Hydrodynamic Drag, Brownian Motion and Diffusion

8/12/2019 Hydrodynamic Drag, Brownian Motion and Diffusion

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http://www.uic.edu/classes/phys/phys461/phys450/MARKO/N004.html 2

Re =10-610-6103

10-3= 10-6

fdraghd v

fdriving= fdrag6 phR v

fdriving= 6 3.14 10-310-610-5= 19 10-14 Newtons

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?

For sufficiently tiny things moving sufficiently slowly, Re


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6 pR hv =4 pR3Dr

3

a

giving v =2 R2Dr a

9 h

v =2 10-120.2 10310

9 10-3= 1.4 10-6 m/sec = 1.4 micron/sec

cequilibrium(z) = C0exp

-Dm a z

kBT

where Dris 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 106g'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 ofwater, then we can compute its sedimentation velocity due to gravity as (mks units)

Note that the particle mass is volume time mass density, or m = 4pR3r/ 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 = 1.4x106Svedbergs or 1.4x106S. 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 onlyoccur until one reaches the thermal equilibrium distribution of concentration of particles,

where Dm is the difference between the mass of the particles and the water that they displace, and z is thedistance above the bottom of the tube. Note that the potential energy of the particles at height z is used in theBoltzmann factor (it is of the form U = m g h you might remember from elementary mechanics). The constant C0is determined if you know the height of the water in the test tube and the total number of particles, since the totalnumber 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 anacceleration well in excess of


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a = kBT

d Dm

4 10

-21

0.5 10-2(43.14/3) 0.2 103(210-9)3= 1.2 105m/sec2

< r> = 0

< |r|2> = 6 D t

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)

or about 12000 g (recall the acceleration due to gravity at the earth's surface, 1 g = 9.8 m/sec2). To achieveeffectiveseparationof 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 ofviscous 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 andengineering 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- almostappearing to hop around discontinuously. ThisBrownian motionis due to collisions with water molecules, whichmakes the particles undergo random-walkmotion.

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

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 theaverage displacement-squared:

where D is called the diffusion constant of the particle. This formula will be often referred to as the `square-root


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law' of Brownian motion. The diffusion constant D depends both on the size and shape of the particle, and on theviscosity 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 experimentalworks beginning with Robert Brown's original experiments of 1827. Much of the crucial quantitative work wasdone 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 thepictures show the trajectory after 1 sec, 3 sec and 10 sec:


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This graph is x(t) for a random walk in one dimension, and could just as well represent how much money youhave 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 adisplacement-squared of exactly6 D t in any one of your trials - this formula represents only the averagethatyou will obtain after many trials. However, it still makes sense for us to say that a particle undergoing Brownian

motion moves a distance (Dt)1/2during a time interval t.

The following figure shows the magnitude of displacement as a function of time for the simulation of Brownianmotion 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 theaverage 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 displacedfrom 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 < x

2

> = < y

2

> = < z

2

> = 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 10-8cm2/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 Physikin 1905 and 1906, Einstein established the theory of Brownian motion. These two papers are more heavily cited


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D =

kBT

kdrag

D =

kBT

6 phR=

4 10-21

6 3.14 10-310-6= 0.2 10-12 m2/sec

than his papers on relativity, due to their wide applicability(biology, chemistry, physics, chemical engineering...) andare the theoretical foundation of chemical and colloid

physics.

There were two immediate and big results of Einstein'sthesis work. First, he provided a new way to estimateAvogadro's number, which in 1905 was only knownroughly. The idea that there were molecules at all was stillcontroversial in 1905.

Second, Einstein showed that the diffusion constant D of asmall particle was simply related to its drag coefficientkdragand absolute temperature:

This impossibly simple formula is fundamental to the description of virtually all transport phenomena - frombiophysics 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 coefficientis reduced, since the water molecules hitting it will bonk it further and faster. Similarly, if absolute temperature isincreased, 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 waterat room temperature?The Einstein relation tells us that the 1-micron particle has a diffusion constant of

or equivalently D = 0.2 microns2/sec. This tells us that a 1-micron-radius particle in water is displaced by 1micron 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 suitableradius. Compare your result with the known diffusion constant obtained from, e.g. the CRC Chemistry andPhysics Handbook.

Problem:A neurotransmitter molecule which is roughly spherical and of radius 0.5 nm is synthesized in the cellbody of a neuron in your lower back. How long will it take to diffuse down the long, skinny axon to the synapse


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c t = D

2

x2+2

y2+2

z2

c

c(r,t) =

C0

t3/2exp-

(x2+ y2+ z2)

4 D t

< X > =

d3r X c(r,t)

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) satisfiesthe diffusion equation:

where D is the diffusion constant of the ink. Another key result of Einstein and researchers following him was thatthe 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 particlesall at once.

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

This solution is a function of only the magnitude of the position vector |r|= (x2+ y2+ z2)1/2, and is thereforespherically 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+, dropsmonotonically with time.

The figure below shows the Guassian solution in one dimension, c(x,t) = exp[-x2/(4Dt) ] / t1/2Note that the three-dimensional solution is just the product of similar x, y and z terms. The figure shows howdiffusion 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 diffusionequation. 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 rof the diffusing particles is zero, and that the average value of |r|2is in fact 6 D t.

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


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d3r c(r,t)

+

- dx e-ax2

=

p

a

1/2

+

-dx x2e-ax2=

p

4 a3

1/2

r= a^n

1+ a

^n

2+a

^n

N

|r|2= a2

^n

1+

^n

2+

^n

N

= a2|

^n

1 |2+ |

^n

2 |2+|

^n

N |2+

^n

i

^n

j

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

The connection between diffusion is precise -diffusion is just the statistical (average) description ofa large number of particles undergoing independent

Brownian motions. The second problem aboveemphasizes that the diffusion equation describes particles whose square-root law is precisely that of Brownianmotion.

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

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

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


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i j

= N a2+ i j

^n

i^n

j

< r2> = Na2

< r2> =a2

t

t

< r2> = 6 D t

D =a2

6 t

-f z

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 oftwo different steps < [^(n)]i[^(n)]j> = 0, since the steps are in random directions. So,

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

To contact with usual diffusion in three dimensions we note that

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

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 ideathat it is moved by 1 distances every picosecond or so. Check your result against the known diffusion constantfrom e.g. the CRC Chemistry and Physics Handbook.

Einstein-Sutherland drag-diffusion relation

The relation D = kBT/kdragis 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 theconcentration 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 theBoltzmann distribution using the potential energy U(z) = m a z, giving the concentration to be


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c(z,t) = C0exp

kBT

ex= 1 + x +x2

2+

c(z,t) = C0

1 -f z

kBT

C0v =C0f

kdrag

-Dc

z=

C0f D

kBT

C0f

kdrag=

C0f D

kBT

D =

kBT

kdrag

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

and keep just the linear term,

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

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

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

Note that the concentration C0and the driving force f cancel out, and are therefore just thought-quantities

invoked to derive the result

Note that the connection to Brownian motion is not quite explicit here since this argument has shown the Einsteinrelation between the microscopic drag coefficient and the macroscopic diffusion constant appearing in thediffusion equation. A complete connection back to microscopic Brownian motion requires some further analysisof the random-walk model for a single particle to show that it is truly statistically described by the diffusionequation. 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


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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 byseniors in physics

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

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Hydrodynamic Drag, Brownian Motion and Diffusion

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