Simultaneous Rheoelectric Measurements of Strongly Conductive Complex Fluids

We introduce an modular fixture designed for stress-controlled rheometers to perform simultaneous rheological and electrical measurements on strongly conductive complex fluids under shear. By means of a nontoxic liquid metal at room temperature, the electrical connection to the rotating shaft is completed with minimal additional mechanical friction, allowing for simultaneous stress measurements at values as low as 1 Pa. Motivated by applications such as flow batteries, we use the capabilities of this design to perform an extensive set of rheoelectric experiments on gels formulated from attractive carbon-black particles, at concentrations ranging from 4 to 15 wt %. First, experiments on gels at rest prepared with different shear histories show a robust power-law scaling between the elastic modulus $G_0^{\prime }$ and the conductivity ${\sigma }_0$ of the gels—i.e., $G_0^{\prime }\sim {\sigma }_0^{\alpha }$, with $\alpha =1.65\pm 0.04$, regardless of the gel concentration. Second, we report conductivity measurements performed simultaneously with creep experiments. Changes in conductivity in the early stage of the experiments, also known as the Andrade-creep regime, reveal for the first time that plastic events take place in the bulk, while the shear rate $\dot{\gamma }$ decreases as a weak power law of time. The subsequent evolution of the conductivity and the shear rate allows us to propose a local yielding scenario that is in agreement with previous velocimetry measurements. Finally, to establish a set of benchmark data, we determine the constitutive rheological and electrical behavior of carbon-black gels. Corrections first introduced for mechanical measurements regarding shear inhomogeneity and wall slip are carefully extended to electrical measurements to accurately distinguish between bulk and surface contributions to the conductivity. As an illustrative example, we examine the constitutive rheoelectric properties of five different grades of carbon-black gels and we demonstrate the relevance of this rheoelectric apparatus as a versatile characterization tool for strongly conductive complex fluids and their applications.

Figure 1

(a) Schematic of the rheoelectric test fixture. The sample is sandwiched between two parallel plates ($R=20\mathrm{mm}$) that form a two-electrode closed electrical circuit. The rotating upper plate is driven by a stress-controlled rheometer that applies stress and measures strain. Liquid metal (EGaIn) in the solvent trap provides a low-friction electrical contact to the rotating shaft. The plates are sputter coated in gold to reduce contact resistance. (b) Photograph of the rheoelectric fixture.

Figure 2

(a) Total measured torque $\mathcal{M}$ resulting from the friction of the electrical connection to the rotating shaft as a function of the angular velocity ${\mathrm{\Omega }}_{\mathrm{rot}}$ of the upper plate. Measurements are conducted in the absence of a sample and for different electrical contacts: a gold-leaf spring, liquid metal (EGaIn), and a salt solution. (Inset) Temporal fluctuations of the frictional torque $\mathcal{M}$ from the EGaIn alloy are represented by the error bar shown for each angular velocity and can, in general, be neglected. (b) Setup dc resistance ${\mathcal{R}}_0$ measured in the absence of a sample as a function of ${\mathrm{\Omega }}_{\mathrm{rot}}$ for two electrical conductors: either a gold-leaf spring or a liquid metal (EGaIn). Error bars are determined by calculating the standard deviation of the temporal signal over 180 s. The use of a salt solution as a conductor does not allow for dc measurements, as the ionic species will polarize only under dc. (c) Measured flow curves showing shear stress $\tau$ vs shear rate $\dot{\gamma }$ for a Newtonian calibration oil and illustrating the friction correction resulting from EGaIn. The red filled circles represent the friction calibration performed with EGaIn in the absence of a sample prior to the flow curve measurement. The green triangles and black squares represent, respectively, the flow curves measured with and without the liquid metal. The open black squares represent the final corrected flow curve.

Figure 3

Nyquist plots of complex impedance for three different classes of conductive complex fluids and their corresponding equivalent circuits. (a) Electronic conductor: suspension of attractive soot particles (carbon black, VXC72R) dispersed in mineral oil at 8 wt %. The conductivity is measured at different shear rates coded in color: $\dot{\gamma }=0$, 10, 30, 50, 100, and $200{\mathrm{s}}^{-1}$ from dark to light color. (b) Ionic conductor (Oakton $2764\mu \mathrm{S}/\mathrm{cm}$ KCl solution) measured under static conditions for $H=750\mu \mathrm{m}$. The linear fit corresponds to the model blocking circuit shown in the inset. (c) Mixed conductor: lithium polysulfide suspension containing $2.5M$ ${\mathrm{Li}}_2{\mathrm{S}}_8$, $0.5M$ LiTFSI, and $1M$ ${\text{LiNO}}_3$, with 1.5-vol % Ketjenblack carbon measured under static conditions. We perform ac tests for $10^2\le \omega \le 10^4\mathrm{Hz}$.

Figure 4

(a) Nyquist plot of the complex impedance for an ionic conductor (Oakton $2764\mu \mathrm{S}/\mathrm{cm}$ KCl solution) measured under static conditions for various gap heights, $H=750$, 500, 400, 330, and $250\mu \mathrm{m}$. This suspension is characterized by an Ohmic resistor in parallel with an imperfect capacitor [$Z^{*}(\omega )={\mathcal{R}}_i+1/(i\omega C_i{)}^{\alpha }$], represented by a constant phase element of phase $\alpha$ and appearing in the Nyquist plot as a line of slope $\tan (\alpha \pi /2)$ that intercepts with the horizontal axis at $Z^{\prime }={\mathcal{R}}_i$. The colored lines correspond to the best linear fit of the data for each gap setting. (b) Ionic resistance ${\mathcal{R}}_i$ as determined in (a) vs the gap height $H$. The present data set shows that the sample resistance is divided into two contributions: a contact resistance ${\mathcal{R}}_c$ which corresponds to the intercept and a bulk resistance associated with the slope [see Eq. (4)]. The black line is the best linear fit of the data and leads to ${\mathcal{R}}_c=0.94$ $\mathrm{\Omega }$ and ${\sigma }_b=(2.76\pm 0.01)\mathrm{mS}/\mathrm{cm}$.

Figure 5

(a) Stress $\tau$ vs apparent shear rate ${\dot{\gamma }}_a$. (b) Apparent conductivity ${\sigma }_a$ vs the apparent shear rate ${\dot{\gamma }}_a$ measured in an 8-wt % VXC72R carbon-black gel during decreasing ramps of stress ranging from $\tau =100\mathrm{Pa}$ to $\tau =0\mathrm{Pa}$. Every test is performed in $N=100$ steps of 1 Pa each, and a different duration per point $\mathrm{\Delta }t$ is used for each experiment. From top to bottom, the values of $\mathrm{\Delta }t$ are 1, 6, 15, 21, 30, and 120 s.

Figure 6

Elastic modulus $G_0^{\prime }$ of carbon-black gels vs conductivity ${\sigma }_0$ measured at rest, 90 min after the end of a stress ramp decreasing from 100 to 0 Pa and composed of 100 steps. Each point corresponds to different step durations $\mathrm{\Delta }t$ during the ramp, from $\mathrm{\Delta }t=1\mathrm{s}$ to $\mathrm{\Delta }t=120\mathrm{s}$, while symbols stand for different concentrations of carbon black (VXC72R): $c=4$ (filled triangle), 6 (filled circle), and 8 wt % (filled square). Open symbols represent an instantaneous quench from $\tau =100$ Pa to $\tau =0$ Pa. Data for the 8-wt % gel corresponds to the experiment reported in Fig. 5. Oscillatory measurements are performed at a strain ${\gamma }_0=0.2\%$ and a frequency $f=1\mathrm{Hz}$. The dashed line corresponds to the best power-law fit of the data for ${\sigma }_0\ge 0.1\mathrm{mS}/\mathrm{cm}$, i.e., $G_0^{\prime }=\widetilde{G_0^{\prime }}{\sigma }_0^{\alpha }$, with $\widetilde{G_0^{\prime }}=400\mathrm{Pa}(\mathrm{cm}/\mathrm{mS}{)}^{\alpha }$ and $\alpha =1.65\pm 0.04$.

Figure 7

Creep experiment in an 8-wt % carbon-black gel (VXC72R) at $\tau =22\mathrm{Pa}$. (a) Shear-rate response $\dot{\gamma }(t)$ and apparent conductivity response ${\sigma }_a(t)$. The shear rate is derived numerically from the strain measured by the rheometer to increase the precision. The jump observed in the curves at $t=3800\mathrm{s}$ marks the onset of a total wall-slip regime followed by the growth of a transient shear band [47] before the sample reaches a homogeneous steady state. (b) Apparent conductivity ${\sigma }_a$ vs strain $\gamma$ for the same data set. In both (a) and (b), the vertical dashed line marks the three different flow regimes discussed in the text: primary creep, heterogeneous flow, and steady-state flow.

Figure 8

Creep experiment in a carbon-black VXC72R suspension ($c=8\mathrm{wt}\%$) at $\tau =30\mathrm{Pa}$ for different gaps ($H=660$, 500, and $330\mu \mathrm{m}$). (a) Shear-rate response $\dot{\gamma }(t)$. (b) Apparent conductivity response ${\sigma }_a(t)$. (Inset) Estimated bulk conductivity ${\sigma }_b(t)$ normalized by the gap $H$. The contribution of the contact resistance is estimated using Eq. (4) and the measurements at multiple gaps for $t=0\mathrm{s}$.

Figure 9

Steady-state flow and conductivity curves in an 8-wt % carbon-black gel (VXC72R). (a) Flow curves showing the effects of the shear-inhomogeneity correction (derivative and single-point methods) as well as the slip correction. The “corrected” flow curves are obtained after correcting for shear inhomogeneity alone, while the true flow curve is obtained after correcting for both shear-inhomogeneity and slip effects. (b) Conductivity curves showing the effects of the shear-inhomogeneity correction as well as the contact-resistance correction. The corrected conductivity curves are obtained after correcting for shear inhomogeneity alone, while the bulk-conductivity curve is obtained after correcting for both shear-inhomogeneity and contact-resistance effects.

Figure 10

Slip velocity $V_s$, bulk conductivity ${\sigma }_b$, and specific contact resistance ${\rho }_c$ obtained from the slip and contact-resistance corrections performed on the steady-state flow curves in an 8-wt % carbon-black gel (VXC72R). The vertical dashed lines indicate the limits of the three regimes discussed in the text. Displayed from large to low stresses, the fully fluidized regime for $\tau \ge 55\mathrm{Pa}$, the plug flow with slip at the walls for $55\mathrm{Pa}\ge \tau \ge 20\mathrm{Pa}$, and the shear-induced structuration (the formation of vorticity-aligned flocs) for $\tau \le 20\mathrm{Pa}$ are shown. (a) Slip velocity $V_s(\tau )$. (b) Bulk conductivity ${\sigma }_b(\tau )$. (c) Specific contact resistance ${\rho }_c(\tau )$.

Figure 11

Sets of corrected (a) flow curves and (b) conductivity curves for carbon-black gels made with carbon-black particles of different grades: Ketjenblack EC600JD, VXC72R, Elftex 8 at weight concentration $c=8\mathrm{wt}\%$, and Monarch 120 at $c=8$ and 15 wt %.

Figure 12

(a) Temporal evolution of the apparent conductivity ${\sigma }_a$ in an 8-wt % carbon-black gel (VXC72R) during the 90-min recovery period that follows a decreasing ramp of stress, from $\tau =100\mathrm{Pa}$ to $\tau =0\mathrm{Pa}$. The stress and conductivity data associated with the ramp are pictured in Fig. 4 in the main text (duration per point, $\mathrm{\Delta }t=6\mathrm{s}$). The conductivity reaches a steady-state value ${\sigma }_0=1.62\mathrm{mS}/\mathrm{cm}$ after about an hour. (b) Viscoelastic moduli $G^{\prime }$ and $G^{\prime \prime }$ vs time, measured with oscillations of small amplitude (${\gamma }_0=0.2\%$) at $\omega =6.3\mathrm{rad}/\mathrm{s}$, after the 90-min recovery period shown in (a). The viscoelastic moduli are constant with values $G_0^{\prime }=805\pm 4\mathrm{Pa}$, and $G_0^{\prime \prime }=114\pm 6\mathrm{Pa}$. The values of ${\sigma }_0$ and $G_0^{\prime }$ are reported in Fig. 6 in the main text, and they are further determined systematically for different ramp durations $\mathrm{\Delta }t$ in carbon-black gels of various concentrations to assemble the data set reported in Fig. 5 in the main text.

Figure 13

Creep experiments in an 8-wt % carbon-black gel (VXC72R) at various stresses: $\tau =45$, 40, 35, and 30 Pa. (a) Shear-rate response ${\dot{\gamma }}_a(t)$ and (b) apparent-conductivity response ${\sigma }_a(t)$. Before each experiment, the gel is presheared at $\dot{\gamma }=200{\mathrm{s}}^{-1}$ for 5 min to erase the previous flow history, then left to restructure for a waiting time $t_w=1200\mathrm{s}$ at $\tau =0\mathrm{Pa}$.

Figure 14

Creep experiment in an 8-wt % carbon-black gel (VXC72R) at $\tau =22\mathrm{Pa}$. Apparent shear-rate response ${\dot{\gamma }}_a(t)$ and apparent-conductivity response ${\sigma }_a(t)$ on doubly logarithmic scales to emphasize the power-law characteristics of the primary-creep regime. The same data is shown as that presented in Fig. 7. The shear rate is calculated numerically from the strain measured by the rheometer to increase the precision. The red lines correspond to the best power-law fits of the data, which leads to ${\dot{\gamma }}_a(t)=A{t}^{\alpha }$, with $A=(2.3\pm 0.1)\times {10}^{-3}{\mathrm{s}}^{-\alpha -1}$ and $\alpha =-0.90\pm 0.02$, and ${\sigma }_a(t)=B{t}^{\beta }$, with $B=1.26\pm 0.01\mathrm{m}\mathrm{S}{\mathrm{s}}^{-\beta }/\mathrm{cm}$ and $\beta =0.035\pm 0.002$. We note that the power-law fit of the conductivity is hardly distinguishable from a logarithmic fit ${\sigma }_a(t)=C\ln (t)+D$ for the data under study, with $C=0.0560\pm 0.0005\mathrm{mS}/\mathrm{cm}$ and $D=1.23\pm 0.01\mathrm{mS}/\mathrm{cm}$.

Figure 15

(a) Flow curves of an 8-wt % carbon-black gel (VXC72R) measured for various gap heights, $H=750$, 500, and $330\mu \mathrm{m}$, together with the true flow curve corrected for slip. The open symbols indicate the region associated with the formation of vorticity-aligned flocs. The boundary for the onset of this regime is given by ${\dot{\gamma }}_c=3650/H^{1.4}$, with $H$ given in micrometers, independently of the CB concentration [18]. (b) Plot of the measured values of apparent shear rate ${\dot{\gamma }}_a$ vs $1/H$ for various imposed stresses: $\tau =68$, 60, 50, 40, 30, and 25 Pa. The colored lines correspond to the best linear fits of the data and allow us to extract the true shear rate ${\dot{\gamma }}_t$ and the slip velocity $V_s$ for each imposed stress $\tau$ using Eq. (6).

Figure 16

(a) Conductivity curves of an 8-wt % carbon-black gel (VXC72R) measured for various gap heights, $H=750$, 500, and $330\mu \mathrm{m}$, together with the bulk-conductivity curve. (b) Inverse of the apparent resistivity $1/{\sigma }_a$ vs $1/H$ for various stresses: $\tau =68$, 60, 50, 40, 30, and 25 Pa. The colored lines correspond to the best linear fits of the data and allow us to extract the bulk resistivity $1/{\sigma }_b$ and the specific contact resistance ${\rho }_c$ for each stress $\tau$ using Eq. (8).