Supplementary Information. Text S1:

Transcription

1 Supplementary Information Text S1: In order to characterize the change in visco-elastic response in the course of a shear thickening transition in a controlled shear stress flow, on a fresh sample of for 5 wt% MWNT in NMP, we superpose an oscillatory shear of low amplitude σ 0 on a rotational shear stress σ DC (σ(t) =σ DC + σ 0 sinωt). Fig S1a shows the variation of G as a function of σ DC (using σ o =0.01Paandω = 10 rad/s) spanning all the three regions. Beyond the yield stress, in the shear-thinning region, G decreases by almost 3 orders of magnitude, followed by a sharp increase by 5 orders of magnitude above the shear thickening stress σ st. Before shear thinning sets in as well as in the ST state, G >G whereas G >G in the shear thinning state (data for G not shown). Fig S1b and Fig S1c shows the storage (G ) and viscous (G ) moduli as a function of angular frequency ω in the jammed state below the yield-stress and the shear-thickened state, respectively which clearly reveals that both the jammed state and the shear-thickened state are solid-like with (G >G ) over the entire frequency range, although, in the shear-thickened state, the sample has a much higher storage modulus. In fact, the shear-thickened state behaves like a strong solid (with G more than two orders of magnitude as compared to the jammed state) and the viscous modulus (G ) being very small compared to G shows large irregular fluctuations. To retain the shear-thickened state, we have applied a constant stress (σ DC ) of 10 Pa during the frequency sweep measurements. The experiments are done in a parallel-plate (PP) geometry with a gap d = 100 µm. Text S2: We describe the metastability of the shear-thickened state in Fig S2 for 5 wt% MWNT in NMP. The shear-thickened state relaxes into the jammed state when the applied stress (σ DC ) is switched off. The shear-thickened state is attained in a stress controlled flow-curve as described in the manuscript in detail. To characterize the relaxation of the shear-thickened state, we apply an oscillatory strain of very low amplitude (amplitude γ 0 = 0.01%) with angular frequency ω = 10 rad/s on the shear-thickened state (as soon as σ DC is switched off) and measure the storage modulus (G ) as a function of time as shown in Fig S2. We find that G drops exponentially (shown by the solid red line) by an order of magnitude over 1

2 500 s and then after 5000 s it approaches the storage modulus of the jammed state. The experiments are done in PP geometry with a gap d = 100 µm. Text S3: We show the variation of viscosity (η) as a function of shear-rate γ. in the controlled shear-rate mode for 2 wt% MWNT in NMP in Fig S3. We got a steady shear-thinning behaviour where the viscosity decreases by more than two orders of magnitude. The occasional appearance of the sharp spikes in viscosity indicates the contacts between the MWNT flocs under flow (described in detail in the manuscript). Because of the absence of percolated structure in a controlled shear-rate run (confirmed by imaging but not shown) no dramatic shear-thickening was observed in this case. The absence of percolated structure in a controlled shear-rate run obtained through a feedback mode in our stress controlled rheometer is not surprising. Here, any jammed state formed by the cluster-cluster contacts will unjam due to the high stress applied by the instrument to maintain the required steady value of the shear-rate. Text S4: Viscosity (η) as a function of shear-stress (σ) is shown for 5wt% flocculated carbon-black suspension in NMP Fig S4. To bring out the role of floc size, weight fraction (Φ) and gap thickness (d) in these systems, we have carried out experiments on carbon black samples formed by primary spherical particles of two different sizes 10 nm (Vulcan XC72R) and 80 nm (N-762). At a weight fraction of 2% in the quiescent state, the average floc sizes in Vulcan XC72R and N-762 was 20 µm and10 µm respectively. As shown in Fig S4, shear-thickening is seen in N-762 at a smaller gap of 20 µm (at Φ= 5 wt%) (Fig S4e) as compared to 30 µm (Φ= 2 wt%) for Vulcan XC72R (Fig S4b). Moreover, the same sequence of transitions shear-thinning shear-thickened jammed state is obtained in these systems (Fig S4a, S4b, S4c, S4d, S4e, S4f) with the decreasing gap size (d) like that obtained in flocculated nanotube suspensions (Fig 3 in the manuscript). 2

3 Text S5: We show the variation of viscosity (η) and normal-stress (σ N ) as a function of the applied shear-stress (σ) for 2 wt% (Fig S5a) and 5 wt% (Fig S5b) MWNT in NMP. In both the cases, σ N remains nearly constant for σ values corresponding to the jammed and shear-thinning state and starts to increase slowly beyond the onset of the shear-thickening transition. The experiments are done in PP geometry with d = 100 µm (Fig S5a) and d = 150 µm (Fig S5b). Text S6: Variation of storage (G ) and loss (G ) moduli as a function of angular frequency (ω) for the jammed-state (below yield stress) of 7 wt% MWNT suspension is shown in Fig S6. From the figure it is clear that G for the jammed state ( 50 Pa) formed by 7 wt% MWNT suspension (Fig S6) is much lower than that of shear-thickened state ( 500 Pa) of 5 wt% MWNT suspension (Fig S1c). Text S7: We show in Fig S7 the interface profile of the 5 wt% MWNT suspension in NMP (imaged in the vorticity - velocity gradient plane) in cone-plate (CP50-2/s) geometry during the gradual transition of the sample from jammed (region A) shear-thinning (region B) shear-thickened (region C) state (as indicated in viscosity (η) vs applied stress (σ) curve). In the jammed region the interface profile remains same as that shown in the figure in panel A. In the shear-thinning region it changes slightly but the sample does not significantly protrude out of the geometry (panel B) but the dilation of the sample and the roughness of the interface in the shear-thickening region (panel C) as compared to the jammed region (panel A) can be clearly seen. Text S8: Variation of viscosity (η) as a function of applied stress (σ) for for 5 wt%mwnt suspension in NMP when the sample cell is underfilled (Fig S8a) and overfilled (Fig S8b), by 3

4 chnging the sample volume by 20% of the cell volume. The experients are done in parallel plate (PP) geometry with a gap d = 150 µm. The figure clearly shows that the presently observed shear-thickening is similar (within the experimental error bars) in both the cases and it is independent of the boundary condition at the outer edge of the sample. Text S9: We demonstrate in Fig S9 the shear-thickening for 2.5 wt% MWNT suspension in silicone oil (viscosity 0.1 Pa-s) as a function of applied stress. the experiments are done in a coneplate geometry (CP50-2/s). Figure Legends Fig S1: (a) Variation of storage modulus (G ) with σ DC in a controlled shear stress flow curve for for 5wt% MWNT in NMP for gap d = 100 µm, the amplitude of the applied sinusoidal stress signal σ 0 = 0.01 Pa. Storage (G ) and Viscous (G ) moduli as a function of angular frequency (ω) for (b) jammed, (c) shear-thickened state The amplitude of the applied sinusoidal stress signal σ 0 = 0.05 Pa for both (b) and (c). Fig S2: Storage (G ) modulus as a function of time for 5wt% MWNT in NMP of the shearthickened state for gap d = 100 µm. The solid line indicates a fit to the exponential function. Fig S3: Viscosity (η) vs Shear-rate (. γ) for 2wt% MWNT in NMP. The experiment is done in cone-plate (CP50-2) geometry. 4

5 Fig S4: Viscosity (η) vs shear stress (σ) for different values of gap thickness (d), (a) d = 50 µm, (b) d = 30 µm and(c)d=20µm for 2wt% Vulcan XC72R carbon-black in NMP. Same for d) d = 50 µm, (e) d = 20 µm (f)d=10µm for N-762 carbon-black in NMP. Fig S5: Viscosity (η) and normal-stress (σ N ) vs shear-stress (σ) for(a)φ=2wt%mwntin NMP and gap thickness (d) = 100 µm,(b)φ=5wt%mwntinnmpandd=150µm. Fig S6: Storage (G ) and Viscous (G ) moduli as a function of angular frequency (ω) for7wt% MWNT suspension in the jammed-state. The experiment is done in cone-plate (CP50-2/s) geometry. Fig S7: Interface profiles corresponding to the different regions (A, B and C) of the flow curve (η vs σ). The vertical black line indicates the position of the outer edge of the cone. The cartoon indicates the geometry for the measurements. Fig S8: Viscosity (η) vs shear-stress (σ) for Φ= 5wt% of MWNT in NMP when the sample cell was (a) underfilled and (b) overfilled by 20% of the cell volume. The experiments are done in PP geometry with d = 150 µm. Fig S9: Viscosity (η) vs shear-stress (σ) for Φ= 2.5wt% MWNT dispersed in silicone oil. The experiments are done in CP50-2/s geometry. 5

6 Figures Fig S1: (a) σ = 0.01 Pa 0 (b) (c) 0 0 DC Fig S2: 6

7 Fig S3: Fig S4: (a) (d) Φ = 5 wt% (b) (e) Φ = 5 wt% (c) (f ) Φ = 5 wt% 7

8 Fig S5: (a) (b) Fig S6: 8

9 Fig S7: 500 µm Fig S8: (a) (b) 9

10 Fig S9: MWNT in silicone oil 10