Thermal expansion volume coefficient

Thermomechanical analysis (TMA). In this technique, information on changes in the size of a sample is obtained, e.g. thermal expansion and coefficient of thermal expansion, cure shrinkage, glass transition, thermal relaxations, any phase transformation involving volume change in the material. We describe the measurement of the coefficient of thermal expansion in detail later in this section. 

Thus the ratio of the coefficient of thermal expansion to coefficient of compressibility is equal to the derivative of the entropy with respect to volume. This ratio is easily measured. It is found cf. table 12.1) that as T->0, a—>0, while k remains finite, so that at low temperatures the entropy becomes independent of volume, in conformity with the Nernst theorem. 

Figure 15 shows measurements of linear thermal expansion and coefficients of thermal expansion at temperatures between 4.2 K and room temperature, which were preformed with unidirectionally reinforced C/C composites parallel to the fibre axis. The composites are fabricated with two different carbon matrix precursors, CT pitch and CT pitch modified by elemental sulfur, as well as two different fibre types, Sigrafil HF (type II) and Sigrafil HM (type I). The C/C composites consist of 50 V/o fibre volume fraction and are densified four times with a final heat treatment of 1000°C. Details of the measuring apparatus are described by Hartwig et al, ... 

Where, is Tg of the wet polymer and a, V, Tg denote coefficient of thermal expansion, volume fraction, and glass transition temperature, respectively. Subscripts p and m refer, in turn, to the polymer and fluid. 

In Figure 9.32 the dependence of the thermal expansion linear coefficient a p on the relative fraction of nanoclnsters, which are considered as nanofiller, for epoxy polymers is addnced. As has been expected [35], an increase in results in a reduction in ttpp, comparable with that observed for polymer composites with the introduction of particnlate fillers. So, an increase in (p from 0 to 0.60 reduces a p by about 1.50 times (Figure 9.32) and with the introduction of calcium carbonate or aluminium powder with volume contents (p = 0.60 in the epoxy polymer the thermal expansion linear coefficient value decreases by 1.70-2.0 times [35]. The dependence a p(expressed analytically by the following empirical equation [62] ... 

Some of polymer properties change at the glass transition in discontinuous way (the coefficient of isobaric thermal expansion a, coefficient of isothermal compressibility kj, specific heat, etc), whereas the other ones change continuously (volume V, enthalpy H, and entropy S). As it is schematically shown in Figure 3, the onset of solidlike rigidity in an amorphous polymer at Tg is accompanied by sharp reductions in heat capacity Cp, thermal expansion coefficient up, and compressibility coefficient /cj (22). 

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. 

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.

No tables of the coefficients of thermal expansion of gases are given in this edition. The coefficient at constant pressure, l/t)(3 0/3T)p for an ideal gas is merely the reciprocal of the absolute temperature. For a real gas or liquid, both it and the coefficient at constant volume, 1/p (3p/3T),, should be calculated either from the equation of state or from tabulated PVT data. 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

This is defined as the increase in volume of unit volume of a substance when its temperature is raised by one degree. It is important in that the coefficient of expansion of LPG in its liquid form is relatively high, so that when filling a storage vessel adequate space must always be provided to allow for possible thermal expansion of the liquid. 

Fig. 3. (a) Thermal expansion coefficients a for the inclusion (f), matrix (m), mesophase (i) and composite (c) of a typical iron-epoxy particulate composite, with 5 percent volume fraction for the inclusions, versus temperature, (b) the reduced longitudinal expansion of the same elements, normalized to the unit-length versus temperature (diameter of inclusions df = 150 pm)... 

The glass transition temperature of a dilute system, according to the free volume changes, is determined by the diluent volume fraction Vd, and changes of the thermal expansion coefficient, a, at Tg by using ... 

Network properties and microscopic structures of various epoxy resins cross-linked by phenolic novolacs were investigated by Suzuki et al.97 Positron annihilation spectroscopy (PAS) was utilized to characterize intermolecular spacing of networks and the results were compared to bulk polymer properties. The lifetimes (t3) and intensities (/3) of the active species (positronium ions) correspond to volume and number of holes which constitute the free volume in the network. Networks cured with flexible epoxies had more holes throughout the temperature range, and the space increased with temperature increases. Glass transition temperatures and thermal expansion coefficients (a) were calculated from plots of t3 versus temperature. The Tgs and thermal expansion coefficients obtained from PAS were lower titan those obtained from thermomechanical analysis. These differences were attributed to micro-Brownian motions determined by PAS versus macroscopic polymer properties determined by thermomechanical analysis. 

The terms p, T, and v are characteristic reducing parameters which may be obtained by fitting pressure-volume-temperature data (density, thermal expansion coefficient, and thermal pressure coefficient) for each pure component in the mixture (3,12). Values of p, v, and T are given in Tables I and II. 

This favors a sample s contraction V is the volume). This attractive force, which will be temperature dependent, is balanced by the regular temperature-independent elastic energy of the lattice Fsiast/V = K/2) 6V/V). Calculating the equilibrium volume from this balance allows us to estimate the thermal expansion coefficient a. More specifically, the simplest Hamiltonian describing two local resonances that interact off-diagonally is... 

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

If a volume expansion is required, then mccisurements in three simultaneous dimensions are needed, a result experimentally difficult to achieve, to say the least. Even a slab of a single crystal does not completely solve the problem since thermal expansion in three dimensions is needed for the volume thermal expansion coefficient. The crystal has three (3) crystallographic axes and may have three (3) linear coefficients of expansion. Only if the crystal is cubic does one have the case where all three values of ol are equal. 

Here, and Fl are the unit cell volumes of the pure HS and LS isomer at 0 K, respectively, and is the coefficient of thermal expansion which is assumed to be equal in both lattices. The expression of Eq. (128) may be rewritten as ... 

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