Enhanced tunneling conductivity induced by gelation of attractive colloids

We show that the formation of a gel by conducting colloidal particles leads to a dramatic enhancement in bulk conductivity, due to interparticle electron tunneling, combining predictions from molecular-dynamics simulations with structural measurements in an experimental colloid system. Our results show how colloidal gelation can be used as a general route to huge enhancements of conductivity, and suggest a feasible way for developing cheap materials with novel properties and low metal content.

Figure 1

(a) Phase diagram for the $\lambda =0.03$ SW system. The binodal and the critical point are represented by open and closed squares, respectively. The gel points, obtained by quenching to state points at low temperature $T_q$, are represented by circles. (b) Average number of bonds $n_b$ for different $\varphi$ as a function of $t$, expressed in units of $D\sqrt{m/u_0}$. Three different stages of aggregation for $\varphi =0.03125$ are shown for (c) $t=0.01$, (d) $t=2.3\times {10}^3$, and (e) $t=6.9\times {10}^4$. The configuration shown in (e) represents an arrested colloidal gel, where the particle positions do not display any subsequent time evolution.

Figure 2

Time evolution of percolation probability $P\left(\delta \right)$ for the connectivity distance $\delta /D$ for (a) $\varphi =0.03125$, (b) $\varphi =0.125$, and (c) $\varphi =0.3125$. The vertical dashed line in (a)–(c) indicates $\delta /D=\lambda =0.03$. For each $\varphi$, each curve represents $P\left(\delta \right)$ calculated at a specific time ranging from $t=0.013$ to $t=6.86\times {10}^4$, illustrated with the color (gray) bar at right. (d)–(f) Conductivity probability $P\left(\sigma \right)$ as function of $-\frac{\xi }{2D}\ln \left(\sigma \right)$ for tunneling decay length fixed at $\xi /D=0.1$.

Figure 3

(a) Time evolution of the critical connectedness distance ${\delta }_c$ for gels simulated at different $\varphi$, with $t$ in units of $D\sqrt{m/u_0}$. (b) Time evolution (in seconds) of the critical distance extracted from the measured spatial positions of PMMA particles in a polymer-colloid system [7]. For times larger than about ${10}^4$ s the system is in an arrested gel state.

Figure 4

Time evolution of conductivity $\sigma$ for $\xi /D=0.1$ during the formation of the colloidal gel. Numerical solution of the tunneling resistor equations shown with symbols; ${\sigma }_{\mathrm{CPA}}$ obtained from Eq. (3) using ${\sigma }_0=0.1$, with solid lines.

Figure 5

Conductivity $\sigma$ as a function of $\varphi$ for the arrested gel state (filled circles), equilibrium SW fluids (filled diamonds), and equilibrium HS fluids (squares). In all cases, $\xi /D=0.01$. Inset: $\varphi$ dependence of $\sigma$ for the arrested gel state calculated for different values of $\xi /D$.

Figure 6

(a) Percolation probability $P\left(\delta \right)$ as a function of the connectivity distance $\delta$ at $\varphi =0.125$ for different time $t$ (in units of $D\sqrt{m/u_0}$) and particle numbers $N$. (b) Fits to the results of (a) using a combination of two error functions.

Figure 7

Finite-size scaling analysis of the ${\delta }_c\left(N\right)$ values extracted from $P({\delta }_c\left(N\right))=1/2$ of Fig. 3. Solid lines are fits to Eq. (A1).

Figure 8

(a) Conductivity probability $P\left(\sigma \right)$ as a function of $-\frac{\xi }{2D}\ln \left(\sigma \right)$ at $\varphi =0.125$ for different values of $t$ (in units of $D\sqrt{m/u_0}$) and particle numbers $N$. The tunneling decay length is $\xi =0.1D$. (b) Fits to the results of (a) using a combination of two error functions.