Droplet rupture under shear flows

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Droplet rupture under shear flows

A key stage in the creation of an emulsion is the deformation of larger drops and their subsequent break up in shear flows. The interfacial force holding a droplet together must therefore be overcome in order to deform a droplet and this can occur if a large enough viscous shear force is applied to the drop. Taylor in 1934 did the first pioneering work to understand these mechanisms using an experimental setup consisting of a “four roller mill” [2]. The apparatus consisted of four independently controllable rotating cylinders submerged in a continuous phase. A droplet was introduced into the center of the assembly, and by varying the cylinders’ rotation, droplet rupture could be observed under a multitude of different shear flows. The possible flow conditions of the device ranged from elongational flow with no vorticity, to pure shear (Couette flow) consisting of equal parts vorticity and elongation. Taylor’s findings can be crudely summarized as follows, albeit with different terminology:
The ratio between the interfacial tension and the counteracting shear forces can be expressed by using the capillary (Ca) number. Small values of Ca represent dominance of interfacial forces over the applied shear and therefore drops are only slightly deformed. Increasing Ca leads to higher deformations and once a critical value is reached the droplet can no longer exist in an equilibrium state and ruptures into two or more droplets. The capillary numbers for pure shear flow and elongational shear flow are calculated as:
Where the variables are: shear rate (γ̇), deformation rate (ε̇), dynamic viscosity of continuous phase (μc), droplet radius (r), and interfacial tension (σ).
Taylor’s predictions for critical capillary numbers, however, were limited as only one viscosity ratio was used in his experiments. In order for a droplet to be ruptured under equilibrium conditions (i.e. neglecting time), it has been found that the critical capillary number is affected by both the type of the shear field and the viscosity ratio ( ) of the emulsion.
Where the viscosity ratio ( ) is the dispersed phase viscosity (ƞd) divided by the continuous phase viscosity (ƞc).
In 1982 Grace radically improved the theory of droplet rupture by doing many experiments using the same four roller mill apparatus but this time using many different viscosities for the fluids [15]. His results can be seen in Figure 5 and to this day are widely used in emulsifying process design to predict required shear flows for emulsions of desired sizes. It is important to note that droplet rupture is promoted when viscosities of both phases are similar.
However, these predictions were valid for emulsions stabilized by surfactant only.
Since the work by Grace, there have been many more studies of drop rupture under similar conditions, with a focus on drop shape, effects of shear cessation during rupture, and how many drops result during and after break-up.
Comprehensive works such as those by Bentley, Stone, Leal, and Rallison [12] – [18] go to great detail in describing the dynamics of droplet rupture and can be briefly generalised as follows: P < 1. Droplets submitted to capillary numbers just short of the critical value are stretched into stable shapes. The lower the viscosity ratio, the more elongated these steady shapes become. When a critical value is reached they rupture into two or more drops, the more elongated the stable shape, the more drops will result from the rupture. As well as the main drops resulting from the rupture, satellite (much smaller drops) drops are often formed in between the main drops. The lower the viscosity ratio, the higher the likelihood and number of satellite drops are formed.
P > 1. Above a viscosity ratio of 3.5, drop rupture cannot occur in pure shear flows (under such conditions droplets rotate instead of changing shape).

Sub and Super critical capillary rupture

When creating an emulsion, predicting the final droplet size is often the most important factor in the process design. The works described in the section above are crucial in understanding how much shear is required to rupture emulsion drops of known size and properties but are impractical in predicting final droplet size for most emulsifying processes. Critical capillary numbers are calculated theoretically or determined experimentally, by assuming a very slow increase of shear until rupture occurs whereby at all lower shear values the droplet maintains a stable equilibrium shape. This description of gradual shear increase is at odds with the majority of emulsification processes where often drops are subject to a sudden increase in shear far above critical values [4].
Under supercritical capillary number conditions, drops do not have time to form equilibrium shapes and instead are stretched into long thin liquid threads. The less the drop resembles a sphere and the thinner the liquid cylinder becomes, the less force is required to further deform the drop and so the drop keeps stretching [11]. Once a drop deforms into a long and slender continually stretching state, capillary waves begin to appear on the surface. At some point the amplitude of this varicose instability becomes too large and the droplet breaks into many (potentially thousands) drops due to Rayleigh-Plateau instability [15], [14].
Understanding and predicting when a stretching liquid thread will rupture into many drops is a very complicated task and not well understood, which makes final droplet size predictions challenging. It is known that viscosity ratio plays a very important part in this process, increasing viscosity ratio results in increasing the stretch before rupture as the higher the viscosity, the more viscous damping occurs on the forming capillary waves. It has been shown that the most efficient viscosity ratio for forming small drops under shear is not equal to one, as the Grace graph suggests, but to use higher viscosity ratios to maximize droplet stretching (and minimize satellite drops) before rupture [4].
Another important phenomenon worth discussing is the fact that drops can also be ruptured under sub-critical capillary number conditions. As mentioned above, droplets can be stretched into very long liquid threads at super-critical conditions which continue to stretch until rupture occurs, however, if the shear flow is suddenly stopped during this stretching, instead of returning to its original drop shape, the liquid thread often ruptures into many drops (more prominent at low viscosity ratios). When considering a very low viscosity ratio system, stable equilibrium drop shapes at sub-critical conditions can be extremely slender and long, and it has been shown that if the shear flow is stopped suddenly (rather than gradually), this long slender shape will rupture into many drops [15], [14], [18]. Another phenomenon associated with sub-critical rupture is “tip-streaming”, this only occurs in surfactant containing continuous phases and at low viscosity ratios. The droplet will obtain a stable slender shape, but instead of entirely rupturing, very small droplets will be formed and released from the pointed ends of the slender drop. This is due to the forced movement of surfactant molecules at the surface of the drop from the center to the ends, thus creating a distribution of interfacial tensions along the slender drop with much lower interfacial tensions at the points [14].

Methods used to create monodisperse emulsions

Controlled shearing of emulsions

It has been shown by Mason and Bibette [6]– [17] that monodisperse emulsions can be created by applying uniform shear stress to a polydisperse coarse emulsion. The reasoning behind this is that if every droplet in an emulsion is subjected to a large enough shear stress that is equal for every droplet, then the droplets will all rupture and reduce in size to a value below the critical capillary number.
In order to subject an emulsion to a uniform shear stress a Couette flow is utilized (Figure 2.2). Couette flow is observed when a liquid is placed between two parallel plates where one is stationary and the other is moving. Due to the non-slip condition of the liquid at the plate boundaries, a viscous shear driven flow is induced where the shear stress at every point in the liquid is equal.
This flow regime can be created using various geometries but the most common is two concentric cylinders where on rotates and the other is stationary (Figure 2.3).
Scheme of a commercialized Couette cell where a coarse emulsion is forced through an area of uniform shear stress to create monodisperse emulsions
It was discovered, however, that monodisperse emulsions are only produced if the initial coarse emulsion is shear thinning [5], whereas shearing Newtonian emulsions yields polydisperse results. The proposed reason for this is the mechanism of droplet rupture. When a polydisperse shear-thinning emulsion is sheared, the diameters of all the liquid threads formed from drop stretching at rupture are very similar (therefore resulting in similar sized drops). However, when a polydisperse Newtonian emulsion is sheared, the diameter of the liquid threads produced at rupture seems to be a function of the initial droplet size and therefore yields polydisperse results [7], [8].
This special case of monodisperse rupture that occurs if the emulsion is shear-thinning may at first seem rather specific in application, as the majority of industrial emulsions do not contain shear thinning materials as their continuous phases. However, when an emulsion’s phase ratio is increased to levels above ≈ 50%, the increased droplet interactions due to the close packing dramatically changes the rheology of the emulsions increasing its viscosity and exhibiting shear-thinning effects [16]. It is for this reason that this method of creating monodisperse emulsions has great potential in being used for a very wide range of materials.

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Emulsion stability

Once an emulsion is formed there are mechanisms by which an emulsion can be destroyed; one being coalescence, another being Ostwald ripening. Coalescence is the mechanism by which when two droplets come together, the thin film of continuous phase separating the two is ruptured and the droplets merge into one bigger droplet. The rate of coalescence can be reduced by using surfactants to coat the droplets and by reducing the size of the droplets.
Ostwald ripening is the mechanism where dispersed phase in smaller droplets dissolves in the continuous phase and precipitates on to larger droplets causing smaller droplets to shrink and larger droplets to grow [22].

Coalescence

The coalescence phenomenon of drops in liquid / liquid systems is reviewed with particular focus on its technical relevance and application. Due to the complexity of coalescence, a comprehensive survey of the coalescence process and the numerous influencing factors is given. Subsequently, available experimental techniques with different levels of detail are summarized and compared. These techniques can be divided in simple settling tests for qualitative coalescence behavior investigations and gravity settler design, single-drop coalescence studies at flat interfaces as well as between droplets, and detailed film drainage analysis.
To model the coalescence rate in liquid/liquid systems on a technical scale, the generic population balance framework is introduced.
Additionally, different coalescence modeling approaches are reviewed with ascending level of detail from empirical correlations to comprehensive film drainage models and detailed computational fluid and particle dynamics [23].

Table of contents :

Chapter 1 Introduction
Aims of the work
Organization of the manuscript
Introduction
1-1-Introduction of Microencapsulation
1-2-Emulsion classification
1-3-Formulation
1-3-1-How Emulsions Are Made
1-3-2- Shearing of Coarse Emulsions
1-4- Droplet dynamics
Chapter 2 Literature Review
2-1- Droplet rupture under shear flows
2-2- Sub and Super critical capillary rupture
2-3- Methods used to create monodisperse emulsions
2-3-1-Controlled shearing of emulsions
2-3-2- Couette instrument
2-4- Emulsion stability
2-4-1 Coalescence
Chapter 3 Emulsions Preparation
3-1- Physical Properties of the Fluids
3-1-A- Density
1- Density values for the system made of Aqueous core / Aliphatic Urethane Acrylate Shell / Aqueous External phase
2- Density values for the system made of Oil PAO40 core/ Epoxy Urethane Acrylate Shell / Glycerol external phase
3-1-B- Rheology
3-1-B.1 Viscosity of Aqueous phase / Aliphatic Urethane Acrylate Shell / Aqueous phase System as function of shear rate
3-1-B.2 Viscosity of Oil phase / Epoxy Urethane Acrylate Shell / Aqueous System VS Shear rate
3-1-C Interfacial Tension
3-1-C.1 Interfacial Tension of Aqueous phase / Aliphatic Urethane Acrylate Shell / Aqueous phase System
3-1-C.2 Interfacial Tension of Oil Phase / Acrylate shell / Aqueous Phase System
3-2 Fragmentation Diagram
3-2-A Shear Rate Applied by the Overhead Mixer
3-2-B Fragmentation Diagram for the First Emulsion
3-2-B.1 First Emulsion: Aqueous Phase in Aliphatic Urethane Acrylate Liquid Polymer
3-2-B.2 First Emulsion: Oil phase in Epoxy Urethane Acrylate Liquid Polymer
3-2-C Maximum Mass Fraction of the First Emulsion
3-2-C.1 Maximum mass fraction for the First Emulsion of aqueous phase in Aliphatic Urethane Acrylate Polymer liquid polymer
3-2-D Double Emulsion Fragmentation Diagram
3-2-D.1 Fragmentation diagram of Double Emulsion: Aqueous phase / Aliphatic Urethane Acrylate polymer / Aqueous Phase
3-2-D.2 Oil / polymer / Aqueous Phase Double Emulsion System
Chapter 4 Visualization of Double Emulsion Fragmentation
4-1 Visualization of Double Emulsion Fragmentation
4-1- Aqueous Phase / Polymer Shell / Aqueous phase Capsules
4-1-A Low viscosity ratio between phase 1 and phase 2
4-1-A.2 Optimum viscosity ratio between phase 1 and phase 2
4-1-A.3 High Viscosity Ratio between Phase 1 and Phase 2
4-1-B Oil / Epoxy Urethane Acrylate Polymer shell / Aqueous Phase Capsules
4-1-B.1 Optimum viscosity ratio
Chapter 5 Shell Polymerisation and Capsules Characterization
5-1 Polymerisation of Acrylate Shell
5-1-A Effects of Initiator Concentration and UV Exposure Duration
5-1-A.1 Initiator concentration 0.0%
5-1-A.2 Initiator concentration 0.01%
5-1-A.3 Initiator concentration 1 %
5-1-A.4 Initiator concentration 10.0%
5-2 Size distribution of Capsules
5-2-A Bright Field Images by Optical Microscopy
5-2-A.1 Aqueous Phase / Aliphatic Urethane acrylate shell / Aqueous Phase
5-2-A.2 Oil / Epoxy Urethane acrylate shell / Aqueous Phase
5-2-B Fluorescence Images Using Dissolved Dyes
5-2-B.1 Aqueous Phase / Aliphatic Urethane Acrylate Shell / Aqueous Phase System
5-2-B.2 Oil / Epoxy Urethane Acrylate Polymer shell / Aqueous Phase
5-2-C Sizes Distribution of capsules through Laser Light Scattering
5-2-C.1 Aqueous Phase / Aliphatic Urethane acrylate shell / Aqueous Phase
5-2-C.2 Oil / Epoxy Urethane Acrylate polymer Shell / Aqueous Phase
5-2-D Transmission electron microscopy
5-2-D.1 Aqueous Phase / Aliphatic Urethane acrylate shell / Aqueous Phase
5-2-D.2 Oil / Epoxy Urethane acrylate shell / Aqueous Phase
5-2-D.3 Aqueous Phase / Epoxy Urethane Acrylate Shell / Oil Phase
5-3 Encapsulation efficiency
5-3-A Aqueous Phase / Aliphatic Urethane Acrylate polymer shell / Aqueous Phase
5-3-B Oil phase / Epoxy Urethane acrylate polymer shell / Aqueous phase
5-4 Methods for triggering the rupture of capsules
5-4-1 Rupture by osmotic stress
5-4-2 Rupture by mechanical pressure
5-4-2. A Capsules contain aqueous core phase, Rubbery polymer shell CN991 aliphatic urethane di acrylate and external phase of water between two glass slides
5-4-2. B Capsules contain aqueous core phase, Glassy polymer shell CN2035 aliphatic urethane di acrylate and external phase of water between two glass slides
5-4-3 Rupture by ultrasound
5-4-4 Rupture by thermal expansion of the core
References

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