Fluctuations and Brownian MotionFluctuations and Brownian Motion

2 2 fluorescent spheres in water (left) and DNA solution (right) fluorescent spheres in water (left) and DNA solution (right)(Movie Courtesy Professor Eric Weeks, Emory University: http://www.seas.harvard.edu/weitzlab/research/brownian.html)

Brownian motion in water Brownian motion of DNABrownian motion in water Brownian motion of DNA

Copyright (c) Stuart Lindsay 2008

FluctuationsFluctuations

One set of collisions 5·105 set of collisions

Simulated distribution of speeds in a population of 23 atoms of an ideal gas: V=1.25·103 nm3 (50·50·50nm); T=300K.

Calculating FluctuationsCalculating Fluctuations

Zln

Z

EexpEE r

rr

rr

r EexpEZ

Z

Z

ZlnZln

1

2

2

2

2 111

Z

Z

Z.

Z

Z

Z

Z

Z

Zln

Taking the second derivative of dlnZ/d:

22

22

2

22

2

expexp11

EEZ

EE

Z

EEZ

Z

Z

Zr

rrr

rr

This is just the mean square thermal average of E:

222222 2 EEEEEEEEE

jj E

jj

E

jjj

jj eE

QeE

QPEE

11 222

Qln

EE

EQQ

1

VB CTkT

EkTE 222

22 ET

EkT

For an ideal gas:

NkCTNkE VB 2

3

2

3

NE

CkT

E

EV 122

The relative size of energy fluctuations scales as The relative size of energy fluctuations scales as N

1

T,V

ZlnkTN

In an open system:

222 NNN

NE

j,N

NE

j,Nj,N

j,N

eeNZ

kTeeN

ZPNN j,Nj,N222 1

2N

NkT

ZlnN

NkTNZ

Z

kT

V,T

NkTN

2

From thermodynamics:T,NV,T V

p

N

V

N

2

22 NV

kTN

T,Np

V

V

1 Isothermal compressibility

For an ideal gas:p

1

NN 2NN

N 12

• The result just obtained for energy holds for all quantities (extensive quantities) that, like energy, grow with N (T, E, P, V, S).

• In general the root-mean-square value of the fluctuations relative to the mean value of a quantity is given by

NX

X RMS 1

NN Relative fluctuationRelative fluctuation

10 31.6%

1000 3.2%

1026 10-13

Copyright (c) Stuart Lindsay 2008

Brownian motionBrownian motion

)t(Fvdt

dvm

Langevin equation:Langevin equation:

α (friction coefficient )

For a sphere of radius a in a medium of viscosity η: =6πηa

Stoke’s forceStoke’s force = friction exerted on the particle by the fluid. For small velocities, it is proportional to the velocity v.

0)( tF )tt(F)t(F)t(F

F(t) is a random force representing the constant molecular bombardment exerted by the surrounding fluid:

Average is zero! Finite only over duration of single “effective” collision

F(t) is independent of the velocity of the particle (v) and varies extremely rapidly compared to the variations in v.

There is no correlation between F(t) and F(t+Δt) even though Δt is expected to be very small.

)()( 2 txFxxxxxdt

dmxmx

Multiplying both sides of the Langevin eqn. by x and using

xv

Re-arranging and taking thermal averages:

)(2 txFxxxmxxdt

dm

Tkxm B2

1

2

1 2 = 0

so

xxmm

Tkxx

dt

d B

substituting

CBtexpAxx

yields

Tk

C Bm

B

To find A, note

2

2

1x

dt

dxx

with <x2>=0 at t=0:

Tk

A B

m

texp

Tkx

dt

d B

12

1 2

d

mdt

m

t

substitute

m

tmt

Tkx B

exp1

22

Integrate with <x2>=0 at t=0:

m

tmt

Tkx B

exp1

22

Long time solution:

m

t

a

tTktTkx BB

3

22

Mean square displacement increases with t!

Copyright (c) Stuart Lindsay 2008

• Now we see why the sphere in a viscous DNA solution moves more slowly!

2 fluorescent spheres in water (left) and DNA solution (right)

a

TtkTtkx BB

3

22

Copyright (c) Stuart Lindsay 2008

(Movie Courtesy Professor Eric Weeks, Emory University: http://www.seas.harvard.edu/weitzlab/research/brownian.html)

The Diffusion EquationThe Diffusion Equation

Flow of solute or heat under action of random forces:

J is flux per second per unit area

xA

txAJtxxJAC

)()(.

0, txx

txJ

t

txC

),(),(

The flux is the change in concentration across a surface multiplied by the speed with which particles arrive:

t

x

x

C

t

xCJ

2

Using: Dt

x

2

x

CDJ

From which

2

2 ),(),(

x

txCD

t

txC

DiffusionDiffusioncoefficientcoefficient

Fick’s first lawFick’s first law

Fick’s second lawFick’s second lawDiffusion EquationDiffusion Equation

In 3D: )t,r(CDr

)t,r(CD

t

)t,r(C 22

2

12 sm

The solution in 1-D for a solute initially added as a point source is:

Dt

x

Dt

AtxC

4exp),(

2

Dtx / 221

A=1

As t→∞, the distribution becomes uniform, the point of ‘half-maximum concentration’ x½ advancing with time according to:

Einstein-Smoluchowski RelationEinstein-Smoluchowski Relation

Dtx / 221

Fa

Fv

6The mobility is defined as the inverse of the friction coefficient:

a

TkD B

6

TkD BEinstein-Smoluchowski relationEinstein-Smoluchowski relation

a

TtkTtkx BB

3

22 Compare to

A surprising relation between thermal motion and driven motion: the diffusion constant is the ratio of kT to the friction constant!

A fundamental relation between energy dissipation and diffusion

Viscosity is not an equilibrium property, because viscous forces are generated only by movement that transfers energy from one part of the system to another.

Ex. Motion of a spherical large particle with respect to a large number of small molecules.

vaF 6

Forceradius

viscosity

speed

Stokes’ lawStokes’ law

The introduction of bulk viscosity requires that the small molecules rearrange themselves on very short times compared with the time scale of the motion of the sphere.

Diffusion, fluctuations and chemical reactionsDiffusion, fluctuations and chemical reactions

1. If the reactants are not already mixed they need to come together by diffusion.

2. Once together, they need to be jiggled by thermal fluctuations into a “transition state”

3. If the free energy of the products is lower than that of the reactants the products lose heat to the environment to form stable end products.

Haber process for AmmoniaHaber process for Ammonia

The transition state

Tk

G

h

Tkk

B

B exp1

Eyring transition state theory

Copyright (c) Stuart Lindsay 2008

The entropy adds a temperature-dependent component to the energy differences that determine the final equilibrium state of a system.

Kramers’ Theory of Chemical ReactionsKramers’ Theory of Chemical Reactions

• Noise driven escape:

Thermal fluctuations allow the particles in the well to rapidly equilibrate with the surroundings.The motion of the particles over the barrier is much slower.

The Kramers ModelThe Kramers Model

Reaction coordinate

Tk

Eexpk

B

bba

2

Microscopic description of the prefactor in terms of potential curvature

c,b,ax

c,b,a x

U

m

2

22 1

Unimolecular reactionsUnimolecular reactions

• Not very common, but can be described as a one step process by the Kramers theory

• Example is isomerism of isonitrile

E1 E2

k-

k+

CH3

NC

CH3

CN

211 EkEk

dt

Ed

Copyright (c) Stuart Lindsay 2008

Thermodynamic Potentials for NanosystemsThermodynamic Potentials for Nanosystems

• The Gibbs free energy in a multicomponent system is:

This equation contains no reference to system size (all quantities are extensive: doubling the volume of a system, doubles its free energy)

• Nanosystems at equilibrium derive their “stable” size from Nanosystems at equilibrium derive their “stable” size from surface and interface effects which are not extensive.surface and interface effects which are not extensive.

Ex. Self-assembly originates from a competition between bulk Ex. Self-assembly originates from a competition between bulk and surface energies of the phases that self-assemble (stability and surface energies of the phases that self-assemble (stability of colloidal systems).of colloidal systems).

NTSPVEG

Copyright (c) Stuart Lindsay 2008

• Hill has generalized thermodynamics to include a “subdivision potential”

dd

dEdNVdpTdSdE

The simplest approach add surface terms to the free energy.

Subdivision potential

number of independent parts of the system

Modeling nanosystems explicitly:Modeling nanosystems explicitly:Molecular DynamicsMolecular Dynamics

irij

ij rr

rUF

2

2 )(

TkVm Bzyxii 2

1

2

1 2,,,

Limited to s timescales by required time step/processor speed.

Interatomic vibrations are rapid (≈1013 Hz) so that the time step in the calculations must be much smaller (≈10-15 s).

Copyright (c) Stuart Lindsay 2008

• The number of calculations scales as N2 (it depends on the problem).

If all possible pair interactions need to be considered , the number of interactions would scale like 2N.

Limitations of MDLimitations of MD

• Timescale is limited to ns or s at best. This is fine for harmonic vibrations but nanosystems like enzymes work on ms timescales.

Ex. Viscously damped motion:

Ex. Activated transitions over barriers:

m

a

61

Tk

Ek

B

bba

exp2

Example of MD

Cyclic Sugar molecule being pulled over DNA

Qamar et al. Biophys J. (2008) 94, 1233

Movie link:13mer.avi

Thermal Fluctuations and Quantum Thermal Fluctuations and Quantum MechanicsMechanics

• The density matrix formulation of quantum mechanics allows description of the time evolution of a system subject to time dependent forces.

Copyright (c) Stuart Lindsay 2008

ˆ,Ht

ˆi

q*nnm aaqn

• Thermal fluctuations can be represented by the “random phase approximation” – quantum interference effects are destroyed.

Uncertainty Principle and fluctuationsUncertainty Principle and fluctuations

• Interaction energy, , will decay exponentially at large distances

• When kT

The time characteristic of a molecular vibration.

Copyright (c) Stuart Lindsay 2008

pseV.

seV

Et 1

0250

104 14