Application of a two-phase model to the viscoelastic data

where ρ is the density at temperature, T and R is the universal gas constant (8.314 J K⁻¹mol⁻¹).

From the measurements shown in Fig. 5.12, G⁰_N for the PC grades (determined at 160 ^◦C) is between 1.6 and 2.5 MPa. Using Eq. 5.6, Me is calculated to be between 1700 and 2700 g mol⁻¹. These values are consistent with literature value of 2500 g mol⁻¹ for PC (Jordan and Richards, 2000).

The shift factors shown in Figs. 5.7 and 5.11 are used to produce reduced frequency-modulus mastercurves for the range of nanocomposites. These are illustrated in Fig. 5.13. These mastercurves exhibit two important differences relative to those of the equivalent matrix polymers: (1) the presence of a second plateau instead of a zone of terminal flow; and (2) a loss tangent that is less than unity throughout the extended frequency range. At low reduced frequencies the second plateau has a modulus of ∼10⁵ Pa (see Fig. 5.13(a)). This behaviour was first observed by P¨otschke et al. (2002) for PC-MWCNT nanocomposites with more than 2 wt% filler content, sufficient to form a percolated network within the matrix system. Similar observations of network percolation for PC-MWCNT were reported in the work of Alig et al. (2008) and of Skipa et al. (2010). This second plateau has a modulus, approximately 1.5 orders of magnitude smaller than the rubbery plateau arising from the entanglement network, and suggests the presence of an interconnected network of nanotubes, and hence of rheological percolation.

5.6.1 Application of a two-phase model to the viscoelastic data

Song and Zheng (2010) proposed a two-phase model to predict the elastic moduli of polymers filled with nanoparticles. This model is parameterised through a strain amplification factor parameter, A_f, storage G^′_CNT and loss G”_CNT moduli parameters representing the filler net-work, and an exponent α related to the frequency dependence of this filler network. The model includes immobilised chains on the rigid filler surface and the interphase, if any, into the filler phase. This is because the chain confinement or chain absorption is localised to the filler surface and generally does not affect the behaviour of the bulk polymer.

(a)

(b)

Figure 5.13: Reduced frequency mastercurves of (a) storage modulus and (b) loss modulus of PC-MWCNT (3 wt%) of varying molar mass (Mw= 33600 - 50500 g mol⁻¹) at a strain

amplitude of 0.5%

A_f describes the hydrodynamic effect of exaggerated strain of the polymer chains in the presence of rigid fillers. Song and Zheng’s approach assumes perfect bonding between the filler and matrix interface, and thus A_f is only dependant on the filler loading content as the relaxation timescale of the filler phase is beyond the range of frequencies that can be experimentally tested. The filler phase is treated as an elastically rigid network structure,

and in accordance to a microrheological model based on fractal concept, G’ of the filler phase is weakly dependent on frequency whereas G” is almost independent of frequency (Song and Zheng, 2010). The contribution of the matrix and the filler phases are combined to estimate G’ and G” of nanocomposites, expressed by (Song and Zheng, 2010)

G’(ω) = A_f

Song and Zheng’s model was applied to the viscoelastic mastercurves of all matrix polymers and nanocomposites in order to identify the nature of the changes in structure arising between polymer and nanocomposite. Since G’ of unfilled PC grades at the lower limit of the reduced frequency rage were deemed to be an artefact, values that deviate from the relationship G^′ ∝ ω² were not considered in the model. In the case of PC-MWCNT, only data in the frequency range corresponding to the unfilled grades were applied to the model. Optimisation of the parameters was carried out with n = 5 using Matlab, and fitted results are shown in Fig. 5.14 for all grades of filled and unfilled PC.

Table 5.1 shows that A_f for all PC grades did not exceed 1.20. The stiffness of the nanotube network G’_CNT ∼10⁵ Pa for all grades, and G”_CNT ∼10⁴ Pa. The frequency dependence of the network was negligible, with α∼10⁻². Thus, the model is consistent with the view that the present systems do not exhibit pronounced strain amplification, and that the nanotubes agglomerate into a mostly elastic network with stiffness ∼10⁵Pa. The stiffness of this network is probably connected to nanotube bending modes (Rubinstein and Colby, 2003).

Table 5.1: Parameters obtained from Song and Zheng’s two-phase model, with n = 5, applied to the data of all PC grades

Material Strain amplification factor, A_f Frequency exponent, α log G^′_CNT log G”_CNT

2205 0.93 0.09 4.94 4.15

2405 1.20 0.04 5.08 4.26

2805 0.93 0.06 4.98 4.07

3105 0.71 0.01 4.95 3.90

The confinement of PC chains, if any, must therefore be localised to the surface of CNTs or CNT-agglomerates. This is consistent with the T_g measurements presented in Chapter

(a) (b)

(c) (d)

Figure 5.14: The polynomial fit of PC-MWCNT (red solid line) and unfilled PC (blue solid line) of grades (a) 2205 (circle), (b) 2405 (up triangle), (c) 2805 (down triangle) and (d) 3105 (square) employed in the two-phase model (Song and Zheng, 2010). Only unfilled PC data that followed the relationship G^′∝ ω², and the corresponding PC-MWCNT of the selected unfilled PC reduced frequency range were considered in the model. The cross-over

frequency is used to determine the characteristic relaxation time

4, and suggests that the earlier onset of deviation from linearity observed during the ampli-tude sweeps in Fig. 5.1 for PC-MWCNT are caused by a destruction of the CNT network.

Such a percolation network is known to exhibit gradual build-up with time in flocculation experiments, as observed in Section 5.4.1 and reported by Richter et al. (2009), arising from diffusion-controlled agglomeration of MWCNTs (Zeiler et al., 2010).

Fig. 5.15 illustrates the ratios of viscoelastic moduli of filled materials to those of pure materials as a function of reduced frequency. A ratio that deviates from unity is a signature that the nanotubes are influencing the viscoelastic response. The influence is greatest at

the lowest reduced frequencies, where the pure polymer undergoes flow but the filled system exhibits the second plateau. Here the ratio of storage moduli exceeds 10³and the ratio of loss moduli exceeds 10. At higher reduced frequencies the ratio is close to unity. A noteworthy feature is that the onset of the deviation from unity occurs at a lower reduced frequency (and hence longer timescale) with increasing molar mass of the matrix polymer. Due to the substantial differences in size between polymer chains and CNTs, it is not unreasonable to assume that polymer mobility is the main driver for any rearrangement of the filler network, and hence that the dominant relaxation mechanism observed is that of the polymer matrix.

Figure 5.15: Ratios of storage modulus (solid line) and loss modulus (dashed line) of filled PC-MWCNT (3 wt%) to unfilled PC of matching matrix molar mass (Mw = 33600 - 50500 g mol⁻¹) as a function of reduced frequency, determined at a strain amplitude of

0.5%