Impact of tunneling anisotropy on the conductivity of nanorod dispersions

While the tunneling conductance between two spherical-like conducting particles depends on the relative interparticle distance, the wave function overlap between states of two rodlike particles, and so the tunneling conductance, depends also on the relative orientation of the rod axes. Modeling slender rodlike particles as cylindrical quantum wells of diameter $D$ and length $L\gg D$, we calculate the matrix element of the tunneling between two rods for arbitrary relative orientations of the rod axes. We show that tunneling between two parallel rods is about $L/\sqrt{D\xi }$ times larger than the tunneling matrix element for perpendicular rods, where $\xi$ is the tunneling decay length. By considering the full dependence of the tunneling conductance on the angle between rod axes, we calculate within an effective medium theory the conductivity of dispersions of rods with different degrees of alignment. We find that for isotropically oriented rods, the effect of orientation in the tunneling processes is marginal for all rod concentrations. On the contrary, for systems of strongly aligned rods, the enhanced tunneling between nearly parallel rods increases significantly the system conductivity in a relatively large concentration range. Next, we consider systems in which short-range attraction between rods is added, as in dispersions of rods with depletion interaction. We find that the strongly anisotropic attraction promotes enhanced tunneling between neighboring parallel rods, increasing the effective medium conductivity by several orders of magnitude compared to the case in which the angular dependence of tunneling is ignored, even for relatively weak attractions.

Figure 1

Schematic representation of two cylinders, $i$ and $j$, whose axes are tilted by an angle ${\gamma }_{ij}$. The axis of cylinder $i$ is directed along $z$, while that of $j$ is directed along $z^{\prime }$. The axis $x$ is directed along the shortest distance, of length ${\Delta }_{ij}$, between $i$ and $j$. $z_i$ and $z_j$ are the positions of the centers of mass of $i$ and $j$ along their respective axes.

Figure 2

Comparison between the dimensionless tunneling matrix element $|I_{ij}|$ of Eq. (24) (solid lines) and the analytical formula of Eq. (27) (dashed lines). The cylinders are at contact (${\Delta }_{ij}=D$) and the tunneling decay length is fixed at $\xi =0.1D$. (a) $|I_{ij}|$ as a function of $sin{\gamma }_{ij}$ for cylinders with aligned centers of mass ($z_i=z_j$) and different aspect ratios $L/D$. (b) $|I_{ij}|$ as a function of $z_j$ for $z_i=0$, $L/D=50$, and different angles ${\gamma }_{ij}$.

Figure 3

Distribution function $F\left(\gamma \right)$ for the angle $\gamma$ between the axes of two rods. $F\left(\gamma \right)$ is calculated numerically from Eq. (43) using Eq. (41) for different values of the nematic parameter $S$. Note that Eq. (41) implies $F\left(\gamma \right)=F(\pi -\gamma )$.

Figure 4

EMA conductance $g^{*}$ (solid lines) calculated from Eq. (38) for the case of isotropic rod orientations ($S=0$) and different values of $L/D$. The tunneling decay length is $\xi =0.1D$ for all cases. Corresponding results for $g_0^{*}$ [Eq. (40)] are shown by dashed lines. Inset: $g^{*}/g_0^{*}$ (solid lines) for $L/D=200$, 100, and 50, from uppermost to lowermost, compared to $R(S=0,0)=4/e\simeq 1.47$ (dashed line).

Figure 5

EMA conductance $g^{*}$ (solid lines) calculated from Eq. (38) for strongly aligned rods with nematic parameter $S=0.9$ and for different values of $L/D$. The tunneling decay length is $\xi =0.1D$ for all cases. Corresponding results for $g_0^{*}$ [Eq. (40)] are shown by dashed lines. Inset: $g^{*}/g_0^{*}$ (solid lines) for $L/D=200$, 100, and 50, from uppermost to lowermost, compared to $R(S=0.9,0)\simeq 7.93$ (dashed line) obtained from numerical calculation of Eq. (46).

Figure 6

EMA conductances $g^{*}$ (solid lines) and $g_0^{*}$ (dashed lines) calculated from Eqs. (38) and (40) as functions of the nematic parameter $S$ and for different volume fractions. $L/D=100$ and $\xi /D=0.1$, for all cases. Inset: $g^{*}/g_0^{*}$ (solid lines) for the same $\varphi$ values shown in the main panel, compared to $R(S,0)$ (dashed line).

Figure 7

Reduced second-virial coefficient $\chi =\frac{8\lambda }{\pi L^2}{\langle sin\gamma {\int }_0^{L}dzzexp[-\beta W^{\mathrm{A}}(z,\gamma )]\rangle }_{\gamma }-\lambda$, with $W^{\mathrm{A}}(z,\gamma )$ given in Eq. (61). $\chi$ is shown as a function of the potential depth $\varepsilon$ for different values of the range $\lambda$, and for aspect ratio fixed at $L/D=100$.

Figure 8

EMA conductances $g^{*}$ (solid lines) and $g_0^{*}$ (dashed lines) obtained from numerical solution of Eqs. (50) and (56), respectively, with the attraction well potential given in Eq. (61). Different pairs of solid and dashed lines are calculated for different reduced second-virial coefficients $\chi$, with potential range fixed at $\lambda =0.2$. $L/D=100$ and $\xi /D=0.1$, for all cases. Inset: enhancement factor $g^{*}/g_0^{*}$ as a function of $\varphi /{\varphi }_0^{*}$, where ${\varphi }_0^{*}$ is given in Eq. (59). $\chi =2$, 1, $0.5$, and $0.2$ from the uppermost to the lowermost curve. Dashed horizontal lines indicate $4/e\simeq 1.47$ and $L^2/\left(2\pi D\xi \right)\simeq 15915$.

Figure 9

EMA conductances $g^{*}$ (solid lines) and $g_0^{*}$ (dashed lines) obtained from numerical solution of Eqs. (50) and (56), respectively, with the attraction well potential given in Eq. (61). The attraction range is $\lambda =0.05$, $0.1$, $0.2$, and $0.4$ from the uppermost to the lowermost solid and dashed lines, while the reduced second-virial coefficient is fixed at $\chi =1$. $L/D=100$ and $\xi /D=0.1$, for all cases. Inset: enhancement factor $g^{*}/g_0^{*}$ as a function of $\varphi /{\varphi }_0^{*}$, where ${\varphi }_0^{*}$ is given in Eq. (59). $\lambda =0.05$, $0.1$, $0.2$, and $0.4$ from the uppermost to the lowermost curve. Dashed horizontal lines indicate $4/e\simeq 1.47$ and $L^2/\left(2\pi D\xi \right)\simeq 15915$.

Figure 10

Dimensionless tunneling matrix element $|I_{ij}|$ of Eq. (23) as a function of the misalignment $z_{ij}=z_i-z_j$ (with $z_i=0$ and $-L/2\le z_j\le L/2$) of the centers of mass of two cylinders with $L/D=100$, ${\Delta }_{ij}=D$, and $\xi /D=0.1$. The Fermi wave numbers are $k_F=\pi n_F/L$ with $n_F=15$ (dashed lines) and $n_F=30$ (dotted lines). The angle between the rod axes is ${\gamma }_{ij}=0$, $\pi /100$, $\pi /50$, $\pi /10$, and $\pi /2$ (panels from top to bottom). $|I_{ij}|$ is identically zero for ${\gamma }_{ij}=\pi /2$ and $n_F=30$ (lowest panel). Solid lines are the approximated expression for $|I_{ij}|$ given in Eq. (24).