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 edit

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  , that are occurring periodically in time with period  .

The i-th random step is  , with equal probability for the positive and negative sign, so that the average over time of the step is  .

The position after N steps is


with an average


meaning that on average the particle is around the same position. However the variance is


since  , and   for uncorrelated events.

Recalling that the number of step is linked to the total elapsed time by  , we have that  , with   a diffusion coefficient whose value is here  .

As a consequence a molecules explores a distance of standard deviation  .

Similarly an assembly of molecules (in a spot of dye for instance) spreads like  

Diffusion coefficient edit

In a liquid the value of the diffusion coefficient of a particule of radius   given by the Stokes-Einstein formula.


The demonstration of this formula comes the solution for the motion of a particle with two main physical ingredients: a fluctuation velocity given by  , and a friction forces of value  .

Small molecules therefore diffuse faster

Table with values??

Many molecules are characterised by a concentration   which is a number per unit volume.

Advection-diffusion equations edit

Ponctual injection in a liquid at rest edit

Concentration front edit

Diffusion is slow in microsystems edit

At rest edit

In motion with two coflows edit

Two streams merging at the same speed, mixing in a growing diffusion zone

The mixing zone grows with time:


The channel is completely mixed when this mixing zone reaches the channel width ( ), so after a time   during which the fluid has moved by a distance  .

The number of channel width required to mix entirely the two fluids defines the Péclet number     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!

In motion a spot undergoes the Taylor dispersion edit

Active mixing using chaos edit

Micromixer "Staggered herringbone" edit

Grooved channel to promote helicoidal flows. Two cross sections are displayed to show alternate directions of the fluid induced by the grooves.

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


with   the number of cycles.

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

Cross micromixer edit

Cross micromixer creates chaotic mixing with an alternate secondary flow.

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)

Going faster than diffusion: liquid chromatography for particle separation edit