Thermodynamic Principles of Self-assembly 계면화학 8 조 최민기, Liu Di ’ nan, 최신혜 Chapter 16

16.1 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 X 1 =  ½ kT log ½ X 2 =  ⅓ kT log ⅓ X 3 =... or  =  N =  0 N + (kT/N) log (X N /N) = constant, N= 1, 2, 3, …, monomers dimers trimers  N : mean chemical potential of a molecule in an aggregate of aggregation number N  0 N : standatd part of the chemical potential (the mean interaction free energy per molecule) X N : concentration (activity) of molecules in aggregates

Rate of association = k 1 X 1 N, Rate of dissociation = k N (X N /N) In equilibrium, both rates are same. k 1 X 1 N = k N (X N /N) ∴ k 1 /k N = (X N /N)X -N Equilibrium constant is also given by, K= k 1 /k N = exp[-N(  0 N -  0 1 )/kT] We can combine two equations to obtain X N = N{X 1 exp[(   0 N )/kT]} N More generally, X N = N{(X M /M)exp[M(   0 N )/kT]} N Law of Mass Action (Alexander and Johnson, 1950)

16.3 Conditions Necessary for the Formation of Aggregates X N = NX 1 N * Aggregates formation is originated from, different cohesive energies between the molecules in the ‘aggregate’ and the ‘dispersed states’ (  N 0 <  1 0 ) ∴ X 1 X N <<X 1 Most of molecules will be in the monomer state {  N 0 decreases progressively  N 0 has a minimum at some finite N * For  1 0 =  2 0 =  3 0 =... =  N 0 ( No cohesive force ) * The exact function of  N 0 => mean size, polydispersity of aggregates

16.4 Variation of  N 0 with N for simple structures of different geometries : RODS, DISCS, and SPHERES * Geometrical shape of the aggregate determines the way  N 0 decreases with N Spheres Discs Rods

N  N 0 = - (N-1) αkT : total interaction free energy  N 0 = - ( 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 ∝ πR 2 ∝ N * the number of unbonded molecules in the rim) ∝ 2πR ∝ N 1/2 2. Two-dimensional aggregates (discs, sheets) R * N  N 0 = - (N–N 1/2 ) αkT : total interaction free energy  N 0 = - ( 1- 1/N 1/2 ) αkT =  ∞ 0 + αkT / N 1/2

3. Three-dimensional aggregates (spheres) * the number of molecules per disc ∝ 4/3πR 3 ∝ N * the number of unbonded molecules in the rim) ∝ 4πR 2 ∝ N 2/3 R * N  N 0 = - (N–N 2/3 ) αkT : total interaction free energy  N 0 = - ( 1- 1/N 1/3 ) αkT =  ∞ 0 + αkT / N 1/3 α : constant dependent on the strength of the intermolecular interactions p : number dependent on the shape or dimensionality of the aggregates  N 0 =  ∞ 0 + αkT /N P

Relation Between Surface energy and intermolecular interactions Consider the droplets of hydrocarbon in water (sphere-shape) N = 4πR 3 /3v v : volume per molecule The total free energy of sphere = N  ∞ 0 + 4πR 2 γ γ: Interfacial energy per unit area Hence,  N 0 =  ∞ 0 + 4πR 2 γ/N =  ∞ π γ (3v/4 π) 2/3 =  ∞ 0 + αkT / N 1/3 ∴ α = 4πγ(3v/4π) 2/3 / kT = 4πr 2 γ / kT Interfacial energy is proportional to intermolecular forces !

16.5 The critical micelle concentration (CMC) (X 1 ) crit = CMC ≈ exp[-(   N 0 )/kT] or (X 1 ) crit = CMC ≈ e -α X N = N{X 1 exp[(   N 0 ) / kT]} N = N{X 1 exp[α(1 – 1/N P )]} N ≈ N[X 1 e α ] N X 1 > X 2 > X 3 >....for all α ∴ X 1 ≈ C ‘At what concentration will aggregates form?’ * For low monomer concentrations X 1, X 1 exp[(   N 0 ) / kT] or X 1 e α << 1 * Since X N cannot exceed unity, X 1 exp[(   N 0 ) / kT < 1 Once X 1 approaches exp[ -(   N 0 )/kT ] or e -α, it cannot increase no further!

16.6 Infinite aggregates (Phase separation) Nature of aggregates depend on shape X N = N[X 1 e α ] N e - α N 1/2 for discs (p=1/2) X N = N[X 1 e α ] N e- α N 2/3 for spheres (p=1/3) Above the CMC (X 1 e α ≈1) X N ≈Ne -α N 1/2 2/3 As N increases above certain limit (N>5), X N 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.

(X 1 )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 CH 2 increment Amphiphiles : 1.7~2.8 kJ/mol per CH 2 increment

Consider the case where p=1, X N = N[X 1 e α ] N e -α : Above CMC, X 1 e α ~1 → X N ∝ N for small N X N ~ 0 for large N Therefore, distribution is Highly polydisperse. p<1 : as occurs for simple discs or spheres abrupt phase transition to one infinitely sized aggregate occurs at the CMC and the concept of a size distribution does not arise p>1 : Impossible to occur How polydispersity comes about? { 16.7 Size Distribution of Self-assembled Structures

C = ∑ X N = ∑ N [X 1 e α ] N e -α = e - α [X 1 e α + 2(X 1 e α ) 2 +3(X 1 e α ) 3 + …. ] = X 1 /(1 - X 1 e α ) 2 By using approximation, ∑ Nχ N = χ/(1- χ) 2 Thus, N=1 ∞ X 1 = (1+2Ce α ) – 1+4Ce α 2Ce 2α X 1 ≈ (1 - 1/ Ce α )e -α ≤ e -α (CMC) for low C This function peaks at ∂X N / ∂N = 0, N max = M = Ce α X N = N(1 - 1/ Ce α ) N e -α ≈ N e -N/ Ce α for large N X N = N[X 1 e α ] N e -α

* The expectation value of N, =∑NX N /∑X N = ∑NX N /C = 1 + 4Ce α ≈ 1 below the CMC ≈ 2 Ce α = 2M above the CMC * The density of aggregates above the CMC X N /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 N0 N0 N Complex molecules Simple molecules * { Hydrocarbons: infinite aggregate formation (phase separation) Amphiphiles: finite aggregate formation (micellization) N=M

σ = kT / 2MΛ CMC ≈ exp[- (   M 0 ) / kT The variation of  N 0 about  M 0 can usually be expressed in the parabolic form:  N 0 -  M 0 = Λ(ΔN) 2, where ΔN = (N-M) X M For the case when  N 0 has a minimum value at N=M, CMC is given by M { } N exp[ -M Λ(ΔN) 2 kT] N/M X N = Gaussian Distribution of Aggregation number N Surfactant concentration X N M N Monomers X 1 =CMC Micelles At the CMC Below the CMC Distribution above the CMC

* Attractive / repulsive forces between aggregates → structural phase 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! H1H1 V1V1 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)  kT log X 1 =  0 M + (kT / M)log (X M / M)log (X M /M) =  0 M + (kT/M)log(X M /M) monomer micelles/vesicles liposomes/superagregates X M /M = {(X M /M)exp[M(  M 0 -  M 0 )kT]} M/M The concentration at which X M = X M is therefore, (X M ) crit = M exp[-M(  0 M -  0 M )/kT] M : aggregation number in micelle or vesicles M : aggregation number in liposomes