Yield stresses of concentrated suspensions of rigid particles in the volume fraction range of 0.62 to 0.78 via steady torsional flow

Abstract

The ubiquitous wall slip behavior of viscoplastic fluids renders the characterization of their yield stress values a challenge but also presents an opportunity. Here, a new process for the determination of the yield stresses of viscoplastic fluids is introduced and demonstrated on concentrated suspensions subjected to steady torsional flow, i.e., parallel-disk viscometry based on the understanding of apparent wall slip. Four viscoplastic suspensions (particles with a maximum packing fraction, ϕ_m, of 0.86 mixed with a Newtonian binder at the volume fraction, ϕ, range of 0.62 to 0.78) were used. It is demonstrated that a step change in the slope of the torque versus apparent shear rate (or the rotational speed) occurs at a critical torque that corresponds to the yield stress of the suspension. Below the critical torque the behavior is governed by apparent slip and plug flow while above the critical torque the behavior is governed by continuous deformation and apparent slip. The yield stresses of the four concentrated suspensions were verified by comparisons with those obtained from other methods including from wall slip velocities at various shear stresses.

Appendices

Appendix 1

Mixing, small-amplitude oscillatory shear data and parameters of Herschel–Bulkley equation

An intensive batch mixer/torque rheometer, with a mixing volume of 300 ml (EU-5 V manufactured by Haake Buchler Instruments, Inc., Saddle Brooke, NJ, USA), was used to mix the particles with the binder. This mixer has two intermeshing counter-rotating rotors and can impart relatively high stress magnitudes (the design of the intensive mixer is that of the Banbury mixer that is widely used for the compounding of elastomers in the rubber industry). Mixing was carried out at ambient temperature. The rotational speed of the rotors was kept at 32 rpm, and the degree of fill of the mixer (volume occupied by the suspension over the total available volume in the mixer) was 0.8. For each suspension, the mixing time was systematically varied (5-30 min). Sets of five specimens each were collected at regular time intervals of 5–10 min for the determination of mixing indices at various mixing times. This way, the mixtures could be prepared under reproducible conditions. Additional information can be found at He et al. (2019).

The linear viscoelastic material functions of suspensions prepared under various concentrations of the solid phase are shown in Fig. 7 (He et al. 2019). The dynamic properties suggest that G′ > > G″ and that the moduli are relatively insensitive to the frequency of the oscillatory shear deformation at relatively high solid concentrations, indicating that gel-like behavior (DeRosa and Winter 1994) will be approached with increasing ϕ.

Fig. 7

A The storage moduli, G′, and b magnitude of complex viscosity, |η^*|, of the dense suspensions as a function of frequency, ω, at the strain amplitude of 0.02% for different ϕ (He et al. 2019)

He et al. determined the parameters of the shear viscosity material function based on the Herschel–Bulkley equation of the four suspensions (He et al. 2019) as provided in Table 2.

Table 2 Parameters of the Herschel–Bulkley-type viscoplastic equation (Eq. 1) for different ϕ

The wall slip velocity ${U}_{s}$ is defined as the difference between the velocity of the fluid at the wall, and the velocity of the wall. The wall slip velocity is negative for the fluid found adjacent to a moving surface and is positive for the fluid adjacent to a stationary surface. He et al. determined the parameters of wall slip velocity, ${U}_{s}$, versus the shear stress, for the four suspensions given in Table 3:

Table 3 The apparent slip layer thicknesses, δ, as obtained from the slip coefficients, β, under steady torsional flow with m_b = 4.9 Pa s and n_b = 1

These parameters were used for the determination of the torque values for each rotational speed and gap for the four suspensions.

Appendix 2

The calculated torque versus rotational speed and gap used in steady torsional flow via parallel disk viscometry

The torque versus Ω at various gaps and concentrations are further shown in Fig. 8. The data for ϕ = 0.72, 0.76, and 0.78 are consistent with the torque versus Ω and torque versus apparent shear rate data discussed in conjunction with Fig. 2. Overall, the torque does not depend on the gap used for the plug flow region as suggested by Eq. 14. Thus, the plug flow dynamics is wholly governed by the thickness of the apparent slip layer and the flow properties of the binder that constitutes the apparent slip layer. In the plug flow region the slope $\frac{d\mathrm{ln}\mathfrak{I}}{d\mathrm{ln}\left(\mathrm{\Omega R}/\mathrm{H}\right)}$ is equal to 1 for all gaps for the different concentrations, accentuating the understanding that in the plug flow region the flow properties of the bulk suspension do not play a role. Since the binder is Newtonian the slope of torque versus rotational speed for the plug flow region is expected to be 1 ( Eq. 14).

Fig. 8

Steady torque, ℑ, versus the rotational speed of the disk, Ω, for different volume fractions: a 0.72, b 0.76, and c 0.78. The critical torque, ℑ_c, corresponds to the yield condition from which the yield stress can be determined

The slope $\frac{d\mathrm{ln}\mathfrak{I}}{d\mathrm{ln\Omega }}$ depends on the gap for the continuous deformation region, i.e., $\left|{\uptau }_{\mathrm{z\theta }}\left(r\right)\right|>{\uptau }_{0}$ for all concentrations. This is expected on the basis of Eqs. 17 and 18, which represent that the torque values associated with the continuous deformation region are affected by both the apparent slip and the bulk deformation behavior of the suspensions. When the shear stress is significantly greater than the yield stress of the suspension, the slope $\frac{d\mathrm{ln}\mathfrak{I}}{d\mathrm{ln\Omega }}$ will approach the power law index, n (the shear rate sensitivity index) of the Herschel–Bulkley fluid model that represents the shear viscosity of the viscoplastic fluid in the continuous deformation region (Eqs. 17 and 18).

Torque versus the apparent shear rate are shown in Fig. 9 for the solid volume fractions of 0.72, 0.76, and 0.78. It should be noted that in the plug flow region, the dependence shown on the gap is superficial and stems from the definition of the apparent shear rate having the gap in it, i.e., ${\dot{\gamma }}_{aR}=\mathrm{\Omega R}/H$. The slopes are all equal to 1 since $\frac{d\mathrm{ln}\mathfrak{I}}{d\mathrm{ln}\left(\mathrm{\Omega R}/H\right)}=\frac{d\mathrm{ln}\mathfrak{I}}{d\mathrm{ln}\left(\Omega \right)}$. These figures are included since the dependencies shown below can be misconstrued.

Fig. 9

Steady torque, ℑ, versus the apparent shear rate at the edge, ΩR/H, for different volume fractions: a 0.72, b 0.76, and c 0.78. The critical torque corresponds to the yield condition from which the yield stress can be determined