^{1,a)}, K. P. S. Parmar

^{2}, B. Schjelderupsen

^{3}and J. O. Fossum

^{3}

### Abstract

Under application of an electric field, suspensions of the synthetic clay Na-fluorohectorite in a silicone oil aggregate into chain/columnlike structures parallel to . This microstructuring results in a transition in the suspensions’ rheology, from Newtonian to a shear-thinning with a significant yield stress. We study this electrorheology (ER) as a function of and of the particle volume fraction on samples with a large clay particle polydispersity. The flow curves under fixed shear rate are well fitted by the Cho–Choi–Jhon model [M. Cho *et al.*, Polymer46, 11484 (2005); H. J. Choi and M. Jhon, Soft Matter5, 1562 (2009)]; proper scaling of and of the measured shear stress provides a collapse of all flow curves onto a master curve. The corresponding dynamic yield stress scales as , while the static yield stress inferred from disruption tests behaves as . The bifurcation in the rheology when letting the flow evolve under constant shear stress is also characterized; the corresponding bifurcationyield stress scales as with . All measuredyield stresses increase with ; for the static yield stress, a scaling law is found. The three mutually consistent types of measurements are compared with previous measurements on laponite suspensions, and the rheologies of these two types of samples are discussed in light of existing theories of the ER effect.

This work was supported by NTNU and by the Research Council of Norway (RCN) through the RCN NANOMAT and FRINAT programs, as well through a RCN Strategic University Program (SUP) project awarded to the COMPLEX Collaborative Research Team in Norway. Y.M. acknowledges the Egide Organization and CNRS for financial support in traveling between France and Norway, under the Egide Aurora Program (Grant No. 18810WC) and PICS program, respectively. J.O.F. acknowledges travel support from the RCN under the same Aurora Program.

I. INTRODUCTION

II. PHYSICAL BACKGROUND—THE RHEOLOGY OF ER FLUIDS

A. Rheology of ER suspensions when no external electric field is applied

B. Rheology when an external electric field is applied

C. Stress-bifurcation in the rheological behavior

D. Relation between the individual properties of the suspended particles and the material’s rheology

III. EXPERIMENTS

A. Sample preparation

B. Rheology measurements

IV. RESULTS

A. Rheology of the suspensions in the absence of external electric field

B. CSR tests

C. CSS tests

D. Bifurcation in the rheological behavior

V. DISCUSSION

A. The electrorheology of Na-Fh suspensions

B. How does the electrorheology of Na-fluorohectorite and laponite suspensions confront each other and the existing models?

1. Scaling as a function of the applied electric field

2. Scaling as a function of the particle fraction

3. Electric properties of the suspensions

4. Magnitude of the yield stress

5. Range of Mason numbers investigated in the CSR tests

VI. CONCLUSION AND PROSPECTS

### Key Topics

- Suspensions
- 91.0
- Yield stress
- 77.0
- Electric fields
- 50.0
- Bifurcations
- 23.0
- Shear rate dependent viscosity
- 17.0

## Figures

Microscopy picture of ER chains obtained from applying a 1 kV/mm electric field to a suspension of Na-Fh clay crystallite aggregates in a silicone oil. The red bands seen in the upper and lower parts of the image are copper electrodes, and the distance between them is 1 mm.

Microscopy picture of ER chains obtained from applying a 1 kV/mm electric field to a suspension of Na-Fh clay crystallite aggregates in a silicone oil. The red bands seen in the upper and lower parts of the image are copper electrodes, and the distance between them is 1 mm.

Viscosity of the Na-Fh suspensions as a function of the volume fraction , for . The viscosity values have been obtained from linear fits (shown as plain lines) onto the flow curves (shown as symbols) in the inset.

Viscosity of the Na-Fh suspensions as a function of the volume fraction , for . The viscosity values have been obtained from linear fits (shown as plain lines) onto the flow curves (shown as symbols) in the inset.

(a) Flow curves of Na-Fh suspensions with , for different strengths of the applied electric field; experimental measurements are plotted as symbols, while corresponding fits of the CCJ model [Eq. (3)] are drawn using solid lines. (b) Corresponding evolution of the viscosity as a function of shear rate.

(a) Flow curves of Na-Fh suspensions with , for different strengths of the applied electric field; experimental measurements are plotted as symbols, while corresponding fits of the CCJ model [Eq. (3)] are drawn using solid lines. (b) Corresponding evolution of the viscosity as a function of shear rate.

Scaling of the data of Fig. 3(a) using Eq. (1) with , for a particle fraction . The plain line represents the asymptotic Bingham model. is a reference electric field value.

Scaling of the data of Fig. 3(a) using Eq. (1) with , for a particle fraction . The plain line represents the asymptotic Bingham model. is a reference electric field value.

(a) Flow curves of Na-Fh suspension for two different values at . (b) Viscosity curves for the same suspensions.

(a) Flow curves of Na-Fh suspension for two different values at . (b) Viscosity curves for the same suspensions.

Log-log plot of the static yield stress as a function of the strength of the applied dc electric field for different volume fractions of the Na-Fh particles.

Log-log plot of the static yield stress as a function of the strength of the applied dc electric field for different volume fractions of the Na-Fh particles.

Log-log plot of the scaled static yield stress versus particle volume fraction. The dashed line (power law with an exponent of 1.70) represents the behavior of our previously reported laponite suspensions (Parmar *et al.*, 2008).

Log-log plot of the scaled static yield stress versus particle volume fraction. The dashed line (power law with an exponent of 1.70) represents the behavior of our previously reported laponite suspensions (Parmar *et al.*, 2008).

Bifurcation in the rheology of a Na-Fh suspension of volume fraction , under applied electric field strengths of (a) 530 and (b) .

Bifurcation in the rheology of a Na-Fh suspension of volume fraction , under applied electric field strengths of (a) 530 and (b) .

Bifurcation yield stress as a function of the applied electric field, for three different particle volume fractions. The fitted power-law exponents are , , and , respectively. This decrease of the exponent with volume fraction is attributed to the growing discrepancy between the measured and applied voltages.

Bifurcation yield stress as a function of the applied electric field, for three different particle volume fractions. The fitted power-law exponents are , , and , respectively. This decrease of the exponent with volume fraction is attributed to the growing discrepancy between the measured and applied voltages.

Evolution of the current density as a function of the electric field strength, for particle fractions and ; the data, reproduced from Rozynek *et al.*, 2010, are rescaled by , so as to make the values at coincide.

Evolution of the current density as a function of the electric field strength, for particle fractions and ; the data, reproduced from Rozynek *et al.*, 2010, are rescaled by , so as to make the values at coincide.

## Tables

Critical stresses (in pascal) measured for various electric field strengths and particle fractions. Leak currents prevent accurate measurement for the two larger electric fields and particle fractions . See also Fig. 9, in which these values are plotted as a function of .

Critical stresses (in pascal) measured for various electric field strengths and particle fractions. Leak currents prevent accurate measurement for the two larger electric fields and particle fractions . See also Fig. 9, in which these values are plotted as a function of .

Exponents and , as defined by Eq. (1), obtained for ER laponite suspensions (results taken from Parmar *et al.*, 2008) and Na-Fh ER suspensions (this study) from our three yield stress estimates: dynamic, static, and bifurcation yield stress (y.s.). For an explanation of and comparison between the various exponent values, see the discussions in Secs. IV D, V A, and V B.

Exponents and , as defined by Eq. (1), obtained for ER laponite suspensions (results taken from Parmar *et al.*, 2008) and Na-Fh ER suspensions (this study) from our three yield stress estimates: dynamic, static, and bifurcation yield stress (y.s.). For an explanation of and comparison between the various exponent values, see the discussions in Secs. IV D, V A, and V B.

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