Dynamics of Polyelectrolyte Solutions

Tutorial: Rouse and Zimm Dynamics Rouse Model: In this model polymer chain is represented by a bead-spring chain Model assumptions: beads have no excluded volume; there are no hydrodynamic interactions between beads. Chain friction coefficient of the chain in Rouse model Chain diffusion coefficient Longest chain relaxation time Terminal modulus Viscosity of solution of Rouse chains

Zimm Model Zimm model takes into account hydrodynamic interactions between monomers inside polymer coil. The hydrodynamic interactions are long-ranged and and decay as 1/r. Thus motion of polymer segments is accompanied motion of the solvent inside polymer coil. Polymer drags all solvent as it moves. R Chain friction coefficient of the chain in Zimm model

Scaling of Zimm Model Chain diffusion coefficient Longest chain relaxation time Terminal modulus Solution viscosity

Semidilute Unentangled Polyelectrolyte Solutions The hydrodynamic interactions between sections of a chain in semidilute solutions are screened at the length scales larger than the solution correlation length x. The dynamics of chain sections at the length scales smaller than the solution correlation length is strongly hydrodynamically coupled – Zimm-like. The relaxation time of section of a chain with gx monomers inside correlation blob Each chain consists of N/gx correlation blobs. The hydrodynamic interactions between blobs are screened and their motion is described by Rouse dynamics x x D D e e

Semidilute Unentangled Polyelectrolyte Solutions The chain self-diffusion coefficient is concentration independent The terminal modulus G of solution of Rouse chains is kBT per chain The viscosity of polyelectrolyte solution in this regime is

Tutorial: Reptation motion In semidilute and concentrated polymer solutions the topological constraints imposed on a given by neighboring chains restrict its motion to tube-like region. entanglements Ne Number of monomers between entanglements Localizing tube around a chain. Tube diameter

Tutorial: Reptation motion Curvilinear diffusion of a chain along the tube In a melt chain diffusion coefficient is Rouse diffusion coefficient The time which requires for the chain to diffusive out of the original tube of length L is the reptation time

Semidilute Entangled Polyelectrolyte Solutions Entanglements are characterized by a tube diameter a (the mesh of temporary entanglement network) The conformation of a polymer strand between entanglements is a random walk of correlation blobs x. x where Ne is the number of monomers between entanglements. The relaxation time of the entangled chain is a Relaxation time of a strand between entanglements Reptation

Semidilute Entangled Polyelectrolyte Solutions The plateau modulus of the entangled solutions is where ne is the density of entanglement strands. The volume per entanglement is equal to . Therefore n-number of entanglement strands inside a3 volume Solution viscosity in the entangled regime Chain self-diffusion coefficient

Summary of Scaling Relations for Polyelectrolyte Solutions     Unentangled Entangled t G h Dself where and

Modulus of Polyelectrolyte Solutions Concentration dependence of the terminal modulus calculated from steady shear relaxation time (filled symbols) and oscillatory shear relaxation time (open symbols) at 250C in ethylene glycol solvent of random copolymer 2-vinyl pyridine and N-methyl-2-vinyl pyridinium chloride (PMVP-Cl) with various charge densities and uncharged neutral parent poly (2-vinyl pyridine) (P2VP) of Mw=364 000. (Provided by R. H. Colby)

Viscosity of Polyelectrolyte Solutions Dependence of the solution specific viscosity on polymer concentration for semidilute salt-free solutions of NaPSS. Filled squares are data of Boris and Colby for M=1.2x106; filled triangles are data of Prini and Lagos for M=3x105; and data of Oostwal for M=398 000 (filled circles) and M=199 000 (inverted filled triangles). from Boris,D.C. & Colby,R.H. Macromolecules 31, 5746-5755 (1998).)

Phase Separation in Polyelectrolyte Solutions

Mean-Field Approach Flory-Huggins lattice model of salt-free polyelectrolyte solutions Counterion contribution small Solution stability region is determined from the following equation Location of a critical point

Mean-Field Approach Phase Diagram N=1000

Mean-Field Approach Flory-Huggins lattice model at high salt concentrations Equation for the spinodal line of the phase diagram Critical point

Mean-Field Approach Comments: The Flory-Huggins lattice consideration of the polyelectrolyte solutions incorrectly describes dilute polyelectrolyte solutions. In the Flory-Huggins approach the monomers are uniformly distributed over the whole volume of the system leading to underestimation of the effect of short-range monomer-monomer interactions and of the intrachain electrostatic interactions. A similar problem appears in the Flory-Huggins theory of phase separation of polymer solutions. This oversimplification leads to the incorrect expression for the low polymer density branch of the phase diagram.

Necklace Model of Phase Separation In the necklace model the phase separation is driven by counterion condensation and bead formation on polymer backbone. (uf2)1/3 (f/u)-1/3 cb3 q-solvent -(uf2)1/3 Bead Controlled Concentrated Solution -(f/u)1/3 String-controlled Dilute necklaces Phase Separation t Phase diagram of polyelectrolytes in poor solvent. Logarithmic scales. from Dobrynin,A.V. & Rubinstein,M. Macromolecules 34, 1964-1972 (2001).

Acknowledgements Collaborations Financial Support J. Jeon (Uconn) V. Panchagnula (Uconn) Prof. M. Rubinstein (UNC-CH) Dr. Q. Liao (Institute of Polymer Science, China) Dr. I. Withers (University of Bristol, England) Prof. S. Obukhov (UF) Prof. R. Colby (PennState) Prof. J.-F Joanny (Institute Curie, France) Financial Support The Petroleum Research Fund The National Science Foundation, University of Connecticut Research Foundation Eastman Kodak Company.