Thermodynamic Principles of Self-assemblyThermodynamic Principles of Self-assembly

계면화학 8 조

최민기 , Liu Di’nan, 최신혜

Chapter 16

16.1 Introduction16.1 Introduction

* Understanding Self-assembly by using statistical thermodynamics

* Association colloids or complex fluids – ‘fluid-like’ micelles, bilayers

* Forces in micelles, bilayers - van der Waals, hydrophobic, hydrogen-bonding,

screened electrostatic interactions

Micelle Inverted micelles

Bilayer Bilayer vesicle

16.2 Fundamental thermodynamic equations of self-assembly

= N + kT log X1= 20 + ½ kT log ½X2 = 3

0 + ⅓ kT log⅓ X3 = . . .

or= N = 0

N + (kT/N) log (XN/N) = constant, N= 1, 2, 3, …,

monomers dimers trimers

N : mean chemical potential of a molecule in an aggregate of aggregation number N

0N : standatd part of the chemical potential (the mean interaction free energy per molecule)

XN : concentration (activity) of molecules in aggregates

Rate of association = k1X1N,

Rate of dissociation = kN(XN/N)

In equilibrium, both rates are same .

k1X1N = kN(XN/N)

∴k1/kN = (XN/N)X -N

Equilibrium constant is also given by,

K= k1/kN= exp[-N(0N- 0

1)/kT]

We can combine two equations to obtain

XN = N{X1 exp[(01- 0

N)/kT]}N

More generally,

XN= N{(XM/M)exp[M(01- 0

N)/kT]}N

Law of Mass Action (Alexander and Johnson, 1950)

16.3 Conditions Necessary for the Formation of Aggregates

XN= NX1N

* Aggregates formation is originated from, different cohesive energies

between the molecules in the ‘aggregate’ and the ‘dispersed states’ (N0 < 1

0 )

∴ X1<1 => XN<<X1

Most of molecules will be in the monomer state

{N

0 decreases progressively

N0 has a minimum at some finite N

* For 10 = 2

0 = 30 = . . . = N

0 ( No cohesive force )

* The exact function of N0 => mean size, polydispersity of aggregates

16.4 Variation of N0 with N for simple structures of different geometries

: RODS, DISCS, and SPHERES

* Geometrical shape of the aggregate determines the way N0 decreases with N

SpheresSpheresDiscsDiscs

RodsRods

NN0 = - (N-1) αkT : total interaction free energy

N0 = - ( 1- 1/N ) αkT = ∞

0 + αkT / N

bond energy α kT

1. One-dimensional aggregates (rods)

* Linear chains of identical molecules or monomer units in equilibrium

with monomers in solution.

‘Bulk’ energy of a molecule in an infinite aggregates

* the number of molecules per disc ∝ πR2

∝ N

* the number of unbonded molecules in the rim) ∝ 2πR

∝ N1/2

2. Two-dimensional aggregates (discs, sheets)

R

* NN0 = - (N–N1/2) αkT : total interaction free energy

N0 = - ( 1- 1/N1/2) αkT = ∞

0 + αkT / N1/2

3. Three-dimensional aggregates (spheres)

* the number of molecules per disc ∝ 4/3πR3

∝ N

* the number of unbonded molecules in the rim) ∝ 4πR2

∝ N2/3

R

* NN0 = - (N–N2/3) αkT : total interaction free energy

N0 = - ( 1- 1/N1/3) αkT = ∞

0 + αkT / N1/3

α : constant dependent on the strength of the intermolecular interactionsp : number dependent on the shape or dimensionality of the aggregates

<General Expression> N

0 =∞0 + αkT /NP

Relation Between Surface energy and intermolecular interactions

Consider the droplets of hydrocarbon in water (sphere-shape)

N = 4πR3/3v v : volume per molecule

The total free energy of sphere = N ∞0 + 4πR2γ

γ: Interfacial energy per unit area Hence, N

0 = ∞0 + 4πR2 γ/N = ∞

0 + 4 π γ (3v/4 π)2/3

= ∞0 + αkT / N 1/3

∴ α = 4πγ(3v/4π)2/3 / kT = 4πr2 γ / kT

Interfacial energy is proportional to intermolecular forces!

16.5 The critical micelle concentration (CMC)

(X1)crit = CMC ≈ exp[-(10 - N

0)/kT]

or

(X1)crit = CMC ≈ e-α

XN = N{X1 exp[(10- N

0) / kT]}N

= N{X1 exp[α(1 – 1/NP)]}N ≈ N[X1eα]N

X1 > X2 > X3 > . . . .for all α∴ X1 ≈ C

‘At what concentration will aggregates form?’

* For low monomer concentrations X1 , X1 exp[(10- N

0) / kT] or X1eα << 1

* Since XN cannot exceed unity, X1 exp[(1

0- N0) / kT < 1

Once X1 approaches exp[ -(10 - N

0)/kT ] or e-α, it cannot increase no further!

16.6 Infinite aggregates (Phase separation)

Nature of aggregates depend on shape

XN = N[X1 eα]Ne- α N1/2 for discs (p=1/2)

XN = N[X1 eα]Ne- α N2/3 for spheres (p=1/3)

Above the CMC (X1eα ≈1)

XN ≈Ne-α N

XN ≈Ne-α N

1/2

2/3

As N increases above certain limit (N>5), XN decreases exponentially.

Then, ‘where do the molecules go above the CMC?’

Infinite size aggregate (N →∞) at the CMC , ‘phase separation’

Such a transition to a separated phase occurs whenever p<1 .

(X1)crit ≈ e-α ≈e- 4πr2 γ / kT

Relation between intermolecular interaction and CMC (solubility)

By measuring CMC (solubility), we could obtain α value.

above which oil will separate out into a bulk oil phase

Ex) Strength of hydrophobic interaction

{Simple hydrocarbon : 3.8 kJ/mol per CH2 increment

Amphiphiles : 1.7~2.8 kJ/mol per CH2 increment

Consider the case where p=1,

XN = N[X1eα]N e-α

: Above CMC, X1eα ~1 → XN N for small N∝

XN ~ 0 for large N

Therefore, distribution is Highly polydisperse.

p<1 : as occurs for simple discs or spheres abrupt phase transition to one infini

tely sized aggregate occurs at the CMC and the concept of a size distribution d

oes not arise

p>1 : Impossible to occur

How polydispersity comes about?

{

16.7 Size Distribution of Self-assembled Structures

C = ∑ XN = ∑ N [X1eα]Ne-α

= e- α[X1eα + 2(X1e α)2 +3(X1e α)3 + …. ]

= X1/(1 - X1eα)2 By using approximation, ∑ NχN = χ/(1- χ)2

Thus,

N=1

∞

X1 = (1+2Ceα) – 1+4Ceα 2Ce2α

X1≈ (1 - 1/ Ceα )e-α ≤ e-α (CMC) for low C

This function peaks at ∂XN/ ∂N = 0, Nmax = M = Ceα

XN = N(1 - 1/ Ceα )Ne-α ≈ N e-N/ Ceα for large N

XN = N[X1eα]N e-α

* The expectation value of N, <N> =∑NXN/∑XN = ∑NXN/C

<N> = 1 + 4Ceα

≈ 1 below the CMC

≈ 2 Ceα = 2M above the CMC

* The density of aggregates above the CMC

XN/N = Const.e-N/M for N > M

16.8 More Complex Amphiphilic Structures

* The value of p is actually not constant for complex molecules (like amphiphiles)

* Complex amphiphilic molecules can assemble into more complex shapes such as vesicles, interconnected rods or three-dimensional ‘periodic structures

{Directionalites of binding forces

Flexible molecule structures N

0

N

Complex molecules

Simple molecules

* { Hydrocarbons: infinite aggregate formation (phase separation)

Amphiphiles: finite aggregate formation (micellization)

N=M

σ = kT / 2MΛ

CMC ≈ exp[- (10 - M

0) / kT

The variation of N0 about M

0 can usually be expressed in the parabolic form:

N0 - M

0 = Λ(ΔN)2 , where ΔN = (N-M)

XM

For the case when N0 has a minimum value at N=M, CMC is given by

M{ }N exp[ -M Λ(ΔN)2 kT]N/M

XN =

Gaussian Distribution of Aggregation number N

Su

rfac

tan

t co

nce

ntr

atio

n X

N

M N

MonomersX1=CMC

Micelles

At the CMC

Below the CMC

Distribution above the CMC

* Attractive / repulsive forces between aggregates → structural phase transitions

<Structural transitions>

1. Strong Repulsive forces (electrostatic, steric or hydration forces)

Phase transition : to get apart within a confined volume of solution,

16.9 Effects of interactions between aggregates : Mesophases and mutilayers

* Interaggregate interactions cannot be ignored at high concentration!

H1

V1

L

Liquid phase + crystal

hexagonal

cubic Ia3d

lamellar

2. Attractive forces

* Between uncharged amphiphilic surfaces

(nonionic, zwitter ionic, for charged headgroups in high salt concentration)

10 + kT log X1 = 0

M + (kT / M)log (XM / M)log (XM/M) = 0M + (kT/M)log(XM/M)

monomer micelles/vesicles liposomes/superagregates

XM/M = {(XM/M)exp[M(M0 - M

0)kT]}M/M

The concentration at which XM = XM is therefore,

(XM)crit = M exp[-M(0M - 0

M)/kT]

M : aggregation number in micelle or vesiclesM : aggregation number in liposomes