Capillary breakup rheometry

Capillary breakup rheometry is an experimental technique used to assess the extensional rheological response of low viscous fluids.

Since no external forcing is exerted in these experiments, the fluid thread can spatially rearrange and select its own time scales.

Quantitative observations about strain rate, along with an apparent extensional viscosity and the breakup time of the fluid, can be estimated from the evolution of the minimal diameter of the filament.

Moreover, theoretical considerations based on the balance of the forces acting in the liquid filament, allow to derive information such as the extent of non-Newtonian behaviour and the relaxation time.

The information obtained in capillary breakup experiments are a very effective tool in order to quantify heuristic concepts such as "stringiness" or "tackiness", which are commonly used as performance indices in several industrial operations.

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.

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.

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.

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 (

[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.

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.

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.

the extensional viscosity, and the term in square brackets represents the non-Newtonian contribution to the total normal stress difference.

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.

In absence of inertia (Ohnesorge number larger than 1) and gravitational effects, the thinning dynamics of a Newtonian fluid are governed purely by the balance between capillary pressure and viscous stresses.

[3] The visco-capillary thinning is described by the similarity solution derived by Papageorgiou, the midpoint diameter temporal evolution may be written as:

Using an upper convected Maxwell constitutive model, the self-similar thinning process is described by an analytical solution of the form

[4] The CaBER experiments employ a liquid bridge configuration and can be thought as a quantitative version of a "thumb & forefinger" test.

In CaBER experiments, a small amount of sample is placed between two measurement plates, forming an initial cylindrical configuration.

The plates are then rapidly separated over a short predefined distance: the imposed step strain generates an “hour-glass” shaped liquid bridge.

The raw CaBER output (Dmid vs time curve) show different characteristic shapes depending on the tested liquid, and both quantitative and qualitative information can be extracted from it.

In terms of quantitative parameters, rheological properties such as the shear viscosity and the relaxation time can be obtained by fitting the diameter evolution data with the appropriate scaling laws.

[6][7] In recent years a number of different techniques have been developed to characterize fluid with very low visco-elasticity, commonly not able to be tested in CaBER devices.