Microfluidics/Mixing

Mixing molecules in a solute is a crucial operation for biological analysis or chemical reactions. We will see in this chapter a specificity of microfluidics: mixing is long.

Origin of diffusion in a liquid: brownian motion
In a liquid all molecules are agitated due to their temperature. A big molecule embedded in a liquid is thus subject to continuous water impacts by neighbour molecules. This impacts are random and produces step motion in random directions.

To understand the effect of this random motion, we consider a 1D model (along one axis) with random steps of finite size $$\ell$$, that are occurring periodically in time with period $$\Delta t$$.

The i-th random step is $$\ell_i=\pm \ell$$, with equal probability for the positive and negative sign, so that the average over time of the step is $$\langle \ell_i \rangle=0$$.

The position after N steps is
 * $$ X= \sum_i^N \ell_i$$

with an average
 * $$\langle X \rangle=\sum_i^N \langle \ell_i \rangle=0$$

meaning that on average the particle is around the same position. However the variance is
 * $$\langle X^2 \rangle=\langle (\sum_i^N \ell_i )^2 \rangle=\langle \sum_i^N  (\ell_i)^2 + 2\sum_{i<j} \ell_i \ell_j \rangle=N \ell^2$$

since $$\langle \ell_i^2 \rangle=\ell^2$$, and $$\langle \ell_i \ell_j \rangle=\langle \ell_i \rangle\langle \ell_j \rangle=0$$ for uncorrelated events.

Recalling that the number of step is linked to the total elapsed time by $$N=t/\Delta t$$, we have that $$\langle X^2 \rangle=D t$$, with $$D$$ a diffusion coefficient whose value is here $$D=\ell^2/\Delta t$$.

As a consequence a molecules explores a distance of standard deviation $$\sigma=\langle X^2 \rangle^{1/2} \sim (D t)^{1/2}$$.

Similarly an assembly of molecules (in a spot of dye for instance) spreads like $$\sigma \sim (D t)^{1/2}$$

Diffusion coefficient
In a liquid the value of the diffusion coefficient of a particule of radius $$R$$ given by the Stokes-Einstein formula.
 * $$D=\frac{kT}{6\pi \mu R}$$

The demonstration of this formula comes the solution for the motion of a particle with two main physical ingredients: a fluctuation velocity given by $$m\langle u^2 \rangle/2=kT/2$$, and a friction forces of value $$F=6\pi \mu R u$$.

Small molecules therefore diffuse faster

Table with values??

Many molecules are characterised by a concentration $$C(\mathbf{r},t)$$ which is a number per unit volume.

In motion with two coflows
The mixing zone grows with time:
 * $$\delta \sim \sqrt{Dt}$$

The channel is completely mixed when this mixing zone reaches the channel width ($$\delta=w$$), so after a time $$t=w^2/D$$ during which the fluid has moved by a distance $$Z=U t=Uw^2/D$$.

The number of channel width required to mix entirely the two fluids defines the Péclet number $$Pe$$ $$\frac{Z}{w}=\frac{U w}{D}\equiv Pe$$ which gives the importance of the ratio advection/diffusion.

For instance a protein flowing at 100 micrometers per second in a 100 micrometer channel displays a Péclet number Pe=250: a very long channel length is necessary to mix these fluids!

Micromixer "Staggered herringbone"
Fluid rolls can be induced in the system by helicoidal grooves. An alternate succession of directions promotes stretching and folding of fluid elements, which in turn lead to chaotic mixing. As in the baker's transform the typical thickness of a layer of fluid decreases exponentially with time
 * $$w_{eff}=w_0/2^N$$,

with $$N$$ the number of cycles.

See reference: Stroock, Dertinger, Ajdari et al., Science 295, 647 (2002)

Cross micromixer
This micromixer is very efficient but necessitates to drive an external alternating flow.

Reference: F. Okkels and P. Tabeling, Physical Review Letters 92, 038301 (2004)