Microfluidics/Physics of fluids at smaller scales

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Contents

Microscopic scales found in fluidsEdit

  • Distance between molecules in a liquid:

 

  • Distance between molecules in a gas:

 

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

 , with   the effective collision cross-section.

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

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

Hierarchy of forcesEdit

  • Size of the object:  
  • Surface:  
  • Volume and mass:  

The surface to volume ratio increases when the size decreases.

Importance of forces, as a function of distance   or object size  

Van der Waals between molecules  
Van der Waals between surfaces  
Capillary force  
Muscular force  
Gravity force  
Magnetic force  
Dielectrophoretic force  

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

Examples in Nature of the hierarchy of forcesEdit

Insects can walk on waterEdit

 
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  , with   the surface tension, a force per unit length whose value is  
  • Weight scales like  

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

 

Below this length capillary forces are preponderant. \\

Smaller but strongerEdit

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  . Each fiber can exert a maximum force  . The number of fibers is  

  • Muscular force exerted by a muscle therefore scales like
 

with   the maximum stress exerted by a fiber. It is a "Natural" constant, independent of the size. It can be evaluated for human beings as  , where we computed the typical force exerted by a muscle divided by its typical section area.

  • The weight scales like
 

The number of people you can lift is therefore

 

with   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 [1].

Dimensionless Numbers in MicrofluidicsEdit

[2]


Reynold's Number:

 

where   is the density, U is the velocity, R is the channel dimension, and   is the viscosity


Peclet Number:

Defines the ratio of diffusion transfer verse convective transfer.

 

where D is the diffusion coefficient.


Knudson Number:

Defines the transition between micro and nanoscale.

 

where   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.

 

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

 

where   is the surface tension,   is the difference in density, and g is the acelleration due to gravity.

ReferencesEdit

  1. Patek, S. N. and Baio, J. E. and Fisher, B. L. and Suarez, A. V. (2006) Multifunctionality and mechanical origins: Ballistic jaw propulsion in trap-jaw ants, Proceedings of the National Academy of Sciences, vol 103, page 12787
  2. Berthier, Jean, Silberzan, Pascal (2010), Microfluidics for biotechnology, Chapter 1: Dimensionless Numbers in Microfluidics - Preview