*3.2. Piezoresistivity Computations*

Firstly, the dependence of the conductivity distribution on the unit cell size and the isotropy of the non deformed composite were investigated checked (see Figure 5). The peak values of the conductivity coincide for different cell sizes, while the distribution becomes broader with the decrease of the cell size. The conductivity distributions for the composites above the percolation threshold are close to lognormal in agreement with literature [35]. We conclude, that the usage of the periodic boundary conditions (PBC) in Equations (3) and (4) allowed us to obtain the independence on the unit cell size. For further computations, the cell with *n* = 4 was used.

The conductivity dependence on the concentration *p* and deformation *k* is presented in Figure 6. The concentration dependencies of *σ* (Figure 6a) follows the power law [36]. After the deformation, *σ* decreases both along *x*− and *z*−directions, and this is supported by the experimental data [37–39]. Significant anisotropy (more than the order of magnitude) was observed for the deformed composite. It was previously shown, that the *z*−direction is preferable for the percolation in the deformed composites in comparison with *x*. But, being percolated, the composite has lower conductivity along *z*, than one along the perpendicular direction. In contrast to the percolation, where the appearance of only one

conductive path is necessary, the conductivity strongly depends on the number of conductive paths through the unit cell and the total tunnelling distance.

**Figure 5.** (**a**) Empirical PDF of the conductivity of the isotropic composite with 5 vol. % of CNTs with different cell size *n*. (**b**) Empirical CDF of the conductivity in different directions for the isotropic composite with 5 vol. % of CNTs with *n* = 4. 1500 realisations were collected.

**Figure 6.** Conductivity dependence (**a**) on the concentration for the initial and deformed composite in different directions, and (**b**) on the deformation for the samples with 6 vol. % of the CNTs. Symbols stands for the mean values, and lines denote the 95 % confidence interval width.

Note that in the microwave and THz frequency regions the partial alignment of the nanotubes will lead to the increase of the conductivity and imaginary part of the permittivity along the deformation [32,40]. But in studied case the situation is opposite. That is related to very different mechanisms of the interaction of the electromagnetic radiation with the media at different frequencies. At microwave frequencies, the mechanism is the polarisation of the CNT [41]. In case of the direct current and quasistatic frequency range, where the tunnelling mechanism is dominant [42], the conductivity decreases due to the tunnelling distance increase.

#### Boundary Conditions Impact

In the introduction part it was mentioned, that there is the contradiction in the modelling results presented in literature, and several authors observe the maximum of the conductivity in the *z*-direction for slightly aligned composites. In our case the maximum is observed, if the conductivity is calculated without the PBC in Equations (3) and (4) (see Figure 7). The maximum is reached for *k* ≈ 1.5. The conductivity along the *x*-direction decreases, similarly to presented in Figure 6b.

**Figure 7.** Conductivity dependence on the deformation for the samples with 4 vol. % of the CNTs, computed without boundary conditions. Symbols stands for the mean values, and lines denote the 95% confidence interval width.

Both models with and without PBC should provide similar results for the infinitely large unit cell. But in the case of the PBC model, the independence of the conductivity on the cell size is demonstrated. Thus, the appearance of the maximum of the conductivity is the cell size related effect of the non-PBC model. Both PBC and non-PBC models are suitable for applications. For the thin films filled with aligned nanotubes the non-PBC model is preferable, while with the composite size increase, the PBC model should be used.
