Composite Nielsen model

FIGURE 2.9 Comparison of theoretical models quantifying the effect of path tortuosity on the permeability of a composite Nielsen model [eq. (2.8)], Friedrickson-Bicraano [eq. (2.10)], modified Nielsen [eq. (2.9)], and Cussla--Aris [eq. (2.11)]. 

Fig. 12 Percent elongation at break (yield) for the incompatible PpCIS/PPO blends. ( ), elongation at break ( O elongation at yield. Values within parentheses indicate the fraction of samples failing in the principal mode in the embrittlement region. Error bars indicate 95% confidence intervals. Curve 1 was drawn from values calculated from the Nielsen model for perfect adhesion composites (Eq. 7).

For diffusion of liquid through rubbery polymer composites, Fickian and non-Fickian diffusion theories are frequently used to describe the mechanism of transport, but for gas or vapour, other models have been developed to fit experimental data of diffusion profiles. The models of gas transport include Maxwell s model," free volume increase mechanism," solubility increase mechanism," nanogap hypothesis," Nielsen model, " " Bharadwaj model, ° Cussler model " " and Gusev and Lusti model, " etc. 

The Nielsen model has been a popular theory, originally used to explain polymer lay nanocomposites. This model is used to describe the tortuosity effect of plate-like particulates of filled rubber polymer composite on the gas permeation. An increase in barrier properties of gas permeation of rubber polymer nanocomposites is a result of the impermeable nature of filler particles which creates a long path of penetrant molecule by directing them around the particle. 

The Nielsen model (31) describes the elastic shear modulus G of a two-phase composite consisting of inclusions having volume fraction suspended uniformly in a continuous matrix. For the case of rubbery inclusions in a glassy matrix (here, a PS-rich matrix), the model takes the form ... 

The Lewis-Nielsen model considers the effect of the shape of the filler and the orientation or type of packing for a two-phase system or single-phase reinforcement composite, resulting in equation (11.8) for effective thermal conductivity (Tavman... 

Aggregated Microcomposites. Agglomeration of clay platelets leads to tac-toid stractures (micro composites) with reduced aspect ratios and, according to the Nielsen model, reduced barrier efficiencies. 

Fig. 3. The predicted effect of maximum packing fraction on the relative thermal conductivity of composites filled with spherical particles in which kf/k 1000, according to the Nielsen model...

All of the assumptions from the Nielsen model remain in the Cussler model except for the regularity of the array. The Cussler model for relative permeability of a composite with monodisperse particles in a regular array is given by Eq. (8.7)... 

Spinels. There are limited experimental data on uranium and thorium partitioning between magnetite and melt (Nielsen et al. 1994 Blundy and Brooker 2003). Both studies find U and Th to be moderately incompatible. Blundy and Brooker s results for a hydrous dacitic melt at 1 GPa and 1025°C give Du and D h. of approximately 0.004. The accuracy of these values is compromised by the very low concentrations in the crystals and the lack of suitable SIMS secondary standards for these elements in oxide minerals. Nonetheless, these values are within the range of Djh of magnetites at atmospheric pressure 0.003-0.025 (Nielsen et al. 1994). It is difficult to place these values within the context of the lattice strain model, firstly because there are so few systematic experimental studies of trace element partitioning into oxides and secondly because of the compositional diversity of the spinels and their complex intersite cation ordering. 

Rare earth element data will also serve as the basis for a forward modeling study to better constrain melting systematics in the Galapagos. The melting model will invoke clinopyroxene-rare earth element partition coefficients, which vary with composition (Gallahan and Nielsen, 1992), and a polybaric or column melting process. 

Mechanical tests indicate that these blends do not behave like conventional blends and suggest that the polystyrene phase is continuous in the substrate. The moduli of the blends as a function of blend composition is plotted in Figure 10.6. The Voigt and Reuss models are provided for comparison (Nielsen, 1978) These are the theoretical upper and lower bounds, respectively, on composite modulus behavior our data follows the Voigt model, suggesting that both the polystyrene and polyethylene phases are continuous. In most conventional composites of polystyrene and HDPE, the moduli fall below the Voigt prediction indicating that the phases are discontinuous and dispersed (Barentsen and Heikens, 1973 Wycisk et al., 1990). 

Figure 11 shows the theoretical permeabilities that are expected for a two-phase blend of polymers. The two solid curves represent calculations based upon Maxwell s equation (24) for an aspect ratio of 1 for the discontinuous phase. The dotted line is a prediction of the permeability using Nielsen s model (25) when a barrier polymer with an aspect ratio of 8 is discontinuous in a nonbarrier matrix. Figure 12 shows the expected result of a phase inversion for a two-polymer blend. The discontinuous phase is assumed to have an aspect ratio of 1. At some critical composition, the composite switches from being continuous in one polymer to being continuous in the other. Figure 12 is really a special case of Figure 11. Selar RB is a blend of polyethylene and nylon-6. Polyethylene is the majority constituent and forms the continuous phase. The product has its best barrier when it can be used in processes that impart orientation to the product. This gives a high aspect ratio to the nylon-6 and enhanced barrier to the article. Blends of polyethylene and EVOH are being developed.

Theory. Basic theories for the prediction of the modulus of a composite from those of the components were derived by, for example, Hashin in 1955 (I), Kerner in 1956 (2), and van der Poel in 1958 (3). Takayanagi (4, 5), and Fujino et al. (6) developed a very promising and instructive model theory which includes calculation of the loss spectra of composites, and it may easily be extended to anisotropic morphologies. Furthermore, Nielsen and coworkers may be cited for fundamental theoretical and experimental contributions (7, 8, 9,10). 

Both brain function and composition are affected by dietary boron (Nielsen 1996, Pen-land 1998). Assessments of both animal models and humans found that boron deprivation results in decreased brain electrical activity similar to that observed in nonspecific malnutrition. Boron deprivation also resulted in poorer performance in tasks of motor speed and dexterity, attention, and short-term memory in humans. Increased copper and calcium concentrations in total brain and increased phosphorus in the cerebellum have been found in boron-deprived rats. Boron reportedly can restore the a-hel-ical conformation of (l-amyloid peptide (1 -40) disrupted by aluminum (Ramakrishna etal. 1997). 

A limit value for the yield strain Cy can be obtained using the simple series model of Nielsen. According to this model, perfect adhesion is assumed, and the actual strain of the polymer (cm) is higher than the strain of the composite (Cq) due to the fact that glass does not elongate the relationship is... 

When gas travels through NCH, the permeability coefficient of the gas can be analyzed using a geometrical model in which silicate layers are dispersed. In NCH, silicate layers are ahgned nearly parallel with the film sinface. According to Nielsen, the diffusion coefficient D of a liquid or a gas can be calculated using Eq. 10 if the liquid or gas is in a composite material in which plate particles are in a planar orientation ... 

For the model shown in Figure 12.5, assuming perfect adhesion, Nielsen calculated the elongation of the plastic per se (in the composite) relative to the overall elongation of the filled specimen to be... 

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