^{1}, R. P. Sahu

^{1}, S. Sinha-Ray

^{1}and A. L. Yarin

^{1,a)}

### Abstract

This work aims to study squeeze flows when the lubrication approximation does not necessarily hold. Strong squeeze flows are defined as the cases in which a sample is compressed by a disk with the initial speed of 40 cm/s, whereas weak squeeze flows are realized when the disk is softly released manually to avoid any impact of the sample at the beginning of compression. Strong and weak squeeze flows of yield-stress materials are studied experimentally and theoretically. In the experiments, disk-like constant-volume samples of Carbopol solutions and bentonite dispersions are compressed between two approaching disks being subjected to constant forces. In addition, experiments with shear flows in parallel-plate and vane viscometers are conducted. Using visualization through the transparent wall of the squeezing apparatus, it is demonstrated that the no-slip conditions hold. It is also demonstrated that during the fast stage of strong squeeze flows, the material response can be explained by deviatoric normal stresses, which elucidates the link of strong squeeze flows to elongational flows. The analysis of the data in the framework of the Newtonian and Herschel–Bulkley models shows that in the present case the nonlinearity of the rheological response at the fast stage of strong squeeze flows is not very significant, and a strain-rate-independent viscosity can be used as a reasonable approximation. On the other hand, at the final stage of squeeze flows, when samples spread significantly under the action of a constant squeezing force, the compressive stresses become small enough, and the dominant role is played by the yield stress. No significant signs of thixotrophy were observed. It is shown that strong squeeze flow in the squeezing apparatus is a convenient tool useful for the measurement of viscosity and the yield stress of complex soft materials.

This work was partially supported by The US Gypsum Corporation (USG).

I. INTRODUCTION

II. THEORETICAL

III. EXPERIMENTAL

A. Materials

B. Preparation of solutions and dispersions

C. Flow curves in simple shear

D. Experimental setup for strong and weak squeeze flows

IV. EXPERIMENTAL RESULTS AND DISCUSSION

A. Flow curves of Carbopol solution and bentonite clay dispersion in simple shear

B. Squeeze flow of pure Newtonian materials: Silicone oils

C. Squeeze flow of Carbopol solution C3 and bentonite dispersion B10: No-slip conditions

D. Squeeze flow of Carbopol solutions and bentonite clay dispersions

E. The effects of thixotropy

V. CONCLUSION

### Key Topics

- Yield stress
- 47.0
- Viscosity
- 40.0
- Shear flows
- 29.0
- Shear rate dependent viscosity
- 29.0
- Solution processes
- 19.0

##### F16N

## Figures

Schematic of the velocity profiles in squeeze flows of (a) Newtonian material, and (b) Bingham plastic. The current sample thickness is denoted by h.

Schematic of the velocity profiles in squeeze flows of (a) Newtonian material, and (b) Bingham plastic. The current sample thickness is denoted by h.

Sketch of a sample with the coordinate axes.

Sketch of a sample with the coordinate axes.

The experimental setup: (a) schematic, (b) image of the squeezing apparatus.

The experimental setup: (a) schematic, (b) image of the squeezing apparatus.

(a) Flow curves of Carbopol solution C1 at two different concentrations of NaOH. Parallel-plate viscometer: 2% NaOH (solid line) and 4% NaOH (dashed-dotted line). Vane viscometer: 2% NaOH (dashed line) and 4% NaOH (dotted line). All lines are practically indistinguishable in this graph. (b) Flow curves of Carbopol solution C3 at two different concentrations of NaOH. Parallel-plate viscometer: 2% NaOH (solid line) and 4% NaOH (dashed-dotted line). Vane viscometer: 2% NaOH (dashed line) and 4% NaOH (dotted line). In panel (b), the viscosity values measured in strong squeeze flow of C3 with 2% NaOH (symbols) are superimposed using the same strain rate axis for both shear rate in simple shear and the uniaxial elongation in the radial direction in squeezing. (c) Flow curves of bentonite dispersion B10. Parallel-plate viscometer: (solid line), vane viscometer (dashed line). In panel (c), the viscosity values measured in strong squeeze flow of B10 (symbols) are superimposed using the same strain rate axis for the shear rate in simple shear and the uniaxial elongation in the radial direction in squeezing.

(a) Flow curves of Carbopol solution C1 at two different concentrations of NaOH. Parallel-plate viscometer: 2% NaOH (solid line) and 4% NaOH (dashed-dotted line). Vane viscometer: 2% NaOH (dashed line) and 4% NaOH (dotted line). All lines are practically indistinguishable in this graph. (b) Flow curves of Carbopol solution C3 at two different concentrations of NaOH. Parallel-plate viscometer: 2% NaOH (solid line) and 4% NaOH (dashed-dotted line). Vane viscometer: 2% NaOH (dashed line) and 4% NaOH (dotted line). In panel (b), the viscosity values measured in strong squeeze flow of C3 with 2% NaOH (symbols) are superimposed using the same strain rate axis for both shear rate in simple shear and the uniaxial elongation in the radial direction in squeezing. (c) Flow curves of bentonite dispersion B10. Parallel-plate viscometer: (solid line), vane viscometer (dashed line). In panel (c), the viscosity values measured in strong squeeze flow of B10 (symbols) are superimposed using the same strain rate axis for the shear rate in simple shear and the uniaxial elongation in the radial direction in squeezing.

Typical deformation curves: (a) Area A versus time t, and (b) the corresponding A4(t) dependences for S1. Curve 1 corresponds to 165 g, curve 2 to 305 g, and curve 3 to 465 g.

Typical deformation curves: (a) Area A versus time t, and (b) the corresponding A4(t) dependences for S1. Curve 1 corresponds to 165 g, curve 2 to 305 g, and curve 3 to 465 g.

Typical deformation curves: (a) Area A versus time t, and (b) the corresponding A4(t) dependences for S2.5. Curve 1 corresponds to 165 g, curve 2 to 305 g, and curve 3 to 465 g.

Typical deformation curves: (a) Area A versus time t, and (b) the corresponding A4(t) dependences for S2.5. Curve 1 corresponds to 165 g, curve 2 to 305 g, and curve 3 to 465 g.

Viscosity obtained from the squeeze flows of S1 (squares) and S2.5 (circles). Horizontal dashed lines depict the viscosity values provided by the manufacturer of these oils.

Viscosity obtained from the squeeze flows of S1 (squares) and S2.5 (circles). Horizontal dashed lines depict the viscosity values provided by the manufacturer of these oils.

Upper row: Carbopol solution C3 being strongly squeezed under a mass load of 230 g. Lower row: Bentonite dispersion B10 being strongly squeezed under a mass load of 465 g. (a) t = 0, (b) 32.5 ms, (c) 39 ms, (d) 45.5 ms. 1—The aggregate of black dye that did not move during the squeezing process. 2—The aggregate of black dye that was smeared because it was deeply embedded into the sample. The dashed circles correspond to the initial circumference of the samples of C3 (in the upper row) and B10 (in the lower row). In the upper row for C3, the bigger dark ring is created by incoming light from the light source being partially refracted due to the sample geometry. Therefore, the inner circumference of the dark ring represents where the material ceases to make contact with the top and bottom disks and the outer circumference represents the leading edge of the sample. In the lower row for B10, the outer circumference of the dark area corresponds to the leading edge of the sample. There is no dark ring in this case because the material is opaque.

Upper row: Carbopol solution C3 being strongly squeezed under a mass load of 230 g. Lower row: Bentonite dispersion B10 being strongly squeezed under a mass load of 465 g. (a) t = 0, (b) 32.5 ms, (c) 39 ms, (d) 45.5 ms. 1—The aggregate of black dye that did not move during the squeezing process. 2—The aggregate of black dye that was smeared because it was deeply embedded into the sample. The dashed circles correspond to the initial circumference of the samples of C3 (in the upper row) and B10 (in the lower row). In the upper row for C3, the bigger dark ring is created by incoming light from the light source being partially refracted due to the sample geometry. Therefore, the inner circumference of the dark ring represents where the material ceases to make contact with the top and bottom disks and the outer circumference represents the leading edge of the sample. In the lower row for B10, the outer circumference of the dark area corresponds to the leading edge of the sample. There is no dark ring in this case because the material is opaque.

Typical deformation curves for C3. (a) A(t), and (b) the corresponding A4(t) dependences. Curves 1 correspond to 230 g, 2-to 305 g, and 3-to 465 g. The inset in panel (a) resolves the initial stage of sample spreading. Weak squeezing: The load was applied softly.

Typical deformation curves for C3. (a) A(t), and (b) the corresponding A4(t) dependences. Curves 1 correspond to 230 g, 2-to 305 g, and 3-to 465 g. The inset in panel (a) resolves the initial stage of sample spreading. Weak squeezing: The load was applied softly.

Typical deformation curves for B10. (a) A(t), and (b) the corresponding A4(t) dependences. Curves 1 correspond to 230 g, 2-to 305 g, and 3-to 465 g. The inset in panel (a) resolves the initial stage of sample spreading. Weak squeezing: The load was applied softly.

Typical deformation curves for B10. (a) A(t), and (b) the corresponding A4(t) dependences. Curves 1 correspond to 230 g, 2-to 305 g, and 3-to 465 g. The inset in panel (a) resolves the initial stage of sample spreading. Weak squeezing: The load was applied softly.

Comparison of deformation curve for Newtonian (S1, S2.5) and Bingham (C3 and B10) materials under a squeezing load of 465 g. (a) Short time evolution. (b) Longer time behavior. Curve 1—S2.5, 2—S1, 3—C3, and 4—B10. Weak squeezing: The load was applied softly.

Comparison of deformation curve for Newtonian (S1, S2.5) and Bingham (C3 and B10) materials under a squeezing load of 465 g. (a) Short time evolution. (b) Longer time behavior. Curve 1—S2.5, 2—S1, 3—C3, and 4—B10. Weak squeezing: The load was applied softly.

Typical deformation curves for (a) C3 and (b) B10 in strong squeezing. Curve 1: 230 g, curve 2: 305 g, curve 3: 465 g, and curve 4: the tangent line corresponding to the instant when the maximum rate of spreading occurs for curve 3.

Typical deformation curves for (a) C3 and (b) B10 in strong squeezing. Curve 1: 230 g, curve 2: 305 g, curve 3: 465 g, and curve 4: the tangent line corresponding to the instant when the maximum rate of spreading occurs for curve 3.

Shear layer thickness relative to the sample semithickness during the entire squeezing process for (a) C3 solution and (b) B10 dispersion. The mass load corresponding to curve 1 was 230 g, and to curve 2 was 465 g. The relative shear layer thicknesses corresponding to the maximum slopes of the dependences A(t) are designated by open circles.

Shear layer thickness relative to the sample semithickness during the entire squeezing process for (a) C3 solution and (b) B10 dispersion. The mass load corresponding to curve 1 was 230 g, and to curve 2 was 465 g. The relative shear layer thicknesses corresponding to the maximum slopes of the dependences A(t) are designated by open circles.

The experimental deformation curves with the crossover asymptotics of Eqs. (20) and (22) . (a) C3 and (b) B10 under a mass load of 465 g. The solid line depicts the experimental data; the short-dashed line, Eq. (20) ; and the long-dashed line, Eq. (22) . The long-dashed lines are plotted according to Eq. (22) with τc = 0.00253 s for C3 and τc = 0.00545 s for B10.

The experimental deformation curves with the crossover asymptotics of Eqs. (20) and (22) . (a) C3 and (b) B10 under a mass load of 465 g. The solid line depicts the experimental data; the short-dashed line, Eq. (20) ; and the long-dashed line, Eq. (22) . The long-dashed lines are plotted according to Eq. (22) with τc = 0.00253 s for C3 and τc = 0.00545 s for B10.

## Tables

pH of several Carbopol solutions with NaOH added.

pH of several Carbopol solutions with NaOH added.

Viscosity of the silicone oils measured in the squeeze experiments vs standard values. The viscosity values listed are the averages of two trials.

Viscosity of the silicone oils measured in the squeeze experiments vs standard values. The viscosity values listed are the averages of two trials.

Viscosity μ and the characteristic time τc for Carbopol solution C3 and bentonite dispersion B10 established in strong squeeze experiments. The characteristic time τc, which is affected not only by the viscous forces but also by the yield stress, is introduced in Eqs. (21) and (22) .

Viscosity μ and the characteristic time τc for Carbopol solution C3 and bentonite dispersion B10 established in strong squeeze experiments. The characteristic time τc, which is affected not only by the viscous forces but also by the yield stress, is introduced in Eqs. (21) and (22) .

Experimental data for the yield stress of C3.

Experimental data for the yield stress of C3.

Experimental data for the yield stress of B10.

Experimental data for the yield stress of B10.

Elongational strain rate during strong squeezing, , elongational strain rate at saturation, calculated using Eq. (23) , and shear strain rate, established in squeezing experiments with C3 and B10.

Elongational strain rate during strong squeezing, , elongational strain rate at saturation, calculated using Eq. (23) , and shear strain rate, established in squeezing experiments with C3 and B10.

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