Hydrodynamic Drag, Brownian Motion and Diffusion
Drag, diffusion,
and Brownian motion are intimately related. Since they all bear on the motion of
small objects in an aqueous environment, they are of paramount importance to
understanding cellular processes. The basic formulae presented below are
applicable from the micron scale down to the nanometer scale and below.

Fluid Viscosity

All fluids have a viscosity, which describes the amount of friction
between nearby regions of the fluid moving at different velocities. Viscosity is
what causes dissipation of energy in hydrodynamic flow: after you stir
your coffee, it eventually stops moving, thanks to frictional losses.

The units of viscosity are pressure×time, or mass/ (length×time). The cgs
unit of viscosity is 1 g/(cm·sec) = 1 Poise; the SI unit of viscosity is 1
Pa·sec.

It is useful to remember that 1 Poise = 0.1 Pa·sec (check this!)

Water, and solutions which are mostly water, have a viscosity close to hwater = 0.01 Poise or 0.001 Pa·sec. This value of
viscosity will be used extensively, since the stuff around biomolecules in vivo
is mostly water. By the way, we should remember that water has a mass density of
rwater = 1 g/cm3 = 103
kg/m3.

Some fluid mechanics people and books talk about `kinematic viscosity' n = h/r. We
will never use kinematic viscosity.

Reynolds Number

Consider hydrodynamic flow characterized by a fluid of some mass density,
flowing at some velocity, past an object of some length. The Reynolds number of
the flow past the object (or `wake') is the dimensionless combination

Re =

velocity×length×mass density

viscosity

Alternately,
the length used to estimate Re might be the width of a channel (or pipe) that
fluid is flowing through.

If Re >> 1 then the flow is turbulent, with complicated vortices
and eddies at a wide range of sizes which continually are reorganizing.
Turbulence is not well understood at all, and will not be discussed further,
except for a simple example:

Example: What is Re in the wake of a sailboat? The fluid velocity is
about 10 m/sec, the rough length scale is 1 m, the density is 103
kg/m3 and the viscosity is 10-3
Pa·sec, thus Re » 107.

Fortunately turbulence is largely irrelevant at molecular and cellular
scales. The velocities and lengths we will worry about end up just being too
small to ever allow Re to get big.

Example: What is Re in the wake of a micron-sized particle moving at a
velocity of 1 micron/sec?

Re =

10-6
×10-6 ×103

10-3

= 10-6

For
sufficiently tiny things moving sufficiently slowly, Re << 1. In general,
the situations discussed in this course will be in this limit. This is great
because for Re << 1, the equations of hydrodynamics may be linearized and
solved, and the solutions are meaningful.

Drag Coefficients for Re << 1

For Re << 1, the hydrodynamic drag force on a moving object is simply
linearly proportional to the object's velocity through the fluid, and being a
frictional (or energy-dissipating) force, it acts to oppose the motion. The drag
force therefore looks just like the law of friction some of you may remember
from elementary physics courses:

fdrag »
hd
v

Here,
h is the viscosity, d is the longest dimension
of the moving object, and v is the object's velocity relative to the fluid. This
formula neglects an overall numerical factor which depends on the precise shape
of the moving object; this factor is usually roughly 10.

The ratio of the drag force to the velocity (fdrag / v =
kdrag) is called the drag coefficient, and is roughly hd.

Berg's book gives a nice tabulation of drag coefficients, but one
particularly important case is that of a sphere of radius R, for which the drag
coefficient is kdrag = 6 phR.

Example: Force needed to keep a bead moving at some velocity

Suppose a spherical bead of radius R moves at constant velocity v through a
fluid of viscosity h. The force that must be supplied
to keep the bead moving at that velocity must just balance the drag force:

fdriving = fdrag 6
phR
v

For a
1-micron diameter bead moving at 10 microns/sec in water, this force must be
(mks units)

fdriving = 6 ·3.14 ·10-3 ·10-6 ·10-5 = 19 ×10-14
Newtons

or
about 0.2 piconewtons (pN).

Example: Sedimentation

A commonly used technique to separate particles (or molecules) of different
sizes used in biology and biophysics labs is to use either gravity, or
centrifugal acceleration (in a centrifuge). Suppose we have a spherical particle
of radius R and mass density r subject to an
acceleration a. Then the velocity with which the particle falls through the
fluid is just given by the balance of the drag force and the force due to the
acceleration,

6 pR hv =

4 pR3
Dr

3

a

giving     v =

2 R2 Dr
a

9 h

where
Dr is the difference in mass density of particle and
surrounding fluid.

Roughly speaking, larger particles fall, or `sediment' faster. Also, larger
accelerations lead to faster sedimentation, hence the use of centrifuges of
accelerations up to 106 g's to separate very small particles (i.e.
large molecules).

If we suppose our canonical 1-micron-radius particle in aqueous solution has
a mass density 1.2 times that of water, then we can compute its sedimentation
velocity due to gravity as (mks units)

v =

2 · 10-12 ·0.2 ×103×10

9 ×10-3

= 1.4 ×10-6    m/sec = 1.4
micron/sec

Note
that the particle mass is volume time mass density, or m = 4pR3 r/ 3.

Molecular biologists usually quote the sedimentation rate normalized by the
acceleration, v/a. The dimensions of this quantity are in seconds.
Molecular biologists often use the Svedberg unit (1 S = 10-13
sec).   The particle discussed above therefore sediments at v/a
= 1.4x10-7 sec = 1.4x106 Svedbergs or 1.4x106
S.  Proteins or complexes of proteins are therefore often described as
being particles with a certain S value.

Example: Role of thermal motion in sedimentation

Remember that sedimentation is opposed by thermal fluctuation! At room
temperature, sedimentation can only occur until one reaches the thermal
equilibrium distribution of concentration of particles,

cequilibrium(z) = C0
exp

é
ê
ë

-Dm a z

kB T

ù
ú
û

where
Dm is the difference between the mass of the particles
and the water that they displace, and z is the distance above the bottom of the
tube. Note that the potential energy of the particles at height z is used in the
Boltzmann factor (it is of the form U = m g h you might remember from elementary
mechanics). The constant C0 is determined if you know the height of
the water in the test tube and the total number of particles, since the total
number of particles should be given by the integral of concentration over the
tube volume.

So, if you want to push particles of mass m to within a distance d of the
bottom of a tube, you had better use an acceleration well in excess of

a =

kB T

d     Dm

For
molecules, this is important; if you have 2 nm-radius particles with mass
density 1.2 times water in a 1 cm-high tube, then to push them down to to the
lower 0.5 cm of the tube, you need an acceleration well in excess of (mks units)

4 ×10-21

0.5 ·10-2 ·(4·3.14/3) ·0.2
·103 ·(2×10-9)3

= 1.2 ×105
m/sec2

or
about 12000 g (recall the acceleration due to gravity at the earth's surface, 1
g = 9.8 m/sec2). To achieve effective separation of particles
using centrifugation requires at least an order of magnitude larger
accelerations.

Further reading on hydrodynamics:

H. Berg, ``Random walks in biology'' Princeton (1993) has an excellent
discussion of the basic concepts of viscous drag discussed above; there is a
useful table of drag coefficients on p. 57

D.J. Tritton, ``Physical Fluid Dynamics'' Oxford (1992) - lots of pictures,
aimed at senior-level physics and engineering students

L.D. Landau and E.M. Lifshitz, ``Fluid Mechanics'' Pergamon (1987) -
beautiful but at a high mathematical level, good for graduate students in
physics or applied math

Philip Nelson, on-line textbook

Brownian Motion

If you observe small (0.1 to 1 micron-radius) particles in water, you will
see that they move erratically - almost appearing to hop around
discontinuously. This Brownian motion is due to collisions with water
molecules, which makes the particles undergo random-walk motion.

If we observe such a particle for a time interval t, we will see it displaced
by a position vector r. If we repeat this over and over again, we can
compute the average displacement during time interval t, and as long as no other
forces (gravity, bulk hydrodynamic flow...) act, we should find that

< r > =
0

i.e. there
is no preferred direction for the random forces exerted by the water molecules
on our particle.

The notation < > represents an average over many repeated and
uncorrelated measurements.

So, we characterize Brownian motion not by the average displacement over some
time, by instead by the average displacement-squared:

< |r|2 > = 6 D
t

where
D is called the diffusion constant of the particle. This formula will be often
referred to as the `square-root law' of Brownian motion. The diffusion constant
D depends both on the size and shape of the particle, and on the viscosity and
temperature of the fluid that it is moving through. The factor of 6 is a
convention, but an important one to remember. Also you should always remember
that diffusion constants have the dimensions of length2/time.

This law, and other quantitative features of Brownian motion, were
established by a long series of experimental works beginning with Robert Brown's
original experiments of 1827. Much of the crucial quantitative work was done by
Jean Perrin in the early 1900s. Believe it or not, interesting experiments on
Brownian motion continue to be done today - you can read about them in Nature,
Science and other top research journals.

Here is a picture of the trajectory of a simulated Brownian particle
projected into the x-y plane, with D = 0.16 micron2/sec.  The x
and y axes are marked in microns.  It starts from the origin (x=0,y=0) at
t=0, and the pictures show the trajectory after 1 sec, 3 sec and 10 sec:

Brownian trajectories are rough (actually `fractal'), and very unlike
the smooth trajectories we learn about in elementary mechanics courses.
The roughness of the trajectories reflects the underlying randomness of
molecular motion.  Here is the x-coordinate versus time for the Brownian
trajectory shown above:

This graph is x(t) for a random walk in one dimension, and could just
as well represent how much money you have in your pocket as you repeatedly play
a game with equal odds.

The square-root-law is quite different from formulae of ordinary mechanics -
you will probably not measure a displacement-squared of exactly 6 D t in
any one of your trials - this formula represents only the average that
you will obtain after many trials. However, it still makes sense for us to say
that a particle undergoing Brownian motion moves a distance » (Dt)1/2 during a time interval t.

The following figure shows the magnitude of displacement as a function of
time for the simulation of Brownian motion shown above (D = 0.16
micron2/sec). The rough curve is the simulation; the smooth curve
shows the average behavior, ( < |r|2 > )1/2 = (6 D t)1/2.
Note that although there are large fluctuations away from the average behavior,
the Brownian motion tends to grow, roughly following the smooth curve.

Problem: A particle has a diffusion constant of D = 0.3
micron2/sec. How far can we expect it to be displaced from its
initial position after 1 second, 10 seconds, 100 seconds, 1 hour and 1 day?

Problem: Consider a particle which undergoes Brownian motion in three
dimensions with < |r|2 > = 6 D t. Explain why < x2
> = < y2 > = < z2 > = 2 D t.

Problem: A particle undergoing Brownian motion is observed in a
microscope, which of course gives only x-y information
about position. From a series of measurements, < x2 +
y2 > is determined to grow linearly in time, with a slope of 3.2
×10-8 cm2/sec. What is the
diffusion constant D for the particle?

Einstein's relation between drag coefficient and diffusion constant

In his Ph.D. work at the ETH in Zurich in 1905, and in two subsequent papers
published in Annalen der Physik in 1905 and 1906, Einstein established the
theory of Brownian motion. These two papers are more heavily cited than his
papers on relativity, due to their wide applicability (biology, chemistry,
physics, chemical engineering...) and are the theoretical foundation of chemical
and colloid physics.

There were two immediate and big results of Einstein's thesis work. First, he
provided a new way to estimate Avogadro's number, which in 1905 was only known
roughly. The idea that there were molecules at all was still controversial in
1905.

Second, Einstein showed that the diffusion constant D of a small particle was
simply related to its drag coefficient kdrag and absolute
temperature:

D =

kB T

kdrag

This
impossibly simple formula is fundamental to the description of virtually all
transport phenomena - from biophysics to astrophysics! This formula is also
sometimes called an `Einstein relation', or a `fluctuation-dissipation formula'.
Remarkably, this result was derived by William Sutherland almost simultaneously
in 1905.

It makes good sense that the rate at which a particle diffuses by Brownian
motion will go up if its drag coefficient is reduced, since the water molecules
hitting it will bonk it further and faster. Similarly, if absolute temperature
is increased, then the amount of momentum transferred per bonk will be larger,
and D should therefore increase.

Example: What is the diffusion constant of a 1 micron-radius particle
and a 1 nm-radius particle, both in water at room temperature?
The Einstein
relation tells us that the 1-micron particle has a diffusion constant of

D =

kB T

6 phR

=

4 ×10-21

6 ·3.14 ·10-3
·10-6

= 0.2 ×10-12
m2/sec

or
equivalently D = 0.2 microns2/sec. This tells us that a
1-micron-radius particle in water is displaced by 1 micron in about 5 seconds.

Since D is inversely proportional to R, the 1 nm-radius particle has a
1000-fold larger diffusion constant, D = 200 microns2/sec. This means
that in one second, our 1 nm-radius particle is displaced about 15 microns.

Problem: Estimate the diffusion constant of a sucrose molecule in
water, modeling it as a sphere of some suitable radius. Compare your result with
the known diffusion constant obtained from, e.g. the CRC Chemistry and Physics
Handbook.

Problem: A neurotransmitter molecule which is roughly spherical and of
radius 0.5 nm is synthesized in the cell body of a neuron in your lower back.
How long will it take to diffuse down the long, skinny axon to the synapse in
your toe where it will be used to transmit a signal to one of your toe muscle?
(this axon is a few microns wide, and about 1 m in length). Hint: this is
one-dimensional diffusion.

Diffusion is due to the independent Brownian motion of many particles (or
molecules)

Even in perfectly still water, a drop of ink will slowly spread by
diffusion. If care is taken to eliminate fluid flow, then the
concentration (number of particles per volume) of ink as a function of position
and time, c(r,t) satisfies the diffusion equation:

¶c ¶t = D

æ
ç
è

¶2

¶x2

+

¶2

¶y2

+

¶2

¶z2

ö
÷
ø

c

where
D is the diffusion constant of the ink. Another key result of Einstein and
researchers following him was that the diffusion constant of the ink is the same
diffusion constant for Brownian motion of individual ink particles.
Therefore diffusion can be understood as the simultaneous and independent
Brownian motion of many particles all at once.

A fundamental solution of the diffusion equation is the `spreading Gaussian'
solution

c(r,t) =

C0

t3/2

exp

é
ê
ë

-

(x2 + y2 +
z2)

4 D t

ù
ú
û

This
solution is a function of only the magnitude of the position vector |r| = (x2 +
y2 + z2)1/2, and is therefore spherically
symmetric. As a function of time, the region where the concentration is
appreciable grows in radius as (Dt)1/2. Note also that the
concentration at the origin, which is the location of the drop at t ® 0+, drops monotonically with time.

The figure below shows the Guassian solution in one dimension, c(x,t) =
exp[-x2/(4Dt) ] / t1/2
Note that the three-dimensional
solution is just the product of similar x, y and z terms.  The figure shows
how diffusion gradually spreads out an initially concentrated
distribution.  The value of D for this figure is 0.16
microns2/sec, and the times shown are 0.2, 0.5, 1 and 2 seconds.

Problem: Verify that the three-dimensional spreading drop solution
above solves the three-dimensional diffusion equation. Then, verify that the
integral of the concentration over all of space - which equals the total number
of particles - does not change with time.

Problem: Using the fact that c(r,t) represents the number of
particles of ink or whatever per unit volume, show that the average value of
r of the diffusing particles is zero, and that the average value of |r|2 is in fact 6
D t.

Hint: averages over space of some quantity X are of the form (why?)

< X > =

ó

õ

d3 r X
c(r,t)

ó

õ

d3 r
c(r,t)

where
X is either r or |r|2;
you will also need two `gaussian'
integrals:

ó
õ

+¥

-¥

dx e-ax2
=

æ
ç
è

p

a

ö
÷
ø

1/2

ó
õ

+¥

-¥

dx x2 e-ax2 =

æ
ç
è

p

4 a3

ö
÷
ø

1/2

The
connection between diffusion is precise - diffusion is just the statistical
(average) description of a large number of particles undergoing independent
Brownian motions. The second problem above emphasizes that the diffusion
equation describes particles whose square-root law is precisely that of Brownian
motion.

This final section is optional reading, and presents simple derivations of two
of the formulae presented above.

Square-root law for distance traveled by a random-walking particle

We introduce a simple model to show roughly where the square-root law comes
from. We imagine a particle to take a step of length a every time interval t, with each step in a random direction.

Then, the displacement after N steps is just

r = a

^
n

1

+ a

^
n

2

+ ¼a

^
n

N

where
the [^(n)]i are unit vectors (|[^(n)]i|2 = 1) in random directions.

So, the distance-squared travelled by our particle in N steps is just

|r|2 = a2

æ
è

^
n

1

+

^
n

2

+ ¼

^
n

N

ö
ø

= a2

æ
è

|

^
n

1

|2 +
|

^
n

2

|2 +
¼|

^
n

N

|2
+

å
i ¹ j

^
n

i

·

^
n

j

ö
ø

= N a2 +

å
i ¹ j

^
n

i

·

^
n

j

Note
that we didn't do any averaging yet.

Now, if we collected data for many N-step Brownian motions, we would find
that the average dot product of two different steps <
[^(n)]i ·[^(n)]j > = 0, since the steps
are in random directions. So,

< r2 > =
Na2

The
time after N steps is just t = N t so the
displacement-squared grows linearly with time:

< r2 > =

a2

t

t

To
contact with usual diffusion in three dimensions we note that

< r2 > = 6 D
t

which
means that for this particular random-walk model of Brownian motion

D =

a2

6 t

This
model (and the calculation) is close to the mathematical description of Brownian
motion used in research, and is at least in spirit the core of most of
statistical physics.

Problem: Estimate the diffusion constant of a sucrose molecule using
this random-walk model, based on the idea that it is moved by 1 Å distances
every picosecond or so. Check your result against the known diffusion constant
from e.g. the CRC Chemistry and Physics Handbook.

Einstein-Sutherland drag-diffusion relation

The relation D = kB T/kdrag is not hard to follow in
Einstein's original papers (or for me, translations of them into English). The
following argument led Einstein to connect the drag coefficient to the diffusion
constant. It requires you to know Fick's law, that the current of diffusing
particles is equal to the diffusion constant times the concentration gradient.

Suppose we have some particles undergoing Brownian motion in water, and also
subject to a weak constant force f in the -z-direction
(you could think of a gravitational force f = m g if you like).

According to statistical mechanics, the equilibrium distribution of particles
with height should be given by the Boltzmann distribution using the potential
energy U(z) = m a z, giving the concentration to be

c(z,t) = C0 exp

æ
ç
è

-f z

kB T

ö
÷
ø

Consider
heights only very close to z = 0, so that we can expand the exponential using
its Taylor series

ex = 1 + x +

x2

2

+ ¼

and
keep just the linear term,

c(z,t) = C0

æ
ç
è

1 -

f z

kB T

ö
÷
ø

This
concentration gradient must be maintained by two opposing forces. First,
particles are falling at their terminal velocity v = f/kdrag; the
number of particles per area falling through z = 0 per unit time is

C0 v =

C0 f

kdrag

At
the same time, random forces cause the particles to diffuse, tending to smooth
out the concentration gradient, which means there is a net upward flux of
particles due to Brownian forces. According to Fick's law for diffusion the
number of particles going up at z = 0, per area and time, is

-D

¶c

¶z

=

C0 f D

kB T

Since
these two fluxes of particles must cancel for there to be equilibrium, we have

C0 f

kdrag

=

C0 f D

kB T

Note
that the concentration C0 and the driving force f cancel out, and are
therefore just thought-quantities invoked to derive the result

D =

kB T

kdrag

Note
that the connection to Brownian motion is not quite explicit here since this
argument has shown the Einstein relation between the microscopic drag
coefficient and the macroscopic diffusion constant appearing in the diffusion
equation. A complete connection back to microscopic Brownian motion requires
some further analysis of the random-walk model for a single particle to show
that it is truly statistically described by the diffusion equation. This kind of
thing can be found in more detail in the literature.

Further reading on Brownian motion, diffusion and Einstein:

Howard Berg ``A random walk in biology'' Princeton (1993) - excellent, short
and cheap in paperback

Albert Einstein ``Investigations on the theory of the Brownian movement''
Dover (1956), edited by R. Furth - this cheap and short book contains Einstein's
original papers translated into English, which are readable by seniors in
physics

Abraham Pais ``Subtle is the Lord: the science and life of Albert Einstein''
Oxford (1982) - for fun - Chapter 5 has an excellent discussion of the history
of Brownian motion, emphasizing the significance of Einstein's contributions

File translated from TEX by TTH, version
2.53.
On 15 Jan 2001, 23:39.