Capillary breakup rheometry

Capillary breakup rheometry and its recent development are based on the original experimental and theoretical work of Schümmer and Tebel and Entov and co-workers. Nonetheless, this technique found his origins at end of the 19th century with the pioneering work of Joseph Plateau and Lord Rayleigh. Their work entailed considerable progress in describing and understanding surface-tension-driven flows and the physics underlying the tendency of falling liquid streams to spontaneously break into droplets. This phenomenon is known as Plateau–Rayleigh instability.

Uniaxial extensional flow created within necked filament by a capillary instability

The linear stability analysis introduced by Plateau and Rayleigh can be employed to determine a wavelength for which a perturbation on a jet surface is unstable. In this case, the pressure gradient across the free-surface can cause the fluid in the thinnest region to be "squeezed" out towards the swollen bulges, thus creating a strong uniaxial extensional flow in the necked region.

As the instability grows and strains become progressively larger, the thinning is governed by non-linear effects. Theoretical considerations on the fluid motion suggested that the behaviour approaching the breakup singularity can be captured using self-similarity. Depending on the relative intensity of inertial, elastic and viscous stresses, different scaling laws based on self-similar considerations have been established to describe the trend of the filament profile near breakup throughout the time.

Experimental configurations used to study free surface flows. From left: liquid bridge, dripping, jetting

Capillary thinning and breakup of complex fluids can be studied using different configurations. Historically, mainly three types of free-surface conformations have been employed in experiments: statically-unstable liquid bridges, dripping from a nozzle under gravity and continuous jets.[1] Even though the initial evolution of the capillary instability is affected by the type of conformation used, each configurations capture the same phenomenon at the last stages close to breakup, where thinning dynamics is dominated by fluid properties exclusively.

The different configurations can be best distinguished based on the Weber Number, hence on the relative magnitude between the imposed velocity and the intrinsic capillary speed of the considered material, defined as the ratio between the surface tension and shear viscosity (${\displaystyle \gamma /\eta }$).[2] In the first geometry, the imposed velocity is zero (We=0), after an unstable liquid bridge is generated by rapid motion of two coaxial cylindrical plate. The thinning of the capillary bridge is purely governed by the interplay of inertial, viscous, elastic and capillary forces. This configuration is employed in the CaBER device and it is at present the most used geometry, thanks to its main advantage of maintaining the thinnest point of the filament approximately located in the same point. In dripping configuration, the fluid leaves a nozzle at a very low velocity (We < 1), allowing the formation of a hemispherical droplet at the tip of the nozzle. When the drop becomes sufficiently heavy, gravitational forces overcome surface tension, and a capillary bridge is formed, connecting the nozzle and the droplet. As the drop falls, the liquid filament becomes progressively thinner, to the point in which gravity becomes unimportant (low Bond number) and the breakup is only driven by capillary action. At this stage, the thinning dynamics is determined by the balance between capillarity and fluid properties. Lastly, the third configuration consists in a continuous jet exiting a nozzle at a velocity higher than the intrinsic capillary velocity (We > 1). As the fluid leaves the nozzle, capillary instabilities naturally emerge on the jet and the formed filaments progressively thin as they are being convected downstream with the flow, until eventually the jet breaks into separate droplets. The jetting-based configuration is generally less reproducible compared to the former two due to different experimental challenges, such as accurately controlling the sinusoidal disturbance.[1]

The temporal evolution of the thinnest region is determined by a force balance in the fluid filament.[2] A simplified approximate force balance can be written as

${\displaystyle \eta _{E}{\dot {\epsilon }}_{min}(t)\approx {\frac {\gamma }{D_{min}(t)/2}}-\left[\tau _{zz}-\tau _{rr}\right]}$

where ${\displaystyle \gamma }$ is the surface tension of the fluid, ${\displaystyle {\dot {\epsilon }}_{mid}}$ the strain rate at filament midpoint, ${\displaystyle \eta _{E}}$ the extensional viscosity, and the term in square brackets represents the non-Newtonian contribution to the total normal stress difference. The stress balance shows that, if gravity and inertia can be neglected, the capillary pressure ${\displaystyle \gamma /D_{min}}$ is counteracted by viscous extensional contribution ${\displaystyle 3\eta _{s}{\dot {\epsilon }}_{min}(t)}$ and by non-Newtonian (elastic) contribution.

Depending on the type of fluid, appropriate constitutive models have to be considered and to extract the relevant material functions. Without any consideration on the nature of the tested fluid, it is possible to obtain a quantitative parameter, the apparent extensional viscosity ${\displaystyle \eta _{E,app}(t)}$ directly from the force balance, among capillary pressure and viscous stresses alone. Assuming an initial cylindrical shape of the filament, the strain rate evolution is defined as

${\displaystyle {\dot {\epsilon }}(t)=-{\frac {2}{D_{min}}}{\frac {dD_{min}}{dt}}}$

Thus, the apparent extensional viscosity is given by

${\displaystyle \eta _{E,app}(t)={\frac {\gamma }{\frac {dD_{min}(t)}{dt}}}}$

Temporal evolution of the midpoint filament diameter, scaling laws and high-resolution images of capillary breakup experiments for different fluid. From left: visco-capillary thinning (silicone oil) and elasto-capillary thinning (PEO in water solution)

The behaviour of the fluid determines the relative importance of the viscous and elastic terms in resisting the capillary action. Combining the force balance with different constitutive models, several analytical solutions were derived to describe the thinning dynamics. These scaling laws can be used to identify fluid type and extract material properties.

There are many processes and applications that involves free-surface flows and uniaxial extension of liquid filaments or jets. Using capillary breakup rheometry to quantify the dynamics of the extensional response provides an effective tool to control processing parameters as well as design complex fluids with required processability. A list of relevant applications and processes includes:

• Ink-jet printing
• Atomization
• Dispensing and dosing of complex fluids
• Electrospinning
• PSA
• Spray and curtain coating
• Fertilizer distribution

• Fluid thread breakup
• Plateau–Rayleigh instability
• Rheometer
• Visco-elastic jets
• Capillary action
• Capillary