Chemistry Reference
In-Depth Information
for Brownian particles will be reviewed. Based upon the Langevin dynamics
formalism, one can relate the self-diffusion of Brownian particles to the viscos-
ity of the solvent made up of the solution. Combining the Langevin dynamics
and the bead-spring model of polymer chains will yield the Rouse and Zimm
models.
6.5.1
Brownian Motion
Brownian motion signifies the incessant movements of particles in random
directions in a solution in which the particles are much larger than the
solvent molecules. It is now known that the reason for Brownian motion is
the random bombardments of the particles by the solvent molecules. At equi-
librium, the average velocity of the particles over a long period of time is
zero, which is a consequence of the particles moving in all directions with
equal probability. The key mathematical concept to model Brownian motion
is the random walk model. Einstein used the random walk model
to relate
Brownian motion to the self-diffusion coefficient
in the limit of sufficiently
long time.
Let's consider a one-dimensional random walk problem. A Brownian particle
starts its one-dimensional random walk journey at the origin of the x-axis and
each step has the same length. Before each step, the Brownian particle flips a
coin. If it is heads, it moves one step forward. Otherwise, it moves one step back-
ward. The coin is absolutely fair, which means that the chance of getting heads or
tails is the same.
After many coin flips, the particle may end up at any spot on the x-axis but
not, obviously, with the same probability. The problem here is to find the proba-
bility of landing at any given spot after a given total number of steps, N. In partic-
ular, it is of interest to determine on average how far away the particle is from
the origin after N steps. Let's define n as the number of forward steps minus the
number of backward steps. Obviously, n can be either positive or negative. The
following relations can be obtained.
n
forward
2
n
backward
5
n
(6-12)
n
forward
1
n
backward
5
N
(6-13)
Based on the definition of n and N, one can easily derive the following two
equations:
1
2
ð
n
forward
5
N
1
n
Þ
(6-14)
and
1
2
ð
n
backward
5
N
2
n
Þ
(6-15)
After N steps, nL is the distance from the origin where L is the length of
each step. The total number of paths of a Brownian particle landing at a
