Pressure effects, elastic constants

ELASTIC PROPERTIES AND PRESSURE EFFECTS Elastic constants... 

As demonstrated, Eq. (7) gives complete information on how the weight fraction influences the blend viscosity by taking into account the critical stress ratio A, the viscosity ratio 8, and a parameter K, which involves the influences of the phenomenological interface slip factor a or ao, the interlayer number m, and the d/Ro ratio. It was also assumed in introducing this function that (1) the TLCP phase is well dispersed, fibrillated, aligned, and just forms one interlayer (2) there is no elastic effect (3) there is no phase inversion of any kind (4) A < 1.0 and (5) a steady-state capillary flow under a constant pressure or a constant wall shear stress. 

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

The specific volume was slightly lower for a sample of very high molecular weight and was reflected by an increase in c to 156° C. This effect was attributed to the very high melt elasticity of this polymer. One consequence of the finite value of c is that the internal pressure is not constant as it is for melted polyethylene but decreases with increasing volume. 

Due to the chain architecture and the large size of the macromolecules, the wetting behaviour of polymer liquids can be different from that of simple liquids. The effect becomes particularly strong when the dimension of the liquid phase, e.g. film thickness and droplet diameter, approaches the dimension of the polymer coil. In addition to the spreading coefficient and the surface pressure effects, entropic elasticity of the polymer chain provides a strong contribution to the free energy for a constant volume V0=Ad ... 

This method has a number of powerful features. First, it is very fast. Thus, one can calculate many structures and also study the effects of pressure and temperature upon the structures. Structures of low symmetry can be considered if covalency effects are small, so as to obtain a full set of elastic constants. Indeed, the most important applications of this method within geophysics have been in evaluating equations of state (volume as function of pressure and/or temperature) and elastic constants that can be related to seismic velocities. 

Elastic constants depend on pressure and temperature because of the anharmonicity of the interatomic potentials. From the dependence of bulk and shear moduli on hydrostatic and uniaxial pressure, third order elastic constants and Griineisen parameters may be determined. Griineisen parameter shows the effect of changing volume, V, on the phonon mode frequencies, co. 

Next we note that there are two physieally different sources of temperature and pressure dependence of the elastic constants of polymers. One, in common with that exhibited by all inorganic crystals, arises from anharmonic effects in the interatomic or intermolecular interactions. The second is due to the temperature-assisted reversible shear and volumetric relaxations under stress that are particularly prominent in glassy polymers or in the amorphous components of semi-crystalline polymers. The latter are characterized by dynamic relaxation spectra incorporating specific features for different polymers that play a central role in their linear viscoelastic response, which we discuss in more detail in Chapter 5. 

That this form does indeed hold had been demonstrated for the cavitation of glassy polypropylene (Mott et al. 1993) in a computational study furnishing the validity of this extension of the universal binding-energy relation to symmetrical bulk response. The additional attraction of this expression is that it points out directly that application of a pressure produces symmetrical elastic compaction in an isotropic solid. However, more interestingly, one notes that, when dilatation is imposed, the bulk modulus monotonically decreases and eventually, at a dilatation of 1 /p, vanishes. This also leads to the observation that, if dilatation results from thermal expansion in response to a temperature increase, the bulk modulus also decreases. This simple observation represents the essence of the temperature dependence of all other elastic constants in anisotropic solids, beyond the mere effect on the bulk... 

A variety of experimental techniques are available for the investigation of the electron-lattice interaction. For static phenomena such as thermal expansion and magnetostriction one can use dilatometric and X-ray techniques. For dynamic effects such as elastic constant measurements, ultrasonic propagation and phonon dispersion the methods of sound velocity and attenuation measurements, and inelastic neutron or light scattering are available. In addition high-pressure work can give valuable information for some quantities. 

The electrical resistivities of the dense Kondo systems CeNiln, CePdln, and CePtln have been measured under hydrostatic pressures up to 19 kbar (Kurisu et al., 1990). The Kondo temperature of CeNiln and CePtln shifts linearly with pressure to higher temperatures at rates of 2.3 and 1.5 K/kbar, respectively. For CePdIn, the pressures were not high enough to reach the CePtln or CeNiln state. Measurements of the elastic properties of CePdln reveal that all elastic constants exhibit softening at low temperatures due to the crystal electric field effect and the antiferromagnetic ordering (Suzuki et al., 1990). 

During their investigation Palmer et al. (1974) examined the effect of a 2.5 Tesla magnetic field on the temperature dependence of the elastic constants of terbium, Monfort and Swenson (1965) and Stephens and Johnson (1969) obtained the pressure dependence of the compressibility. 

Figure 7 displays the diastolic P-V relations in the control state and following methoxamine infusion. Note the dramatic parallel shift in these relations following the infusion. Also shown are the computed pressure-volume curves resulting from variations in the various factors namely (a) an increase in elasticity constants by 30%, (b) assuming a spherical geometry for the LV and (c) an increase of 30% in the LV wall mass to simulate the erectile effect due to increased coronary perfusion. In these computations, external pressure was assumed to be zero. It is... 

This rheometer is also similar to the one described in section 3.2.1 except for two differences. Firstly, the capillary used is of very short length and secondly, the polymer is extruded by the use of dead weights (i.e. constant pressure) rather than constant plunger speed. This instrument, popularly known as the Melt Flow Indexer, is very popular in the thermoplastics industry due to its ease of operation and low cost, which more than compensates for ite lack of sophistication. The parameter measured through the melt flow indexer contains mixed information of the elastic and viscous effects of ttie pol)nner. Further, no end loss corrections have been developed for this capillary equipment nor can the melt flow index be easily related to the Weissenberg-Rabinowitsch shear rate expression. 

It is known that if bending moments act on a deformed membrane, the Laplace law is modified [1]. The problem is even more complicated in the case of a fluctuating membrane. The membrane s out-of-plane fluctuations change its effective elasticity [2,3] and renormalize the tension of the membrane [4], The aim of the present work is to find the relationship between the difference of the hydrostatic pressure Ap inside and outside a fluctuating giant vesicle, its tension a, and its elastic constants. For this purpose, the free energy of the vesicle is calculated. 

The history of a-U is a long and complicated saga that merits a special review (Fisher, Lander, and Bader, to be published). Reviews on the elastic-constant anomalies (Fisher 1974), superconductivity (Smith and Fisher 1973), and structural effects discovered up to 1984 (Smith and Lander 1984) are already available. Briefly, all physical-property measurements show an anomaly at 43 K, and many show subsequent anomalies at 37 and 22 K. The phase diagram under pressure of a-U is given in fig, 2. At low temperature, i.e. below the solid line, a-U is a superconductor. Assigning the phases a, a2, as we have done in the caption in fig. 2, may not be correct since, strictly speaking, the phase boundaries have been determined at zero pressure... 

Because perfect dislocations are observed in high-stress conditions where a hydrostatic component is present in the stress tensor, it is of interest to check the effect of such a hydrostatic pressure on their core structure configuration and mobility. One can expect three kinds of effects due to pressure (i) the material is usually stiffer (this is the case for silicon), with an increase of elastic constants that affect the strain field around the core, (ii) the core structure and its stability could be modified, and (iii) pressure could favor dislocation core mobility along certain directions. One may then wonder whether theoretical investigations of non-dissociated perfect dislocations are really representative of experiments. 

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