Halpin-Tsai equations nanocomposites

The effect of polymer-filler interaction on solvent swelling and dynamic mechanical properties of the sol-gel-derived acrylic rubber (ACM)/silica, epoxi-dized natural rubber (ENR)/silica, and polyvinyl alcohol (PVA)/silica hybrid nanocomposites was described by Bandyopadhyay et al. [27]. Theoretical delineation of the reinforcing mechanism of polymer-layered silicate nanocomposites has been attempted by some authors while studying the micromechanics of the intercalated or exfoliated PNCs [28-31]. Wu et al. [32] verified the modulus reinforcement of rubber/clay nanocomposites using composite theories based on Guth, Halpin-Tsai, and the modified Halpin-Tsai equations. On introduction of a modulus reduction factor (MRF) for the platelet-like fillers, the predicted moduli were found to be closer to the experimental measurements. 

The close fit of the experimental data and the values predicted by the constitutive modified Halpin-Tsai equations I and II (24) and (25), as seen in Fig. 43 (for NR) illustrates the appropriate definition of the IAF. Table 10 also confirms that newly devised equations (24) and (25) provide astounding results because their predictions conform to the experimental data. The introduction of IAF imparts a definitive change to the predicting ability of the constitutive equations for polymer/filler nanocomposites (Fig. 43 Table 10). 

Relative tensile modulus of OMMT-polypropylene nanocomposites as a function of inorganic volume fraction. The solid line represents the fitting by using unmodified Halpin-Tsai equation. (Reproduced from Mittal, V., /. Thermoplast. Compos. Mater., 22, 453, 2009. With permission from Sage Publishers.)... 

Deng et al. [31] obtained PLA/cdHAp nanocomposites by solvent casting. SEM observations confirmed close contact between the polymer matrix and the filled nanocrystals, and the homogeneous dispersion of nanocrystals in the polymer matrix at a microscopic level. The tensile modulus for the nanocomposites increased with cdHAp loading. Theoretical predictions of the modulus (by assuming that the nanocomposites behave as a short-fibre-filled system) based on the Halpin-Tsai equation show excellent agreement with the experimental results. 

The expression for the Halpin-Tsai equation that is applicable to polymer-clay nanocomposites that have the clay particles oriented in the direction of the applied stress includes the aspect ratio (A) of the particle in the equation. 

Although there is enormous variability in mechanical properties because of local concentration and isotropy variation, the Halpin-Tsai model has been shown to provide reasonably good approximations of the elastic modulus of randomly distributed CNT-based nanocomposites [322,323]. The Halpin-Tsai model (Equations 5.7, 5.8) is specifically for glassy matrix polymers, but it has also shown to be fairly effective for semicrystalline polymers as well [324]. 

By virtue of the strong anisotropy of the shape of particles of Na -montmorillonite mentioned above, for theoretical estimation of the degree of reinforcement of nanocomposites filled by them the models of Halpin-Tsai and Mori-Tanaka are used [19]. For the case of isotropic (spherical) filler particles EJE estimation can be carried out according to the equation, obtained in paper [35] ... 

Figure 7.11 The dependences of the reinforcement degree EJE on the filling degree W for nanocomposites filled by Na -montmorillonite. 1-5 - the theoretical dependences corresponding to Halpin-Tsai (1, 2) and Mori-Tanaka (3, 4) equations at L/d i = 100 (1, 3) and 50 (2,4) and to Equation 7.7 (5). 6-13 -the experimental data for nanocomposites on the basis of epoxy polymer at T < (6), polyamide-6 (7), poly(butylenes terephthalate) (8), polycarbonate...

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