Microstructure and yielding of capillary force induced gel

Abstract

We have investigated the rheology and structure of a gel formed from a mixture of non-Brownian particles and two immiscible liquids. The suspension of particles in a liquid undergoes gelation upon the addition of a small content of second, wetting liquid which forms liquid bridges between particles leading to a sample spanning network. The rheology of this gel primarily exhibits a yield stress at low shear rates followed by a linear variation of shear stress at high shear rates. The apparent yield stress extracted from the flow curves increases rapidly with volume fraction of the second liquid before saturation, while it exhibits a monotonic increase with increasing particle concentration. Rescaling of the yield stress curves using suitable shift factors results in an empirical expression for the yield stress showing squared dependence on liquid fraction and a rapid increase with particle fraction above a certain value, both combined in a highly non-linear manner. The microstructural variations with changing secondary liquid content and particle fractions are captured using three-dimensional X-ray tomography technique. The microstructure is observed to show increased local compactness with increased liquid content and increased spatial homogeneity with increased particle fractions. The images from X-ray tomography are analysed to obtain the distributions of particle-particle bonds (coordination number) in the system which serve to explain the observed yield stress behaviour in a qualitative manner.

Appendix

A.1 Rheology dependence on shear gap

Stress ramp measurements were performed for three combinations of particle fraction (ϕ_p) and secondary liquid fraction (ϕ_sl) in concentric cylinder (cup-bob) geometry with outer cylinder (cup) of two different radii, viz. 11.33 mm and 14.33 mm. This realises two shear gaps 3 mm and 6 mm since the inner cylinder (bob) is with radius of 8.33 mm. Given the particle mean diameter of D = 350 ± 50 microns, the two shearing gaps then correspond to approximately 8–9 and 16–18 particle diameters. The variation of stress versus shear rate for two shear gaps and different particle-liquid fraction combination is shown in Fig. 11. The consistently smaller values of stress for the smaller gap compared to those for the larger gap (by a factor or 50% or less) over the entire shear rate range suggest the presence of a small amount of slip. Further, the effect of slip, even if small, seems to be more pronounced for the data below yielding. At the same time, the near similar behaviour for both gaps indicates that the overall results shown for 3-mm shear gap in the main text and the conclusions drawn from them are not affected qualitatively by the presence of a small amount of slip. Morevoer, the values of yield stress which is the main focus of this work remain nearly the same for both the shear gaps.

Fig. 11

Effect of shear gap on stress ramp rheology behaviour. aϕ_p : 0.15,ϕ_sl : 0.002, bϕ_p : 0.20,ϕ_sl : 0.003 and cϕ_p : 0.30,ϕ_sl : 0.002. Circles and triangles, respectively, represent the data acquired for shear gap of 3 mm (∼ 10 particle diameters) and 6 mm (∼ 20 particle diameters)

A.2 Stress strain relationship

Stress ramp rheology experiments were performed for three combinations of ϕ_p and ϕ_sl and for two different acquisition times, viz. t_aq = 10 s and t_aq = 100 s. The variation of the applied stress and corresponding recorded strain experienced by the sample at the end of the acquisition time is shown in Fig. 12. The stress-strain data for both the acquisition time shows reasonable superimposition below the yielding, which points towards a possible elastic behaviour, though the data shown in Fig. 13 suggests plastic deformation. Further, the curves, above the yield point, are distinctly different for both the acquisition times suggestive of predominantly plastic deformation of the sample. The yielding behaviour, however, remains unchanged by the acquisition times as seen from the near identical yield stress for both cases. The exact mechanism of this apparent plastic as well as possible elastic deformation below yielding is not clear and will require flow imaging in the limit of low shear rates not within the scope of this work.

Fig. 12

Variation of shear stress with strain across the entire stress ramp change. aϕ_p : 0.20,ϕ_sl : 0.003. bϕ_p : 0.20,ϕ_sl : 0.004. cϕ_p : 0.30,ϕ_sl : 0.002. Triangles and circles, respectively, represent the data acquired for acquisition time t_aq = 10 s and t_aq = 100 s

Fig. 13

Cumulative evolution of strain with time at two different acquisition times (t_aq) during the step-stress ramp experiment for ϕ_p : 0.2 and ϕ_sl : 0.003. The applied stress values increase in a stepwise manner from 0.00865 to 8.65 Pa. Inset: Data at early times (t < 80 s) shown on a linear scale

A.3 Time-dependent strain evolution

The cumulative evolution of strain (γ) with time (t) during a step-stress ramp experiment is shown in Fig. 13 for two different acquisition times (t_aq). Different gel samples, albeit with same ϕ_sl = 0.003 and ϕ_p = 0.2, are used to obtain data for each acquisition time. Stress was increased in stepwise manner from 0.00865 to 8.65 Pa and the time interval between two consecutive stress levels was equal to the acquisition time. The data acquired with t_aq = 100 s was re-scaled by dividing time (t) by a factor of 10. This is done to enable direct comparison of the strain evolution data between the experiments with t_aq = 10 s and t_aq = 100 s. In both acquisition time experiments, the data shows a yield event in which the strain increases rapidly beyond a certain stress (or strain) level. Before the yield event, the strain values are small suggestive of a solid-like behaviour. However, for each level of applied stress, the strain jumps slightly and then increases more slowly with time, which suggests that the material in its pre-yield stage actually creeps slowly. Thus, the pre-yield state is not truly elastic but comprises plastic deformation to certain extent, perhaps, arising due to rearrangement of particle-particle bonds, i.e. breakage of bonds and reforming with different particles resulting in a local rearrangements. Because of the creep flow, the strain values with t_aq = 100 s are always higher than those with t_aq = 10 s. Interestingly, the sample with t_aq = 10 s tends to yield (i.e. the strain starts increasing rapidly) at a slightly lower stress value (0.162 Pa) than the sample with t_aq = 100 s which tends to yield at 0.216 Pa. The yield event for the sample with t_aq = 100 s is also more distinct than that for the sample with t_aq = 10 s. These differences are likely because the sample with t_aq = 100 s is also ageing to a greater extent during the acquisition time compared to the sample with t_aq = 10 s. Ageing probably results in stronger network, which resists yielding and also breaks more distinctly. From the behaviour described above, the exact mechanism of yielding is not entirely clear. It seems that the solid-like material in the pre-yield state actually creeps slowly until it yields. However, the yield strains for both cases are not identical (about 48% strain with t_aq = 10 and about 58% with t_aq = 100 s). Further investigations of the rheology and yielding behaviour of the suspension are needed which will require rheo-visualization, not within the scope of the present work.

A.4 Refractive index and density matching details

The refractive index of CHB-DEC and TG-1P liquid mixtures was measured using refractometer for varying fractions of CHB (ϕ_CHB) and TG (ϕ_TG), in their respective mixtures (see Fig. 14a). The refractive index for CHB-DEC mixture is nearly constant with varying ϕ_CHB while that for TG-1P mixture shows a linear variation with ϕ_TG. The horizontal dashed line represents the reported value for the refractive index of PMMA. In the vicinity of this value, ϕ_CHB and ϕ_TG were varied in small increments, respectively, in CHB-DEC and TG-1P mixtures in order to determine the optimal binary composition of each mixture at which the refractive index was closest to PMMA. For each small variation, the suspension (of PMMA particles in CHB-DEC or TG-1P liquid mixture) was placed in a transparent walled rectangular cell and a thin laser line, of thickness about one-tenth that of the particle diameter, was transmitted through one of the faces of the cell, about 10 particle diameters away from the other orthogonal face of the cell. The laser line was imaged on a planar surface on the other side of the cell placed orthogonal to the incident laser line. The values of ϕ_CHB and ϕ_TG in their respective mixtures, which resulted in minimum scatter (measured as the variation of intensity with the width of the scattered image) yielded the optimal value of the liquid mixture composition (to the order of third decimal place) for each constituent. This matching is good enough to obtain near transparency up to about 20–30 particle diameters inside the sample.

Fig. 14

Variation of the measured values of (a) refractive index and b density of the liquid mixture with volume fraction of one of its constituents (TG or CHB). Circles represent data for ϕ_CHB in its mixture with DEC. Squares represent values for ϕ_TG in its mixture with 1P. The horizontal dashed line in (a) and (b), respectively, represents the reported values of refractive index and density for PMMA. Inset: An expanded view of the variation of mixture density for small changes to (c) ϕ_CHB and (d) ϕ_TG in their respective mixtures. The solid vertical line in (c) represents the value of ϕ_CHB desired for accurate density matching, while the solid vertical line in (a) provides the closeness of refractive index matching using the same value of ϕ_CHB. The dashed vertical line in (d) represents the value of ϕ_TG desired for accurate density matching, while the dashed vertical line in (a) provides the closeness of refractive index matching using the same value of ϕ_TG

The densities of the CHB-DEC and TG-1P liquid mixtures measured using mass-volume method are shown in Fig. 14b for varying ϕ_CHB and ϕ_TG, in their respective mixtures. The horizontal dashed line represents the reported value of density for PMMA (≈ 1.193). In the vicinity of this value, ϕ_CHB was varied (from 0.66 to 0.71) in very tiny increments (inset (c) in Fig. 14b). As for the refractive index matching procedure, for every small variation, the suspension of PMMA particles in CHB-DEC liquid mixture, placed in a transparent walled rectangular cell was illuminated using a thin laser line. Given the negligible variation of refractive index of CHB-DEC mixture with its composition and its closeness to the refractive index of particles, the sample is rendered nearly transparent. A small amount of fluorescent dye (carboxy-X-rhodamine succinimidyl ester procured from Sigma-Aldrich) added to the CHB-DEC liquid mixture fluoresces a plane in the sample showing (un-dyed) particles in that plane as black shadows on a bright background. The negligible variation of positions of different particles in any plane over a time period of 3 h provides the composition (ϕ_CHB = 0.686) which matches the density of CHB-DEC mixture with that of the particle to the accuracy of three decimal places. Similar procedure (using Nile Red from Sigma-Aldrich as a fluorescent dye) was followed for TG-1P mixture which yielded the composition (ϕ_TG = 0.865) having the density same as the particles to the accuracy of three decimal places. Any further improvement in the accuracy is not possible due to practical limitations in varying the concentration accurately enough.

It is to be noted that while the density matching composition for CHB-DEC mixture is also close to the refractive index matching composition for the same mixture (solid line in Fig. 14a), the same does not hold for TG-1P mixture. The density matching composition induces significant mis-match in the refractive indices between TG-1P mixture and the particles (dashed line in Fig. 14a). The primary importance during the gel preparation is, however, given to accurate matching of densities of liquid mixture and particles. Thus, suspensions of PMMA particles in CHB-DEC mixture of composition ϕ_CHB = 0.686, which is the primary liquid, and containing tiny quantities of TG-1P mixture as the secondary liquid of composition ϕ_TG = 0.865 are exactly density matched and are also transparent enough to be viewed to a depth of about 20–30 particle diameters inside the suspension. All the measurements pertaining to density and refractive index matching and laser imaging were performed in a room with temperature maintained at 25 ± 1^∘C.