^{1,a)}, G. Petekidis

^{2}, L. Isa

^{3}, W. C. K. Poon

^{4}and R. Besseling

^{4}

### Abstract

We present a comprehensive study of the slip and flow of concentrated colloidalsuspensions using cone-plate rheometry and simultaneous confocal imaging. In the colloidalglass regime, for smooth, nonstick walls, the solid nature of the suspension causes a transition in the rheology from Herschel–Bulkley (HB) bulk flow behavior at large stress to a Bingham-like slip behavior at low stress, which is suppressed for sufficient colloid-wall attraction or colloid-scale wall roughness. Visualization shows how the slip-shear transition depends on gap size and the boundary conditions at both walls and that partial slip persist well above the yield stress. A phenomenological model, incorporating the Bingham slip law and HB bulk flow, fully accounts for the behavior. Microscopically, the Bingham law is related to a thin (subcolloidal) lubrication layer at the wall, giving rise to a characteristic dependence of slip parameters on particle size and concentration. We relate this to the suspension’s osmotic pressure and yield stress and also analyze the influence of van der Waals interaction. For the largest concentrations, we observe nonuniform flow around the yield stress, in line with recent work on bulk shear banding of concentrated pastes. We also describe residual slip in concentrated liquid suspensions, where the vanishing yield stress causes coexistence of (weak) slip and bulk shear flow for all measured rates.

The authors thank K.N. Pham, J. Arlt, and N. Pham for advice and help with the experiments, A.B. Schofield for particle synthesis and sizing and M.E. Cates, A. Morozov, and D. Marenduzzo for useful discussions. R.B. and W.P. acknowledge funding through EP/D067650 and EP/D071070/1. L.I. was funded by the EU network MRTN-CT-2003-504712. G.P. and P.B. acknowledge EU funding from ToK “Cosines” (MTCDCT-2005-029944), NMP Small “Nanodirect” (CPFP7- 213948-2), and NoE “SoftComp.”

I. INTRODUCTION

II. SAMPLES AND METHODS

A. Colloidalsuspensions

B. Measurement system

C. Wall properties

III. MAIN EXPERIMENTAL RESULTS

IV. MODEL FOR THE SLIP AND YIELD BEHAVIOR

A. Infinite parallel plates

B. Cone and plate

V. COMPARISON WITH EXPERIMENT

A. Global rheology

B. Local shear rate

VI. ANALYSIS OF BULK FLOW AND SLIP PARAMETERS

A. dependence for nonstick walls

B. Effect of wall interaction

VII. SHEAR LOCALIZATION

VIII. SLIP BELOW THE GLASS TRANSITION

IX. DISCUSSION AND CONCLUSIONS

### Key Topics

- Colloidal systems
- 47.0
- Suspensions
- 43.0
- Solvents
- 25.0
- Lubrication
- 20.0
- Yield stress
- 20.0

##### B01J13/00

##### G01N21/00

## Figures

(a) Cone-plate rheometer with transparent plate and optics connected via an adjustable arm to the confocal scanner. The cone angle and radius are shown. (b) Confocal image of an RI-matched suspension with the *a* = 652 nm fluorescent tracers showing as bright spots. Scale bar: 10 μm. (c) AFM image of the sintered layer of *a* = 652 nm colloids on the glass slide. The color scale marks height variations of ∼500 nm. Scale bar: 2.5 μm. Inset: confocal image of a coated slide on a larger scale, 50 × 50 μm.

(a) Cone-plate rheometer with transparent plate and optics connected via an adjustable arm to the confocal scanner. The cone angle and radius are shown. (b) Confocal image of an RI-matched suspension with the *a* = 652 nm fluorescent tracers showing as bright spots. Scale bar: 10 μm. (c) AFM image of the sintered layer of *a* = 652 nm colloids on the glass slide. The color scale marks height variations of ∼500 nm. Scale bar: 2.5 μm. Inset: confocal image of a coated slide on a larger scale, 50 × 50 μm.

Measured shear stress versus applied shear rate for *a* = 300 nm colloids at in RI-matching solvent (□) and decalin (•), using a smooth glass slide and coated cone. Full line: fit of the low branch to Eq. (1), giving , . The data for decalin are multiplied by a factor 1.5 for comparison, the difference with the RI-matching solvent is due to slightly different .

Measured shear stress versus applied shear rate for *a* = 300 nm colloids at in RI-matching solvent (□) and decalin (•), using a smooth glass slide and coated cone. Full line: fit of the low branch to Eq. (1), giving , . The data for decalin are multiplied by a factor 1.5 for comparison, the difference with the RI-matching solvent is due to slightly different .

(a) and (b) Measured stress versus for coated cone and plate (○) and with (a) uncoated plate (□) and and *a* = 138 nm, and (b) uncoated cone and uncoated plate (□) with and *a* = 150 nm. Regime I in (a) and I_{a} in (b) represent full slip along one boundary; regime I_{b} in (b) represents full slip along two boundaries; and regime II in (a) and (b) mark slip plus bulk flow. In (a) and (b) the dashed-dotted curves are Hershel–Bulkley fits with *n* = 0.5, Eq. (3), giving , in (a) and , in (b); dashed curves are fits to the Bingham form Eq. (1). In regime I in (a), where Eq. (10) applies, this gives , . In regime I_{a} or I_{b} in (b), where Eqs. (10) and (11) apply, the parameters are , , ; full lines in regime II are the global flow curves from Eq. (A2) in the Appendix A 1 using the above parameters. (c) Normalized velocity profiles for the suspension in (a) at *r* = 3 mm with coated surfaces (filled symbols) and at *r* = 2.5 mm with uncoated plate (open symbols) for various . (d) for the data in (b) for uncoated cone and plate, at *r* = 5.5 mm and various . Full lines: linear fits. Dotted lines: behavior without slip.

(a) and (b) Measured stress versus for coated cone and plate (○) and with (a) uncoated plate (□) and and *a* = 138 nm, and (b) uncoated cone and uncoated plate (□) with and *a* = 150 nm. Regime I in (a) and I_{a} in (b) represent full slip along one boundary; regime I_{b} in (b) represents full slip along two boundaries; and regime II in (a) and (b) mark slip plus bulk flow. In (a) and (b) the dashed-dotted curves are Hershel–Bulkley fits with *n* = 0.5, Eq. (3), giving , in (a) and , in (b); dashed curves are fits to the Bingham form Eq. (1). In regime I in (a), where Eq. (10) applies, this gives , . In regime I_{a} or I_{b} in (b), where Eqs. (10) and (11) apply, the parameters are , , ; full lines in regime II are the global flow curves from Eq. (A2) in the Appendix A 1 using the above parameters. (c) Normalized velocity profiles for the suspension in (a) at *r* = 3 mm with coated surfaces (filled symbols) and at *r* = 2.5 mm with uncoated plate (open symbols) for various . (d) for the data in (b) for uncoated cone and plate, at *r* = 5.5 mm and various . Full lines: linear fits. Dotted lines: behavior without slip.

(a) Flow curves for two uncoated surfaces, *a* = 150 nm, and various . The lower branches of the full lines are fits to Eqs. (10) and (11), the upper branch is Eq. (A2) for slip at two surfaces. The parameters are , , , for and , , , for [the parameters are as in Fig. 3(b)]. (b) Ratio of the slip threshold stresses at the bottom and top surface, , versus , the dotted line is the average value.

(a) Flow curves for two uncoated surfaces, *a* = 150 nm, and various . The lower branches of the full lines are fits to Eqs. (10) and (11), the upper branch is Eq. (A2) for slip at two surfaces. The parameters are , , , for and , , , for [the parameters are as in Fig. 3(b)]. (b) Ratio of the slip threshold stresses at the bottom and top surface, , versus , the dotted line is the average value.

Shear stress minus slip stress divided by the Bingham viscosity for *a* = 138 nm and . and were extracted from fits of the small behavior to Eq. (1). Dashed line: .

Shear stress minus slip stress divided by the Bingham viscosity for *a* = 138 nm and . and were extracted from fits of the small behavior to Eq. (1). Dashed line: .

(a) Velocity profiles for fixed shear rate , , and *a* = 138 nm with uncoated glass and coated cone, at selected distances *r* from the center. (b) Corresponding local shear rate versus *r*. Full line: Eq. (5) with , using Eq. (6) and rheological parameters as in Fig. 3(a). Inset: corresponding *r*-dependence of the normalized slip velocity , along with the prediction using Eqs. (4)–(6) with (full line).

(a) Velocity profiles for fixed shear rate , , and *a* = 138 nm with uncoated glass and coated cone, at selected distances *r* from the center. (b) Corresponding local shear rate versus *r*. Full line: Eq. (5) with , using Eq. (6) and rheological parameters as in Fig. 3(a). Inset: corresponding *r*-dependence of the normalized slip velocity , along with the prediction using Eqs. (4)–(6) with (full line).

(a) Full line: HB flow curve with , , and . Other curves: calculations for different geometries with slip at the bottom plate using , : cone-plate with , [dashed line, Eqs. (10) and (A2)]; parallel plate with (dotted) and (dashed-dotted), using Eqs. (3), (5), and (6). (b) Zoom in on (a).

(a) Full line: HB flow curve with , , and . Other curves: calculations for different geometries with slip at the bottom plate using , : cone-plate with , [dashed line, Eqs. (10) and (A2)]; parallel plate with (dotted) and (dashed-dotted), using Eqs. (3), (5), and (6). (b) Zoom in on (a).

(a) Flow curves for an RI-matched suspension with *a* = 138 nm, , using different coated cones and smooth plates. (b) versus velocity at the geometry edge; symbols as in (a).

(a) Flow curves for an RI-matched suspension with *a* = 138 nm, , using different coated cones and smooth plates. (b) versus velocity at the geometry edge; symbols as in (a).

Velocimetry results corresponding to the data in Fig. 3. (a) and (b) Measured normalized local shear rate versus for (a) *r* = 2.5 mm (□) and *r* = 4 mm (○), and (b) *r* = 2.5 mm (□) and *r* = 5.5 mm (○). (c) and (d) Normalized slip velocity at the glass plate corresponding to data in (a) and (b). In (e), the normalized slip velocity at the cone is shown. Data in (a) and (c) are for *a* = 138 nm, coated cone and uncoated plate. Data in (b), (d), and (e) are for *a* = 150 nm, uncoated cone and uncoated plate. Full lines in (a) and (b) for are given by Eqs. (5) and (6) and Eqs. (5) and (7), respectively, with parameters given in the caption of Fig. 3. In (d) and (e), the curves in regime I_{b} are given by Eq. (A5); the transition occurs at given in Eq. (A6). The bulk (slip) velocity at the second plateau (corresponding to in regime II) is given by Eq. (A7); In (c) (d), and (e), the curves for the largest (where ) follow from those in (a) and (b) via Eq. (4).

Velocimetry results corresponding to the data in Fig. 3. (a) and (b) Measured normalized local shear rate versus for (a) *r* = 2.5 mm (□) and *r* = 4 mm (○), and (b) *r* = 2.5 mm (□) and *r* = 5.5 mm (○). (c) and (d) Normalized slip velocity at the glass plate corresponding to data in (a) and (b). In (e), the normalized slip velocity at the cone is shown. Data in (a) and (c) are for *a* = 138 nm, coated cone and uncoated plate. Data in (b), (d), and (e) are for *a* = 150 nm, uncoated cone and uncoated plate. Full lines in (a) and (b) for are given by Eqs. (5) and (6) and Eqs. (5) and (7), respectively, with parameters given in the caption of Fig. 3. In (d) and (e), the curves in regime I_{b} are given by Eq. (A5); the transition occurs at given in Eq. (A6). The bulk (slip) velocity at the second plateau (corresponding to in regime II) is given by Eq. (A7); In (c) (d), and (e), the curves for the largest (where ) follow from those in (a) and (b) via Eq. (4).

(a) Local shear rate extracted from the measured (*z*), versus for different , *a* and *r*, see symbols in (b), for slip at the bottom plate only (*a* = 138 nm data) and at both surfaces (*a* = 150 nm data). The full lines represent Eq. (5) with the rheological parameters entering via Eq. (6) or Eq. (7) with . Dotted line: . (b) Normalized local shear rate versus normalized applied rate for various *r*, and two particle sizes. The full line shows Eq. (8).

(a) Local shear rate extracted from the measured (*z*), versus for different , *a* and *r*, see symbols in (b), for slip at the bottom plate only (*a* = 138 nm data) and at both surfaces (*a* = 150 nm data). The full lines represent Eq. (5) with the rheological parameters entering via Eq. (6) or Eq. (7) with . Dotted line: . (b) Normalized local shear rate versus normalized applied rate for various *r*, and two particle sizes. The full line shows Eq. (8).

Bulk HB parameters for *a* = 138 nm and *a* = 150 nm versus . (a) Normalized yield stress. Line: with *p* = 3 and . (b) The HB exponent *n*. (c) The HB parameter . Lines: with *A* = 10, *n* = 0.45, *p* = 3, and and the Brownian times.

Bulk HB parameters for *a* = 138 nm and *a* = 150 nm versus . (a) Normalized yield stress. Line: with *p* = 3 and . (b) The HB exponent *n*. (c) The HB parameter . Lines: with *A* = 10, *n* = 0.45, *p* = 3, and and the Brownian times.

(a) Normalized lubrication parameter versus . Symbols: data for different particle sizes [see (b)]; full line: result based on Eq. (17). Dotted line: result based on Eq. (14) and the explanation in the text. Inset: (•) versus from a numerical evaluation of Eq. (16). The full line represents . (b) Normalized slip stress versus , the dotted line is , the full line represents Eq. (18) with *A* = 0.005 and *m* = 2.5.

(a) Normalized lubrication parameter versus . Symbols: data for different particle sizes [see (b)]; full line: result based on Eq. (17). Dotted line: result based on Eq. (14) and the explanation in the text. Inset: (•) versus from a numerical evaluation of Eq. (16). The full line represents . (b) Normalized slip stress versus , the dotted line is , the full line represents Eq. (18) with *A* = 0.005 and *m* = 2.5.

(a) Measured stress versus for a suspension with *a* = 138 nm, in RI-matching solvent, immediately after loading, going from small to large , followed by large to small , again small to large (○) and finally large to small (◊). (b) The power *n* and (c) , both obtained from a fit of the low branch to , versus run number.

(a) Measured stress versus for a suspension with *a* = 138 nm, in RI-matching solvent, immediately after loading, going from small to large , followed by large to small , again small to large (○) and finally large to small (◊). (b) The power *n* and (c) , both obtained from a fit of the low branch to , versus run number.

Mean squared displacement of *a* = 652 nm particles versus time in dilute suspensions in a decalin-tetralin mixture far from the glass (▪), close to the glass (•) and close to the glass in decalin (▴). The lines represent diffusive behavior with , and from top to bottom. Inset: trajectories at *z* ∼ *a* of particles in decalin (the stuck particle and short trajectory) and a single long particle trajectory in decalin-tetralin (•).

Mean squared displacement of *a* = 652 nm particles versus time in dilute suspensions in a decalin-tetralin mixture far from the glass (▪), close to the glass (•) and close to the glass in decalin (▴). The lines represent diffusive behavior with , and from top to bottom. Inset: trajectories at *z* ∼ *a* of particles in decalin (the stuck particle and short trajectory) and a single long particle trajectory in decalin-tetralin (•).

Temperature dependence of (a) the normalized Bingham slip stress and (b) Bingham viscosity normalized by the temperature dependent solvent viscosity . Data are for , *a* = 138 nm using a coated cone and smooth glass. The inset to (b) shows the unrenormalized data .

Temperature dependence of (a) the normalized Bingham slip stress and (b) Bingham viscosity normalized by the temperature dependent solvent viscosity . Data are for , *a* = 138 nm using a coated cone and smooth glass. The inset to (b) shows the unrenormalized data .

(a) Velocity profiles for , *r* = 5 mm and both surfaces coated, for various . (b) Same as (a) but for coated cone, smooth plate and *r* = 5.5 mm.

(a) Velocity profiles for , *r* = 5 mm and both surfaces coated, for various . (b) Same as (a) but for coated cone, smooth plate and *r* = 5.5 mm.

(a) Flow curve for an RI-matched suspension with *a* = 138 nm, , with smooth (□) and rough walls (full line). (b) (*z*) for smooth walls at *r* = 5.5 mm and for various .

(a) Flow curve for an RI-matched suspension with *a* = 138 nm, , with smooth (□) and rough walls (full line). (b) (*z*) for smooth walls at *r* = 5.5 mm and for various .

(a) Normalized flow profiles at *r* = 12.5 mm for at various using a smooth cone and smooth glass. Inset: measured flow curve. (b) Local shear rate versus slip velocity at various *r*. Full line: fit to a power law with . Dotted line: with . Magenta discontinued line: with (see text).

(a) Normalized flow profiles at *r* = 12.5 mm for at various using a smooth cone and smooth glass. Inset: measured flow curve. (b) Local shear rate versus slip velocity at various *r*. Full line: fit to a power law with . Dotted line: with . Magenta discontinued line: with (see text).

Slip-length versus applied rate for , , , and . Full line is the exact form using Eqs. (A8), (5), and (6) in Sec. IV, dotted line represents Eq. (A9).

Slip-length versus applied rate for , , , and . Full line is the exact form using Eqs. (A8), (5), and (6) in Sec. IV, dotted line represents Eq. (A9).

## Tables

Particle size *a*, density of the index-matched solvent , and density of the random close packed sediment for the different samples.

Particle size *a*, density of the index-matched solvent , and density of the random close packed sediment for the different samples.

Dielectric permittivity and index of refraction *n* for the solvents, glass, and PMMA.

Dielectric permittivity and index of refraction *n* for the solvents, glass, and PMMA.

van der Waals interactions for different particle sizes *a*, surfaces and solvents: particle-particle interaction in decalin , particle-wall interaction in decalin , particle-particle interaction in decalin-tetralin , particle-wall interaction in decalin-tetralin , as well as the particle-steel cone interaction for decalin and for decalin-tetralin .

van der Waals interactions for different particle sizes *a*, surfaces and solvents: particle-particle interaction in decalin , particle-wall interaction in decalin , particle-particle interaction in decalin-tetralin , particle-wall interaction in decalin-tetralin , as well as the particle-steel cone interaction for decalin and for decalin-tetralin .

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