The rule of mixtures is a general approximation of the description of the mechanical behavior (especially elastic behavior) of composites. It gives upper (Eq. (6)) and lower

(Eq. (7)) bounds to their mechanical response:

where Ec is the modulus of the composites, Em and Ef denote the moduli of the matrix and the filler, and <fim and symbolize the volume fraction of the matrix and the filler, respectively. As a matter of fact, Eqs. (6) and (7) are based on a strong interfacial adhesion and rigid spherical reinforcement, respectively. The main reinforcing mechanism is attributed to the efficiency of the load transfer from the matrix to the filler. It was found that the moduli of most particulate filled polymers lie between the two limits [206]. On the other hand, Kerner's equation was derived from a basic consideration of mechanical properties of a composite at moderate concentration [207]. For low filler content, Nielsen's modification of Kerner's equation is more applicable because the maximum packing of the fillers, interfiller interactions, and the mechanical coupling of the matrix and filler are taken into consideration [208, 209]:

where A accounts for the contribution of the interfacial interaction and factors like filler geometry, B represents the relative stiffness of the two components, and fi is dependent on the filler packing fraction.

Predictions of nanocomposite moduli based on the above equations were made by Petrovicova et al. [177]. The experimental results indicated that all of the nano-SiO2/nylon 11 nanocomposites exhibited a modulus higher than that calculated by the rule of mixtures. Although the data for most of the nanocomposites closely fit Eq. (8) for A = 3 — 4, there was a large deviation for the 2.4 vol.% filled composites, manifesting the extraordinary reinforcing tendency of the nanoparticles. It was believed that the lower values of A were due to an improved spatial distribution of the silica in the matrix.

As a result of the unsatisfactory agreement between the Kerner equation and the measured data, Kryszewski and Bak [189] assumed that some volume of the surrounding

Table 12. Nonisothermal crystallization and melting data of PP (MI = 8.5 g/10 min) and its nanocomposites. Content of SiO2

Samples (vol.%) Tma (°C) Tcnb (°C) ATc (°C) Xcd (%)

Table 12. Nonisothermal crystallization and melting data of PP (MI = 8.5 g/10 min) and its nanocomposites. Content of SiO2

Samples (vol.%) Tma (°C) Tcnb (°C) ATc (°C) Xcd (%)

Neat PP |
— |
164.7 |
115.3 |
49.4 |
44.6 |

Nanocomposites filled |
1.96 |
165.8 |
117.8 |
48.0 |
46.4 |

with untreated SiO2 |
3.31 |
166.6 |
118.9 |
47.7 |
45.1 |

4.68 |
164.4 |
118.6 |
45.8 |
44.4 | |

6.38 |
163.3 |
118.9 |
44.4 |
46.7 | |

Nanocomposites filled |
1.96 |
164.7 |
116.5 |
48.2 |
45.8 |

with SiO2-g-PS |
3.31 |
166.2 |
118.9 |
47.3 |
43.9 |

4.68 |
164.9 |
118.7 |
46.2 |
46.9 | |

6.38 |
165.3 |
119.1 |
45.3 |
44.6 |

aTm denotes the peak melting temperature.

bTcn denotes the peak crystallization temperature recorded during cooling. cAT = Tm — Tc, denotes the supercooled temperature. dXc denotes the crystallinity of PP.

Source: Reprinted with permission from [107], M. Z. Rong et al., Polymer 42, 167 (2001). © 2001, Elsevier Science.

aTm denotes the peak melting temperature.

bTcn denotes the peak crystallization temperature recorded during cooling. cAT = Tm — Tc, denotes the supercooled temperature. dXc denotes the crystallinity of PP.

Source: Reprinted with permission from [107], M. Z. Rong et al., Polymer 42, 167 (2001). © 2001, Elsevier Science.

Samples |
Ta (°C) |
AH» (J/g) |
tfc (min) |
tmaxd (min) |
t1/2e (min) |
n |
Kg (min-") |
Tm (°C) | |

Neat PP |
130 |
95.6 |
14.6 |
5.7 |
6.1 |
2.69 |
5.30 |
< 10-3 |
166.5 |

132 |
97.9 |
23.8 |
9.8 |
10.1 |
2.75 |
1.18 |
< 10-3 |
167.1 | |

SiO2 as-received/PPh |
130 |
92.3 |
10.2 |
4.1 |
4.3 |
3.10 |
7.16 |
< 10-3 |
166.8 |

132 |
94.7 |
17.1 |
7.5 |
7.9 |
2.94 |
1.64 |
< 10-3 |
167.3 | |

SiO2-g-PS/PPh |
130 |
93.4 |
11.3 |
4.9 |
5.0 |
2.84 |
7.15 |
< 10-3 |
168.6 |

132 |
95.2 |
18.7 |
8.9 |
8.8 |
2.99 |
1.07 |
< 10-3 |
167.2 | |

SiO2-g-PMMA/PP' |
130 |
94.6 |
12.4 |
5.1 |
5.2 |
3.02 |
5.87 |
< 10-3 |
167.0 |

132 |
94.0 |
18.2 |
7.7 |
8.1 |
2.80 |
1.98 > |
< 10-3 |
168.0 |

a Tci denotes the present isothermal crystallization temperature.

b AH denotes the enthalpy of crystallization.

c if denotes the time at which the crystallization is completed.

d imax denotes the time at which the crystallization rate is the maximum.

e h ¡2 denotes the time at which the crystallization is carried out for a half.

f n denotes the Avrami index.

g K denotes the rate constant of crystallization.

Source: Reprinted with permission from [107], M. Z. Rong et al., Polymer 42, 167 (2001). © 2001, Elsevier Science.

a Tci denotes the present isothermal crystallization temperature.

b AH denotes the enthalpy of crystallization.

c if denotes the time at which the crystallization is completed.

d imax denotes the time at which the crystallization rate is the maximum.

e h ¡2 denotes the time at which the crystallization is carried out for a half.

f n denotes the Avrami index.

g K denotes the rate constant of crystallization.

matrix becomes immobilized as a result of interfacial binding, giving rise to an effective increase in the size of the nanoparticles. The effective volume fraction of the partially immobilized material ^ is given by [210]

where R stands for the radius of nanoparticles and AR is the increment of the effective particle size. By replacing with in Kerner's equation for a Poisson ratio of 0.5, the elastic modulus of nanocomposites can be described as

A comparison between the testing data of nano-palladium (1-2 nm)/PMMA and Eq. (10) indicated a good agreement, provided AR = 2.25 nm. This suggests that the effective radius of the immobilized matrix environment could be much greater than that of the nanoparticles. The stronger the interfacial interaction is, the thicker is the interlayer [171]. Similarly, Sumita et al. studied the energy dissipation of PP-based nanocomposites under the circumstances of dynamic mechanical testing [159]. According to the concept of the effective volume fraction of the dispersed phase, which assumes that the "immobilized" matrix associated with the interphase does not contribute to energy loss, they yielded an expression for the relative loss modulus,

where E" and E^ represent the loss moduli of the composites and the matrix, respectively. By substituting the measured values of E^'/Em into Eq. (11), one can obtain The data showed that the effective volume fraction increased with increasing filler content and decreasing filler size.

In fact, most of the current theories valid for conventional composite systems assume an isotropic and homogeneous matrix filled with spherical particles. In this context, Young's moduli of the composites should be independent of the dimension of the dispersed phase. The local stresses in these composites under load are only dependent on the ratio of the distance between the particles of the dispersed phase and the dimensions of these particles. This ratio is often a constant in the case of a certain volume fraction of the dispersed phase and a certain spatial distribution of the particles. Evidently, this is not the case, especially when nanoparticles, characterized by high surface area, are incorporated. Therefore, the commonly accepted theories have to be modified in the case of nanocomposites, as shown by the above-mentioned works. Vollenberg and Heikens treated this problem in another way [172]. They clearly demonstrated the considerable dependence of Young's modulus on filler size (ranging from 35 nm to 100 /¿m); it was indicated that effects like adsorption of additives, particle size distribution, dewetting, and matrix morphology cannot explain the phenomenon well. The remaining explanation is that the solidification of the matrix is locally activated at a free filler particle surface, leading to a particular morphology of the polymer surrounding the filler particles, such as segmental orientation and improved packing in the neighborhood of the filler surface. For nanoparticles, the interparticle distance is so small that a homogeneous matrix material of a higher modulus polymer is reasonably assumed to be created, as supported by the results of annealing and solid-state nuclear magnetic resonance experiments [211].

It is also of importance to note that the formation of a relatively compliant layer at the interface tends to hinder a complete stress transfer under low stress and thus masks the stiffness of the nanoparticles. Table 11 gives evidence of such a masking effect provided by grafting polymers (PBA, PVA, and PEA, for example) on various nanoparticles. In this or a similar case, the above-discussed relationship between composite modulus and effective volume fraction of the interphase, consisting of an immobilized matrix polymer, is no longer applicable.

To avoid complicated geometric effects in composite materials, Shang and co-workers proposed a thermodynamic model based on the work of adhesion [170]. It was hypothesized that the interfacial bond strength can be quantified by this parameter. The rationality of this assumption lies in the breakdown of the interface interaction required to do certain work. Accordingly, Young's modulus of a composite at a given filler loading is written as a function of the work of adhesion, Wa:

where Eco is the modulus of the composite with the weakest interfacial bond, Wao is the work of adhesion of the composite with the weakest interfacial bond, and KEc is a constant, independent of the filler volume content and experimentally determined from the slope of log Ec versus 1/Wa. When the Young's modulus of a composite with its corresponding Wa is known, Eq. (12) can be used to predict the Young's modulus of another composite with another Wa but with the same filler content. Results from SiO2/EVA composites verified that Eq. (12) leads to good agreement for both, composites filled with micro- and nanosized particles. Furthermore, the data indicated that the composites that incorporated smaller particles have a higher KEc value. Since the KEc values show how the Young's modulus changes with Wa, the dependence of KEc on particle size proves that the effect of the interfacial bond on Young's modulus depends on filler particle size.

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