Supplementary Informations Spatial cooperativity in soft glassy flows

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1 doi:.38/nature76 Supplementary Informations Spatial cooperativity in soft glassy flows J. Goyon, A. Colin, G. Ovarlez, A. Ajdari, L. Bocquet I. SUPPLEMENTARY METHOD. Static properties of the emulsions The material used in this study is a soft glassy material, namely a nearly monodisperse concentrated emulsion. The emulsion is composed of silicone droplets in a water-glycerine mixture (5%wt-5%wt), stabilized by trimethyl tetradecyl ammonium bromide ( wt %), and obtained by shearing a mixture of oil, water, glycerine and surfactant in a narrow-gap Couette cell. The surfactant concentration within the aqueous phase is set to wt %. The emulsion is non adhesive. The droplet mean diameter measured by light scattering (Malvern Mastersizer) is a 6.5 microns, with a slight polydispersity to prevent crystallization. The emulsion is optically transparent. Two sets of emulsions with polydispersity of % and 36% have been considered, with volume fractions in the range 4-85%. The polydispersity of the droplet diameters of the emulsions is measured by light scattering (Malvern Mastersizer), as shown in Supp. Fig. II.. Bulk rheological properties of the emulsions The emulsion s properties are tuned by its volume fraction φ: While the emulsion behaves like a liquid for low and intermediate droplet concentration, a jamming transition occurs at a critical volume fraction φ c. Above φ c the emulsion is solid-like at rest and flows only above a finite external stress σ (the so-called yield stress ). The jamming point φ c is defined as the volume fraction above which σ is non-vanishing: φ c =.64 and φ c =.68 for the emulsions with % and 36% polydispersity respectively. We measure the bulk flow behavior using a commercial rheometer (ARG, TA Instruments), with a cone-andplate geometry (4mm diameter and 58 3 angle). The bulk flow curve, shear stress σ versus shear rate γ, is very well fitted in all cases by a Herschel-Bulkley-type of rheological model: σ = σ + A γ n. The values for the emulsion paramaters, σ, A, n are given in Supplementary Table for the two emulsions (with % and 36% polydispersities) as a function of the emulsion volume fraction φ. 3. Microchannels and surface characteristics The microchannels are made of transparent smooth glass rectangular capillary tube (Composite Metal Services) or in home made glass microdevices. All devices have rectangular cross sections, with aspect ratio (height divided by width) larger than 8 (except the 5 µm width microchannel), so that the flow at middle height of these channels can be assimilated to the flow between two infinite planes. The rectangular capillary tube bought to Composite Metal Services are smooth and optically flat. The microdevices with rough surfaces are homemade. Two glass slides (mm thick) are glued to the first glass side with an optical adhesive (NOA 8, Norland Products) to form a channel. A spacer allows us to control the width of the outlet channel. Access holes are made in a second glass side with a sand blaster. The channels are then sealed by this glass slide using the same optical adhesive. To clog the holes that appear on the edge, an optical adhesive is placed on each hole. The distance between the two glass walls is measured under a microscope to get the real dimension of the channel. The rugosity of the wall is due to the rugosity of the edge of the two mm thick glass slides. Typical pictures of the glass side are displayed in Supp. Fig. II. The capillary tube are smooth, see Supp. Fig. IIb. While it is difficult to identify precisely a characteristic rugosity length scale in Supp. Fig. IIa, it seems that scales between and 5 µm are involved in the rough surfaces. The main characteristics of the microchannels used in this study are summarized in Supplementary Table. 4. Velocimetry techniques The local velocity profiles are measured using a magnetic resonance imaging (MRI) technique for the Couette cell, and a particle imaging velocimetry (PIV) procedure for the flows in the microchannels. In the Couette cell, MRI measurements are conducted by inserting a velocity-controlled Couette rheometer in a.5t vertical MRI spectrometer (4/8 DBX by Bruker). This permits the measurement of both the local orthoradial velocity and the local droplet concentration of the material anywhere in the gap, with a radial resolution of.5 mm. Note that in this geometry, sandpaper is glued to the walls in order to avoid slip at the walls. In the microchannels, velocity profiles in this geometry are measured using Particle Imaging Velocimetry (P.I.V). In this purpose the fluid is seeded with small particles (Invitrogen Fluorespheres µm diameter at a concentration

2 doi:.38/nature76 of.% wt). Images in the x-z plane of the channel width w are acquired using an inverted fluorescent microscope, at a 4 X magnification, at middle height of the channel (depth of field micron), and far enough from the inlet of the channel to measure fully developped profiles. A CCD camera coupled to an intensifier (Hamamatsu and RD Vision) allows us to record couples of images. The images of a same couple are taken at a fixed time interval dt (dt [3µs, s]) and their intensity cross-correlation is determined for a variable translation dx along the flow (Mathwork procedure). The correlation maximum gives the velocity dx/dt. Note that in our experiments dx is a scalar since the flow is unidirectionnal but depends upon z. This experimental set up allows us to access local velocities up to m/s with a spatial resolution dz = µm in the plane of the flow. All devices have rectangular cross sections, with aspect ratio (height divided by width) larger than 8, so that the flow at middle height of these channels can be assimilated to the flow between two infinite planes. No variation of the profiles is observed along the velocity direction in our channel confirming that steady flow is achieved rapidly. Measurements are performed 5 cm after the entrance of the channel in order to eliminate entrance effects Measurement of local droplet concentration In all the geometries no variation of local droplet concentration was measured within the gap. This quantity is directly measured by MRI in the Couette cell. In microchannel flows, rhodamine was added to the continuous phase and the local density profile is obtained by the analysis of the fluorescence after calibration. As discussed in Supplementary Figure, these measurements revealed no local variation of the measured intensity, suggesting no variation in the local droplet concentration.

3 doi:.38/nature76 3 II. SUPPLEMENTARY FIGURES 5 a Volume (%) Volume (%) 5 b Diameter (µm) Diameter (µm) Supplementary Figure SI-: Droplets diameter distribution of the two different emulsions measured by light scatterring (Malvern Mastersizer).(a) ( ) The mean diameter is 6.49 µm and the standard deviation %. (b) ( ) The mean diameter is 6.4 µm and the standard deviation 36%. Supplementary Figure SI-: Pictures of the surfaces used in the study: (a) glass slides exhibiting a rough surface; (b) broken piece of a capillar tube from Composite Metal Service displaying a smooth surface at the micron scale. Scales are identical for both figures. 3

4 doi:.38/nature76 4 V V s (mm.s ) 3 Supplementary Figure SI-3: Velocity profiles of the emulsion in the fluid state, φ < φ c. The volume fraction is φ=4%, with % polydispersity. The microchannel is rough with a width w = 5 µm. The different curves correspond to different pressure drops P, from top to bottom, 5,, 6 and mbar. Velocity profiles are corrected for the slip velocity. Solid lines are velocity profiles predicted by a shear thinning model (σ =. γ.65 Pa). As the volume fraction of the emulsion is below the jamming concentration, we could not measure finite size effects in the flow profiles, so that ξ=. 4

5 doi:.38/nature76 5 σ (Pa) γ (s ) Supplementary Figure SI-4: Local flow curve extracted from the velocity profiles for an emulsion with volume fraction φ =.75 and % polydispersity. The various symbols correspond to the various studied geometries. wide Couette cell, smooth microchannel with w = 56 µm, rough microchannel with w = 5 µm, smooth microchannel with w = µm, rough microchannel with w = 5 µm. 5

6 doi:.38/nature V V s (mm s s ) a 3 V V (mm s s ) 3 b c d.5.5 Supplementary Figure SI-5: Velocity profiles in the jammed state, φ > φ c : (a) rough microchannel with w = 5 µm, (b) rough microchannel with w = 5 µm, (c) rough microchannel with w = 85 µm, (d) smooth microchannel with w = µm. Solid lines are velocity profiles predicted by the non local model of Eq. () with ξ = µm. The volume fraction is φ = 75%, with 36% polydispersity. The different curves correspond to different pressure drops P tuned to get the same range of wall shear stress in the various geometries. From top to bottom the different curves correspond to σ wall equal to 34, 5, 63, 74, 95, 3 Pa (a), 3, 48, 64, 8, 97 Pa (b), 57, 69, 8, 95 Pa (c), 9, 44, 59, 74, 9 Pa (d). Velocity profiles are corrected for the slip velocity. V V (mm.s s ) a V V (mm.s s ) b.6.4. c d.5.5 Supplementary Figure SI-6: Velocity profiles of the emulsion in the jammed state, φ > φ c : a) rough microchannel with w = 5 µm, (b) rough microchannel with w = 85 µm,(c) smooth microchannel with w = µm (d) smooth microchannel with w = 56 µm. Solid lines are velocity profiles predicted by the non local model of Eq. () with ξ = 5.8µm. The volume fraction is φ = 8%, with 36% polydispersity. The different curves correspond to different pressure drops P tuned to get the same range of wall shear stress in the various geometries. From top to bottom the different curves correspond to σ wall equal to 54, 68, 87, 6 Pa (a) 87, 98,, 3 Pa (b) 53, 67, 84,, 7 Pa (c) and 78, 94, 9 Pa (d). Velocity profiles are corrected for the slip velocity. 6

7 doi:.38/nature76 V s (mm.s - ).5 a 5. γ wall b (s - ) σ wall (Pa) 5 σ wall (Pa) Supplementary Figure SI-7: Surface rheology: (a) Slip velocity at the wall as a function of the wall shear stress for various microchannels; (b) Shear rate at the confining wall γ wall as a function of the wall shear stress σ wall for various microchannels. The dashed line is the bulk flow curve (Herschel-Bulkley model with σ =.65 Pa, A =.5 SI). Data gather different pressure drops P and confinements w. Symbols are identical to those in Figure??. Open (filled) symbols correspond to rough (smooth) confining surfaces. The volume fraction is φ = 75%, with % polydispersity. 7

8 doi:.38/nature76 8 V V s (mm/s) V V s (mm/s) Supplementary Figure SI-8: Comparison of the predictions for the Herschel-Bulkley model with the experimental flow profiles, in a 5µm (Left) and 5 µm (Right) wide channel. Experimental profiles (symbols) are obtained for various pressure gradients such that the wall shear stresses are from bottom to top: (a) 45, 75, 9 Pa; (b) 7, 55, 8 Pa. The emulsion is 75% in volume fraction and % polydispersity. The Herschel Bulkley model reads σ bulk ( γ) = σ + A γ /, with A =. Pa.s / and σ =.6Pa (see Supplementary Table ). 8

9 9 doi:.38/nature76 4 V Vs (mm s ) a 3 3 V Vs (mm s ) b c V Vs (mm s ) d V Vs (mm s ) Supplementary Figure SI-9: Comparison of the experimental velocity profiles with an alternative non-local rheological model for various confinements, pressure drops and emulsion volume fractions φ: (a) 5µm wide microchannel, φ = 75% (same parameters as in Fig. SI-5-a); (b) 5 µm wide microchannel, φ = 75% (same parameters as in Fig. SI-5-b); (c) 5µm wide microchannel, φ = 8% (same parameters as in Fig. SI-6-a); (d) 85µm wide microchannel, φ = 8% (same parameters as in Fig. SI-6-b). In all figures the polydispersity of the emulsion is 36%. In the alternative non-local rheological model, a second derivative of the shear rate is added to the flow curve, in line with the suggestions by Dhont4 : σ = σbulk (γ) κ zγ, with σbulk (γ) = σ + Aγ /. The parameters A and σ are / / A = Pa.s and σ = Pa for φ = 75% and A = 4.5 Pa.s and σ = 3Pa for φ = 8% (see Supplementary Table ). We fix the value of the non-locality parameter κ by trying to find the best compromise between the experimental and predicted velocity profiles for a given pressure drop. This provides κ = 3. Pa.m.s to fit the profiles in Fig. SI-9-a (volume fraction φ = 75%); and κ =.5. Pa.m.s to fit the profiles in Fig. SI-9-c (volume fraction φ = 8%). These values are then used to predict the velocity profile, without any free parameters, in the other confinement: Fig. SI-9-a and Fig. SI-9-d. While the tendency is seen to be correct, a good agreement can not be obtained for all values of pressure drop for the two channel widths for both volume fractions. The agreement obtained with this alternative model is also seen to be worst than the predictions of the non-local fluidization model, given by Eq. in the main text, and shown under the same conditions in Figs. SI-5.a-b and Figs. SI-6.a.b. 9

10 doi:.38/nature76 φ (%).5 w Supplementary Figure SI-: Left: Spatial dependence of the droplet volume fraction profile relative to its value at rest, for various pressure drops P : φ = φ( P ) φ( P = ). The different profiles correspond to various pressure drops: P = 4 (thin solid line), 6 (thin dashed -dotted line), 7 (thin dashed line), 8 (thin dotted line), 9 (thick dotted line), Pa (thick solid line). The mean volume fraction of the emulsion is φ = 75% with 36% of polydispersity. The microchannel is rough and w = 5µm in width. This figures shows that the modulation of volume fraction are below ±% and insensitive to pressure drop. We emphasize that these modulations are much below the spatial variation of the volume fraction which would required, if one was to explain the local rheological behavior (and flow profiles) in terms of spatial variations of the droplet concentration. For the present conditions, such an assumption would indeed require a volume fraction change of up to %, depending on pressure drop: this estimate is obtained on the basis of the density dependence of the bulk rhelogical law, which is given in supplementary table. Volume fraction variations up to 5% would be required for smaller confinement. Therefore, since one does not measure such variations, this rules out density variations as the origin of non-local effects in the rheology. The droplet volume fraction, φ = φ( P ) φ( P = ), is measured experimentally by adding rhodamine to the continuous phase and measuring the fluorescence intensity profiles. The local volume fraction is obtained by calibration of the experimental set up, providing φ/ I (here, φ/ I =.36 in the arbitrary units for the fluorescence intensity given by the camera). The fluorescence intensity is averaged on 45 images. Laser beam intensity and frame rate are kept constant for all the measurements. Fluctuations of the mean intensity are measured independently to be I/I =.5%. The mean intensity is 6 (A.U). This yields an uncertainty on the measurement of φ equal to ±.6%, as represented by the vertical bar in the figure. Right: Confocal picture of an emulsion flowing in a microchannel (Zeiss live microscope) The liquid fraction is φ = 75% with 36% polydispersity. The microchannel is rough with w = 5µm. The drop of pressure is equal to P = 6P a. No ordering is exhibited near the boundaries.

11 doi:.38/nature76 σ (Pa) σ (Pa) 8 σ (Pa) 8 σ (Pa) γ (s ) 4 6. γ (s ). 4 6 γ (s ). γ (s ) 4 6 Supplementary Figure SI-: Local flow curves for various confinement ratio ξ/w. From left to right, ξ/w = (ξ =, φ = 4%), ξ/w =.5 (ξ = 4µm, φ = 65%), ξ/w =.4 (ξ = µm, φ = 7%), ξ/w =. (ξ = 3µm, φ = 75%). Note that the vertical scale has been changed in the left figure for readability. Dashed lines are the predictions of the model in Eq. for the various experimental conditions. As the ratio ξ/w, the flow curves increasingly converge to the corresponding HB bulk flow curve, and non-local effects disappear.

12 doi:.38/nature76 III. SUPPLEMENTARY TABLES φ σ (Pa) A (Pa.s n ) n φ σ Pa) A (Pa.s n ) n Supplementary Table : Bulk rheological flow curve as a function of volume fraction φ for the emulsions with % (left) and 36% (right) polydispersities. In all cases, the flow curves are very well described by a Herschel-Bulkley model, according to σ = σ + A γ n. L (cm) w(µm) h (mm) SN 4 5 R R R S 56 S Supplementary Table : Main characteristics of the microdevices used in this study. L is the length, w the thickness and h the height of the microchannel. SN is the surface nature of the microdevice. S stands for smooth and R for rough.

13 doi:.38/nature76 3 IV. SUPPLEMENTARY NOTES P.Guillot, P.Panizza, J.B.Salmon, M.Joanicot, A.Colin, C.H.Bruneau,T.Colin, A viscosimeter on a microfluidic chip, Langmuir, (6). Ovarlez, G., F. Bertrand, and S. Rodts, Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging, J. Rheol. 5, 59-9 (6). 3 M. Collins, W.R. Schowalter, Behavior of non-newtonian fluids in the inlet region of a channel, A.I. Ch. E. Journal 9, 98- (963). 4 J. K. G. Dhont, A constitutive relation describing the shear-banding transition Phys. Rev. E 6, (999). 3