# Percolation and Transport Properties in
## 3 Results and Discussion  
### Percolation Computations  
The comparison of the nanotubes angular distribution for the isotropic and heavily deformed composite is presented on Figure 2. For the visualisation purposes, the ellipsoids with small aspect ratio of 5 and the cubic unit cell with side length of 60 nm were used.  
Figure 1: Probability density function (PDF) for the zenith angle for different deformations (\(k\)) for the carbon nanotubes (CNTs).  
For the computations of the percolation threshold, the cubic unit cell with \(n=3\) was considered. Firstly, the system with uniform and fully aligned (\(\theta=0\), \(k=\infty\)) angular distribution of the ellipsoids was studied. The empirical cumulative probability distribution (CDF) for the percolation concentration \(p_{c}\) in non-deformed and aligned composites is presented in Figure 3. As it was expected, the distribution of the uniform composite coincides in different directions. Upon deformation, the percolation concentration increases in both directions, and the slope of CDF changes as well.  
Obtained distribution follows the Weibull law [34]:  
\[CDF_{W}(p_{c},\lambda,m)=1-e^{-(p_{c}/\lambda)^{m}} \tag{5}\]  
Figure 3: Empirical cumulative probability function for the percolation concentration in deformed and non-deformed composites (symbols), and Weibull cumulative probability distributions (CDFs) (5) (solid curves).  
Figure 2: Visualisation of the CNT-filled composite with \(k=1\) and \(k=10\).  
where \(\lambda\) is the scale factor, and \(m\) is the slope or Weibull modulus. The mean value of percolation concentration is \(<p_{c}>=\lambda\Gamma(1+1/m)\approx\lambda\), where \(\Gamma\) is the Euler gamma function. The values for \(\lambda\) of 0.12 and 0.079, and \(k\) of 53.69 and 6.00 were obtained for fully oriented composite in the \(x-\) and \(z-\)directions, correspondingly.  
The dependencies of \(\lambda\) and \(m\) parameters on the deformation coefficient were also studied. The results were collected in Figure 4. For both directions, the Weibull scale factor \(\lambda\) increases with the deformation, so the minimal concentration is achieved for non deformed composite. This conclusion is supported with the measurement results [22; 23; 24].  
Anisotropic properties of the deformed composite are confirmed by distinct behaviour of \(\lambda\) and \(m\) along the \(x-\) and \(z-\)directions. For low values of \(k\) (up to 3), the dependencies of \(\lambda\)_versus_\(k\) demonstrate linear increase both along \(x\) and \(z\) with the lower slope along the \(z-\)direction. In contrast to the scale factor \(\lambda\), the Weibull modulus \(m\) demonstrates the decrease in the direction of the deformation and increase in the perpendicular one. This can be understood in terms of very different dimensionality of the inclusions network. Being isotropically distributed, the network of ellipsoids has 3D dimensionality, and, upon the deformation, the dimensionality changes toward 1D along the \(z-\)direction and 2D along the perpendicular one. As a result, a very small number of the particles (three, in the studied situation) may percolate with non-zero probability along \(z\) for fully aligned system. It is expected that both \(\lambda\) and \(k\) will asymptotically approach the above mentioned values for \(k=\infty\).