Multi-faceted nature of equilibrium self-association phenomena

Karl F. Freed & Jacek Dudowicz

James Franck Inst., U. Chicago

Jack F. Douglas

Polymers Div., NIST

Self-assembly Phenomena

• Occurs in diverse systems:

• Polymerization (chem., vdW, H-bond,…)

• Actin, microtubules, platelets, blood

• Polymer coated colloids

• Gels (electrochm. E storage; asphaltine)

• Dipolar & ionic fluids

• Irreversible: kT << Esticking

• Covalent: kinetically controlled

• Non-covalent: reversible: kT ~ Esticking

Categories of Equilibrium Clustering

Polymerization Universality Classes

1) Linear chains

2) Branched Chains

3) Compact Clusters

+kf

kd

Dynamic Equilibrium

+k2 = kf/kd , k2 = exp (-fp / kBT)

fp = hp - T sp

enthalpy hp, entropy sp of association

Lattice Model of Equilibrium, reversible Self-association:

Flory-Huggins Type Model

• Formulate in terms of free energy all thermodynamic properties

• Include polymer-solvent interaction ( = FH/T)• o

mon = initial monomer concentration• Aim: distinguish between various mechanisms• Competition: assembly vs. phase separation

F/kBT = fFH + fAssoc

Models: Free association (F), activated (A, low, high, med), initiated (I)

Average Chain Length

• Low T L: diverges for Alow

• Saturates for I model• Diverges for T 0 in F

• 1.2 as T 0 for Ahigh

cpc 0 for I & Alow models

cpc = 0 for other models

Literature: only 01/2 scaling

Extent of Polymerization

•

• is fraction of monomers converted into polymers

• = 1 complete polymerization

• Sharp change of for I, Alow, & Aint models

• Gradual change for F model• Very limited polymerization

in Ahigh at low T (where L 1)

Specific heat & multiple critical points

• Sharp transition: I & Alow

• Limiting 2nd order transition

• Very broad for F & Ahigh

• Maximum in Cv Tp

High Tc: monomer/solvent Tc

Other Tc on polymerization lineAppears for sharp transitions

Fit to experiment for G-actin polymerization

G-actin monomer: PDB ID2HMP

Theory (lines) and experiment (points)

Greer et al., JCP 123,194906 (2005).

Treat a single multi-lattice site “bead” in FH model, but include volume changes on assembly.

Recent studies: equilibrium self-assembly

• Hierarchical self-assembly

• Assembly in polymer matrix

• Influence of crowding

• Assembly on surface vs. in bulk

• Mutual A+B (ApBq) (ApBq)n assembly

• Cooperativity in self-assembly

• Entropy-enthalpy compensation

• Monomer structure & “sticky interactions

Hierarchical self-assembly

• Self-assembly is cascade of ordering transitions increasing structural complexity.

• Roundedness different generations coexist at equilibrium

• Can tune properties by varying thermodynamic conditions.

• Self-assembly sharpens with increasing m.

• Sublinear concentration dependence of < M >, any m.

mM1 Mm ,mMm(j) Mm(j+1)

Example for m = 6

Sierpinski gasket

Mutual association

• Hierarchical assembly

• pA + qB ApBq

• nApBq (ApBq)n

• Clustering in lipid membranes, peaked for specific compositions

• Appears in liquid mixtures, polymers that form charge transfer complexes

Mutual association

• composition at various temperatures

allylthiocarbimide-diethylamine and allylthiocarbimide-methylaniline binary mixturesJaeger F. M. Second Report on Plasticity; Nordemann Publ.: New York, 1938; pp 81-82, Chapter II

x10 p x103 p

Shear viscosity vs. at various T

Self-assembly in polymer matrix

•Polymer matrix can affect viscoelastic, optical, glass, etc., properties

•Self-assembly transforms phase boundary:dilute solution blend as Nmatrix increases

•Figures: self-assembly on heating

Crowding & self-assembly: Entropy-enthalpy compensation

•Relative solubility of self-assembly in presence of crowding by polymers•Balance attractive & repulsive interactions (Minton)•Balance entropy vs. enthalpy

Self-assembly in bulk vs. on surface

• Surface adsorption promoted by cooling• Assembly (bulk, surface) promoted by heating

• Control surface activity (∆hp on surface) surface vs. bulk assembly

Lattice Cluster Theory & Monomer Structure

Self-association Phenomena: Summary• Diverse systems:

pico nano micro• Bio materials & technology• Several categories of universal

behavior• Simple Flory-Huggins theory• geometric details secondary• Control of assembly, competition• Extension to “sticky” interactions

and molecular details (LCT)