All organisms, including ourselves, live in some form of liquid or gaseous fluid. The intuitions that we have developed over the course of our lifetimes regarding our interactions with our fluidic environments are subject to a size scale dependency that becomes apparent when examining a radically different scale, such as the microscale. For example, in swimming, we produce a forward force by some form of stroke, in which we cause a turbulent, overall directional movement of fluid around us, propelling us forward even for some time after the stroke has been performed. Due to our relatively large mass, the force we generate with a stroke is large enough to greatly overcome the resistance offered by the fluid around us to being deformed (i.e. viscous forces), giving rise to turbulence and inertia. The way in which we swim would be fundamentally different if we were the size of a microorganism, as we would no longer be able to generate forces that out-compete other inherent forces in the system, such as viscosity and surface tension. This scenario exemplifies how fluid flow changes from the macro to the microscale, due to a relativistic difference in the magnitude of the different forces that govern a given fluidic system.
The Reynolds number describes the ratio of inertial to viscous forces, and can thus be used to predict the scale dependencies of fluid flow depicted in the analogy. As the scale decreases, or as channels become microfluidic channels, viscous forces start to dominate inertial forces118. This viscosity is given by the velocity of flow caused by a
given shear stress, such as a pressure gradient, and is a measure of the internal friction within the particles that constitute a fluid. Newtonian fluids display a linear relationship profile between shear stress and velocity, whilst the viscosity of Non- Newtonian fluids varies depending on the rate of shear stress applied141.
37 The Reynolds number (Re) is given by the following equation:
𝑅𝑒 = 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒𝑠 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠=
𝜌𝑣𝐿 𝜇
Where 𝜌 is density (kg m-3), 𝑣 is velocity (m s-1), 𝐿 is characteristic length (commonly
channel diameter), and 𝜇 is dynamic viscosity (N s m-2).
Low Re flows are governed by viscous forces and are described as being laminar, due to a tendency of fluid to flow in layers parallel to the overall direction of flow. Higher Re numbers give rise to turbulent flow, governed by inertial forces (Figure 1.9).
Figure 1.9 Representation of laminar (left) vs. turbulent (right) flow in a situation where a fluid
(blue) is flowing in a channel. Flow is represented by arrows. At low Re numbers, fluid flows in parallel layers. As Re increases, flow becomes turbulent and is characterised by a chaotic regime with swirls and eddies.
For a straight channel, a transition from laminar to turbulent typically occurs around Re = 2000 – 2500. From the equation described above, different parameters can be modified in order to attain low Re fluidic flows, including fluid density, viscosity, velocity, and small channel diameters, with the latter being the obvious primary method within microfluidics. Laminar flows have the virtue of being of a more orderly and predictable nature141, and can be exploited to create concentration gradients142, efficient reagent
technologies143, and to generate microdroplets144.
Re < 2000
Re > 2000
38
1.3.2.1 Multiphase Microfluidics
Multiphase flow refers to the flow of two (or more) immiscible phases in contact with each other. When two immiscible phases, such as water and oil, flow in the same channel, different fluidic regimes can occur (Figure 1.10).
Figure 1.10 Representation of common fluidic conformations in multiphase microfluidics. a)
Sub-streaming b) droplets and c) slugs. Red and blue represent different immiscible phases, where the red fluid preferentially wets the channel surfaces (black).
Due to the laminar nature of microfluidics, the fluids may flow in streams parallel to each other. However, this creates a large contact surface area between one fluid and the other, which can be energetically unfavourable. Also, the surface of the channel may preferentially wet one fluid over the other, causing a drive of the non-wetting fluid to minimise its contact with channel walls145. Furthermore, viscous instabilities may
arise from the flow of one immiscible fluid in another, and pressures may build up when one fluid occludes the channel from another146. For all of the above reasons, it is
common in the microfluidic scenario described above for one fluid to segment in the other, forming droplets or channel-occluding slugs. The process of droplet formation
b)
c)
a)
oil water
39 can be controlled in order to give rise to the rapid production of monodisperse droplets encouraged by droplet-generating flow geometries147.
1.3.2.1.1 Physics of droplet formation
In a liquid-liquid multiphase system, droplet formation is governed by the shear force generated from one fluid onto another and the interplay of surface tensions between the fluids and also the channel walls145. Surface tension is defined as energy per unit
area and is an important driver in the formation of droplets, as it drives a segment of fluid to adopt a spherical shape in order to minimise its surface area to volume ratio. In droplet formation, the wettability of the different fluids with the channel walls will define which of the fluids will form droplets in the other, where the fluid with the highest surface tension with the channel walls forming droplets, and the other fluid constituting a “carrier” phase. For example, hydrophobic channels will produce water in oil (W/O) droplets and not vice versa, to give rise to the most energetically favourable scenario. The interplay of surface tensions can be understood via the measurement of contact angles (figure 1.11), which quantifies the degree at which a liquid wets a solid surface148. The angle of a droplet on a surface represents the relative strength of forces
between the liquid, solid and the surrounding air. Strategies exist to modify the surface tension between the different fluids, such as the use of surfactants149, and also to
modify the surface energy of the channel walls via surface modification techniques150, 151 (Section 2.2.1.4).
40
Figure 1.11 Droplet of water on a surface depicting the contact angle between the droplet and
the surface. The relatively low angle here indicated a relatively high wettability of the droplet against the surface.
Whilst surface forces encourage the formation of droplets in certain scenarios, viscous forces provide a resistance to fluid deformation required to break a fluid stream into segments. Thus the relative strength between these forces determines the manner in which droplets are formed in a particular droplet-generating scenario. This interplay is described by the Capillary number (Ca), which is given by the following equation:
𝐶𝑎 = 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒𝑠 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑓𝑜𝑟𝑐𝑒𝑠=
𝜇𝑣 𝜎
Where 𝜇 is dynamic viscosity (N s m-2), 𝑣 is velocity (m s-1) and 𝜎 is surface tension (N
m-1).
The weber number can also be of importance when analysing the formation of droplet formation, and relates to the interplay between fluidic inertia and surface tension along an interface152. Although inertia is usually negligible for microfluidic purposes, this can
differ for high flow rate values and/or large diameters153. It is given by the following
equation:
𝑊𝑒 =𝑖𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒𝑠 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑓𝑜𝑟𝑐𝑒𝑠=
𝜌𝑣2𝑙
𝜎
Where 𝜌 is density (kg m-3), 𝑣 is velocity (m s-1), 𝑙 is characteristic length (typically
41
1.3.2.1.2 Droplet Coalescence
A dispersion of droplets in an immiscible carrier phase is a thermodynamically unfavourable system due to the total interfacial area between the droplets and the carrier fluid, in comparison to a system where the fluids are separated by a continuous interface. This drives smaller, miscible droplets present in a system to coalesce with each other when in close proximity in order to minimize the total surface area between both fluids, in a mechanism called Ostwald ripening154. In a system without any
surface-active molecules, the only resistance to coalescence is the thinning and evacuation of the immiscible fluid between one droplet and another, which is affected by the proximity of the droplets and the viscosity of the carrier phase155. Therefore, in
order to avoid the coalescence of droplets into larger droplets, surfactant molecule solutions are used, which adsorb at immiscible interfaces (i.e. water and oil) due to their amphiphillic structure. Surfactants decrease the surface tension between the fluids which decreases the drive for coalescence, as well as providing electrostatic or steric repulsion between droplet interfaces149. This has an overall effect of droplet
stabilization, and droplet microfluidics makes extensive use of surfactants for this particular purpose.