1

Theories of Polyelectrolytes Theories of Polyelectrolytes in Solutions in Solutions

Andrey V. DobryninPolymer Program, Institute of Materials Science

& Department of Physics

University of Connecticut

2

OutlineOutline

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

OutlineOutline 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 BiopolymersCharged Polymers and Biopolymers

Poly(styrene sulfonate) Poly(methacrylic acid)

CH-CH2

SO3Na

CH2-C

CH3

COOH

PolyelectrolytesPolyelectrolytes – polymers with positively or negatively charged groups

DNA

5

Charged Polymers and BiopolymersCharged Polymers and BiopolymersPolyampholytes

Polyampholytes - polymers with positively and negatively charged groups

Gelatin Histone

6

Physical Model of Charged Physical Model of Charged MacromoleculesMacromolecules

Bead-spring model

+

-

-

---

+

+

+

-

f – fraction of charged monomers

7

Polyelectrolytes in Dilute SolutionsPolyelectrolytes in Dilute Solutions

8

Intrachain InteractionsIntrachain Interactions

Consider a polyelectrolytechain with the degree of polymerization N , fractionf of charged groups and bond length b.

+

--

---

+

+

+

-r1

xy

z

r2

r3

The potential energy of the polyelectrolyte chain with monomers located at positions r1, r2, r3,…, rN and carrying

charges eq1, eq2, …, eqN is

iU r

Tk

Uqql

bTk

U

B

jish

ji

N

i ij ji

jiBN

iii

B

irr

rrrr

rrr

exp

2

3

1

1

1

212

Elastic energy Electrostatic energy Short-range interactions

Bjerrum length Debye screening length

Tkel BB 2222 4 s

ssBD qclr

9

Short-range InteractionsShort-range Interactions

Lennard-Jones 6-12 potential

612

4)(rr

rU LJLJ

where LJ is the interaction parameter and is the monomer

diameter.

10

Flory’s ApproachFlory’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 chainof size Re

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Flory Theory of Polyelectrolyte ChainFlory Theory of Polyelectrolyte Chain

2/1

2

ln)(

bN

eR

R

fNl

Tk

RF e

e

B

B

eelectr

Consider a polyelectrolyte chain of size Re

The contribution of the intrachain electrostatic interactions is

Nb

RTkRF e

Beconf 2

2

These interactions will try to increasechains size. The elastic contribution tochain free energy is

The total free energy of a chain in Flory approximation is

2/1

2

2

2

ln)(

bN

eR

R

fNl

Nb

R

Tk

RFe

e

Be

B

eFlory

Re

12

Flory Theory of Polyelectrolyte ChainFlory Theory of Polyelectrolyte Chain

0ln2

)(2/12

2

2

bN

R

R

fNl

Nb

R

Tk

RF

Re

e

Be

B

eFlory

e

3/13/223/23/1 )(ln ufeNfbNuR Fe

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.

NbR /Dependence of free energy on

Re/bN1/2

0 5 10 15 20

F/k

BT

0

200

400

600

800

1000

100

500

1000

20002/32 Nuf

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Flory Theory of Polyelectrolyte ChainFlory Theory of Polyelectrolyte Chain

4/12/12/1

2

1)(

NufNbN

fNlB

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

3/23/13/43/2 exp fufuN

For longer chains the chain size is proportional to bN

Example: For polyelectrolyte chain with u=2 and fraction of chargedmonomers f=0.2 the crossover degree of polymerization to a fullystretched chain regime is about 50 Kuhn segments.

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Conformations of a Polyelectrolyte Conformations of a Polyelectrolyte ChainChain

Gaussian Coil

4/12/1 NufN 3/23/13/13/24/12/1 exp fufufNNu

Elongated Conformation

Rod-like Conformation

fNfufu 3/23/13/13/2 exp

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Scaling Model of Polyelectrolyte Scaling Model of Polyelectrolyte ChainChain

The scaling approach to the polyelectrolyte chain conformation is based on the assumption of separation of different length scalesand concept of electrostatic blob.

Electrostatic blob: the conformation of the chain inside a blob is unperturbed by electrostatic interactions

00ee gbD

Relation between blob size and number of monomers

The energy of electrostatic interactions betweenall charged monomers inside a blob is kBT

12/302

0

20

ee

eB gufD

fgl

De0

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Scaling Model of Polyelectrolyte Scaling Model of Polyelectrolyte ChainChain

Solving for the number of monomers and electrostatic blob size

3/1203/220 ,

ufbDufg ee

At the length scales larger than the electrostatic blob size, the electrostaticinteractions lead to elongation of the polyelectrolyte chain into array ofblob.

3/1200

ufbNDg

NR e

e

blobe

De0

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Non-uniform Chain StretchingNon-uniform Chain Stretching

Rez

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 conformationsthat minimizes chain potential energy

- Distribution of the electrostatic potential along the chain.

1

)(

4ln

)()(

2e

22

2e

zD

zR

b

zfDlz eB

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Non-uniform Chain StretchingNon-uniform Chain StretchingMechanical analogy

The trajectory of an object moving in the external potential isanalogous to polymer conformation in strong stretching approximation

Object velocity Chain tension

TimeCurvilinear coordinate

along polymer backbone

dt

tdt

)()(

rv ds

sds

)()(

rt

v(t) v(0)=0

Potential -Potential

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Non-uniform Chain StretchingNon-uniform Chain Stretching

Rez

De(z)

Chain conformation – balance of electrostatic and chain elasticity.

dz

zdfzD

dz

dbe

)()(

2

3 22

3/1

0

220 1

2

4ln)(

ee

eee

DR

zRDzD

Solution forthe blob size

dz

zdf

dn

nzd

b

)()(32

2

2

Strong stretching

approximation dz

d

zD

b

dz

d

zg

zD

dn

d

ee

e

)()(

)( 2

20

Non-Uniform Chain StretchingNon-Uniform Chain Stretching

Monomer Density Distribution Along the End-to-End VectorMonomer Density Distribution Along the End-to-End Vector

3/12202

4/ln)(

)()(

)(

BzRb

zDzDzg

z ee

e

e

Logarithmic correction to linear stretching

Re

0 10 20 30 40 50 60 70 80

0

1

2

3

4

5

6

N=187, f=1

c3

1.5x10-4

1.5x10-5

1.5x10-6

z/

21

Size of a Polyelectrolyte ChainSize of a Polyelectrolyte Chainin Dilute Solutionsin Dilute Solutions

10 100 40010

100

200

c3

5.0x10-3

1.5x10-3

5.0x10-4

1.5x10-4

1.5x10-5

<R

2 e2 >

1/2

N

10 100 40010

100

400

1

c3

5.0x10-3

1.5x10-3

5.0x10-4

1.5x10-4

1.5x10-5

ReD

e[ln(

e R

e/De)]

-1/3

/ 2

N

3/1

02

02/2/

0

ln)(

)(2

0

e

eee

DR

e

e

D

eR

b

DR

zD

dzzgN

ee

Logarithmic correction to linear stretching

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Intra-Chain Correlation FunctionIntra-Chain Correlation Function

2 10 100 20010-6

10-5

10-4

10-3

10-2

10-1

-2

N=187, f=1, c = 1.5x10-5-3

Simulation results eq.16 eq.17

g intr

a(r)c

3

r/

ji

ijrrcN

rg )(1

)(intra

rRR

rR

cr

Ne

e

e

222

Non-uniform chain stretching is not important for intra-chain correlation function.

Scaling theory predictsgintra(r) ~ r -2

2/

2/2

)()(2

1 e

e

R

R

rzzdzcNr

Numerical integrationAnalytical expression

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Polyelectrolyte Chain in Poor SolventPolyelectrolyte Chain in Poor Solvent

pH dependence of the reduced viscosity for poly(methyacrylic acid)

R necL 3

Katchalsky&Eisenberg ‘51

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Tutorial:Tutorial: Collapse of a Polymer ChainThe chain collapse is caused by two-body monomer-monomer attractive interactions.

RThe density inside a globule is stabilized by monomer-monomer repulsive interactions.

/T-1 where,6

36

3

23

2

2

R

Nb

R

Nb

Nb

R

Tk

F

B

Free energy of a polymer chain:

two-body and three-body interactions

1 2 3 4

-10

10

20

NbR /Dependence of free energy on

=0

=-0.05

=-0.1

=-0.2

Tk

F

BN=1000

The chain size in a collapsed (globular) state is

3/1

N

bR

In very poor solvent conditions

1

3/1bNR

25

Tutorial: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 numberof neighbors for each blob inside globule and atthe 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.

22

23/2 R

b

RNF surfsurfsurf

26

Instability of a Charged Liquid DropletInstability 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.

R

Q

Rcrit22

+

27

Charged GlobuleCharged Globule

Rayleigh’s stability condition:electrostatic repulsion is balanced by surface energy

R

22

RR

Nefcrit

For polymeric globule

3/1bNR

Critical charge of polymeric globule:2/1eNNefQ critcrit

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Chain Size vs ChargeChain Size vs Charge

Normalized charge Q/Qcrit

Nor

mal

ized

siz

e R

2 /N

2/3

u=2, LJ=1.5

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Cascade of TransitionsCascade of Transitions

f=0f=0.1

f=0.2

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Necklace GlobuleNecklace Globule

Db

Lnec

lstr mb

efmb

Rayleigh’s stability condition of a bead:electrostatic repulsion between cargedmonomers in a bead is balanced by bead surface energy

22

bb

b DD

efm

Beads are small globules with size 3/1bb bmD

Number of monomers in a bead 2 fmb

31

Necklace GlobuleNecklace Globule

lstr

efmbefmb

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:

2/1bbstr bmbfml

bl

l

efmstr

str

b

2

2 where, fmb

bfNmNlL bstrnec /Necklace length: