Polyelectrolyte

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Theories of Polyelectrolytes
in Solutions
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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

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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
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Charged Polymers and Biopolymers
Polyelectrolytes
Polyelectrolytes – polymers with positively or negatively charged groups
DNA
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Charged Polymers and Biopolymers
Polyampholytes
Polyampholytes - polymers with positively and negatively charged groups
Gelatin
Histone
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Physical Model of Charged Macromolecules
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Polyelectrolytes in Dilute Solutions
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Intrachain Interactions
Consider a polyelectrolyte
chain with the degree of
polymerization N , fraction
f of charged groups and bond length b.
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Short-range Interactions
Lennard-Jones 6-12 potential
where eLJ is the interaction parameter and s is the monomer diameter.
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Flory’s Approach
Flory-like calculations of chain properties separate entropic
(conformational) and energetic contributions to chain free energy.
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Flory Theory of Polyelectrolyte Chain
Consider a polyelectrolyte chain of size Re
The contribution of the intrachain electrostatic interactions is
Re
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Flory Theory of Polyelectrolyte Chain
The equilibrium chain size is obtained by minimizing the total chain free energy with respect to chain size Re
Solving this equation for chain size Re
we have
The chain size grows faster than linear with the chain’s degree of polymerizations.
Dependence of free energy on
100
500
1000
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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
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.
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Conformations of a Polyelectrolyte Chain
Gaussian Coil
Elongated Conformation
Rod-like Conformation
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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.
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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
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Non-uniform Chain Stretching
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.
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Non-uniform Chain Stretching
Mechanical analogy
The trajectory of an object moving in the external potential is
analogous to polymer conformation in strong stretching approximation
Object velocity
Chain tension
Time
Curvilinear coordinate
along polymer backbone
v(t)
v(0)=0
Potential
-Potential
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Non-uniform Chain Stretching
Re
z
De(z)
Chain conformation – balance of electrostatic and chain elasticity.
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Non-Uniform Chain Stretching
Monomer Density Distribution Along the End-to-End Vector
Logarithmic correction to linear stretching
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Size of a Polyelectrolyte Chain
in Dilute Solutions
Logarithmic correction to linear stretching
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Intra-Chain Correlation Function
Non-uniform chain stretching is not important for intra-chain correlation function.
Scaling theory predicts
gintra(r) ~ r -2
Numerical integration
Analytical expression
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Polyelectrolyte Chain in Poor Solvent
pH dependence of the reduced viscosity for poly(methyacrylic acid)
Katchalsky&
Eisenberg ‘51
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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
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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.
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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.
+
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Charged Globule
Rayleigh’s stability condition:
electrostatic repulsion is balanced
by surface energy
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Chain Size vs Charge
u=2, e LJ=1.5
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Cascade of Transitions
f=0
f=0.1
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Necklace Globule
Rayleigh’s stability condition of a bead:
electrostatic repulsion between carged
monomers in a bead is balanced
by bead surface energy
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Necklace Globule
The length of a string is determined by the balance of the electrostatic repulsion between neighboring beads and the surface tension of string