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

2. 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

3. 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

4. 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

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

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

7. Polyelectrolytes in Dilute Solutions

8. 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

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

10. 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

11. 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

12. 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.

13. 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.

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

15. 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

16. 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

17. Non-uniform Chain Stretchingz
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.

18. 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

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

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

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

22. 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

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

24. 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

25. 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.

26. 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.

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

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

29. Cascade of Transitions

30. 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

31. 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: