Volume fraction characteristic

The HLB system has made it possible to organize a great deal of rather messy information and to plan fairly efficient systematic approaches to the optimiza-tion of emulsion preparation. If pursued too far, however, the system tends to lose itself in complexities [74]. It is not surprising that HLB numbers are not really additive their effective value depends on what particular oil phase is involved and the emulsion depends on volume fraction. Finally, the host of physical characteristics needed to describe an emulsion cannot be encapsulated by a single HLB number (note Ref. 75). 

In an amorphous material, the aUoy, when heated to a constant isothermal temperature and maintained there, shows a dsc trace as in Figure 10 (74). This trace is not a characteristic of microcrystalline growth, but rather can be well described by an isothermal nucleation and growth process based on the Johnson-Mehl-Avrami (JMA) transformation theory (75). The transformed volume fraction at time /can be written as... 

The polyamides are soluble in high strength sulfuric acid or in mixtures of hexamethylphosphoramide, /V, /V- dim ethyl acetam i de and LiCl. In the latter, compHcated relationships exist between solvent composition and the temperature at which the Hquid crystal phase forms. The polyamide solutions show an abmpt decrease in viscosity which is characteristic of mesophase formation when a critical volume fraction of polymer ( ) is exceeded. The viscosity may decrease, however, in the Hquid crystal phase if the molecular ordering allows the rod-shaped entities to gHde past one another more easily despite the higher concentration. The Hquid crystal phase is optically anisotropic and the texture is nematic. The nematic texture can be transformed to a chiral nematic texture by adding chiral species as a dopant or incorporating a chiral unit in the main chain as a copolymer (30). 

Because of the rotation of the N—N bond, X-500 is considerably more flexible than the polyamides discussed above. A higher polymer volume fraction is required for an anisotropic phase to appear. In solution, the X-500 polymer is not anisotropic at rest but becomes so when sheared. The characteristic viscosity anomaly which occurs at the onset of Hquid crystal formation appears only at higher shear rates for X-500. The critical volume fraction ( ) shifts to lower polymer concentrations under conditions of greater shear (32). The mechanical orientation that is necessary for Hquid crystal formation must occur during the spinning process which enhances the alignment of the macromolecules. 

Droplet Dispersion. The primary feature of the dispersed flow regime is that the spray contains generally spherical droplets. In most practical sprays, the volume fraction of the Hquid droplets in the dispersed region is relatively small compared with the continuous gas phase. Depending on the gas-phase conditions, Hquid droplets can encounter acceleration, deceleration, coUision, coalescence, evaporation, and secondary breakup during thein evolution. Through droplet and gas-phase interaction, turbulence plays a significant role in the redistribution of droplets and spray characteristics. 

A method for the estimation of composite material performance from the characteristics of fillers and the matrices and from the configuration of filler is generally called the law of mixture. In the most basic form of the law of mixture, the characteristics of a composite material are represented as a function of characteristics of constituent components and their volume fractions, as shown in Fig. 3. For a composite material (characteris-ticsiA f) that consists of component A (characteristics Xa, volume fraction ) and component B (characteristics Xf, volume fraction b), the basic formulae of the law of mixture are as follows ... 

The mechanical behaviour of a two-phase composite system depends partly on the filler characteristics, such as the geometry of inclusions, their size, the size distribution, the orientation of inclusions, the filler volume-fraction, the relative positions between the inclusions, the physical state of the filler, etc. and partly on the matrix characteristics, which are related to the physico-chemical state of the matrix, the degree of its polymerization, the crystallinity, the degree of cross-linking, etc. 

In TPE, the hard domains can act both as filler and intermolecular tie points thus, the toughness results from the inhibition of catastrophic failure from slow crack growth. Hard domains are effective fillers above a volume fraction of 0.2 and a size <100 nm [200]. The fracture energy of TPE is characteristic of the materials and independent of the test methods as observed for rubbers. It is, however, not a single-valued property and depends on the rate of tearing and test temperature [201]. The stress-strain properties of most TPEs have been described by the empirical Mooney-Rivlin equation... 

Figure 18.1 is the typical stress-strain curves of the filled rubber (SBR filled with fine carbon black, HAF),

One characteristic of shear banded flow is the presence of fluctuations in the flow field. Such fluctuations also occur in some glassy colloidal materials at colloid volume fractions close to the glass transition. One such system is the soft gel formed by crowded monodisperse multiarm (122) star 1,4-polybutadienes in decane. Using NMR velocimetry Holmes et al. [23] found evidence for fluctuations in the flow behavior across the gap of a wide gap concentric cylindrical Couette device, in association with a degree of apparent slip at the inner wall. The timescale of these fluctuations appeared to be rapid (with respect to the measurement time per shear rate in the flow curve), in the order of tens to hundreds of milliseconds. As a result, the velocity distributions, measured at different points across the cell, exhibited bimodal behavior, as apparent in Figure 2.8.13. These workers interpreted their data... 

The interior of the living cell is occupied by structural elements such as microtubules and filaments, organelles, and a variety of other macromolecular species making it an environment with special characteristics [97], These systems are crowded since collectively the macromolecular species occupy a large volume fraction of the cell [98, 99]. Crowding can influence both the... 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

An experimental test of the scaling model requires a selective variation of the two scaling variables of the model, i.e. the lateral chain distance and the chain stiffness. The Kuhn length /K depends on temperature via the characteristic ratio Cw the lateral chain distance s can be varied via the volume fraction 4>. 

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

Example. Although the surface area itself very often represents a desirable information about the system morphology, it can sometimes be used as a measure of the characteristic lengthscale in the system. Thus, if the volume fraction of the domains (/m) separated by the surface is constant (or nearly constant), the following geometrical relation holds regardless of the domain size and arrangement ... 

Figure 40. Time evolution of the Euler characteristic density for different average volume fractions, 4)0 — 0.5, 0.4, 0.375, 0.36, 0.35, and 0.3, quenched from the homogeneous state binary mixture. The negative Euler characteristic corresponds to the bicontinuous morphology, while the... 

Figure 41. The percolation threshold determination for polymer blends undergoing the phase separation. Minority phase volume fraction, fm, is plotted versus the Euler characteristic density for several simulation runs at different quench conditions, /meq- = 0.225,..., 0.5. The bicontinuous morphology (%Euier < 0) has not been observed for fm < 0.29, nor has the droplet morphology (/(Euler > 0) been observed for/m > 0.31. This observation suggests that the percolation occurs at fm = 0.3 0.01.

Euler characteristic and the minority domain volume fraction. Because the Euler characteristic is the direct measure of the domain connectivity, we identify the domain percolation at the point where the Euler characteristic attains zero value. 

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