The theories for the strength properties of filled systems are less developed than those for the modulus. In particular, only a few reports deal with the exploration of fracture mechanisms of nanocomposites where changes in the nature of crack propagation are believed to be one of the main mechanisms responsible for the strengthening effect [212]. In the case of poor bonding between the filler particles and the matrix, the tensile strength of a composite is usually reduced, with the filler content following a power law [213-216],

where ac and am are the strengths of the composite and the matrix, respectively; the constants a and n depend on the shape of the particles and their arrangement in the composite. This means that the strength of the composite cannot be greater than that of the unfilled version because the filler particles do not bear any fraction of the external load. However, this contradicts the results of the nanocomposites described in previous sections.

When the bonding between fillers and the matrix is strong enough, as suggested by Jancar and co-workers [217], the tensile strength of a composite can be 1.33 times as high as that of the matrix polymer:

where F(c) is proportional to the average yielded area per particle. In this case, an overlapping of stress concentration fields around the particulate fillers is responsible for shear yielding of the matrix [218]; the latter can be considered the dominant energy dissipation mechanism. Figure 32

Figure 32. Tensile strength of SiO2-g-PS/PP nanocomposites as a function of SiO2 content. The lines represent the results of curve fitting in terms of Eqs. (14) and (15). Reprinted with permission from [3], M. Z. Rong et al., Polymer 42, 3301 (2001). © 2001, Elsevier Science.

Figure 32. Tensile strength of SiO2-g-PS/PP nanocomposites as a function of SiO2 content. The lines represent the results of curve fitting in terms of Eqs. (14) and (15). Reprinted with permission from [3], M. Z. Rong et al., Polymer 42, 3301 (2001). © 2001, Elsevier Science.

illustrates good regression results in terms of Eqs. (14) and (15), demonstrating that a "perfect" adhesion is associated with the PP-based nanocomposites when > 0.65 vol.% [3]. It should be noted that the equilibrium relative strength is about 1.15 but not 1.33, which can be attributed to the low ductility of the matrix polymer employed. When the filler content exceeds the critical value of 0.65 vol.% (i.e., F(c)faj = 1), the corresponding F(c) can be estimated to be around 2.4 x 104. Considering the fact that F(c) characterizes the rate of increase in connectivity of yielded microzones around individual particles [75], one might conclude that the reinforcing effect generated by the nano-SiO2 particles here is far superior to that induced by conventional particles. For example, the F(c) values for CaCO3 (10 /m in diameter)-filled PP composites range only from 3 to 5 [217]. In addition, the greater value of F(c) is also indicative of a strong interfacial interaction in the nanocomposites.

Pukanszky et al. developed, alternatively, a semiempirical correlation for describing the composition dependence of the tensile yield stress in heterogeneous polymer systems [219]:

where B is a parameter related to the components' interaction. It is given by

where Af and pf are the specific surface area and the density of the filler, and l and are the thickness and yield stress of the interphase. By using Eq. (16) and the measured mechanical performance, Fekete et al. calculated a series of B values [220]. It was shown that the correlation of B versus Af is linear up to a specific surface area of about 7-8 m2/g as predicted by Eq. (17), but it becomes independent of Af at higher values. The deviation from the prediction was attributed to the formation of particle agglomerates. With agglomeration, the surface area for interphase formation no longer changes with increasing Af. Clearly, an examination of the linear relation between B and Af based on Eq. (17) can reveal the homogeneity of the nanocomposites of interest.

Shang et al. also took the interfacial interaction into consideration when discussing the tensile strength of SiO2/EVA composites [170]. Similar to the description of the modulus in Eq. (12), the dependence of tensile strength on the work of adhesion can be written as

where ac is the tensile strength of a composite with Wa at the interface. aco is the tensile strength of the composite with the weakest interfacial bond Wao. When the tensile strength of a composite and its corresponding Wa are known, Eq. (18) can predict the tensile strength of another composite of a certain filler loading, even if the fillers have different Wa values. is the slope of log ac versus 1 /Wa and depends on both silica particle size and volume fraction. The experimental results reflected that the K„ values increased as the uc filler content increased, especially for the composites with coarser particles. It was thus suggested that for the tensile strengths of the nanosilica composites, the interfacial bond strength has a more pronounced effect on the composites filled with larger filler particles than on the composites with fine particles. The effect of Wa on the tensile strength is dependent on both particle size and volume fraction, but the Young's modulus depends only on filler size.

Sumita et al. observed that the shear yield stress for the composites filled with nanoparticles increases with increasing filler content and with decreasing filler size, whereas the value for the composites filled with micron-sized particles decreases with filler content [155, 156]. Considering that the dependence of reinforcing and antireinforcing effects on filler size and volume fraction can be explained by the dispersion strength theory, Sumita and co-workers proposed a modified equation as follows:

tc = Tm(1 - f) + Gb/{dk(d)[(4w/3fa)1/3 - 2]/2} (19)

where Tc and Tm are the shear yield stresses of a composite and its matrix, G is the shear modulus, b is the Burger's vector, d is the diameter of a particle, and k(d) is an aggregation parameter. It suggests that under the same filler content, smaller nanoparticles result in a more remarkable reinforcing effect. For the micron-sized particle-filled composites, the second item of Eq. (19) is negligibly small. The experimental results of nano-SiO2 incorporated PP and nylon 6 exhibited a good agreement with the calculation with Eq. (19) [155, 156].

Xiong et al. concluded that the influence of particle size on the composite strength can be depicted more clearly by a simple empirical expression. The authors studied the mechanical performance of ultrafine Al2O3/PS composites prepared through bulk polymerization in the presence of filler particles [221]:

where the units of ac and R are MPa and /m, respectively. The authors showed that particles with radii smaller than 0.5 / m can bring about significant reinforcing effects.

Based on the methodology of structural elements, Ovchin-skii et al. developed a computer simulation model and algorithms for simulating the processes of failure of fibrous nanocomposites [212]. An account was made of the scale effect of the strength of the superthin fibers, which was associated with a change in not only their lengths but also their diameters. Analysis of the modeling results showed that the increase in the strength of the examined composites was obtained by employing superthin fibers, which was more efficient than increasing the fiber volume content.

The fracture toughness of CaCO3 (70 nm)-filled PP was studied by Levita et al. [142]. It was found that the critical stress intensity factor remained almost unchanged in the lower filler concentration range (5 vol.%). Different behaviors were observed at higher loadings, depending on the surface treatment. There was a monotonic decrease in the case of untreated fillers, whereas for modified fillers a well-defined maximum at 10 vol.% was observed. As a coupling agent, stearic acid was more efficient than titanate. No difference between the two treatments was found at higher concentrations. Usually the fracture properties of particulate composites can be explained by the model of crack pinning [222]. It is assumed that the crack front is characterized by a line tension (energy per unit length), so that lengthening this line increases the energy associated with the crack. When the crack encounters an array of impenetrable particles it is arrested (pinning) and a higher tension is required to release the crack. Lange and Redford related the fracture energy of a composite, Gc, with the line tension, T [223]:

where Gm is the fracture energy of the matrix. To account for the influence of particle size, an empirical factor, F(d) (0 < F(d) < 1), has been introduced [224]. At low concentration Eq. (21) thus becomes

By examining the F(d)T values as a function of particle size, Levita et al. pointed out that F(d)T is very small for finely sized particles. That is, the pinning contribution proved to be negligible. A particle can only interact with a crack if its size is greater than a certain length that characterizes the crack. Instead, interfacial debonding that reduces the local stress was attributed to the main factor responsible for the toughening effect.

Studies on the filler loading dependence of the toughness of SiO2/PP nanocomposites showed that the maxima in impact strength and fracture toughness appeared already at very low filler loadings (<1 — 3 vol.%) [107]. This is the opposite of the optimum loading conditions known for conventional particle-filled systems. Although the typical size of the modified nanoparticles at the above-mentioned filler content range varies between 100 and 150 nm (as found by TEM observation), the distances between individual particles and therefore the interparticle ligament thicknesses are still much larger than those found in toughness-optimized conventional particle-filled systems [225]. Therefore, the considerable toughening effect perceived at such a low filler loading in the nanocomposites implies that the mechanisms involved should be different from those used as a basis for the single-percolation concept of Wu [226], which has been proved to be well applicable to conventional particulate-filled polymer composites.

stress volume constructed by stress volume constructed by

nanoparticle stress volume constructed by individual nanoparticles nanoparticle stress volume constructed by individual nanoparticles

Figure 33. Hypothetical model of a nanocomposite with superposed stress volumes.

To overcome this problem, a hypothetical model of double percolation was suggested [3]. It was assumed that a double percolation, characterized by the appearance of connected shear-yielded networks throughout the composite, might be responsible for the performance enhancement at low nanofiller loading. This means one is dealing with (i) a percolation of shear yielded zones inside the dispersed phases (i.e., SiO2-g-PS agglomerates, consisting of nano-SiO2, the grafting PS on SiO2, and the homopolymer derived from the grafting styrene monomer) due to the superposition of stress volumes around the nanoparticles, and (ii) a percolation of shear-yielded zones throughout the matrix resin due to the superposition of stress volumes around the dispersed agglomerates (Fig. 33). Provided these two percolation processes take place almost simultaneously, the eventual percolation threshold should be equal to the product of the individual percolation thresholds, in accordance with the scaling theory [227]. As a result, a rather low filler content is sufficient to bring about a significant improvement of the nanocomposites' performance. In fact, the concept of a so-called double percolation or multiple percolation has been successfully used in designing very low filler-loaded

conductive polymer composites [228]. It should also be applicable to the description of critical transitions in other disordered systems with randomly geometric structure. In addition to the quantitative analysis provided in [3], frac-tography of tensile-fractured surfaces of the nanocomposites also give supporting evidence, as illustrated by the closely packed concentric matrix-fibrillated circles around particlelike objects (Fig. 34).

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