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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