Dynamic Structure Factor and Diffusion

Outline

Dynamic structure factor

Diffusion

Diffusion coefficient

Hydrodynamic radius

Diffusion of rodlike molecules

Concentration effects

Dynamic Structure Factors

g1(τ) ~S(k,τ)=1nP

exp[ik⋅(rm(0)−rn(τ))]n,m=1

nP

∑

S(k,τ) = exp[ik⋅(r1(0) −r1(τ))] + (nP −1)exp[ik⋅(r1(0)−r2(τ))]

single-particlestructure factor

is zero at low concentrations≡S1(k,τ)

Dynamic Structure Factor and Transition Probability

The particle moves from r’ at t = 0 to r at t = with a transition probability of P(r, r’; ).

S1(k,τ) = exp[ik⋅(r1(0)−r1(τ))] = drV∫ exp[ik⋅(r − ′ r )]P(r, ′ r ;τ)

g1(τ) =S1(k,τ)S1(k,0)

S1(k,) is the Fourier transform of P(r, r’; ).

DLS gives S1(k,).

Diffusion of Particles

P(r, ′ r ;t) =(4πDt)−3/2exp−(r − ′ r )2

4Dt

⎛

⎝ ⎜ ⎞

⎠ ⎟ transition probability

diffusioncoefficient

Mean Square Displacement

t in log scale

<(r

 – r

´)2>i

n lo

g s

cale

slope = 1

D=(r − ′ r )2

6t

r − ′ r =0

(r − ′ r )2 =6Dtmean squaredisplacement

Diffusion Equation

∂P∂t

=D∇2P =D∂2P∂r2 =D

∂2

∂x2 +∂2

∂y2 +∂2

∂z2⎛

⎝ ⎜ ⎞

⎠ ⎟ P

c(r,t) = P(r, ′ r ;t)c(∫ ′ r ,0)d ′ r concentration

at t = 0, P(r, ′ r ;0)=δ(r − ′ r )

∂c∂t

=D∇2c

Structure Factor by a Diffusing Particle

S1(k,τ) = exp[ik⋅(r − ′ r )]∫ (4πDτ)−3/2exp−(r − ′ r )2

4Dτ

⎛

⎝ ⎜ ⎞

⎠ ⎟ dr

=exp(−Dτk2)

g1(τ) =exp(−Γτ)

Γ =Dk2 decay rate

How to Estimate Diffusion Coefficient

1. Prepare a plot of as a function of k2.

2. If all the points fall on a straight line, the slope gives D.

It can be shown that Γ =Dk2is equivalent to

∂P∂t

=D∇2P

(diffusional)

Stokes-Einstein Equation

Nernst-Einstein Equation

D=kBTζ

Stokes Equation

Stokes-Einstein Equation

ζ =6πηsRS

D=kBT

6πηsRS

Stokes radius

frictioncoefficient

Hydrodynamic Radius

D=kBT

6πηsRHhydrodynamic radius

A suspension of RH has the same diffusion

coefficient as that of a sphere of radius RH.

Hydrodynamic Interactions

The friction a polymer chain of N beads receives from the solvent is much smaller than the total friction N independent beads receive.

The motion of bead 1 causes nearby solvent molecules to move in the same direction, facilitating the motion of bead 2.

Hydrodynamic Radius of a Polymer Chain

1RH

=1

rm−rn

For a Gaussian chain,1RH

=823π

⎛ ⎝

⎞ ⎠

1/2 1bN1/2

polymer chain RH/Rg RH/RF RF/Rg

ideal / theta solvent 0.665 0.271 2.45

real (good solvent) 0.640 0.255 2.51

rodlike 31/2/(ln(L/b)−γ) 1/[2(ln(L/b)−γ)] 3.46

Hydrodynamic Radius of Polymer

PS in o-fluorotoluene -MPS in cyclohexane, 30.5 °C

good solvent theta solvent

Diffusion of Rodlike Molecules

γ ≅0.3D||=

32DG

D⊥ = 34DG

DG =13D||+

23D⊥ =

kBT[ln(L /b)−γ]3πηsL

RH =L /2

ln(L /b) −γ

Concentration Effects

If you trace the red particle, its displacement is smaller because of collision.

The collision spreads the concentration fluctuations more quickly compared with the absence of collisions.

Self-Diffusion Coefficients andMutual Diffusion Coefficients

mutual diffusion coefficients

self-diffusion coefficients

Self-Diffusion Coefficients

ζ =ζ0(1+ζ1c+L )

Ds =

kBTζ

=kBTζ0

(1−ζ1c+L )

DLS cannot measure Ds.

As an alternative, the tracer diffusion coefficient is measured for a ternary solution in which the second solute (matrix) is isorefractive with the solvent.

Mutual Diffusion Coefficients

Dm =D0(1+kDc+L ) kD =2A2M−ζ1 −vsp

∇μ=kBT[c−1 +(2A2M−vsp) +L ]∇c

In a good solvent, A2M is sufficiently large

to make kD positive.

specific volume

DLS measures Dm in binary solutions:

kD =2A2M−ζ1 −2vspwith backflow correction