Theories of Polyelectrolytes in Solutions Andrey V. Dobrynin Polymer Program, Institute of Materials Science & Department of Physics University of Connecticut

Outline What are polyelectrolytes? Polyelectrolytes in dilute solutions Flory theory and scaling model of a polyelectrolyte chain Polyelectrolyte chain in a poor solvent for polymer backbone Polyelectrolyte chains at finite concentrations and counterion condensation Electrostatic persistence length

Outline Semidilute Polyelectrolyte Solutions Overlap concentration Scaling model of semidilute polyelectrolyte solutions Osmotic pressure and scattering function Dynamics of polyelectrolyte solutions Semidilute polyelectrolyte solutions in a poor solvent for polymer backbone Phase separation in polyelectrolyte solutions Mean-Field approach Microphase separation Necklace model of phase separation

Charged Polymers and Biopolymers Polyelectrolytes Polyelectrolytes – polymers with positively or negatively charged groups Poly(styrene sulfonate) Poly(methacrylic acid) CH-CH2 SO3Na CH2-C CH3 COOH DNA

Charged Polymers and Biopolymers Polyampholytes Polyampholytes - polymers with positively and negatively charged groups Histone Gelatin

Physical Model of Charged Macromolecules Bead-spring model + e - f – fraction of charged monomers

Polyelectrolytes in Dilute Solutions

Intrachain Interactions + - r1 x y z r2 r3 Consider a polyelectrolyte chain with the degree of polymerization N , fraction f of charged groups and bond length b. The potential energy of the polyelectrolyte chain with monomers located at positions r1, r2, r3,…, rN and carrying charges eq1, eq2, …, eqN is Elastic energy Electrostatic energy Short-range interactions Bjerrum length Debye screening length

Short-range Interactions Lennard-Jones 6-12 potential where eLJ is the interaction parameter and s is the monomer diameter.

Flory’s Approach Flory-like calculations of chain properties separate entropic (conformational) and energetic contributions to chain free energy. Elastic deformation of an ideal chain up to size Re Interactions between monomers within a volume occupied by a chain of size Re

Flory Theory of Polyelectrolyte Chain Consider a polyelectrolyte chain of size Re The contribution of the intrachain electrostatic interactions is These interactions will try to increase chains size. The elastic contribution to chain free energy is Re The total free energy of a chain in Flory approximation is

Flory Theory of Polyelectrolyte Chain Dependence of free energy on The equilibrium chain size is obtained by minimizing the total chain free energy with respect to chain size Re 1000 Solving this equation for chain size Re we have 500 100 The chain size grows faster than linear with the chain’s degree of polymerizations.

Flory Theory of Polyelectrolyte Chain Onset of elongation of a polymer chain is at the value of its electrostatic energy of the order of thermal energy kBT Upper bound for chain deformation: ReF should be smaller than the size of fully stretched chain bN For longer chains the chain size is proportional to bN Example: For polyelectrolyte chain with u=2 and fraction of charged monomers f=0.2 the crossover degree of polymerization to a fully stretched chain regime is about 50 Kuhn segments.

Conformations of a Polyelectrolyte Chain Gaussian Coil Elongated Conformation Rod-like Conformation

Scaling Model of Polyelectrolyte Chain The scaling approach to the polyelectrolyte chain conformation is based on the assumption of separation of different length scales and concept of electrostatic blob. Electrostatic blob: the conformation of the chain inside a blob is unperturbed by electrostatic interactions Relation between blob size and number of monomers The energy of electrostatic interactions between all charged monomers inside a blob is kBT De0

Scaling Model of Polyelectrolyte Chain Solving for the number of monomers and electrostatic blob size At the length scales larger than the electrostatic blob size, the electrostatic interactions lead to elongation of the polyelectrolyte chain into array of blob. De0

Non-uniform Chain Stretching z De(z) – size of electrostatic blob with ge(z) monomers (De(z)2~ b2ge(z) ). De(z) In the case of strong deformation of the polymer chain the main contribution to the chain free energy comes form conformations that minimizes chain potential energy - Distribution of the electrostatic potential along the chain.

Non-uniform Chain Stretching Mechanical analogy v(t) v(0)=0 The trajectory of an object moving in the external potential is analogous to polymer conformation in strong stretching approximation Curvilinear coordinate along polymer backbone Time Object velocity Chain tension Potential -Potential

Non-uniform Chain Stretching De(z) Re z Chain conformation – balance of electrostatic and chain elasticity. Strong stretching approximation Solution for the blob size

Non-Uniform Chain Stretching Monomer Density Distribution Along the End-to-End Vector Re Logarithmic correction to linear stretching

Size of a Polyelectrolyte Chain in Dilute Solutions Logarithmic correction to linear stretching

Intra-Chain Correlation Function Numerical integration Analytical expression Non-uniform chain stretching is not important for intra-chain correlation function. Scaling theory predicts gintra(r) ~ r -2

Polyelectrolyte Chain in Poor Solvent Katchalsky& Eisenberg ‘51 pH dependence of the reduced viscosity for poly(methyacrylic acid)

Tutorial: Collapse of a Polymer Chain The chain collapse is caused by two-body monomer-monomer attractive interactions. The density inside a globule is stabilized by monomer-monomer repulsive interactions. Free energy of a polymer chain: two-body and three-body interactions Dependence of free energy on a t=0 t=-0.05 t=-0.1 t=-0.2 N=1000 The chain size in a collapsed (globular) state is In very poor solvent conditions

Tutorial: Collapse of a Polymer Chain The globule has an additional contribution to the free energy due to polymer-solvent interface. Origin of surface energy is the different number of neighbors for each blob inside globule and at the globule surface. The surface energy of a globule can be estimated as the number of monomers at the globule surface times the energy per monomer inside a globule.

Instability of a Charged Liquid Droplet Lord Rayleigh ‘82 + Q < Qcrit Q > Qcrit For the surface charge larger than the critical value charged liquid droplet splits into two smaller droplets.

Charged Globule R Rayleigh’s stability condition: electrostatic repulsion is balanced by surface energy R For polymeric globule Critical charge of polymeric globule:

Chain Size vs Charge Normalized size R2/N2/3 Normalized charge Q/Qcrit u=2, e LJ=1.5

Cascade of Transitions

Necklace Globule lstr mb Db Lnec efmb Rayleigh’s stability condition of a bead: electrostatic repulsion between carged monomers in a bead is balanced by bead surface energy Beads are small globules with size Number of monomers in a bead

Necklace Globule efmb lstr The length of a string is determined by the balance of the electrostatic repulsion between neighboring beads and the surface tension of string String length: Necklace length: