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 << E sticking Covalent: kinetically controlled Non-covalent: reversible: kT ~ E sticking

Categories of Equilibrium Clustering Polymerization Universality Classes 1) Linear chains 2) Branched Chains 3) Compact Clusters + kfkf kdkd Dynamic Equilibrium + k 2 = k f /k d, k 2 = exp (-  f p / k B T)  f p =  h p - T  s p enthalpy  h p, entropy  s p 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/k B T = f FH + f Assoc Models: Free association (F), activated (A, low, high, med), initiated (I)

Average Chain Length Low T L: diverges for A low Saturates for I model Diverges for T  0 in F  1.2 as T  0 for A high cpc  0 for I & A low models cpc = 0 for other models Literature: only  0 1/2 scaling

Extent of Polymerization   is fraction of monomers converted into polymers  = 1  complete polymerization Sharp change of  for I, A low, & A int models Gradual change for F model Very limited polymerization in A high at low T (where L  1)

Specific heat & multiple critical points Sharp transition: I & A low Limiting 2nd order transition Very broad for F & A high Maximum in C v  T p High T c : monomer/solvent T c Other T c on polymerization line Appears 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, (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  (A p B q )  (A p B q ) 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, any m. mM 1  M m, mM m(j)  M m(j+1) Example for m = 6 Sierpinski gasket

Mutual association Hierarchical assembly pA + qB  A p B q nA p B q  (A p B q ) 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 mixtures Jaeger F. M. Second Report on Plasticity; Nordemann Publ.: New York, 1938; pp , Chapter II x10 px10 3 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 N matrix 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 (∆h p 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)