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.