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Capillary Breakup Rheometry

Capillary Breakup Rheometry is an experimental technique used to assess the extensional rheological response of low viscous fluids. Unlike most shear and extensional rheometers, this technique does not involve active stretch or measurement of stress or strain, but exploits only surface tension to create a uniaxial extensional flow. Hence, although it is common practice to use the name rheometer, capillary breakup techniques should be better addressed to as indexers.

Capillary breakup rheometry is based on observing the breakup dynamics of a thin fluid thread, governed by the interplay of capillary, viscous, inertial and elastic forces. Quantitative observations about axial stress and 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 behavior and the relaxation time spectrum. 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 indeces in several industrial operations. At present, the only commercially available device based on capillary breakup technique is the CaBER.

Theoretical framework
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. They 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.

The linear stability analysis introduced by Plateau and Rayleigh can be employed to select a wavelenght 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 growths and strains becomes progressively larger, the thinning is governed by non-linear effects. Theroretical considerations on the fluid motion suggested that the behaviour approaching the breakup singularity can be captured using self-similarity. Depending on the relative importance of inertial, elastic and viscous stresses, different scaling laws based on self-similar considerations have been establisehd to describe the temporal evolution of the filament profile near breakup.

Experimental geometries


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 continous jets. 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 are 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. 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 maintaing 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 are determined by the balance between capillarity and fluid properties. Lastly, the third configuration consits in a continous 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 accaurately controlling the sinusoidal disturbance.

Scalings and constitutive models
The evolution in the midpoint profile is governed by a force balance on the  fluid filament. This can be written compactly in the following form (Yarin, 1993; Renardy, 1995):

These scaling laws can be used to identify material classes and extract other material properties of the fluid besides an apparent extensional viscosity.

The breakup behaviour of viscoelastic fluids is very distinct from that of Newtonian liquids. Adding a small amount of flexible polymers to the liquid drastically alters the thinning dynamics of the jet. As the local radius of the instabilities decreases, elastic stresses are growing and the jet consists of a series of small droplets joined by small threads, that become increasingly thinner with distance [68]. The breakup length of the jet is considerably enhanced and this effect can thus be considered as strain-hardening of the polymer solution in an extensional flow field. These viscoelastic effects can be used to alter the atomisation characteristics and to suppress the formation of unwanted satellite drops [15, 16].

CaBER
The CaBER (Capillary Breakup Extensional Rheometer) is the only commercially available instrument based on capillary breakup. Based on the experimental work of Entov, Bazilevsky and co-workers, the CaBER was developed  by  McKinley  and  co-workers  at  MIT   in collaboration with   the   Cambridge   Polymer   Group in  the  early  2000s. Currently, it is manufactured by Thermo Scientific with the commercial name HAAKE™ CaBER™ 1.

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 intial cylindrical configuration. The plates are then rapidly separeted over a short predefined distance: the imposed step strain generates an “hour-glass shaped” liquid bridge. The necked sample subsequently thins and eventually breaks under the action of capillary forces. During the surface-tension-driven thinning process, the evolution of the mid-filament diameter (Dmid(t)) is monitored via a laser micrometer.

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. The time-to-breakup is the most direct qualitative information that can be obtain. Although this parameter does not represent a property of the fluid itself, it is certainly useful to quantify the processability of complex fluids. 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 constitutive models. The second quantitative information that can be extracted is the apparent extensional viscosity. Assuming that the stress is equal to the capillary pressure and calculating the strain rate evolution as

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

the apparent extensional viscosity can be then defined as

$$\eta_{E,app}(t)=\frac{\gamma}{\frac{dD_{mid}(t)}{dt}}$$

Despite the great potential of the CaBER, this technique also presents a number of disadvantages, mainly related to the susceptability to solvent evaporation and the characterization of very low visco-elatic fluids, for which creating a statically-unstable bridge is extremely challenging from the experimental point of view. Different modifications of the commercial instrument have been presented to overcome these issues. Amongst others: the use of surrounding media different than air and the Slow Retraction Method (SRM).

Other techiniques
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.
 * The Cambridge TrimasterTM a fluid is symmetrically stretched to form an unstable liquid bridge. This instrument is similar to the CaBER, but the higher imposed stretch velocity of 150 mm/s prevents sample breakup during the stretching step in case of low visco-elastic sample.


 * The ROJER (Rayleigh Ohnesorge Extensional Rheometer) is a jetting-based rheometer, developed on the basis of earlier works of Schümmer and Tebel and Christanti and Walker. This device exploits the spontaneous capillary instabilities developing on a liquid jet issuing from a nozzle to evaluated very short relaxation times. A piezoelectric transducer is used to control the frequency and the amplitude of the imposed perturbation.


 * The DoS (Dripping-onto-Substrate) technique allows to characterize the extensional response of a variety of complex fluids as well as accessing very short relaxation times not measureble in CaBER experiments. In DoS experiments, a volume of fluid is deposited on a substrate, so that an unstable liquid bridge is formed between the nozzle and the sessile drop.

Applications
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
 * Medical application
 * Fertilizer distribution