Microfluidics/Capillary effects

Capillary effects are very important at small scales (see previous chapter [Physics_at_smaller_scales]).

They appear at interfaces between a liquid and another material liquid, gas, or solid.

The interfaces give rise to a surface tension that rules the dynamics of this interface.

Creating a new interface costs a work
 * $$E=\sigma \, A,$$

with $$\sigma$$ the surface tension (in N/m) and $$A$$ the area of the interface. For water/air interface the value of surface tension at 20°C is $$\scriptstyle \sigma=73\times 10^{-3}$$N/m.

This surface tension gives rise to a common phenomenon: liquid droplets tend to become spheres in order to minimize their surface.

Origin: internal adhesion energies
Because of Van der Waals forces, liquid molecules are attracted by their neighbours, which gives rise to an adhesion energy of the liquid to itself.

The value of surface tension in between two immiscible liquids (labeled 1 & 2) can be understood by dividing the creation of a new interface in steps It costs
 * Step 1: separation of free surfaces within each liquids
 * $$\Delta E=\gamma_{11}\frac{A}{2}$$ in fluid 1, with $$\gamma_{11}$$ the adhesion energy of the fluid to itself.

The origin of this adhesion in a liquid is Van der Waals attraction. We cut here only half of the final surface. Cutting the same surface in the other liquid costs:
 * $$\Delta E =\gamma_{22}\frac{A}{2}$$ in fluid 2.

Flattening the free surfaces produces a new surface $$2 \times A/2=A$$. Their assembly generates a work
 * Step 2: assemble free surfaces
 * $$\displaystyle \Delta E =-\gamma_{12}A,$$

where $$\gamma_{12}$$ is the adhesion energy of liquid 1 on liquid 2.

In total we obtain a total work $$\Delta E=\gamma_{11}\frac{A}{2}+\gamma_{22}\frac{A}{2}-\gamma_{12}A.$$

The value of surface tension is thus
 * $$\sigma=\frac{1}{2}\gamma_{11}+\frac{1}{2}\gamma_{22}-\gamma_{12}$$

Young-Laplace law: how surface tension creates pressures
At mechanical equilibrium a small change in sphere radius $$dR$$ does not create any work (forces are balanced):
 * $$\delta W=-\Delta P dV+\sigma dA=0$$

From this equation we obtain that pressure is higher in a sphere by
 * $$\Delta P=\frac{2\sigma}{R}$$

As a numerical example, we compute the capillary pressure in a bubble of radius 1 micrometer in water: it is 1.4 atm!

This formula can be generalized for arbitrary curved surfaces:
 * $$\Delta P=\sigma (\frac{1}{R_1}+\frac{1}{R_2}),$$

with $$R_1$$ and $$R_2$$ the radii of curvature in the principal planes.

Surface tension can break cylindrical jets
Write a section about the capillary instability ??

Wetting on solid surfaces


Near the contact line, three interfaces meet:
 * the liquid-gas interface, which has a surface tension $$\sigma_{liquid-gas}$$, which was noted $$\sigma$$ previously
 * the solid-liquid interface, of surface tension $$\sigma_{solid-liquid}$$
 * the solid-gas interface, of surface tension $$\sigma_{solid-gas}$$

The equilibrium of forces, is vectorial sum of forces that vanishes. Along the horizontal axis it provides the Young-Dupré relation at the contact line:
 * $$\sigma_{solid-liquid}+\sigma \cos \theta=\sigma_{solid-gas},$$

where we have introduced the contact angle $$\theta$$. Along the vertical axis we have
 * $$\sigma \sin \theta= N,$$

$$N$$ being the reaction from the solid. This shows clearly that the droplet is pulling the surface. This effect can be seen if a droplet is deposited on fresh paint: the contact line will pull a rim that will remain.

The contact line exists only if
 * $$-1<\cos \theta =\frac{\sigma_{solid-gas}-\sigma_{solid-liquid}}{\sigma}<1$$

There are therefore three cases:

Total wetting
The liquid spreads, and form a film coating the surface. It happens when $$(\sigma_{solid-gas}-\sigma_{solid-liquid})/{\sigma}>1$$ then $$\sigma_{solid-gas}>\sigma_{solid-liquid}+\sigma$$,  it costs a lot of energy to have a bare solid.

Partial wetting
The contact line existence condition is satisfied, $$-1<(\sigma_{solid-gas}-\sigma_{solid-liquid})/{\sigma}<1$$. Two cases can be distinguished:
 * hydrophilic surface: $$0<\theta <90^\circ$$
 * hydrophobic surface: $$90^\circ<\theta <180^\circ$$

No wetting
The liquid droplet pearls, and assumes a perfectly spherical shape. It occurs on surfaces such that $$(\sigma_{solid-gas}-\sigma_{solid-liquid})/{\sigma}<-1$$ then $$\sigma_{solid-liquid}>\sigma_{solid-gas}+\sigma$$, the solid does not like to be covered.

These surfaces can be found in nature: observe what happens to a water droplet falling on a lotus leaf. Microscopically these surfaces are covered with little pillars.

Driving droplet on surfaces with surface tension gradient
External modifications of the solid-liquid surface energy will provide forces to move small droplets. A droplet direction can be imprinted using gradients in surface energy.

Thermowetting
The solid-liquid surface energy increases with temperature.
 * $$\frac{d\sigma_{solid-liquid}}{dT}>0$$

If the heating is very local, the hotter side of droplet will become more expensive in terms of surface energy. The drop will tend to minimize its surface energy by going to the cooler side.

Electrowetting
A voltage can be applied on a droplet, if the surface is an electrode and an other electrode is in contact with the liquid droplet. A voltage difference $$V$$ changes the surface energy by:
 * $$\sigma^{EW}_{solid-liquid}=\sigma_{solid-liquid}-c\frac{V^2}{2},$$

where $$c$$ is the surface capacitance.

If only one side of the droplet is sitting on an electrode, the droplet will move towards the electrode, since surface energy is lower there.

Note that electrodes can be insulated to prevent electrolysis of liquids: the technique is called electrowetting on a dielectric (EWOD).

Driving droplets or bubbles in channels acting on surface tension
It is possible to move droplet or bubbles squeezed in microchannels, even if there liquid is totally wetting the solid. The method is to modify the liquid-gas surface tension (instead of the solid-liquid surface tension).

Thermocapillary forcing


Heating a liquid decreases its surface tension (it is the reverse with the solid-liquid surface tension).
 * $$\frac{d\sigma_{liquid-gas}}{dT}<0$$

A long drop or bubble with a side hotter than the other will move towards the hotter places to reduce the total energy.

Electrocapillary forcing


If the droplet/bubble is charged on its surface after having been in contact with an electrode, it can be moved afterwards if it is submitted to a voltage gradient. The electric potential of the droplet decreases when charges are closer to the electrode carrying the opposite charge.

Molecules
Surfactants are molecules that have an affinity with surfaces. They change the surface tension, and are therefore named tensio-active molecules. Usually they are amphiphilic molecules. Amphi- is a Greek prefix, also found in the word amphitheater, meaning dual. Indeed they have both an hydrophilic part and an hydrophobic part. For instance soap molecules have a carboxylic (hydrophilic head) and a long lipidic (hydrophobic chain). At low concentration they all go at the interface, while at high concentration they tend to form micelles.

Draw a cartoon with surfactants at surface and in a micelle ??

Effect on surface tension
They lower the surface tension compared to that of water. A free surface has a tension $$\sigma$$=73 mN/m. Surface tension decreases down to a plateau, when the surface is fully covered with surfactant molecules, presenting their lipidic tail to the gas. At this point surface tension is close to that of a lipid surface, around 30 mN/m. The saturation occurs at the critical micellar concentration (CMC), when surface is full and addition of molecules creates new micelles in the liquid.

Detergent
Surfactants have a detergent effect: they cover the surface of lipid droplets, which helps the detachment by favoring formation of a water film in between the droplet and substrate (cloth!).

Stabilizer
Draw a cartoon showing a liquid film in between two droplets/bubbles ?? Surfactants prevent the break-up of thin liquid films separating droplets in an emulsion or bubbles in a foam. They stabilize the thin liquid films of a soap bubble!

Droplet and bubble formation in microfluidic circuits
The inkjet printer involves the formation of tiny droplets of inks. Tiny droplet or bubbles are promising nanoliter containers: they do not have real walls, contain a very small amount of liquid. It is possible to create them, transport them and finally coalesce them with other.

A first basic operation is droplet formation.

Human scale


We all have seen the formation of a liquid drop at the exit of a tap. If you slowly open the tap, the droplet volumes increases until its weight is larger than the capillary force. The droplets then detaches and falls. The forces in presence are: The droplet detaches when the radius $$r_{droplet}$$ is such that gravity force is at least equal to capillary force. We obtain that:
 * the capillary force of magnitude $$2\pi R_{tap} \sigma,$$ with $$R_{tap}$$ the tap exit radius
 * the gravity force of magnitude $$4\pi r_{droplet}^3/3,$$ with $$r$$ the droplet size.
 * $$r_{droplet}\sim \left( \frac{R_{tap} \sigma}{\rho g}\right)^{1/3}$$

reducing the tap radius does not decrease quickly the drop radius, and droplets are getting very large in front of the tap radius at small scales. We see again that gravity forces become ineffective in front of capillary forces at small scales.

Another strategy, that does not involve gravity, should be considered.

Assisted formation in a T junction
(Drawing)

Formation can be actively achieved by using hydrodynamic forces, i.e. viscous forces.

When a fluid A arrives from a perpendicular branch in a main channel, droplets of this fluid can be detached if a fluid B is flowing. The size is given by a balance between: The corresponding droplet radius is therefore:
 * capillary pressure necessary to form a droplet of radius $$R$$, of magnitude $$\sigma/R$$
 * viscous stresses from the fluid B flowing in a channel of width $$h$$, of magnitude $$\mu \nabla v\sim \mu U/h$$
 * $$R\sim \frac{\sigma}{\mu U_0}h$$

Assisted formation by hydrodynamic flow focusing
(Drawing)

Another geometry is helpful, to produce very calibrated droplet sizes: the fluid A to break is focused by  a stream of fluid B through a tiny orifice of size $$d$$. Both fluids do not flow simultaneously through the orifice (it is the case only a very large velocities). Break-up involves two steps, when flow rates of fluids are imposed This last step takes a characteristic time $$\tau \sim d^3/Q_B$$, with $$Q_B$$ the flow rate of fluid B. The resulting volume of fluid A inflated during this time gives the droplet volume
 * blocking: when fluid A is in the orifice it makes a plug to fluid B
 * squeezing: a pressure builds up in fluid B and pinches fluid A until rupture, releasing a droplet
 * $$V\sim \tau Q_A \sim \frac{Q_A}{Q_B}d^3$$

Since the flow is laminar (small Reynolds number) it is very reproducible and a clock-like mechanism produces very accurate sizes.

Assisted by capillary instability
This method consists in the formation of a cylindrical jet. It is not really possible in flat microchannel whose height is small, but can be used in ink-jet printers.