Particle contact dynamics as the origin for noninteger power expansion rheology in attractive suspension networks

Abstract

In medium amplitude oscillatory shear (MAOS), the cubic scaling of the leading order nonlinear shear stress (σ3 ∼ γ m 3 0 , m3 = 3) is the typical expectation. Expanding on the work by Natalia et al. [J. Rheol., 64(3): 625 (2020)], we report non-cubical, non-integer power law scalings m3 for particle suspensions in two immiscible fluids with a capillary attractive interaction, known as capillary suspensions. Here, we show that distinct power law exponents are found for the storage and loss moduli and these non-integer scalings occur at every secondary fluid concentration for two different contact angles. These compelling results indicate that the non-integer scalings are related to the underlying microstructure of capillary suspensions. Weshow that the magnitude of the third harmonic elastic stress scaling m 3,elastic originates from Hertzian-like contacts in combination with the attractive capillary force. The related third harmonic viscous stress scaling m3,viscous is, therefore, associated with adhesive controlled friction. These observations, conducted for a wide range of compositions, can help explain previous reports of non-integer scaling for materials involving particle contacts and offers a new opportunity to study the importance of the particle bonds and friction in the rheological response under low deformation instead of at very high shear rates.

Effects of particle roughness on the rheology and structure of capillary suspensions

“…The results of the power law fitting of σ ′ 3 and σ ′′ 3 for both the samples with ϕ sec = 1 vol% and adjusted secondary fluid can be seen in Figure 6. First, both the scaling of m 3,elastic and m 3,viscous is noncubic and noninteger for all samples, in accordance with the works of Natalia et al [43,42] The power law exponents do not differ significantly between sample groups, with m 3,elastic values falling mostly between 1.5 and 2.5 and m 3,viscous values between 1 and 2. The spread on the data is quite large due to the limited number of datapoints above the noise floor available for the fitting owing to the 8 mm parallel plate geometry.…”

“…If the region in between the G ′′ maximum and the crossover point is included as well, this relaxes the scaling of m 3,elastic to values between 1.5 and 1, which is more similar to the pendular capillary suspension system in the work of Natalia et al [42] Important to note is that the scaling that can be observed is limited by the sensitivity window of the rheometer, shown as gray shading in Figure 5. As a result of the smaller parallel plate geometry, the noise floor corresponding to the sensitivity limit of the rheometer is shifted to higher stress values, which could mask an initial cubic scaling, as already discussed by Natalia et al [42] For the interval shown in Figure 5, the elastic and viscous exponents are 1.7 ± 0.1 and 1.6 ± 0.2 using the 8 mm. The dashed lines show the effect on the fit after including or excluding an extra datapoint.…”

“…For capillary suspensions, the normal force is given by the capillary bridge force combined with the influence of the applied deformation. [42] This leads to a relationship between the elastic and viscous third harmonic power law scalings with either for load-controlled friction, or (increased signal to noise ratio) would be required to confirm this conclusion.…”

“…By definition, the MAOS regime is the nonlinear region where only the first and third harmonic stress signals appear. Previous research has shown that the scaling of the third harmonic is non-integer and non-cubic [43] and that it is sensitive to particle collisions [42]. Since the particle collisions are expected to be indirectly affected by the roughness through the strength of the bridges and directly affected via the frictional interactions, the third harmonic response should vary between the smooth and rough particles.…”

“…[41] Natalia et al found that this combination of Hertzian contact and capillary force explains the non-integer power law scaling of the third harmonic elastic stress in the medium amplitude oscillatory shear regime. [42] Typically, the third harmonic elastic and viscous stresses σ 3 are expected to scale with an exponent of m = 3 with the applied strain, σ 3 ∝ γ 3 0 , but for capillary suspensions, as well as for some other particulate systems, this power law scaling m was shown to be noncubic. [43] In that work, Natalia et al linked the scaling of the third harmonic viscous stress to particle-particle friction.…”