Dynamic Structure Factor
and Diffusion
Outline
Dynamic structure factor
Diffusion
Diffusion coefficient
Hydrodynamic radius
Diffusion of rodlike molecules
Concentration effects
Dynamic Structure Factors
1
g1 ( ) ~ S(k,  ) 
nP
nP
 exp[ik (rm(0)  rn( ))]
n,m1
S(k,  )  exp[ik (r1(0)  r1( ))]  (nP 1)exp[ik (r1(0)  r2 ( ))]
 S1 (k, )
single-particle
structure factor
is zero at low concentrations
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,) is the Fourier transform of P(r, r’; ).
S1 (k, )  exp[ik  (r1 (0)  r1 ( ))]   dr exp[ik  (r  r)] P(r, r;  )
V
DLS gives S1(k,).
S1 (k, )
g1 ( ) 
S1 (k,0)
Diffusion of Particles
3 / 2
P(r, r;t)  (4 Dt)
 (r  r )2 
exp
 4Dt 

diffusion
coefficient
transition probability
r  r   0
mean square
displacement
(r  r)2  6Dt
D
(r  r )2
6t
<(r – r´)2> in log scale
Mean Square Displacement
slope = 1
t in log scale
Diffusion Equation
2
2
2
2 

P

P



2
 D P  D 2  D 2  2  2 P
t
r
x
y
z 
at t = 0, P(r, r;0)   (r  r)
concentration
c(r,t)   P(r, r ;t)c(r ,0)dr 
c
2
 D c
t
Structure Factor by a Diffusing Particle
2 

(r

r

)
S1 (k, )   exp[ik  (r  r)](4 D ) 3 / 2 exp
dr
 4D 
 exp( Dk )
2
g1( )  exp( )
  Dk
2
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.
P
2

D
P


Dk
It can be shown that
is equivalent to
t
2
(diffusional)
Stokes-Einstein Equation
Nernst-Einstein Equation
D
Stokes Equation
kBT

friction
coefficient
  6s RS
Stokes-Einstein Equation
kBT
D
6 s RS
Stokes radius
Hydrodynamic Radius
kBT
D
6 s RH
hydrodynamic 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
1
1

RH
rm  rn
For a Gaussian chain,
1/ 2
1
2
1


8
3  bN1 /2
RH
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
1/[2(ln(L/b))]
3.46
rodlike
1/2
3
/(ln(L/b))
Hydrodynamic Radius of Polymer
good solvent
PS in o-fluorotoluene
theta solvent
a-MPS in cyclohexane, 30.5 °C
Diffusion of Rodlike Molecules
1
2
k BT[ln( L / b)   ]
DG  D||  D 
3
3
3 s L
  0.3
D|| 
D 
3D
2 G
3D
4 G
L/2
RH 
ln(L / b)  
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 and
Mutual Diffusion Coefficients
mutual diffusion coefficients
self-diffusion coefficients
Self-Diffusion Coefficients
   0 (1  1c  )
Ds 
kBT


k BT
0
(1  1c 
)
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
DLS measures Dm in binary solutions:
Dm  D0 (1  kDc 
)
k D  2A2 M  1  vsp
specific volume
  kBT[c1 (2A2 M  vsp )  ]c
with backflow correction
k D  2A2 M  1  2vsp
In a good solvent, A2M is sufficiently large
to make kD positive.