LS part4 diffusion
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Transcript LS part4 diffusion
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,m1
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( Dk )
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
6s 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[c1 (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.