The effect of branching on the shear rheology and microstructure of wormlike micelles (WLMs): Supporting Information

Transcription

1 The effect of branching on the shear rheology and microstructure of wormlike micelles (WLMs): Supporting Information Michelle A. Calabrese, Simon A. Rogers, Ryan P. Murphy, Norman J. Wagner University of Delaware, Department of Chemical and Biomolecular Engineering I. STATIC SCATTERING FROM DIFFERENT SAMPLE CONFIGURATIONS In order to compare the flow-sans results in different planes of shear, static SANS measurements were taken during each experiment to verify that the static material structure was isotropic between planes and independent of sample cell and configuration. The good agreement between the static scattering of each sample in the static quartz cells, rheometer configurations, and the 1-2 shear cell helps verify the absence of significant multiple scattering due to different cell path lengths or other unintended geometrical effects. Figure 1 below shows the comparison of the static structure measured for both the low branched 0.01% wt NaTos solution and the highly branched 0.10% wt NaTos solution between the standard quartz cells, two rheometer configurations with different path lengths, and the 1-2 shear cell. The static quartz cells and first rheometer configuration have path lengths of 2 mm, the second rheometer configuration has a path length of 1 mm, and the 1-2 shear cell has a path length of 5 mm. The scattering shown in Figure 1 shows excellent agreement in both samples across all configurations, validating that the static material structure is independent of plane and sample cell. Figure 1: Static scattering from four sample cells/configurations used in this work: static quartz cells with 2mm path length, two rheometer configurations with a 1mm and 2mm path length, and the 1-2 shear cell with a 5 mm path length. The static scattering shows excellent agreement across cells for both samples, verifying that effects from multiple scattering or geometry are insignificant. 1

2 II. ADDITIONAL CRYO-TEM IMAGES While performing cryo-tem, multiple images were taken for each sample in different grid locations and at different magnifications. Additional images for the 0.01% wt NaTos (low branching, red) solution and 0.10% wt NaTos solution (high branching, blue) are shown in the top two rows of Figure 2 to validate the trends reported in the main paper. To further confirm the qualitative branching trends observed between the 0.10% wt NaTos solution and 0.01% wt NaTos solution, cryo-tem was also performed on a 0.15% wt NaTos solution (very high branching). The 0.15% wt NaTos solution is highlighted in purple to correspond to the purple 1D scattering shown in Figure 2 of the main paper. Results are shown in the bottom row (purple) of Figure 2, where the 0.15% wt NaTos solution exhibits structures similar to those observed in the highly branched case (0.10% wt NaTos), including loops and three-fold junctions. Conversely, the three images of the 0.01% wt NaTos solution confirm that the self-assembled morphology of this sample is of a linear, wormlike micelle. When compared, the images at the lowest magnification (right hand column) show qualitative differences between the branching levels on longer length scales: the low branched sample appears predominantly linear, whereas the two branched solutions appear network-like. While branching is a dynamic process and not all micelles may be branched at all times, these results confirm increasing branching with increasing salt concentration. III. CONSIDERATIONS OF FLOW INSTABILITIES The recent work by Lerouge and Berret (2010), Fardin et al. (2010), and Perge et al. (2014), among many others, analyzes in great detail macroscopic instabilities and turbulence in WLM solutions. We consider both inertial and elastic instabilities by calculating the inertial and elastic Taylor numbers for the results presented based on geometry. The inertial Taylor number is defined as T a = (d/r i ) 0.5 Re, where Re is the Reynold s number. The critical Taylor number for inertial instability is T a = 41 [Fardin et al. (2014)]. In the results presented, inertial Taylor numbers are as follows for the high salt solution: T a inertial < 0.1 in the 1-2 shear cell and T a inertial < 1 in the ARES G2. The inertial Taylor numbers for the low salt solution are T a inertial < 0.01 in the 1-2 shear cell and T a inertial < 0.1 in the ARES G2. As all of these Taylor numbers are significantly less than 41, we rule out inertial instabilities. The critical condition or the elastic Taylor number, T a, for the onset of elastic instability is defined as T a = (d/r i ) 0.5 W i [Larson et al. (1994)]. For the Upper Convected Maxwell Model, T a c 6, although when fluid rheology is taken into account, shear thinning greatly increases the critical condition [Larson et al. (1994)]. Experimentally, in other shear thinning WLMs (N = 0.45) where both inertia and elasticity are important, the onset of inertioelastic instability occurred at an elastic T a c = 22 [Perge et al. (2014)]. The elastic T a c = 22 agrees well with the predictions of Larson et al. (1994) for N = At T a c = 33, Perge et al. (2014) saw small deviations from the base flow, where δv = According to the predictions of Larson et al. (1994), the critical condition for solutions with a power law index N = 0.7 is T a c 500, and an exponential increase in critical condition is seen with shear thinning index. Experimentally, in shear thinning polymer solutions, they saw an onset of instability 20 to 45% lower than the predicted values, giving a minimum T a c = 220 when N = 0.7. However, no experimental work has been performed on more highly shear thinning WLM solutions similar to our solutions, where N = In the high salt, 0.10% wt NaTos solution, we have calculated the elastic Taylor number in various geometries and instruments (ARES G2, MCR501 and 1-2 shear cell) to verify the interpretation of our results. In the 1-2 shear cell, the three shear rates examined result in elastic Taylor numbers in the range of 1 T a 3.5, which are well below model predictions and experimental results [Larson et al. (1994), Perge et al. (2014)]. Therefore, in the 1-2 shear cell, we do not expect any elastic turbulence in the high salt solution. At the onset of the shear thickening regime, 12 Ta 20 depending on the geometry used (see Methods for geometry details). The Taylor number at the end of the shear thickening regime ranges from 25 Ta 45 based on geometry. According to the predictions and experimental results on shear thinning polymer solutions performed by Larson et al. (1994), it appears that the shear thickening regime is below the critical condition for elastic instability. 2

3 Figure 2: Cryo-TEM images of the low branched (0.01% wt NaTos) solution (top row, red), highly branched (0.10% wt NaTos) solution (middle row, blue) and very highly branched (0.15% wt NaTos) solution (bottom row, purple). Images were taken at different magnifications and grid locations. The 1D static scattering of the 0.15% wt NaTos solution is also highlighted in purple in Figure 2 of the main paper. Similar to the highly branched, 0.10% wt NaTos solution, a variety of irregular structures, junctions and loops are observed in the 0.15% wt NaTos solution; however, the 0.01% wt NaTos solution shows primarily linear structures in all images. 3

4 However, the results from Perge et al. (2014) that show only small deviations from the base flow up to T a = 33 gives confidence that the flow will not be significantly affected by turbulence should elastic instabilities arise in this regime. In the shear banding, 0.01% wt NaTos solution, the stress plateau spans Wi = 1 to Wi 300. Due to the high Weissenberg numbers aross the stress plateau, the elastic Taylor number in the 1-2 shear cell results ranges from 15 T a 50. While we cannot discern elastic instabilities during shear banding using our SANS methods, elastic turbulence may be present in the high shear band due to the high Weissenberg numbers. However, T a = 15 at the first shear rate in the 1-2 shear cell, which is less than the transition of T a c = 22 observed in the work of Perge et al. (2014). While the other two shear rates in the 1-2 shear cell are above T a c = 22, the trends observed that indicate shear banding are the same at all three shear rates. Furthermore, 1-2 shear cell results for this sample at T a = 9 (data not shown) also show the same alignment trend, giving further evidence that the presence or absence elastic instabilities does not affect the interpretation of the 1-2 plane SANS. We also consider endwall and Ekman layer effects due to the short aspect ratio of the 1-2 shear cell (Γ = 5). For a Newtonian fluid, the RMS error of the Couette velocity profile u(r, z) for Γ=5 is 0.01 in comparison to the infinite cylinder profile [Wendl (1999)]. Larson et al. (1994) among others found that shear thinning in viscoelastic fluids generally increases the critical dimensionless groups for the onset of instabilities or deviations from the expected flow, giving evidence that this RMS value is on the correct order of magnitude (and thus fairly insignificant). While the shear cell is a short cylinder, the stationary end walls minimize the size and strength of the Ekman vortices as compared with moving endwalls [Sobolık et al. (2000), Czarny et al. (2003)]. Paired with the low shear rates in used in the 1-2 plane studies and resulting inertial Taylor numbers, the resulting Ekman layer should not interact with the primary flow and the axial and radial components of the velocity gradient should be effectively zero [Sobolık et al. (2000), Czarny et al. (2003)]. IV. SHEAR THICKENING STRESS FLUCTUATIONS % WT NATOS SOLUTION To address the possibility of elastic turbulence in the shear thickening regime of the 0.10% wt NaTos solution, long time startup measurements were performed to determine the fluctuations of the stress response in time. For a shear banding CTAB solution in the semi-dilute regime (10-fold more concentrated than our solutions), Fardin et al. (2010) observed stress fluctuations on the order of 3% in the laminar, highly aligned region exiting the stress plateau, where the aligned structures are expected to fill the entire gap. These fluctuations were stable in the amplitude over the course of one hour. In the transition region (stress upturn, shear thickening may occur), the stress never appears stable in time. The fluctuations were as high as 30%, and the most significant deviations began in less than 20 minutes (unstable in amplitude). Past the transition region, the fluctuations settled (stable in amplitude) but were larger, on the order of 15%. Using these regimes as a guide, the stability of the 0.10% wt NaTos solution at and around the shear thickening transition was assessed using startup measurements in both the ARES G2 (strain controlled) and the NIST MCR 501 rheometer (stress controlled operating in strain controlled mode), the results of which are shown in Table 1. Measurements in the ARES G2 were performed for 30 minutes, and measurements in the MCR501 were performed for 20 minutes during SANS measurements. The fluctuations around the mean value of the stress at a variety of shear rates (before, at and after the shear thickening transition) are reported for both instruments in Table 1. These long-time startup tests show that the fluctuations are stable in amplitude over the course of the test (t>10s) and are roughly the same between the two instruments. The fluctuations in the recorded stress are similar to those observed in the laminar flow regime (maximum 3.54% for each instrument) from Fardin et al. (2010), and are also independent of shear rate. As expected for a WLM system undergoing a structural transition during shear thickening, the fluctuations around the stress are detectable, but not large. These results further support that the shear thickening likely corresponds to a structural transition that is not significantly affected by elastic turbulence. Interestingly, the stress of the 0.10% wt NaTos solution is unstable over long times beginning at γ 2000 s -1. It is likely that elastic turbulence begins to significantly affect the flow at these higher shear rates. 4

5 Table 1: Stress fluctuations in the shear thickening regime for the branched 0.10% wt NaTos solution γ (s -1 ) ARES G2 σ (%) MCR 501 σ(%) N/A N/A 2.23 References Czarny, O., E. Serre, P. Bontoux, and R. M. Lueptow, Interaction between Ekman pumping and the centrifugal instability in Taylor Couette flow, Physics of Fluids 15, (2003). Fardin, M.-A., D. Lopez, J. Croso, G. Grégoire, O. Cardoso, G. McKinley, and S. Lerouge, Elastic turbulence in shear banding wormlike micelles, Physical Review Letters 104, (2010). Fardin, M., C. Perge, N. Taberlet, and S. Manneville, Flow-induced structures versus flow instabilities, Physical Review E 89, (2014). Larson, R., S. Muller, and E. Shaqfeh, The effect of fluid rheology on the elastic Taylor-Couette instability, Journal of Non-Newtonian Fluid Mechanics 51, (1994). Lerouge, S. and J.-F. Berret (2010), Shear-induced transitions and instabilities in surfactant wormlike micelles, in Polymer Characterization,, Springer, pp Perge, C., M.-A. Fardin, and S. Manneville, Inertio-elastic instability of non shear-banding wormlike micelles, Soft Matter 10, (2014). Sobolık, V., B. Izrar, F. Lusseyran, and S. Skali, Interaction between the Ekman layer and the Couette Taylor instability, International Journal of Heat and Mass Transfer 43, (2000). Wendl, M. C., General solution for the Couette flow profile, Physical Review E 60, 6192 (1999). 5