Two-step electrical percolation in nematic liquid crystals filled with multiwalled carbon nanotubes

Percolation of carbon nanotubes (CNTs) in liquid crystals (LCs) opens the way for a unique class of anisotropic hybrid materials with a complex dielectric constant widely controlled by CNT concentration. Percolation in such systems is commonly described as a one-step process starting at a very low loading of CNTs. In the present study the two-step percolation was observed in the samples of thickness 250 $\mu \mathrm{m}$ obtained by pressing the suspension between two substrates. The first threshold concentration, $C_n^{p_1}\sim {10}^{-4}$ wt.%, was sensitive to temperature and phase state of LC, while the second one, $C_n^{p_2}\sim {10}^{-1}$ wt.%, remained practically unchanged in the temperature tests. The two-stage nature of percolation was explained on a base of mean-field theory assuming core-shell structure of CNTs.

Figure 1

Microphotographs of the layers of 5CB-CNT suspensions filled in the cells with a thickness of 50 $\mu \mathrm{m}$. Concentration of CNTs in the samples 1, 2, 3, and 4 is $0.005$, $0.05$, $0.25$, and $0.5$ wt. %, respectively. Series (a) and (b) correspond, respectively, to C and P series of samples. It is evident that the amount of nanotubes in both series is comparable up to $C_n=0.05$ wt.% but begins to differ at higher $C_n$. The data correspond to the nematic phase, $T={25}^{\circ }C$.

Figure 2

Electrical conductivity $\sigma$ versus CNT concentration $C_n$ curves for the layers of 5CB-CNT suspensions with a thickness of 50 $\mu \mathrm{m}$ (a) and 250 $\mu \mathrm{m}$ (b). Lines are added to guide the eyes. Curve 1 corresponds to the C series, while curves 2 and 3 correspond to the P series. Temperature of measurements was ${25}^{\circ }C$ for curves 1 and 2 and ${55}^{\circ }\mathrm{C}$ for curve 3. The numbers I and II in (b) mark concentration ranges of the first and second percolation processes. The insets present curve 2 (P series, ${25}^{\circ }\mathrm{C}$) or its parts in double logarithmic scale along with the fitting curves according to Eq. (2).

Figure 3

Effective dielectric constant of 5CB-CNT suspension ${\epsilon }^{{}^{\prime }}$ as a function of CNT concentration $C_n$ for the samples of thickness $250\mu \mathrm{m}$. Lines are added to guide the eye. Curve 1 corresponds to the C series, while curves 2 and 3 correspond to the P series. The temperatures of the measurements were ${25}^{\circ }\mathrm{C}$ for curves 1 and 2 and ${55}^{\circ }\mathrm{C}$ for curve 3. The inset represents curve 2 in a double logarithmic scale along with the best fit to it with a function ${\epsilon }^{{}^{\prime }}\propto (C_n-C_n^{p_1})$.

Figure 4

The percolation concentration $C_n^p$ and the transport index $t$ as the functions of temperature $T$ for the first (a) and second (b) percolation stages. The dashed line represents the clearing temperature of LC 5CB, $T_c=35.4^{\circ }\mathrm{C}$.

Figure 5

Relative electrical conductivity $\sigma /{\sigma }_n$ (a) and its derivative $dln(\sigma /{\sigma }_n)/d{\phi }_n$ (b) versus volume fraction of CNTs ${\phi }_n$ calculated for different values of parameter $\delta$ at ${\phi }_n^{p_c}=0.25$ vol.% ($\approx 0.5$ wt.%). The calculation was performed assuming that ${\sigma }_0/{\sigma }_n={10}^{-6}$ and ${\sigma }_s/{\sigma }_n={10}^{-3}$. The inset to (a) presents model of CNT coil with the core-shell structure.

Figure 6

Exemplary dependencies of percolation concentration through shells, ${\phi }_n^{p_s}$, versus relative width of the shell $\delta$ calculated for two different values of percolation concentration through cores, ${\phi }_n^{p_c}=0.25$ vol.% and ${\phi }_n^{p_c}=33.3$ vol.%. The calculations were performed assuming ${\sigma }_0/{\sigma }_n={10}^{-6}$ and ${\sigma }_2/{\sigma }_n={10}^{-3}$.