Institute of Solid State Physics (IFF) Soft Matter Division

Dynamics of Dispersion Colloids

Gerhard Nägele

Research Centre Jülich

Institute of Solid State Physics (IFF) Soft Matter Division

E-mail: g.naegele@fz-juelich.de

International graduate school Konstanz – Strasbourg - Grenoble

Grenoble, February 28 – March 2, 2005

2

Overview

1. Introduction & motivation 2. Salient static properties 3. Dynamic light scattering

4. Theoretical description of colloid dynamics

5. Hydrodynamic mobility problem of many spheres 6. Basic properties of Smoluchowski dynamics

7. Short-time dynamics

8. Long-time dynamics

9. Concluding remarks

1. Introduction & Motivation

- Examples of colloidal dispersions

- Modell dispersions and direct particle forces - Hydrodynamic interaction (HI)

- Dynamics on colloidal time scales

2. Salient static properties

- Pair distribution function

- Static light scattering and structure factor - Methods of calculation

3. Dynamic light scattering

- Basic scattering theory - Dynamic correlation functions

4. Theoretical description of colloid dynamics

- Single-particle dynamics in very dilute dispersions - Colloidal time and length scales

- Generalized Smoluchowski diffusion equation for dense systems

5. Hydrodynamic mobility problem of many spheres

- General properties of hydrodynamic interaction - Method of induced forces

- Motion along a liquid-gas interface

6. Basic properties of Smoluchowski dynamics

- Fundamental solution and time correlation functions - Backward operator and eigenfunction expansion - Brownian dynamics simulations

- Projection operators and memory equations

7. Short-time dynamics

- Methods of calculation - Hydrodynamic function - Sedimentation

- Smoluchowski equation with incident fluid flow - Collective and gradient diffusion

- Rotational self-diffusion

8. Long-time dynamics

- Generalized hydrodynamics

- Mode coupling approximation of memory functions

- Application: non-exponential decay of dynamic structure factor

9. Concluding remarks

4

General literature

1. W.B. Russel, D.A. Saville, and W.R. Schowalter, Colloidal Dispersions (Cambrigde University Press, 1989)

2. J.K.G. Dhont, An Introduction to Dynamics of Colloids ( Elsevier, Amsterdam, 1996)

3. R. Pecora, Dynamic Light Scattering, (Plenum Press, New York, 1985)

4. G. Nägele, The Physics of Colloid Soft Matter: Lecture Notes 14 (Polish Academy of Sciences Publishing, Warsaw, 2004)

5. G. Nägele, On The Dynamics and Structure of Charge-Stabilized Colloidal Suspensions (Physics Reports 272, pp. 215-372, 1996)

6. R.M. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications (Clarendon Press, Oxford, 2002)

7. R. Zwanzig, Non-Equilibrium Statistical Mechanics (Oxford University Press, 2001)

1. Introduction & motivation

- Examples of colloidal dispersions

- Modell dispersions and direct particle forces

- Hydrodynamic interaction (HI)

- Dynamics on colloidal times scales

6

Examples of colloidal dispersions

Definition: 1 nm <

∅ < 10 µ m

• Particle properties/interactions

Brownian erratic motion

Examples: micellar systems Beispiele:

microemulsions

Beispiele:

proteins

Beispiele:

viruses

Beispiele:

inorganic particles (goldsol, Silica, ...)

Solvent: H₂O, ...

Industrial products: dispersion paints Industrielle Produkte:

pharmaceuticals

Industrielle Produkte:

food stuff

Industrielle Produkte:

cosmetics

Industrielle Produkte:

waste water

• Transport properties:

- diffusion coefficients - viscosities

- conductivities Theoretical

task:

Model dispersions in 3D

Charge-stabilized dispersions :

Sterically stabilized dispersions :

{ }

2

u (r)

el

∝ Q exp −κ r / r , r > σ

u(r) ≈ u (r)

el

^I¾U

Q / e >> 1

• Range and strength of u(r) is tunable

r

σ

k T

u(r)

B

u (r)

el

u

vdW

(r)

r / σ

r / σ

u(r)

8

• Present lectures restricted on colloidal systems in fluid one-phase region

Phase behavior of colloidal hard-sphere dispersion

Pusey & van Megen Nature 320, 1986

Quasi-two-dimensional model dispersions

Colloidal spheres between two walls

•

Yukawa-type model potential (Chang & Hone, EPL 5, ´88)

Magnetic spheres at water-air interface

2

r

h 2

r r

r

u(r) e

Q , ( )

,

−κ π

< σ

> σ κ = ε

 

= 

 

∞

₂

0 eff 2

4

3

u(r) B

r

µ χ

= π

•

Well-characterized dipolar potential (Zahn & Maret, PRL 85, ´00)

10

Experimental realization: Brownian forces

Glass walls

Video microscopy

100 µ

^m

σ ^{= 1} µ^{m and} σ₂ ^{= h = 2} µ^m, φ_Area ^{= 0.022}

Courtesy: J.L. Arauz-Lara, Univ. of San Luis Potosi

Polystyrene spheres ( σ ^{= 1} µ ^m)

Spacer spheres (h > σ ⁾

σ ^{= 1} µ

Colloidal particle dynamics determined by interplay of:

^{m and} σ₂ ^{= h = 2} µ^m, φ_Area ^{= 0.022}

1. Direct particle interactions (DI)

2. Brownian erratic forces due to solvent molecules‘ bombardement

3. Hydrodynamic interactions through solvent (HI)

12

Hydrodynamic interaction (HI)

B

N

N

k T

1

1 ( )

=

= ∑

^{i j}

⋅

i j

j

D r

v F

B

0

k T

= D

i i

v F

( generalized Stokes‘ friction law )

hydrodynamic mobility tensor (non-linear in r^N)

V

i

F

i

V

i

F

i

F

j

B

0 0

k T

D 6

πη a

=

• Long-range dynamic many-body force

• Quasi-instantaneous & inertia-free on colloidal time scales

• Small-Reynolds-number creeping solvent flow

r

⁻1

∼

HI: Non-Brownian sedimentation of two small spheres

gravity

14

HI: Non-Brownian sedimentation of two small spheres

gravity

- fixed distance vector r

r

15

z/ σ

(-1.1 ,0 ,1.15) (-1.1, 0, 1.20)

x/ σ x/ σ

Asymmetric initial configuration of three spheres at z = 0

g

• Sensitive dependence on initial configuration for N > 2 → chaotic trajectories

16

HI: Non-Brownian sedimentation of many spheres

- meandering trajectories & fluctuating velocities

- hydrodynamic „diffusion“ due to HI

- existence & form of final stationary particle distribution P

_stat

(r

^N

) still under debate

-

slowly sedimenting colloidal particles →→→→ Brownian trajectories, P_stat(r^N) existent

Dynamics on colloidal time scales

t

t

³

t

t

8

[ ^{(t) (0)} ]

²

6

−

r r

t / sec

D

S

D

L

B

0

M τ = 3

πη σ

2

0

I D

τ = σ

VKRUWWLPH ORQJWLPH

τ

τ '/6

'/6W RYHUGDPSHG/DQJHYLQ(T W JHQHUDOL]HG6PROXFKRZVNL(T

τ t τ_I t τ_I

PRPHPWXP UHOD[DWLRQ

18

Dynamics on colloidal time scales

• quasi-inertia free motion on coarse-grained colloidal time- and length scales

8

t

B

10

⁻

∆ >> τ ≈ ^VHF

4

B 0 B

x l D 10

⁻

∆ >> = τ ≈ σ

Rhodospirillum bacteria (length σ ≈ 5 µ^m)

“stopping distance”

Dynamics on colloidal time scales

Generalized Smoluchowski Eq.

DI H R

i i i i

M V (t) = = 0 F + F + K

N

N N N

i i

i 1

P( , t) ( ) P( , t) 0

t

₌

∂ + ∇ ⋅     =

∂ ^r ∑ ^{v r r}

Random force (solvent collisions)

PDF coarse-grained velocity from force balance

Dynamic simulations Overdamped Langevin-Eq.

Theoretical calculations

• pure configuration-space description for t >> τ

_B

20

2. Salient static properties

- Pair distribution function

- Static light scattering and structure factor

- Methods of calculation

21

Pair distribution function

Canonical NVT-ensemble: N >> 1 spherical particles in volume V at temperature T

( ) ( ) ( ) ( )

^N

(n)

1 n n 1 N

N

N

e

U

,..., N N -1 ... N - n 1 d ...d

+

Z

−β

ρ

^{r r}

=

+ =

∫

^{r r r}

n out of N

probability d.f. of rN

joint d.f. for n << N particles at

{

1

}

n

,..., n =

r r r

( )

^N

N 1 N ßU ßU

Z V, T = ∫ d ...d

^{r r}

e

⁻

= ∫ d

^r

e

⁻

( )_N¹

( ) ^{r N}

ρ = V = ρ

: average number density

( ) ( )

g r r , = g r − r = g (r)

Define pair distribution function as:

( ) ( )

( ) ( )

(2)

1 2

N 1 2 (1)N (1)

1 2

N N

g , ₌ ^ρ , _→ 1

ρ ρ

r r r r

r r ^{for r = r}

¹²

^{→ “∞”}

Isotropic fluid state (no crystal or external field) :

( )

^{N N}

⁽

^{i j}

⁾

^N

^{( )}

^{i j}

i j i j

U u u r

< <

= ∑ − = ∑

r r r

assume pair-wise additivity

: NVT radial distribution function

r

12

r

2

r

1

22

( )

⁽²⁾

( ) ( )

^N

N N 2 2 3 N

N

N N -1

U

(r) e

g r d ...d

Z ρ

−β

= =

ρ ρ ∫

^{r r r}

Thermodynamic limit for macroscopic system :

General behavior of g(r) :

( )

id N

g r 1 1

= − N

ideal gas : U = 0 and Z_N = V^N

•

^g_N measures pair correlations relative to ideal gas

( ) ( )

( )

^N

N 12 3 N N

N N -1

U

d g r d d ...d e

^−β

/ Z N 1

ρ = = −

∫

^r

ρ ∫

^r

∫

^{r r r}

1 2

1

V

∫

d d^{r r}

•

4πr²ρ g_N is average particle number in shell [r,r+dr] ( i.e. g(r) is conditional pdf )

( )

N

N ,V

g r lim g ( r )

=

→ ∞ _ρ = N/V fixed

• g(r) ≥ 0, g (r → ∞ ) = 1

• g(r) ≅ 0, when β u ( r ) >> 1

• g(r) = exp [- β u ( r ) ] for ρ → 0

• g(r) continuous where u(r) continuous

r

dr

2. coordination shell

( )

g r

1

1 2

r/ σ

correlation lengthξ(T)

1. coordination shell

-

^{jump at}

^{r =} σ

- ^{g (r <} σ ) = 0 since u (r < σ ) = ∞

- g (r ) = exp [- β u (r)] = θ (r - σ ) for ρ = 0 r/ σ

1

1 2

• Sketch of g(r) for soft pair potential • Sketch of g(r) for hard-sphere dispersion

• Undulations in g(r) more pronounced for higher density and lower temperature

24

Static light scattering and structure factor

• Scattering in homodyne mode: λ

_L

= λ

_vac

/n

_s

= O( σ , ρ

^-1/3

)

Detector

λ

_L

k

_i

ϑ L

q 4 sin 2 π ϑ

= λ laser

q k

f

• Assumptions in basic scattering theory (more rigorous: Maxwell equations)

→ point scatterers (Rayleigh) : σ << λ_L

→ single scattering (1. Born approximation)

→ no absorption

j

A 1

B ϑ

Quasi-elastic : _{f i L ph}

2

k k 2 / ( h m M)

c ν

≈ = π λ = <<

Phase difference between j and 1 :

B1 A1 + =

k_f

⋅ − ⋅

r_j1 k r_{i j1}

= ⋅

q r_j1

Scattered el. field amplitude at detector : ^s 2 ^Nj 1

{ (

1

) }

L

j

E ( ) 1 exp i

R

⁼

∝ λ ∑

^{q r}⋅ −^r q

blue sky R >> scatt. volume size

r

_j1

Time-averaged measured intensity : ^{s s*} _T ^N

{ (

^{l j}

) }

l, j 1 T

I(q) E E exp i

=

= ∝ ∑

^{q r}

⋅ −

^r

Ergodic system (fluid state) :

T N 1

T " "

lim ... ...

_>>

→ ∞

=

l j 12 12 N

N i i i (N 2) U( )

12 N

l j

I(q) N e

^⋅

N N(N 1) e

^⋅

N N(N 1)V d e

^⋅

d

⁻

e

^−β

/ Z

≠

∝ + ∑

^{q r}

= + −

^{q r}

= + − ∫

^{r q r}

∫

^{r r}

N 12 2

1 g (r )

[ ]

V

{

^{i N}

}

³

N 1 d e

^⋅

g (r) 1 N (2 ) ( )

= + ρ ∫

^{r q r}

− + ρ π δ

^q

forward contribution (no correlation information)

Static structure factor in T-limit :

^{S(q) 1 d e}

ⁱ

[ ^{g(r) 1} ] ^{lim 1 ( ) ( ) 0}

N

⋅

= + ρ ∫

^{r q r}

− =

∞

δρ

^q

δρ −

^q

≥

l

N N

i i 3

l

l 1 l 1

( ) d e

^⋅

( ) e

^⋅

(2 ) ( )

= =

 

 

δρ =



δ − − ρ =



− π ρδ

 



∑



∑

∫

^{q r q r}

q r r r q

Compressibility equation : q 0

( )

idT^T B T

lim S q k T

→

p

χ



∂ρ



= χ =



∂



p p

_s

(N → ∞, V → ∞ , ρ = N/V fixed)

26

Remarks about ordering of T - limit and q → 0 limit

Periodic replication of system (scattering) volume V = L³ :

L

f (

r

+

n

L) = f ( )

r

vector with

components 0, ±1, ±2, ...

f ( )

r

= e

i^{q r⋅} ⇒

2 L

= π

q n

( )

³

i ,0

V

V δ

^q

= ∫ d e

^{r q r⋅}

→ 2 π δ ( )

^q

( )

^{3 3}

1 1

F( ) d F( )

V 2

→ π

∑ ∫

q

q q q

»

Applications :

( )

⁶ 2 2

( )

³

,0 ,0

2

2 2

( ) ( )

V V

π π

δ

q

→ δ

_q

= δ

_q

→ δ

q

2

N N

S (q) 1 ( )

= N δρ

q ^l

N

,0 l 1

( ) e

i

N

=

δρ

q

= ∑

^{q r}⋅

− δ

_q

{ }

N idT N

q 0 q 0

T

lim S(q) lim limS (q) lim S (q 0) 0 0

→ → ∞ ∞

= = χ ≠ = = =

χ

•

First T-limit in NVT ensemble then q → 0

•

Several wavelengths λ=2π/q should fit into V

q_y

q_x 2 / Lπ

( ) ²

³

V

= π

density of q points regular at q = 0

27

S(q) and g(r) for a charge-stabilized dispersion

sphere

NV

φ =

V

m m

q ≈ π 2 / r r

_m

low compressibility

particle cage

28

Interpretation: diffusive Bragg scattering from particle density waves

4 n

s

q sin

2

π ϑ

= λ

s

2d sin

n 2

λ = ϑ

1. order constructive interference

d 2

q

≈ π

typical spatial resolution for selected q

ϑ

/ 2

ϑ

/ 2

λ

d

Point scatterers → → → → weakly scattering colloidal spheres with a = O( λ )

r

l

x

[

l

]

N i

s

l 1 x a

E ( ) d e

^{⋅ +}

( )

= <

∝ ∑ ∫

^{q r x}

ψ

q x x

i l

x a

N i l 1

d e ^⋅ ( )

e

<

⋅

=

 

 ψ 

 



∝

∫

^{x q x x} 

∑

^{q r}

p s

( ) ( ) ψ

x

= ε

x

− ε

P(q

I(q) ∝ N ) S(q )

a 2 i 0

a

0

d e ( )

d (

P( )

)

q

⋅

ψ

=

ψ

∫

∫

x q x x

x x

local

scattering strength

Single-particle form factor

→

isotropic spheres

→

Rayleigh-Gans-Debye (RGD) regime :

no phase difference of light through particle and solvent, respectively, i.e

Inherent assumptions :

p s

2 n π − n a / λ < 0.1

P(q = = 0) 1

P(q = ∞ = ) 0

30

Example: optically uniform colloidal spheres with Ψ ^{= const}

( )

2 1 2

3j (qa)

P(q) b(q)

qa

 

=   =

 

q σ ≈ 9

•

Large particles scatter more strongly in forward direction Formamplitude

Methods of calculation

• Computer simulations (Monte-Carlo, MD and BD)

• Ornstein-Zernike-type integral equation schemes

32 Introduce total correlation function :

h(r ) : g(r ) 1

₁₂

=

₁₂

− h(r )

₁₂

→ 0 ,

for

r

₁₂

→ ∞

Define direct correlation function c(r) through Ornstein-Zernike equation :

12 12 3 13 23

h(r ) = c(r ) + ρ ∫ d ^r c(r )h(r )

total correlations of 1 and 2

direct correlations

indirect correlations of 1 and 2 through particles 3,4,…

2 4

12 12 3 13 23 3 4 13 24 34

h(r ) = c(r ) + ρ ∫ d ^r c(r ) c(r ) + ρ ∫ d d c(r ) c(r ) c(r ) ^{r r} + 0(c )

+ +

= + ...

1 2

3 4

c

1 2

3 h c

1 2

c 2 1

i

ρ d

i

• = ∫ ^r

indirect correlations of 1 and 2 through two intermediate particles 3 and 4

General properties of c(r) :

c ) (r = h(r) = e

^−βu(r)

− + ρ 1 O( )

c ) (r = −β u(r) , r →∞

valid for all densities

- Fourier-transformed OZ equation from convolution theorem :

S(q) = + ρ 1 ∫ d e ^r

i^{q r⋅}

h(r) = + ρ 1 h(q)

ⁱ

0

h(q) d e h(r) 4 dr r sin(qr) h(r) q

⋅

π

∞

= ∫ ^r

^{q r}

= ∫

h(q) = c(q) + ρ c(q) h(q)

- Solving for Fourier-transformed c(r) gives S(q) in terms of c(q) :

S(q) 1 0

1 c(q)

= ≥

− ρ

Closed integral equation for g(r) = h(r) –1 Ornstein-Zernike equation

+ approximate closure relation : c(r) = Functional of h(r’) & u(r’) + zero overlap condition (HC) : g(r < σ) = 0

- c(r) is more amenable to approximations than g(r)

34

Examples of important closure relations

• Rescaled mean spherical approximation (RMSA) :

u(r) ,

eff

c(r) = −β r > σ > σ g(r = σ

eff⁺

) = 0

• Hypernetted chain approximation (HNC)

u(r) (r)

c (r) = e

^−β

⋅ e

^γ

− γ (r) 1 − γ (r) : h(r) c(r) = −

• Percus-Yevick approximation (PY)

[ ]

e

u(r)

1 (r) (r) c(r) =

^−β

⋅ + γ − γ − 1

• Rogers-Young mixing scheme (RY): thermodynamically partially self-consistent

{ }

u(r) exp (r)f (r) f (r) 1 1

c ( r ) = e

^−β

⋅   

+ ^{γ −}

   − γ (r ) − 1

f (r) = − 1 e

^−αr Compr

Virial

T T

χ = χ

determines α

: 0 : α → ∞ α →

HNC PY

& MSA

36

Example:

strongly coupled charge-stabilized colloidal spheres ( φ ^{= 0.185)}

• For each u(r), closures should be tested first against simulations

Results by Banchio & Nägele :

see in Lecture Notes 14, Polish Academy of Science Publishing, Warsaw (2004)

3. Dynamic light scattering

- Basic scattering theory

- Dynamic correlation functions

38

Basic scattering theory

L

q 4 sin 2 π ϑ

= λ

f i

= −

q k k

→ Maxwell equations for non-magnetic and linear dielectric media

→ Single scattering (1. Born approximation)

→ RGD regime (superposition + s-waves)

quasi-elastic

• Homodyne scattering set-up :

Photomultiplier at distance R >> V_S

k

i

i V

ˆ = ˆ n n

k

f

ˆn

V

ˆn

H Laser

Polarizer

Analyzer

ϑ

39

• Scattered electric field strength at detector place :

p s

x a l

s

1

i

d e ( )

( )

≤

⋅  

= ε ∫

^{x q x} ^{ε x}

− ε

¹ B q

r

l

ε

p

ˆu

l

i

l

N

S 2 0 f f f i

l 1

l

ik R i

e ˆ ˆ ˆ ˆ

( ) e ( )

( )

R

₌

⋅ ⋅

∝ ⋅ − ⋅ ⋅

λ ∑

^{q r} B

E q E n 1 k k q n

far-field electric field ⊥ propagation direction

form amplitude tensor of sphere l

• VV – geometry : measuring non-depolarized el. field comp.

• VH – geometry : measuring depolarized el. field comp

→→→→ rotational diffusion

( ) ( ) ^{( )}

{ }

a p s l l p s l l

s l

V b ˆ ˆ ˆ ˆ

( )

= (q) ε − ε + ε − ε

^⊥

−

ε

u u 1 u

B q u homog. sphere with single optical axis

pointing along unit vector uˆ _l

form amplitude of isotrop & uniform sphere

• Optically isotropic sphere : ε = ε 1

→→→→ zero VH – singe scattering

40

Siegert relation: polarized scattering from isotropic & monodisperse colloidal spheres

Homodyne DLS: determines intensity autocorrelation function (IACF)

* *

I s s s s s s

g (q, t) = I ( , 0)I ( , t)

q q

= E ( , 0) E ( , 0) E ( , t) E ( , t)

q q

⋅

q q

* *

I s s s s

g (q, t) = E ( , 0) E ( , 0)

q q

E ( , t) E ( , t)

q q

* *

s s s s

E ( , 0) E ( , t) E ( , 0) E ( , t)

+

q q q q

* *

s s s s

E ( , 0) E ( , t) E ( , 0) E ( , t)

+

q q q q

E

s

t

Central limit theorem: - each cell includes many scatterers

- E_s is complex central Gaussian random variable

> Correlation length

2

I s s s

g (q, t) = I ( )

q

+ 0 + E ( −

q

, 0) E ( , t)

q

translational invariance of homog. system time-indep. mean

in equilibrium isotropic

fluid phase

time average

1 1 2 2 n n 1 1 2 2 n n

1 2 n

i i i i i i

... ,0

e

^{k r⋅}

e

^{k r⋅}

... e

^{k r⋅}

= δ

_{k + + +k k}

e

^{k r⋅}

e

^{k r⋅}

... e

^{k r⋅}

*

E s s

g (q, t) = E ( , 0) E ( , t)

q q

2 1/ 2

E I s

g (q, t) =



g (q, t) − I ( )

q 

E E s

ˆg (q, t) = g (q, t) / I ( )

q →

ˆg (q, 0) 1

_E

=

2

I I s

ˆg (q, t) = g (q, t) / I ( )

q →

ˆg (q, 0)

_I

= I ( )

_s q ²

/ I ( )

_s q ²

[ ]

²

I E

ˆ ˆ

g (q, t) = + 1 g (q, t)

ˆg (q, 0) 1

I

≥

spatial homogeneity

j l j l j l

i i a i 2i

e

^{q r r⋅ + }

= e

^{q r⋅ +a+r+ }

= e

^{q⋅ +r r}

e

^q⋅a

= 0

Siegert relation for stationary, ergodic system ( time average → ensemble average ) el. field autocorrelation function (EFAC)

2 2 2

s s s s

0 ≤



I ( ) I ( )



= I ( ) − I ( )

 q

−

q  q q → in general, only valid that :

t

ˆg (q, t)I

1

2

42

Dynamic correlation functions

j l

N i (t) (0)

,0 j,l 1

1 1

lim ( , 0) ( , t) lim e N

S(q, t)

N N

 

⋅ − 

∞ ∞ =

 

 

= δρ − δρ =



− δ



 



∑

^{q r r q} 

q q

l

N i

,0 l 1

( ) e

^⋅

N

=

δρ

^q

= ∑

^{q r}

− δ

^q

[ ]

S(q, 0)

i

S(q)

= = + ρ 1 ∫ d e

^{r q r⋅}

g(r) 1 −

[

^{1 2}

] ( )

i (t) (0) i

d ,0 1 2

S (q, t) lim (N 1)e

^{⋅ −}

N d e

^⋅

lim V (t) (0 1

∞ ∞

 

 

= − − δ = ρ



δ − + −



 

 

∫

q r r q r

q r r r r

{ }

d i d

S (q , t) = ρ ∫ d e

^{r q r⋅} g (r,t)

− 1 g (r, 0)

d

= g(r)

• Homodyne DLS experiment + Siegert relation :

g (q, t)

E

∝ N P (q) S( q, t)

dynamic structure factor (t > τ_DLS ≈ 1 µsec)

S(q, t) = G(q, t) S (q, t) +

d split in self- and distinct parts

g (r, t):d distinct van Hove real-space function (2d – experiments)

- Self-dynamic scattering function G(q,t) :

[ ]

d 2 2 4

1 1

1

G(q, t) 1 1 q x (t) x (0) O(q )

2

_α= ^{α α α}

= − ∑ − +

[

r1

(t) −

r1

(0) ]

²

/ d

[

1 1

]

²

W(t) 1 (t) (0)

= 2d

r

−

r

{

^2W(t)

}

^O(q4

G(q, t) = exp − q ⋅



1 +

QRQ*DXVVLDQWHUPVRI

)



( )

^{d / 2 2}

s W(t)

r

g (r, t) 4 ex

p

(

W t)

4

−  

= π



−



+

 

  QRQ*DXVVLDQWHUPV - Small-q expansion for isotropic system :

[

1 1

]

i (t) (

s

0) i

G(q, t) lim e

^{⋅ −}

d e

^⋅ g (r, t)

=

∞ ^{q r r}

= ρ ∫

^{r q r g (r, ts )}

^{= li}

_∞

^{m δ − (}

^{r r1}

^{(t) +}

^r1

^{(0 )}

self van Hove function (single-particle conditional pdf)

Mean squared displacement

d = 1, 2 and 3 (dimensionallity)

- Expansion in cumulant form :

Small in fluid phase but:

dynamic heterogeneities In colloidal glasses

44

Example: Quasi-2D dynamic structure factor of charged colloidal spheres

q

σσσσ

( Courtesy: J.L. Arauz-Lara, Univ. of San Luis Potosi )

S(q, t = = 0) S(q)

S (q ,t )

S(q, t → ∞ = ) 0

(LQIOXLGSKDVH

Example: Dynamic structure factor of 3D charge-stabilized dispersion

• Density autocorrelation function S(q,t) decays monotonically in t

• Valid for any regular autocorrelation function at colloidal time scales

(Banchio, Nägele & Bergenholtz, J. Chem. Phys. 113 (2000)

46

Example: Dynamic structure factor of 3D charge-stabilized dispersion

BD : Gaylor et al., J. Phys. A (1980)

Theory : Banchio, Nägele & Bergenholtz, J. Chem. Phys. (2000)

How to measure G(q,t) and W(t) using DLS ?

• Large-q measurement :

S(q, t) = G(q, t) S (q, t) +

d

≈ G(q, t)

S (q, 0)

d

= S(q) 1 − G(q, 0) = 1

• Binary mixture of, w/r to interactions, identical tracer and host spheres

g (q, t)

E

∝ G(q, t)

• DLS: typical resolution of correlation times

1 µ

^VHF

< t <

^VHFWRGD\V

(q > q )

m

q

m

q

1

S(q)

T H

( ρ ρ )

48

Example: Silica tracer spheres in index-matched PMMA sphere solution

(van Megen et al., J. Chem. Phys. 85 (1986))

• Plot - log G(q,t)/q² versus q² and extrapolate to q → 0 (then zero non-Gaussian contr.)

• Strong many-body influence of HI for

φ >

0.2

φ =

4. Theoretical description of colloid dynamics

- Single particle dynamics in very dilute dispersions

- Colloidal time and length scales

- Generalized Smoluchowski diffusion equation for dense systems

50

Single particle dynamics in very dilute dispersions

• Langevin equation for a single colloidal sphere :

F (t)

_α F

= 0

t / B

F o

(t) = e

^{− τ}

v v

B B

t o

0

t / (t u) /

(t) = e

^{− τ}

+ ∫ du e

^{− − τ}

( ) / M

v v F u

8 B 10⁻ sec

τ ≈

2

(t) F

v

2

v0

t

3k TB

M

B 0

Γ = k T ζ

Fluctuation-dissipation relation

( )

B B

0

2 2

2 2

F 0

t / 3 t /

(t) e

^{− τ} M^Γ

1 e

^{− τ}

= +

ζ

−

v v ^{2 B}

eq

3 k T

=

M

v equipartition

theorem

(t τ

B

) (t τ

s

)

0

d

M

d t

^v (t) = −ζ ^v (t) + ^F (t)

Gaussian & Markovian random solvent forces

0

6

0

a

ζ = πη

v

−ζ

0v

F (t) F (t ')

_{α β} F

= Γ δ δ − 2

_αβ

(t t ')

isotropic white noise

B

M /

0

τ = ζ

relaxation time^momentum

51

0 0 eq 0, 0 F

... = ∫ ∫ d ^r d ^v P ( ^{r v} ) ...

- Ensemble of independent particles in equilibrium →→→→ stationarity in time

B 20

B

3/ 2 M

2k T eq 0, 0

1 M

V 2 k T

P ( ) e

−

π

 

=    

v

r v

v

(| t ' t '' |) 1 (t ') (t '')

φ − = d v ⋅ v

velocity autocorrelation function (VAF)

B t / B

v 0

1 k T

3 M

(t) (t) e

^{− τ}

φ = v ⋅ v =

t

v 0

W(t) = ∫ du (t − φ u) (u)

(

^{t / B}

)

^{B 2}

0

0 B

k T t 2M

W(t) D t 1 1 e t

D (t )

τ − τ

 

 

=   − −   →    − τ

B

0 B

D = k T ζ

single-sphere VAF

t 0

0

(t) − = ∫ dt ' (t ')

r r v (

0

)

^{2 t t} v

0 0

1 1

2d 2

W(t) = ^r (t) − ^r = ∫ ∫ dt ' dt ' ' φ (| t ' t '' |) −

Ornstein-Fürth formula for stationary system

s B

( τ t τ )

( τ

B

t)

single-sphere translational Stokes-Einstein relation

52

τ

_D

W(t)

t

D ₀

t ² t

τ B

( ) l

B ²

D t

0

Single-sphere mean squared displacement (MSD)

- Displacement

∆

r = r(t) – r₀ is central Gaussian random variable w/r to

< ... >

( )

^{3/ 2}

^{( )}

²

4 W(t) P( , t) 4 W(t)

⁻

exp  ₋ ^∆ 

∆ = π  

 

r

r

- Pdf is fundamental solution of diffusion-like equation :

( )

P( ∆ r , t = = δ ∆ 0) r

* 2

P( , t) D (t) P( , t) t

∂ ∆ = ∇ ∆

∂ r r

^*

^d

₀ ^{t / B}

D (t) = d t W(t) = D   1 e −

^{− τ}

 

( )

²

3 1/ 2

2

1 x

x

( x ) 2

P( , t) 2 exp

α α

− α

α= ∆

− ∆

∆

σ

 

 

∆ = πσ  

 

 

∏

r

²x

( x )

²

2W(t)

∆ α α

σ = ∆ =

- Single-particle self-dynamic structure factor :

0 2

2 D t

i q W(t) q

G(q, t) = ∫ d( ∆ ^r )e

^{q r⋅∆}

P( ∆ ^r , t) = e

⁻

→ e

⁻

& 0 B

t x D

∆ τ_B ∆ τ

54 Overdamped (positional) Langevin equation for a single particle

- Inertia-effects non-observable for times t >>

τ

_B so that dv/dt

≈

⁰

- Time required to diffuse a distance comparable to a = 100 nm :

2 0

3 I

a

D

10

⁻ sec

τ = ≈ τ

_I

τ

_B

Momenta relaxed long before observable change in particle configuration

v (t)

B_α

= 0

& 0 B

( t ∆ τ

_B

∆ x D τ )

B 0

(t) = 1 (t) ≡ (t)

v ζ F v

Gaussian & Markovian (i.e. local in time) random velocity

B B

v (t) v (t ')

_{α β}

= 2 D

0

δ δ −

_αβ

(t t ')

white additive noise v

(t) 2 D

0

(t)

φ = δ

Overdamped (positional) Langevin equation for a single free particle

t

B B

0 0

(t) − = ∫ dt ' (t ') ≡ ∆ (t)

r r v r ∆ r

^B

(t) = 0

B B

(t) (t) 2 D

0

(t)

∆ r ∆ r = 1 δ

( ) 1

_αβ

= δ

_αβ

( u w )

_αβ

= u w

_{α β}

( )

^{3/ 2 B 2}

B

0

0

( )

P( , t) 4 D t exp 4 D

t

−

 − ∆ 

∆ = π  

 

r

r

B 2

W(t) = ∆ ( r ) / 6 = D t

0

B 2 B

P( , t) D

0

P( , t) t

∂ ∆ = ∇ ∆

∂ r r

B

0 2D t

B i B q

G(q, t) = ∫ d( ∆ ^r )e

^{q r⋅∆}

P( ∆ ^r , t) = e

⁻

-

∆ r

^Bis Gaussian of zero mean and variance

2 D

₀

1 δ (t)

MSD for t >> τ_B

Smoluchowski eq. for free particle

statistically equiv. to positional Langevin eq.

: = ⋅

T

u w u w

dyadic notation

56 delta

distribution

O t

τ

B

τ

_I

O t

v

(t) φ

- Green-Kubo relation for free particle : _{0 v}

0

D dt (t)

=

∞

∫ φ

τ

B

O t

τ

B

G(q, t)

O t

τ

I

τ

B coarse-graining

57 Retarded free particle Langevin equation

v

i

H

= −

i

F

i

F (t τ

s

)

t R

d

d t

(t u

M (t) du ) (u ) ( t)

= − ∫

−∞

γ − +

v v F

Retarded solvent response R

F (t)

_α F

= 0

R R

F B

F (t) F (t ')

_{α β}

= k T δ γ −

_αβ

(| t t ' |)

Gaussian & non-Markovian random force

Friction force on sphere by surrounding solvent

- γ(t) from linearized Navier-Stokes equation of incompressible fluid

s 0 2

( , t)

p( , t) ( , t) t

ρ ∂ = − ∇ + η ∇

∂

u r r u r ∇⋅ u r ( , t) = 0 (t τ =

_c

a / c ≈ 10

⁻¹⁰ sec)

s th 0

inertial force viscous force

Re v

ρ σ 1

= =

η

( )

^T

( , t) = − p( , t) + η ∇ + ∇

0

   

σ r r 1 u u

s

( , t)

( , t) t

ρ ∂ =∇⋅

∂

u r σ r

_σσσσ(r,t) : Stress tensor of incompr. Newtonian fluid

58

0 s

2

t ( , t)

^η

q ( , t)

ρ

∂

∂ ω q = − ω q

t

H

(t) du (t u) (u)

−∞

= − ∫ γ −

F v

ˆ ( , t)⋅ ( ) σ r n r ˆ ( )

n r

[ ]

⁰

[ ]

t ( , t)

^ηs

( , t)

ρ

∂

∂ ∇ × u r = ∆ ∇ × u r

1

i

( , t) = 2 ∫ d e

^{q r⋅}

∇ × ( , t) ω q r u r

∇ → − i q q

2

∆ → −

1 t /

( , t

q = a

⁻

→ ω q ) / ( ω q , 0) = e

^{− τη}

s p s

0 2

B

9

a

2

η ρ

ρ

ρ

η

τ = =     τ

 

H

S

ˆ (t) = ∫ dS ( , t) ⋅ ( )

F σ r n r

fluid force/area on dS at r

Stokes-Boussinesq solution

vorticity (shear-wave) diffusion due to unsteady particle motion

viscous hydrodynamic relaxation time for unsteady particle disturbances

B 2

2

3/ 2 v

k T M

d 1 t

d t 9

(t) W(t)

η

−

π τ

 

φ = ≈  

 

1/ 2

0 0

2 t

W(t) D t 1 D t

η

−

π τ

   

 

≈ −   →

   

 

- Hydrodynamic long-time tails from Boussinesq solution for

t τ

_η

Tail resolvable using DLS for micron-sized spheres

• Non-retarded Langevin-eq. applicable to colloids only for

• Non-retarded Langevin-eq. applicable to aerosols for

t τ ≈ τ

B _η

t τ

B

τ

_η

60

Colloidal time scales

τ

s

τ

_c

τ ≈ τ

_{B η}

^τ

_DLS

τ =

_I

a / D

² ₀

10

⁻13

10

⁻¹⁰

10

⁻⁹

10

⁻⁶

10

⁻³

[ ] ^sec

t

molecular view

solvent compr.

matters: ∇⋅u≠ 0

τ

_c ^{= a/c}

- velocity relax. resolved - unsteady solvent flow:

- retarded HI ∂/∂t u ≠0

quasi-inertia free (pure config.-space picture) :

- Many-particle Smoluchowski Eq.

- Overdamped Langevin Eq.

- Quasi-static creeping flow Eq.

structural relaxation time

B

M /

0

τ = ζ

transl. momentum relaxation time

( ζ = πη

₀

6

₀

a)

( )

2

s 0 s p B

a / 4.5 /

τ = ρ η =

η

ρ ρ τ

viscous relaxation time (vorticity diffusion)

(for aerosols: non-retarded Langevin-eq. for )

τ

η

τ

B

⇒ t ∼ τ

_B

Interacting particles:

a → ξ φ ( ,...)

(interaction/correlation length)