Microstructure of fluid systems

1

Microstructure of fluid systems

Gerhard Nägele

Research Centre Jülich, IFF, Soft Matter Division University of Duesseldorf

Modified version of my lecture at IFF Spring School „Soft Matter“ , Jülich, March 4, 2008 Adapted to Theoretical Soft Matter Course, U of Düsseledorf, summer term 2011

Outline

1. Introduction

- Model systems

2. Pair Distribution Function - Basic properties

- Potential of mean force

- Relation to scattering & thermodynamics 3. Ornstein-Zernike Integral Equations

- Direct correlations - Critical opalescence

- Various closure relations 4. Literature

3

1. Introduction

- Model systems: atomar and colloidal

Model systems: atomar

Lennard - Jones 6 - 12 potential of simple liquids: Ar, Xe, …

r

12 6

r r

u(r) = ε 4          ^σ −       ^σ   

electronic repulsion van der Waals attraction

-ε

σ

2

^1/6

r [nm]

u(r)

σ

5

Colloidal Dispersions

Definition:

1 nm < ∅ < 5 µm _Brownian erratic motion

Examples:

proteins Beispiele: viruses (fd)

Beispiele: inorganic particles:

(plexiglass, ...)

solvent: H₂O, ...

Industrial products: paint / ink dairy products

Theory: particle interactions

⇒

microstructure

⇒

thermodynamics

 Colloids, proteins and most bacteria share: - inertia-free, laminar hydrodynamics - strong Brownian motion

granular media

atoms

1 nm 10 nm 100 nm 1 µ m

radius

bacteria, protozoa Colloidal dispersions (including proteins & viruses)

molecules

(0.1 – 800 µm)

human cell: ∼10 µm

Colloidal length scales

7

Length scales in micro - biology

P. Nelson, Biological Physics: Energy, Information, Life W.H. Freeman & Company, N.Y. (2008)

Model systems: colloidal

Charge-stabilized particles:

-20 0 20

1 2 3

40

4

r/ σ

u(r) ≈ u (r)

el _{for |}

Z | >> 1

B

2

el 2

= B

exp[ a] exp[ r]

k T L 1 a r (r 2a

u (r)

Z ^{κ −κ} )

+ κ

 

 >

 



( )

2

B s

4 L n | Z | 2n

κ = π +

B

2

L = e /( k T) ε

B

≈ 0.7nm

r a

( )

vdW eff

1 6

2a 2a

r 2a r

u (r) A

r r

, ,

−

−

 − ≈

−  

 



u(r)

u (r)

el

u

vdW

(r)

vdW-attraction repulsion

[ ] ^{µ m}

r

9

r / σ

1 2 3

u(r)

r 2a

u(r) 0 r

, ,

< σ =

> σ

=  



∞

Model systems: colloidal

Tuning of strength and range by changing solvent and salt content

Sterically stabilized particles:

• Pairwise additive N-particle potential energy (approximation for CS, not HS)

( )

^{N N}

⁽

^{i j}

⁾

^N

^{( )}

^{i j}

i j i j

U u u r

< <

= ∑ − = ∑

r r r

^r

^N

⁼ { ^r

₁

^{,..., r}

_N

}

„hard spheres“

Phase behavior of colloidal hard - sphere dispersion

Pusey & van Megen, Nature 320, 1986

fcc

cp

π 18

φ =

sphere system

Nv φ= V

Kepler: 1611 Hales: 1998

11

2. Pair Distribution Function

- Basic properties

- Potential of mean force

- Relation to scattering & thermodynamics

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

^r

ⁿ

⁼ { ^r

₁

^{,..., r}

_n

}

( )

^N

N 1 N ßU ßU

Z V, T =

∫

d ...d^{r r} e⁻ =

∫

d^r e⁻

 _N¹

  ^{r N}

  V  

: average number density

   

N 1 2 N 1 2 N

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 = r12 → “∞”}

Isotropic fluid state (no crystal or external field) :

: NVT radial distribution function

r

12

r

2

r

1

Basic properties

13

( )

(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 ^idN

( )

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 )

r

dr

Thermodynamic limit for macroscopic system :

General behavior of g(r) :

( )

_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)

piecewise continuous

2. neighbor shell

( )

g r

1

1 2

r/σ

Correlation length ξ(T)

1. neighbor shell

Radial distribution function g(r) for soft pair potential (LJ, Ar, Xe, charged colloid)

2

N(r, r r) 4 r r

g (r) ^{+ ∆} 1

ρ π ∆ →

=

ideal gas

fluid near-field order

gas liquid crystal (T>0)

T = ε2 / kB T = ε/ k_B T = 0.2 / kε _B 0.026

φ = φ = 0.419 φ = 0.471

r /σ g(r)

uLJ(r)

g(r) ≈ e^−β

Radial distribution function of a 2D Lennard - Jones system (MD simulation)

sphere

Nv φ = V

17

-

^{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

•

g(r) for soft pair potential

•

g(r) for hard-sphere dispersion



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

2. neighbor shell

r/σ

Correlation length ξ(T)

( )

g r

1

1 2

1. neighbor shell

( )

^{N 2}

N

)

12 B 12 B 3 N ß U( V

w r k T ln g(r ) k T ln d ...d e⁻ lnZ 

= − = −  +  

 



∫

^{r r r} 

r

12

r

2

r

1

Potential w(r) of mean force

derivative with respect to position vector of particle 1:

( )

³

^{( )}

1,

N 1 N

1 12 1

3 N 2

ßU ßU

d ...d U e

w r U( )

d ...d e

−

−

−∇ =

∫

−∇ = −∇

∫

^fixed

r r

r

r r

•

Potential of mean force between two particles held fixed at r₁₂

•

Reversible work (free energy) to move two particles from infinity to distance r₁₂ :

( )

^{12 12}

w r = F(r ; N −2) − F( ; N∞ −2)

w(r)→ u(r) i.e. g(r) →e^{− β}u(r) for ρ →0

Hard-sphere colloidal fluid

Effective attraction (“depletion force”)

r r/σ

g(r)

w(r) = - kT ln g(r)

1

0 1

Born-Green integral equation determining g(r) for given u(r)

N

1 1 1j

j 1

U( ) u(r )

>

−∇ ^r = ∑ ∇

assuming pairwise additivity of U

(3)

1 2

1 12 1 12 3 3 1 1

3 12

g ( ,

w(r ) u(r ) d 1 u(r )

g(r )

 , ) 

−∇ = −∇ − ρ  − ∇ 

 

 

∫ ^{r r r r}

Yvon–Born–Green

hierarchy:

2

12 1

(3)

1 2 3

g(r ) g(r ) g(r

3 23

g ( , r r r , ) = ) + O( ρ )

•

Non-linear integro-differential equation relating g(r) to u(r)

probability for particle 1 at r₁

given that 2 at r₂ and 3 at r₃ force on 1 due to 2 particle

(n) (n 1)

g ↔ g

⁻

21

0 46 . φ =

PY

Born- Green MD



BG equation gives poor predictions for dense fluids

22

• Scattering in homodyne mode: λ_L = λ_vac/n_s = O(σ, ρ^-1/3)

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 ^N_{j 1}

{ (

¹

) }

L

j

E ( ) 1 exp i

R ⁼

∝ λ

∑

^{q r}⋅ −^r q

r

_j1

Relation to scattering & thermodynamics

23

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 ei}

[

^{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

Osmotic compressibility equation :

( )

_id^T _B

q 0

T T

lim S q k T

→ p

χ ∂ρ

= χ =  ∂ 

p p

_s

(N → ∞, V → ∞ , ρ = N/V fixed)

 Point scatterers →

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 :

2 nπ −n a /λ <0.1

P(q =0) =1 P(q = ∞ =) 0

I(q) ∝ N S(q)

 Weakly scattering colloidal spheres with a = O(λ) :

25

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

•

q > 4.5/a not accessible by light scattering Formamplitude

Interpretation: diffusive Bragg scattering from particle density waves

4 ns

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

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

r

m

Summary: relation to scattering & thermodynamics

2 / qπ

i (t) r

photomultiplier

k

i

k

f

laser

q ϑ

{ } ^{( ) [ ]}

N

, 1

S(q) 1 cos (0) (0) 1 d cos g(r) 1

N ₌  

=

∑

⋅ ⁱ − ^j  = + ρ

∫

⋅ −

i j

q r r r q r

P(q

I(q) ∝ N ) S(q )

: for single and quasi-elastic scattering

static

structure factor

1 S(q) 1 g(r) 1 FT⁻ −

ρ

 

= +  

4

q ^π sin ^ϑ2 λ

=    

( )

_id^T _B

q 0 T T

lim S q k T

→ p

χ ∂ρ

= χ =  ∂ 

29

Thermodynamic properties

•

Energy equation for internal energy E

•

Pressure equation for particle pressure p

( )

^{N N}

^{( )}

^{i j N 3}

⁽

^{i j i k}

⁾

i j i j k

U u r u , ...

< < <

= ∑ + ∑ +

r r r ^r

dr

2 (3)

B 3

3

2 2! 3!

E

N

= k T +

^ρ

∫ d

^rg(r)

u(r) +

^ρ

∫ ∫ d

^r

d ' g

^r

( , ') u ( , ')

^{r r r r}

+ ...

2 (3) 3

id 3 2! 3 3!

p p

u ( , ')

1 d

(r)

u(r) d d ' g ( , ') .. .

r r

ρ g ρ

⋅ ⋅

β∂

= − β∂ − +

∂ ∂

∫

^r

∫ ∫

^{r r r r r r}

2 0

N

U

2

u(r)

g(r) 4 r dr

∞

ρ

= ∫

π

2-body



Valid for state-independent, i.e., (

ρ

,T) – independent U only (one-comp.)

# particles in shell

Thermodynamic properties

•

(Osmotic) compressibility equation (Kirkwood and Buff, 1951)

( ) [ ]

{ }

B idT

q 0 T,N, ' T

lim S q 1 d g(r) 1 k T

→ µ

∂ρ χ

 

= + ρ ∫ ^r − =   ∂Π   = χ

Π

: osmotic pressure of N large particles

relative to reservoir of small particles & solvent



Valid also for non-pairwise additive & state-dependent U



T id T

0, 0, , χ   χ ≈   ∞

fluid near triple point ideal crystal

fluid at critical point

31

3. Ornstein - Zernike Integral Equations

- Direct correlations

- Critical opalescence

- Various closure relations

12 12 3 13 23

h(r ) = c(r ) + ρ ∫ d ^r c(r ) h(r )

•

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 :

total correlations of 1 and 2

direct correlations

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

+ +

= + ...

2 1

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)

− + 1 O( ) ρ

Direct correlations

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 ^r d ^r c(r ) c(r )c(r ) + O(c )

33

•

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

•

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

ρ T

fluid crystal

2-phase coex.

ρl

ρg ρ_l

critical point

triple point

Critical opalescence

c

c

T

T

1 p

∂ρ ρ ∂

 

χ =   = ∞

 

1

1 c(q 0)

S(q 0) 1 d h(r)

− ρ →

→ = + ρ ∫

^r

= → ∞

c(q 0) d c(r) 1

→ =

∫

^r →ρ

•

System with attractive part in u(r)

•

Critical opalescence near T_c

→ fluid turns cloudy, opaque

→ slow dynamics

remains short-ranged

35

Hexane – Methanol mixture (0.67 mol C

₆

H

₁₄

: 0.44 mol CH

₃

OH)

≈

^o

T 18 C T ≈ 46 C

^o

=

_c

≈

^ο

T T 42 C

hexane–rich phase

demixed 2-phase state mixed fluid state

critical opalescence

http://www.physicsofmatter.com/NotTheBook/CriticalOpal/Frame_Index.html

strong scattering near forward direction

•

^{Near T}_c, assume c(q) expandablein truncated series at q = 0 (Ornstein & Zernike, 1914)

2 4

0 2

c(q;T T )c c c q O(q )

ρ ≈ = − +

2 2 2

0 2 2

1 1 1

1 c c q c q

S(q) = ₋

− + ξ +

 

≈  

 

( )

1/ 2 2 T id

T

c

c 0.62

(T)

^χ

T T

⁻

χ

 

ξ =   ∝ − → ∞

 

( )

₃

[ ]

2

i 1 r /

2 c

1 e

2 r

h(r) d e S(q) 1

⋅ − ξ

π ρ πρ

=

∫

^{r q r} − ≈

2 2

2

1 c

S(q)

1 q

I(q)

=  − 

∝ ζ + 

Correlation length

ξ

(T)

( )

_{u u}

q R ^-1 r R

valid for  and 

Deviations from OZ - mean field close to T_c:

Tc 2

1 q S(q) ∝ _−η

slope:

Intercept:

c2

2

c /2 ξ

η = 0.04 (3D-Ising)

∝ 1 I(q)

→∝q² Tc =150.8 K

Argon

Tc

37 Deviations from OZ - mean field behavior very close to T_c:

Tc 2

1 q

S(q) ∝ _−η η =0.04 (3D-Ising)

∝ 1 I(q)

→∝ q²

Various closure relations

Ornstein - Zernike equation

h(r) = c(r) ρ d 'c(| - '|)h(r') + ∫ ^{r r r}

+ approximate closure relation : + zero overlap condition (HC) :

c(r) =

Functional of

u(r)

&

h(r) g(r < σ = ) 0

Closed integral equation for

g(r) = h(r) 1 −

g(r) S(q) u(r)

p, E,

⇒  



micro-structure thermodynamics

&

…

39

Examples of important closure relations

•

Percus-Yevick approximation (PY):

( )

_ind

1 d ' c(| ' |) g(r ') 1

c(r ) = g(r) −   + ρ ∫ ^{r r} − ^r −   = g(r) − g (r )

g(r) = e

^−βw(r)

[

^{w(r) u}

]

nd

(r)

i

e

g (r) ≈

^{−β −}

c(r) ≈ g(r) 1 e   −

^βu(r)

  = g(r) − y(r)

y(r) = g(r) e

^βu(r)

u(| |) u(r )

y(r) = + ρ 1 ∫ d ^r'   e

^{− β r r'−}

y(| ^r − ^r' |) 1 −     e

^{− β '}

− 1 y(r )   ^'

Analytic solution for c(r) and S(q) in the case of hard spheres Good approximation for short - range pair potentials only

is exact

PY approximation for c(r)

cavity function, continuous at r = σ

integral equation for y(r) or c(r < σ) = - y(r) :

OZ equation

σ

Analytic PY solution for the c(r) of a hard-sphere fluid

1 2

3 3

r

r

c(r ) ( ) 1 ( )

2

^σ

   

< σ = − λ φ   + φ  + λ φ 

  σ  

 

•

See my lecture notes for details:

FFT

c

PY

(r > σ = ) 0

=0.49

φ

0.4 0.3

0.1

0.3

r / σ

1 2 3

u(r)

PY hard - sphere g(r) and S(q) excellent for

φ

^{< 0.4}

φ

0.4

φ = 0.49

0.3

0.1

Analytic PY solution for the S(q) of a hard-sphere fluid

Additional approximate closure relations

•

Hypernetted chain approximation (HNC)

•

Rescaled mean spherical approximation

[ ] [ ]

PY

w(r) u(r)

c (r) = g(r) e −

^{−β −}

≈ g(r) − − β 1 w(r) + β u(r)

[ ]

c

HNC

(r) = − β u(r) ln g(r) − + g(r) 1 −

RMSA eff

c (r) = − β u(r) , r > σ > σ ^{g(r = σ}

_eff⁺

^{) = 0}

Good for soft, long-ranged pair potential such as Yukawa & Coulomb

Analytic solution for screened Coulomb potential



Accuracy of OZ integral schemes depends on u(r) and dimensionality

43

Thermodynamic inconsistency of integral equations

•

PY, HNC, RMSA etc. approximate the exact g(r):



Hybrid methods partially reinstall thermodynamic consistency Virial Compr. Energy

p ≠ p ≠ p

Example: hard spheres

id

p - 1 p

φ

Rogers-Young (RY) scheme: mixing of PY and HNC

[ ]

( )

RY

h(r) c(r)

u(r) f (r)

1

1

c (r) e e 1

f (r)

−β



+ −



= ⋅   −  

Compr Virial

T T

χ = χ

determines α

, r r

: 0 : ,

α α

→ ∞

→

HNC PY

f (r) = − 1 e

^−αr

Screened Coulomb potential:

Hard spheres near freezing:

45

• 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

Summary: Various closure relations

4. Literature

1. G. Nägele. Theories of Fluid Microstructures, appeared in Soft Matter: From Synthetic to Biological Materials, 39th IFF Spring School 2008, FZ Jülich Verlag (online verfügbar)

2. J.-P. Hansen and I.R. McDonald, Theory of Simple Liquids, 3rd Edition, Elsevier Academic Press London (2006)