Bulk Elasticity

General physical chemistry texts4 have made us familiar with the concept of the heat capacity being the rate of change of internal energy with temperature  

in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas  

As we know the structure and the dimensions of the atoms/ions it is relatively easy to write Equation (2.13) in terms of the energy minimum, Vm, i.e. the energy at the equilibrium separation. For the three cases we may write6 the bulk modulus as  

Where B is a constant for the type of atomic interaction and includes the Madelung constant  

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Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

Many other interesting examples of spontaneous reflection symmetry breaking in macroscopic domains, driven by boundary conditions, have been described in LC systems. For example, it is well known that in polymer disperse LCs, where the LC sample is confined in small spherical droplets, chiral director structures are often observed, driven by minimization of surface and bulk elastic free energies.24 We have reported chiral domain structures, and indeed chiral electro-optic behavior, in cylindrical nematic domains surrounded by isotropic liquid (the molecules were achiral).25... 

Bulk elastic modulus, of binary compound semiconductors, 22 145, 146-147t Bulk enzymes, from genetically engineered microbes, 22 480 Bulk erosion, 9 78 Bulk fluid velocity method, 16 688 Bulk gallium nitride, supercritical ammonia solution growth of, 14 96-97 Bulk gases... 

For coarse-grained models of linear biopolymers—such as DNA or chromatin— two types of interactions play a role. The connectivity of the chain implies stretching, bending, and torsional potentials, which exist only between directly adjacent subunits and are harmonic for small deviations from equilibrium. As mentioned above, these potentials can be directly derived from the experimentally known persistence length or by directly measuring bulk elastic properties of the chain. 

T. D. Dimitrova and F. Leal Calderon Bulk Elasticity of Concentrated Protein-Stabilized Emulsions. Langmuir 17, 3235 (2001). 

In diamond C, it is the fact that the structure consists of sp covalent bonds only that accounts for its unique physical properties, such as the highest hardness of any material (Mohs hardness 10), small compressibility (1.7X10 cm /l ), the highest elasticity among any known material (4 6 X10 dynes/cm, bulk elasticity), large thermal conductivity (9 26 W/(deg cm)), and small thermal expansion coefficient (0.8 0.1 X10 at 20 °C, comparable to the value of invar). As a result. 

Evidence for wall slip was suggested over thirty years ago [9,32,63]. One of the first attempts at a slip mechanism was the performance of a Mooney analysis by Blyler and Hart [32]. Working in the condition of constant pressure, they explicitly pointed out melt slip at or near the wall of the capillary as the cause of flow discontinuity. On the other hand, they continued to insist that bulk elastic properties of the polymer melt are responsible for the flow breakdown on the basis that the critical stress for the flow discontinuity transition was found to be quite insensitive to molecular weight. Lack of an explicit interfacial mechanism for slip prevented Blyler and Hart from generating a satisfactory explanation for the flow oscillation observed under a constant piston speed. 

It should be noted that the formula about the modulus of bulk elasticity of a foam refers to deformation at both compression and expansion. At large deformations, however, their effects differ significantly. When the foam is compressed the gas volume can be reduced so that to become comparable to the liquid volume. The expansion of a foam cannot be unlimited depending on its initial expansion ratio, the volume of the foam can increase only until the border pressure reaches a critical value (see Section 6.5.2). The latter is related to foam dispersity and surfactant adsorption, and decreases with the increase in surface area. 

Accounting for the liquid volume in the foam and knowing that the liquid is virtually incompressible, the modulus of bulk elasticity of the foam can be expressed by... 

Thus, the real modulus of bulk elasticity would be higher than Ev and the ratio between the modulus of elasticity of wet and dry foam would be... 

The condition for observation is that the phonon coherence length is larger than the layer thickness. Low frequency acoustic modes fulfill this condition because they are an in-phase motion of a large number of atoms and are not strongly influenced by the disorder-instead reflecting the average bulk elastic properties of the materials. 

Laser ultrasonic transducers are truly non-contact devices which effectively avoid acoustic coupling problems (e.g. damping in the transducer and couplant reflection and transmission losses at the interface). Most laser ultrasonic devices have been used for excitation and detection of bulk elastic waves in point source or planar geometry, but also surface acoustic (Rayleigh or Brillouin) waves. Unlike the bulk wave regime, only one sample side is needed for excitation and detection when surface waves are used. This not only renders the measurements easier, but also avoids the need for an accurate knowledge and uniformity of the sample thickness. In addition, the excitation laser can be focused using cylindrical lenses in order to obtain an excitation line. 

STEPANTEX 6530B will give household fabric softeners essential properties pleasant to the touch, wool-like bulk elasticity, good lubricating effect, excellent rewetting, antistatic property, easy drying and ironing. 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

The surface Fuchs-Kliewer modes, like the Rayleigh modes, should be regarded as macroscopic vibrations, and may be predicted from the bulk elastic or dielectric properties of the solid with the imposition of a surface boundary condition. Their projection deep into the bulk makes them insensitive to changes in local surface structure, or the adsorption of molecules at the surface. True localised surface modes are those which depend on details of the lattice dynamics of near surface ions which may be modified by surface reconstruction, relaxation or adsorbate bonding at the surface. Relatively little has been reported on the measurement of such phonon modes, although they have been the subject of lattice dynamical calculations [61-67],... 

In this review, the term elastic is applied to inter-atomic forces of chemical bonding treated in harmonic approximation, to be distinguished from bulk elasticity in solids (see Kanamori [27]). 

Ultrasonic interferometry, in which the travel time of high-frequency elastic waves through a sample is measured, also yields elastic moduli. Because it is a physical property measurement, rather than an optical spectroscopy, it can be used equally well on poly-crystalline samples as single-crystals, although polycrystalline measurements only yield the bulk elastic properties, bulk modulus and shear modulus, G. High-pressure ultrasonic interferometry techniques were initially developed in the piston cylinder... 

Figure 47. Semilogarithmic dependence of the shear elasticity modulus (a) and the bulk elasticity modulus (b) on the iteration number n for p = 0.2088 (/), 0.2092 (2), and 0.2098 (2).

We will consider below isotropic media, for which, just as Eq. (361), the concept of a complex bulk elastic modulus K (co) can be introduced [131]. 

We compare the variations of the pair potentials for particles with bulk elasticity (l)-(2) and surface elasticity (3)-(4), for star polymers (5), and for hard sphere particles in Fig. 5. The different potentials are normalized in a way that allows direct... 

Fig. 5 Variations of the interaction energy between particles versus the center-to-center distance, (a) Hertzian potentials for particles with bulk elasticity the dashed line represents the usual Hertz potential (1), and the solid line the generalized Hertzian potential (2). (b) Potentials for emulsions with surface elasticity the dashed line denotes the approximate solution for small compression ratios (3), and the solid line the general solution (4). (c) Ultrasoft potentials for star polymers (where a 1.3Rg, with Rq being the radius of gyration of the star, following [123]) (5) dashed line f = 256 solid line f = 128 dashed and dotted line f = 64. (d) Hard-sphere potential...

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