product of the specific heat and Griineisen parameter, 

From the perspective of LBA, the a(f) for the representative bond can be derived from the differential of the thermal expansion relation [13], 

The present form is much simpler and straightforward without the parameters of B, V, or yq being involved. The current approach covers the general trend for T- 

To make the model quantitative, imagine that at a particular time a given bond has the length R given by eqn (9.10)  

Here R) Re = 2SR is the thermal expansion of the bond and, by expanding the left-hand side of eqn (9.13) as a power series in AR and ignoring higher-order terms, one gets eqn (9.14)  

As the temperature increases, the mean square amplitude of vibration, SR ), increases, and with it the average bond length. Recognizing that SR ) in eqn (9.14) is the same as (A ) in eqn (9.7), the two equations can be combined to give the expression for the thermal expansion of a bond shown in eqn (9.15)  

For small displacements, the bond shrinkage is A )/2R From eqn (9.9) this leads to a thermal contraction of 

High-temperature lattice parameters of N-saturated a-Th are nearly the same as those of pure Th metal up to nearly 1000°C [7]. The reported data for pure a-Th as a function of temperature can be represented to within 1% by the following equations  

Flgure3.11. The energy troughof graphite in ( a abdirectionsand ( btlcdirection.l Q 

As a result, the thermal expansion of the graphite crystal has a marked anisotropy. It is low in the ab directions (lower than most materials) but higher by an order of magnitude in the c direction, as shown in Fig. 3.12.P][i3][iei 

The increase with temperature is not linear. In the c direction, it increases slowly and gradually. At 0°C, the coefficient of thermal expansion averages 25 x 10 / C and at 400°C, It reaches 28 x 

In the ab directions, the themieil expansion is actually negative up to approximately 400°C with a minimum at 0°C. It is possible that this observed negative expansion is due to internal stress (Poisson effect) associated with the large expansion in the c direction and it has been suggested that, if it were possible to measure the ab thermal expansion of a single atomic plane, this expansion would be positive.i i 

The large thermal expansion anisotropy often results in large Internal stresses and structural problems such as delamination between planes as will be seen in Ch. 5. Sec. 3. 

Most solids expand when heated. Their length increases according to 

Along axis a Along axis c Along axis a, Nowotny phase Along axis c, Nowotny phase Along axis a, [1117] 

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The oil density at surface is readily measured by placing a sample in a cylindrical flask and using a graduated hydrometer. The API gravity of a crude sample will be affected by temperature because the thermal expansion of hydrocarbon liquids is significant, especially for more volatile oils. It is therefore important to record the temperature at... 

Glass has a very low thermal expansion coefficient the materials joined with glass have to be similar in expansion or must be duetile, while staying vacuum tight. Even with best-matched materials skilled craftsmanship is asked for the joining process. 

Estimate, by means of Eq. III-41, the surface tensions of CCI4. CHCI3 and of water at 20°C. Look up the necessaiy data on thermal expansion and compressibility. 

Fig. X-14. SEM picture of a drop ot cooled glass on Femico metal (which has the same coefficient of thermal expansion). xl30. (From Ref. 183.)...

Easier V, Schweiss P, Meingast C, Obst B, Wuhl H, Rykov A I and Tajima S 1998 3D-XY critical fluctuations of the thermal expansivity in detwinned YBa2Cu30y g single crystals near optimal doping Phys. Rev. Lett. 81 1094-7... 

The refractory industry has found chromite useful for forming bricks and shapes, as it has a high melting point, moderate thermal expansion, and stability of crystalline structure. 

However, this approach is of limited predictive usefulness due to the difficulty in predicting Tg accurately. Methods have been proposed for computing the molar volume at 298 K and thus extrapolation to other temperatures, which results in some improvement. These use connectivity indices. Note that it is necessary to employ different thermal expansion equations above and below Tg. 

The corrections to be made on the reading are as follows (1) Temperature, to correct for the difference in thermal expansion of the mercury and the brass (or glass) to which the scale is attached. 

The electronic configuration for an element s ground state (Table 4.1) is a shorthand representation giving the number of electrons (superscript) found in each of the allowed sublevels (s, p, d, f) above a noble gas core (indicated by brackets). In addition, values for the thermal conductivity, the electrical resistance, and the coefficient of linear thermal expansion are included. 

Tensile yield strength, 103 lb in-2 Thermal Burning rate, mm min Coefficient of linear thermal expansion, 10 °C 50-90 0.5-2.2 50-90 50-80 50-60 10-13 Self- extinguishing 40-55 46... 

Coefficient of linear thermal expansion, 10 °C Deflection temperature under 110-170 110-170 100-200 80-120 6.6 11-50 20-60... 

This table lists values of /3, the cubical coefficient of thermal expansion, taken from Essentials of Quantitative Analysis, by Benedetti-Pichler, and from various other sources. The value of /3 represents the relative increases in volume for a change in temperature of 1°C at temperatures in the vicinity of 25°C, and is equal to 3 a, where a is the linear coefficient of thermal expansion. Data are given for the types of glass from which volumetic apparatus is most commonly made, and also for some other materials which have been or may be used in the fabrication of apparatus employed in analytical work. 

Below Tg the material is hard and rigid with a coefficient of thermal expansion equal to roughly half that of the liquid. With respect to mechanical properties, the glass is closer in behavior to a crystalline solid than to a... 

Figure 4.14 Behavior of thermodynamic variables at Tg for a second-order phase transition (a) volume and fb) coefficient of thermal expansion a and isothermal compressibility p.

By an assortment of thermodynamic manipulations, the quantities dn/dp and [N (d G/dp )o] can be eliminated from Eq. (10.48) and replaced by the measurable quantities a, /3, and dn/dT the coefficients of thermal expansion, isothermal compressibility, and the temperature coefficient of refractive index, respectively. With these substitutions, Eq. (10.48) becomes... 

Material Properties. The properties of materials are ultimately deterrnined by the physics of their microstmcture. For engineering appHcations, however, materials are characterized by various macroscopic physical and mechanical properties. Among the former, the thermal properties of materials, including melting temperature, thermal conductivity, specific heat, and coefficient of thermal expansion, are particularly important in welding. 

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