Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 23

where s i is a random step, which can take a value of either þ s or - s . The squared value of the position

x ( n ) is:

X

n

2

s i ¼ ns 2

xðnÞ ¼ fDt: (2.24)

i ¼ 1

The proportionality comes from the assumption that the same amount of time is taken to make
a step . The proportionality factor determines how fast the particle moves, which in reality is the

diffusion coefficient. Brownian motion is the random walk with very small steps. Figure 2.7 shows the
random walk of a particle calculated for 10,000 steps in two and three dimensions.
Jan Ingenhousz, a Dutch doctor, was the first to observe the irregular motion of coal dust particles
on the surface of alcohol in 1785 [5] . However, the botanist Robert Brown, who observed, in 1827,
pollen particles floating in water under the microscope [6] , is credited for the discovery of this motion.
The Brownian motion of particles in a liquid is due to the instantaneous imbalance in the force exerted
by the small liquid molecules on the particle. Thus, the diffusion coefficient of this particle can be
derived from the force balance equation.

ðtfnÞ

2.2.2 Stokes-Einstein model of diffusion
The time evolution of the position of the Brownian particle itself can be described approximately by
the force balance equation where the random force of the liquid molecules represents one term in the
balance. This equation is called the Langevin equation:
m dv

d t ¼

b v þ F ð t Þ (2.25)

FIGURE 2.7
Random movement of a particle: (a) two-dimensional and (b) three-dimensional.