Microfluidics/Physics of fluids at smaller scales

Microscopic scales found in fluids

• Distance between molecules in a liquid:

$d\sim 0.1\;\mathrm {nm}$

• Distance between molecules in a gas:

$d\sim (1/\rho )^{1/3}\sim (kT/P)^{1/3}\sim 3\;\mathrm {nm}$

• Mean free path between collision in a gas, air at ambient pressure:

$\lambda =kT/{\sqrt {2}}\pi \delta ^{2}P\simeq 61\;\mathrm {nm}$, with $\pi \delta ^{2}$ the effective collision cross-section.

Size of objects of interest embedded in a liquids, example of biological elements

• Protein, lipid molecule of the membrane: 1 nm
• Virus: 10 nm
• Cell: 1-10 μm

Hierarchy of forces

• Size of the object: $l^{1}$
• Surface: $l^{2}$
• Volume and mass: $l^{3}$

The surface to volume ratio increases when the size decreases.

Importance of forces, as a function of distance $d$ or object size $l$

Van der Waals between molecules	$d^{-7}\,$
Van der Waals between surfaces	$d^{-3}\,$
Capillary force	$l^{1}\,$
Muscular force	$l^{2}\,$
Gravity force	$l^{3}\,$
Magnetic force	$l^{3}\,$
Dielectrophoretic force	$l^{3}\,$

The effects at the beginning of the table become increasingly present when down-sizing.

Examples in Nature of the hierarchy of forces

Insects can walk on water

The Water strider uses surface tension to walk

Surface tension acts on the perimeter of legs

A contact line occurs on the legs of these insects. The leg surface is hydrophobic, and therefore the surface is curved downwards, which creates an upward tension force.

The typical leg diameter is l, and we can estimate the intensity of the forces:

• Capillary force scales like $\sigma \pi l$, with $\sigma$ the surface tension, a force per unit length whose value is $\sigma =70.10^{-3}\mathrm {N/m}$
• Weight scales like $\rho gl^{3}$

Therefore weight is comparable to the capillary force at the characteristic leghtscale:

$l^{*}={\sqrt {\frac {\sigma }{\rho g}}}\simeq 3\,\mathrm {mm}$

Below this length capillary forces are preponderant. \\

Smaller but stronger

How many times can you lift people of your size?

The structure of muscles is universal in the animal kingdom, with similar fibers of diameter $l_{0}$. Each fiber can exert a maximum force $f$. The number of fibers is $\scriptstyle l^{2}/l_{0}^{2}$

• Muscular force exerted by a muscle therefore scales like

$F\sim Pl^{2}\,$

with $\scriptstyle P=f/l_{0}^{2}$ the maximum stress exerted by a fiber. It is a "Natural" constant, independent of the size. It can be evaluated for human beings as $\scriptstyle P\sim F/l^{2}=100\;\mathrm {N} /10\;\mathrm {cm} ^{2}\simeq 10^{4}\;\mathrm {N.m^{-2}}$, where we computed the typical force exerted by a muscle divided by its typical section area.

• The weight scales like

$W\sim \rho gl^{3}\,$

The number of people you can lift is therefore

$N={\frac {F}{W}}={\frac {P}{\rho gl}}={\frac {H}{l}}$

with $H\simeq 1\;\mathrm {m}$ a constant independent of size.

Humans (l~1m) can lift 1 people Small ants (l~1mm) can lift 1000 people!

Note, that this force is used by a particular species of ants to jump. These ants strike their mandibles on the ground with a force that is 300 times its own weight, and propel themselves up to 10cm in height .

Dimensionless Numbers in Microfluidics

Reynold's Number:

$Re={\frac {\rho UR}{\mu }}$

where $\rho$ is the density, U is the velocity, R is the channel dimension, and $\mu$ is the viscosity

Peclet Number:

Defines the ratio of diffusion transfer verse convective transfer.

$Pe={\frac {RU}{D}}$

where D is the diffusion coefficient.

Knudson Number:

Defines the transition between micro and nanoscale.

$Kn={\frac {\lambda }{L}}$

where $\lambda$ is the mean free path of a particle/molecule and L is the distance in the system.

Bond Number:

Defines the dominance of gravitational or surface tension forces.

$Bo={\frac {R^{2}}{\lambda ^{2}}}$

where R is the radius of curvature of the interface between to fluids or channel dimension.

$\lambda ={\sqrt {\frac {\sigma }{\Delta \rho g}}}$

where $\sigma$ is the surface tension, $\Delta \rho$ is the difference in density, and g is the acelleration due to gravity.