Volume change modulus

Then y equals the so-called volume-change modulus, as introduced by Weekman and Gorring [8],... 

A X M) matrix of stoichiometric coefficients (-) volume change modulus defined by Equation (11.87) (-) fraction of active sites occupied by species i (—) fraction of vacant active sites (-) density (kg/m ) density of B (kg/m )... 

The parameters which characterize the thermodynamic equilibrium of the gel, viz. the swelling degree, swelling pressure, as well as other characteristics of the gel like the elastic modulus, can be substantially changed due to changes in external conditions, i.e., temperature, composition of the solution, pressure and some other factors. The changes in the state of the gel which are visually observed as volume changes can be both continuous and discontinuous [96], In principle, the latter is a transition between the phases of different concentration of the network polymer one of which corresponds to the swollen gel and the other to the collapsed one. 

Hardness measures the resistance of a material to a permanent change of shape. That is, the resistance to shear deformation (not the resistance to a volume change). The precursor to a permanent shape change is a temporary elastic shape change, and a shear modulus determines this. Therefore, the first necessity for high hardness is a high shear modulus. 

For simple power law rate equations the effectiveness can be expressed in terms of the Thiele modulus, Eq 7.28. In those cases restriction is to irreversible, isothermal reactions without volume change. Other cases can be solved, but then the Thiele modulus alone is not sufficient for a correlation. 

Now, 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 ... 

When there is no volume change, as when an elastomer is stretched, Poisson s ratio is 0.5. This value decreases as the Tg of the polymer increases and approaches 0.3 for rigid solids such as PVC and ebonite. For simplicity, the polymers dealt with here will be considered to be isotropic viscoelastic solids with a Poisson s ratio of 0.5, and only deformations in tension and shear will be considered. Thus, a shear modulus (G) will usually be used in place of Young s modulus of elasticity E Equation 14.2) where E is about 2.6G at temperatures below Tg. 

Modulus of elasticity is the stress required to produce unit strain to cause a change of length (Young s modulus), or a twist or shear (shear modulus), or a change of volume (bulk modulus). It is expressed as dynes/cm. ... 

Delaminations can occur during cure as a result of high internal stresses. These stresses develop due to resin shrinkage and thermal volume changes. The level of stresses depend on several material properties, such as the Young s modulus, Poisson s ratio, and thermal expansion coefficients of both resin and fibers. In addition, the level of stresses also depends on several conditions, such as fiber orientation, fiber volume fraction, and part geometry. 

Material Properties. The elementary form of the analysis used requires the following properties of propellant during cure tensile modulus, effective bulk modulus, and propellant volume change. Each of these properties changes with temperature and time elapsed since casting. Because of the unusual nature of the material (sticky, wet, explosive, gravel ), special tests, equipment, and techniques were developed for these measurements. 

This same apparatus permits estimates to be made of the so-called jacketed compressibility (the reciprocal of the effective bulk modulus) during early cure. This is accomplished by applying momentary pressure to the bed alone (ram pressure) and observing the corresponding bed volume change in the water meniscus. 

A. Heydweiller 81 sought to measure the cohesion between a solute and solvent by means of the volume changes which occur on soln. If m be the mass of an ion, and u its mobility, the product kmu is called the Icnenmodulus, and 1c is a constant 0 00112 for univalent ions, and 006372 for bivalent ions. The modulus was calculated in various ways, and the result was considered to represent the cohesion... 

As discussed in this section, the contribution of the shear relaxation is not trivial. This is due to the fact that the diffusion occurs in all three dimensions for the spherical gel, two dimensions (radial direction) for the cylindrical gel, and only one dimension (thickness direction) for a slab gel. Because of the existence of shear modulus, the volume change caused by diffusion is shared by the remaining dimensions through the shear relaxation process. A detailed discussion is given by Li and Tanaka [93],... 

A jumpwise volume change in the transition correlates with a jumpwise change in the shear equilibrium modulus, the refractive index, the stress-optical coefficient and in the components of complex permittivity e and complex modulus G. ... 

For the PVN-PEO polyblends, volume changes at melting temperature (Figure 6) as well as x-ray data at room temperature (2) show that the 25% (PEO) blend is completely amorphous, and that the 50 and 75% blends contain significant amounts of amorphous PEO. Calculations based on specific volume data indicate that the crystalline part of both the 50 and 75% blends consists of PEO, whereas the amorphous part contains 46% PEO and 54% PVN. Another important result is that the unusual phenomenon of a well in the modulus temperature curves (Figure 1) was observed only for the blends which exhibit crystallinity. Based on these observations, the behavior of blends could be interpreted by postulating that the amorphous PEO forms a complex phase with PVN in the ratio of 3 to 1 monomer units (i.e., 46 wt. % PEO to 54 wt. % PVN), respectively. 

Bulk moduli and pressure derivatives. Results for the bulk modulus and its pressure derivative for all three HMX polymorphs obtained from fitting simulation-predicted isotherms to the equations of state discussed above are summarized in Table 7. For all data sets, we include fits to the Us-Up form (Eq. 18) and both weighting schemes for the third-order Birch-Mumaghan equation of state (Eqs. 20 and 21). In the case of the experimental data for /THMX, values for the moduli based on Eqs. 18 and 20 were taken from the re-analysis of Menikoff and Sewell. Two sets of results are included in the case of Yoo and Cynn, since they reported on the basis of shifts in the Raman spectra a phase transition with zero volume change at 12 GPa. Simulation data of the /T HMX isotherm due to Sorescu et al. were extracted by hand from Fig. 3b of their work. 

Isotropic (hydrostatic) compression Bulk modulus Ka Hydrostatic pressure p pV() Volume change per unit volume AV/Va AV (13.4)... 

The problem of Example 1 was reformulated in terms of Equations 16, 17 and 20, and the generalized Thiele modulus was found to be M = 11.40 corresponding to an effectiveness factor of n = 0.0877. Thus the error introduced by the constant diffusivities assumption amounts to less than 5% for this example. Had there been a substantial volume change accompanying the reaction, agreement between the two methods of computation would not have been as good. 

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