Prevention of edge fracture using a nontoxic liquid metal sealant

Rotational rheometers equipped with either a parallel plate or a cone-and-plate fixture are among the most widely used laboratory instruments to characterize the rheological properties of fluids. They can apply almost uniform shear flow fields to the fluid sample, facilitating the analysis of rheological data. However, viscoelastic fluids, such as epoxy and paste, can develop normal stresses when subjected to a sufficiently high shear rate $γ̇$⁠, leading to edge fracture, a flow instability characterized by the sudden indentation of a viscoelastic fluid's free surface.^1–13 The fracture can invade the fluid sample, decreasing its contact area with the rheometer fixture, and leading to a decrease in the measured torque signal. Simulations have shown that edge fracture can also induce apparent shear banding in the bulk of the fluid, further complicating the interpretation of rheological measurement results.¹⁰ 

A basic physical understanding of edge fracture is essential to developing effective strategies to mitigate the phenomenon. Because of this, different theories have been proposed to predict the onset of edge fracture. Based on fracture mechanical principles, the Tanner–Keentok criterion⁴ predicts that fracture for a second-order fluid will occur if

where $N2≡Tzz−Trr$ is the second normal stress difference, T_zz and T_rr are the axial and radial normal stresses, σ is the interfacial tension, and a is the radius of a pre-assumed semi-circular crack on the fluid sample's free surface. Meanwhile, by linear stability analysis, the Hemingway–Fielding criterion⁹ predicts that fracture will occur for the Johnson–Segalman¹⁴ and Giesekus fluids^15,16 if

where τ is the shear stress, $Δτ$ is the jump in the shear stress between the fluid and the outside medium, and H is the gap size of the rheometer fixture. In the low shear rate limit, the Hemingway–Fielding criterion reduces to the Tanner–Keentok criterion in the form of $|N2|>2πσ/H$⁠, with the dominant wavelength of the instability $H/2π$ replacing the pre-assumed crack radius a. For typical rheological experiments, the Tanner–Keentok and Hemingway–Fielding criteria [Eqs. (1) and (2)] can be expressed in terms of the Tanner number^12,13

where L is the characteristic length scale. $Tn$ characterizes the relative importance of N₂ and the Laplace pressure $σ/L$⁠. If $Tn≥O(1)$⁠, edge fracture is likely to occur. Otherwise, the fluid's free surface is stable.

The Laplace pressure can be increased by decreasing the fixture gap size H, which has proved to be effective in preventing the occurrence of edge fracture.^1–3 Another way is to increase the interfacial tension σ between the fluid sample and its surrounding medium. In their rheological study of a liquid crystalline hydroxypropyl cellulose aqueous solution, Grizzuti et al.¹⁷ sealed their fluid sample using mercury, whose interfacial tension with water is 374 mN m⁻¹, a factor of $∼5$ higher than the aqueous–air interface.¹⁸ They observed that fracture could be delayed for around two decades of shear rate. Promising as it was, the method did not get much attention from the rheology community, potentially due to the toxicity of mercury.¹⁹ 

This Letter revisits the idea of Grizzuti et al. in using liquid metal to prevent edge fracture. To solve the toxicity problem, mercury is replaced with galinstan (Conductonaut, Thermal Grizzly), a nontoxic eutectic alloy of gallium, indium, and tin,^20–25 commonly used by computer enthusiasts as a thermal interface for hardware cooling. The density ρ_g and shear viscosity η_g of galinstan are 6440 kg m⁻³ and 2.4 mPa s, respectively.²⁵ The fluid sample is sealed with galinstan in a stainless steel cone-and-plate fixture of cone angle $ϕ=1°$ and radius $Rp=12.5$ mm [Fig. 1(a)]. A rubber gasket, fixed on the rheometer plate with double-sided tape, is used to contain the galinstan. The effectiveness of the galinstan-sealed cone-and-plate (CP-G) fixture in mitigating fracture is compared to two other fixtures: the conventional cone-and-plate (CP) fixture without galinstan, and a stainless steel cone-partitioned plate (CPP) fixture,^26,27 also without galinstan. The CPP fixture consists of a $ϕ=5.73°$ cone, an inner disk of radius R_i = 5 mm, and an outer ring of radius $Rp=12.5$ mm [Fig. 1(b)]. The inner disk is connected to the stress transducer, while the outer ring is mounted to the rheometer frame. Hence, only the fluid in the center 5 mm radius is involved in the rheometric measurement; the outer 7.5 mm radius of excess fluid acts as a liquid guard ring to protect the center area from being invaded by edge fracture.^28–30 All rheological experiments are conducted at room temperature 25 °C with the fixtures installed on an ARES-G2 strain-controlled rotational rheometer (TA instruments). When the liquid metal is not used, it is stored in a glass container filled with 0.1 M hydrochloric acid at 4 °C to suppress oxidation.³¹ No protection is applied to the rheometer fixture as galinstan does not react with stainless steel.³² The rubber gasket can be removed and cleaned using hydrochloric acid after experiments, allowing most galinstan to be collected and reused.

The fluid sample of interest is a weakly viscoelastic silicone oil (Dow Corning 200, Sigma-Aldrich). Its interfacial tensions with air and galinstan are $σa=20.6$ m N m^–1 and $σg=496$ m N m^–1, respectively. The values are obtained at 25   °C using a pendant drop tensiometer (Theta Attension, Biolin Scientific). An oil-in-air pendant drop configuration is used to measure σ_a. In contrast, a galinstan-in-oil configuration is used for σ_g. Figure 2 shows the small amplitude oscillatory shear (SAOS) test results of the silicone oil under a strain amplitude of 5%. The ARES-G2 rheometer equipped with a 10 mm radius, 0.5 mm gap size stainless steel parallel plate fixture is used for the SAOS test. The storage and loss moduli $G′$ and $G″$ are obtained in a temperature range of −65–25 °C. The moduli are then superimposed into a master curve for 25 °C spanning six decades of oscillatory frequency ω using the time-temperature superposition principle.^33,34 The master curve is well-described by the three-mode Giesekus model,^15,16 giving $ηi=38,8, 12.6$ Pa s, $λi=5,0.8,0.3$ ms, and $αi=α=0.49$⁠, where $ηi,λi$⁠, and α_i are the polymer viscosity, relaxation time, and mobility parameter of the ith mode, respectively. These parameters give a zero-shear viscosity $η0=∑iηi=58.6$ Pa s and an average relaxation time $λ=∑iηiλi/η0=3.42$ ms. The three-mode Giesekus model also accurately predicts the normal stress differences in the fluid as a function of the shear rate,¹² providing a zero-shear normal stress ratio $Ψ0≡−N2/N1|γ̇→0=α/2$⁠, where $N1≡Tθθ−Tzz$ is the first normal stress difference and $Tθθ$ is the azimuthal normal stress. The Giesekus model was used to simulate the edge fracture dynamics of liquid bridges made of the same silicone oil used in the current study, the prediction of which showed excellent agreement with experiments.¹² Hence, the model can accurately describe the silicone oil's nonlinear flow behavior. Because of this, its prediction is used as a benchmark to compare the results obtained by the CP-G, CP, and CPP fixtures mentioned earlier. Based on the Giesekus model, a characteristic Tanner number can be defined as^12,13

Here, the zero-shear parameters η₀ and $Ψ0$ are used. At high shear rates, however, N₂ does not scale quadratically with $γ̇$ but tends to plateau.¹¹ Nonetheless, for the shear rate range considered in the current study, Eq. (4) offers a good enough approximation for performing order-of-magnitude analysis.

The shear startup test is employed to characterize how edge fracture affects rheological experiments in the three rheometer fixtures. Step shear rates of different magnitude $γ̇$ are applied to the silicone oil; the shear stress responses τ are recorded as functions of time t. For the CP fixture [Fig. 3(a)], the experimentally measured $τ(t)$ shows a steady-state response for $γ̇≤100$ s⁻¹, agreeing well with the Giesekus model prediction. For $γ̇=200$ s⁻¹, $τ(t)$ reaches a steady state temporarily for around 1 s and then drops over time, deviating from the model prediction. Accompanying the observation that the silicone oil is expelled out of the rheometer gap, the drop in $τ(t)$ signifies the occurrence of edge fracture.^1–3 Further increasing the applied shear rate to $γ̇=500$ and $γ̇=1000$ s⁻¹ causes the fracture to penetrate the fluid sample faster and $τ(t)$ to drop sooner.

For the CPP fixture [Fig. 3(b)], the measured $τ(t)$ agrees with the model prediction for $γ̇≤50$ s⁻¹. For $γ̇=100$ s⁻¹, $τ(t)$ is able first to reach a steady state for roughly 10 s before edge fracture can sufficiently invade the fluid sample to affect the measurement, showing how the liquid guard ring protects the inner active measuring zone of the fixture. Further increasing $γ̇$ to higher values reduces the protection time of the liquid guard ring. For instance, $τ(t)$ drops within 1 s for $γ̇=500$ s⁻¹. It is worth noting that fracture occurs at a lower shear rate than in Fig. 3(a). This is because the cone angle $ϕ$ for the CPP fixture is around five times larger than the CP fixture. By Eq. (4), for the same Tanner number $Tn$⁠, the critical shear rate at which fracture occurs follows $γ̇c∝ϕ−1/2$⁠. Hence, $γ̇c$ for the CPP fixture can be expected to be half of that for the CP fixture, agreeing with the experimental observation.

For the CP-G fixture [Fig. 3(c)], $τ(t)$ is larger than the model prediction for $γ̇=5$ s⁻¹, which is likely because of the elastoplastic oxide film forming on the liquid metal's free surface.^22,24 $τ(t)$ approaches the model prediction as $γ̇$ is increased to 50 s⁻¹. Increasing $γ̇$ to 500 s⁻¹, $τ(t)$ stays steady even after 100 s; its magnitude agrees with the model prediction, implying that edge fracture did not occur. Such a behavior can be rationalized again via the Tanner number $Tn$⁠. According to Eq. (4), for the same $Tn$⁠, the critical shear rate for fracture follows $γ̇c∝σ1/2$⁠. As the silicone–galinstan interface has an interfacial tension around 25 times higher than that of the silicone-air interface, $γ̇c$ for the CP-G fixture can be expected to be around five times that for the conventional CP fixture. Further increasing $γ̇$ to 1000 s⁻¹, $τ(t)$ can be seen to vary with time. It drops slightly in the first 10 s; however, a steady state is reached afterward, the magnitude of which still agrees reasonably well with the prediction of the Giesekus model. Such a slight drop in $τ(t)$ is likely caused by edge fracture as $Tn∈O(10)$⁠. However, inertial secondary flows in the liquid metal may also be relevant,³⁵ as the characteristic Reynolds number $Re=ρgγ̇(Rpϕ)2/ηg$ corresponding to $γ̇=1000$ s⁻¹ is of O(100). It will be interesting to see if inertial effects can suppress edge fracture; the centrifugal force asserted by the surrounding liquid metal may help pull the fracture backward as it develops. Nonetheless, this is outside the scope of the current study and shall be left as future work. The observed difference in $γ̇c$ between the CP-G and CP fixtures is less than what Grizzuti et al.¹⁷ reported for their hydroxypropyl cellulose solution with and without mercury sealing. This is explained by the contrasting normal stress differences of the fluid samples used. The normal stress differences of the hydroxypropyl cellulose solution used by Grizzuti et al. are complex non-monotonic functions of the shear rate; in contrast, those of the silicone oil used here increase monotonically.

The experimental observations described above illustrate that sealing the fluid sample with galinstan can effectively delay the occurrence of edge fracture, extending the measurable shear rate range of rotational rheometry. Figure 4 shows the flow curve of the silicone oil constructed using the CP-G (gold symbols) and CP (blue symbols) fixtures to showcase the advantage of the galinstan-sealing method further. The experimental data are plotted with the Giesekus model prediction for comparison. The CP fixture can measure the shear stress τ and first normal stress difference N₁ of the silicone oil up to a shear rate of $γ̇=100$ s⁻¹, above which edge fracture occurs, preventing data at higher shear rates from being reliably obtained. By sealing the silicone oil's free surface with galinstan, τ and N₁ can be measured up to $γ̇=1000$ s⁻¹, the trend and magnitude of which agree well with the Giesekus model prediction. Regarding the error in N₁, the normal force measurement at high $γ̇$ using rotational rheometry can be sensitive to inertial effects, which tend to cause a downward force pulling the cone and plate toward each other. The inertial contribution to N₁ assumes the form $δN1=−3ρgγ̇2(Rpϕ)2/20$⁠.³⁶ For $γ̇=1000$ s⁻¹, $|δN1|/N1∈O(10−3)≪1$⁠; hence, $δN1$ is negligible. Considering the shear viscosity η, the shear-thinning behavior of the silicone oil can be seen. Typically, measuring the nonlinear rheological response of fluids that suffer from edge fracture would require using specialized instruments such as the CPP fixture,^26,27 specially designed guard ring,^37,38 capillary rheometer,^39,40 or diffusing wave spectroscopy.⁴¹ The latter two cannot measure normal stress differences. The galinstan-sealing method offers a cheap yet effective alternative to those specialized instruments.

To conclude, sealing the fluid sample's free surface in a rheometer fixture using liquid metal galinstan can delay edge fracture onset. The method does not require special instruments except the standard rheometer fixtures. It is relatively cheap and quick to perform. No calibration procedure is needed. Also, galinstan is widely used as a heat-conducting agent to cool computer parts; it can be easily purchased in computer hardware stores and online shops. Unlike the cone-partitioned plate fixture, which could only delay the fracture from penetrating the fluid sample,^28–30 the galinstan-sealing method prevents fracture occurrence, allowing actual steady states to be reached in rheological experiments. This is particularly useful for situations where edge effects must be eliminated, such as the study of shear banding in polymer solutions.^42–48 

Multiple research directions can be derived from the current work. An obvious one is to employ the galinstan-sealing method to the parallel plate and the cone-and-plate fixtures with different gap size H, cone angle $ϕ$⁠, and radius R_p; it is expected that decreasing H or $Rpϕ$ of the rheometer fixture can further delay the onset of edge fracture. Changing H and $Rpϕ$ can also help systematically vary the characteristic Reynolds number $Re$ of the rheological experiment, which is essential to understanding if inertial effects from the liquid metal can help suppress fracture. It will also be interesting to employ the method on fluids such as particulate suspensions, whose N₂ is known to scale linearly with the applied $γ̇$⁠.^49,50 For such fluids, the galinstan-sealing method is expected to be even more effective in preventing edge fracture.