Polyelectrolytes Michael Rubinstein (UNC-CH)

Prog. Polym. Sci. 30, 1049-1118 (2005) Outline

1. Electrostatic interactions, counterion distribution

2. Dilute solutions – solvent quality

3. Semi-dilute solutions – chain conformations What are polyelectrolytes? Polyelectrolytes are polymers with ionizable groups. Common polyelectrolytes: Poly(styrene sulfonate) Poly(methacrylic acid) CH-CH 2 CH3

CH2-C COOH SO3Na Biopolymers – DNA Terminology Polyelectrolytes – chains with charged groups of the same sign (e.g. polystyrene sulfonate, polylysine) + + ______+ + + + _ _ + + + _ polyanion+ polycation Counterions –small charges that dissociate from polyelectrolytes with the sign opposite to that of polyelectrolytes (e.g. Na + for PSS ) Polyelectrolytes can be strong (constant charge) or weak (with pH-dependent charge) depending on the pKa of charged groups. Polyampholytes – chains with charged + + _ _ _ groups of both signs (e.g. proteins).

+ +

In addition to polyelectrolytes (polyions) _ _ +

and counterions solution may contain + _

+ small ions – salt (e.g. Na +, Cl -, Ca 2+ , H+) _ + _ Classes of Charged Systems

+ - - + + + + + ------+ - + + + - - + + + - - - + - - + + + + - + + - + - - - - + - + + - + - + - - - - + + + + - + ionic solution polyelectrolyte solution of (simple salt) solution oppositely charged polyelectrolytes Polyampholytes Ionomers

+ + + + - + + - - - - + + - - + + + + - + - - - + - + + + - + + + - - - - + + - - + + + - - - - + + - - + - - + - - - + random block Basic Concepts

+ q Coulomb law – electrostatic interaction energy r 1 in medium with dielectric constant ε q q U = 1 2 _ ε q2 Interplay of electrostatic energy U r and thermal energy kT is the main theme of the course.

E.g. two elementary charges q1=q 2=e separated by r=7Å in water (dielectric constant ε=80 ) interact with energy U=kT (at room T) e2 Bjerrum length l = is 7Å in water at room T. B εkT For r > l B electrostatics between two elementary charges is weak and thermal energy wins. For r < l B electrostatics wins. In non-polar medium ε is smaller (e.g. ε = 2 ) and electrostatic interactions are much stronger. This prevents dissociation of charges. Most charges stay associated in dipoles and multipoles (e.g. ionomers ). Lecture 1 Small Ions

Simple Salt and Counterions Debye Length

Consider a dissociated 1-1 salt solution with concentration cs

3 _ n ~ c r is the average number of positive _ _ _ + s _

3

charges in volume r with radius r _ _ r _ _ _ If electrostatics is not important –

fluctuations of local charge concentrations are random ( n+ ≈ n -)

Net valence of this volume n+ − n− ~ n+ e2 (n − n )2 e2c r 3 e2c r 2 Electrostatic energy of this volume U ≈ + − ≈ s ≈ s εr εr ε If this energy U ≈ kT concentration fluctuations become correlated

+_ + rD Debye length

2 2 1 e c r 2 ≈ s D ≈ l r c ≈ 1 rD kT B D s l c 1 ε B s r = D π -3 8 lBcs E.g. rD = 1 nm in 0.1M solution and rD = 10 nm in 10 M Fluctuations on Length Scales Larger than Debye Length On length scales larger than Debye length neighboring Debye volumes have opposite charge (are correlated) leading to slower growth of net charge __ + +_ + rD _ _ + +

The energy of charge fluctuations on length scales larger than Debye length saturates at thermal energy kT. Electrostatic Screening

For r> r D electrostatics dominates and fluctuations of Debye volumes are correlated, leading to screening at Debye length.

_ _ + Slightly more negative Debye _ r _ rD + _ + volumes surround a positive _ + + + probe charge, screening its field.

A second probe charge several rD away q q  r  U = 1 2 exp −  “feels” a significantly screened field ε   of the first charge. r  rD 

For r >> r D electrostatic interactions are completely screened. ε For r << r D electrostatic interactions are pure Coulomb U=q 1q2/( r) . Electro-neutrality of screening volume is needed for complete (exponential) screening. -2 rD = 3 nm in 10 M solution -4 and rD = 30 nm in 10 M Counterions

Solution is overall electroneutral.

In order for solutes (polymers, colloids, etc.) to be charged, oppositely charged counterions need to be dissolved in solution.

Electric field around charged objects controls the distribution of counterions.

Distribution of counterions depends on the symmetry of the field and is determined by the balance of electrostatic energy of counterions and the entropic part of their free energy. Counterion Distribution Plane-like charge Entropy of free counterions S ~ ln r Electrostatic potential Energy wins: most r counterions are bound and U ~ r stay near charged surface within Gouy-Chapman length. Line-like charge Electrostatic potential Balance depends on coefficient Onsager-Manning counterion r U ~ ln r condensation at high linear charge density.

Point-like charge Electrostatic potential Entropy wins: at low r concentrations (large r) U ~ 1/r counterions unbind.

Counterions near Charged Plane

_ σ σ -2 σ_ – number of charges per unit area e.g. =1 nm

_ _ _ σ ε

E0 ≈ e / – electric field due to charged plane _ _

≈ σ ε _ _ V = Ez ze / – electrostatic potential near charged plane _ λ Electrostatic attraction of a counterion to the plane U = eV is z on the order of thermal energy kT at Gouy-Chapman length λ. λe2σ/ε ≈ kT λλλ≈ 1/(l σ) for σ = 0.1 nm-2 λ=2.3 nm B c ≈ σ/λ σ λ 0 To screen part of surface charge within ≈ σ222l ≈ λ c λλλ≈ σσσ B counterion concentration near the plane ( z < ) is 0 λλλ222))) 1/(lB λ ≈ Relation between Gouy-Chapman length and c0 1/ lBc0 is the same as between Debye length and c ≈ s rD 1 lBcs λ At high salt concentrations (c s>c 0) surface is screened by salt (r D< ). λ At low salt concentrations (c s ).

PFH:_ Self – Similar Solution σ_

_ _ ≥ λ _ Field is screened by counterions on all length scales

_ _ Screening length at distance z > λ from the surface is _ _ _ ln c z ≈ 1/ l c(z) λ Counterion concentration c(z) ≈ 1/(l z2) B B c0 ≈ ≈ ( λ2 ) λ near the surface c c0 1/ lB z < c 1  λ 2 0 At all distances c = = c   -2 l ()λ + z 2 0  z + λ  ln E B λ λ ln z σ Electric field e eff e z λ E0 = = σ − ()  = E  c z' dz ' E0 ε ε ∫0 + λ E0 -1   z Far from the plane ( z >λ) 2 λ λ ln z distribution of counterions c ≈ 1/(l Bz ) ε σ and electric fieldE ≈ e/( lBz) is independent of the surface charge σ λ Effective charge density “seen” from distance z is eff ≈1/(l B (z+ )) ~ E(z)

Gouy-Chapman Distribution of Counterions

_ _ λλλ 2π2π2πσ <λ) σ ε σ = 1/( lΒΒΒ ) Near the plane ( z E ≈ E0 ≈ e /

_ _

_ For σ=1 nm-2 E =2×10 7 V/cm _ 0

_ _ _ Half of counterions are within G-C layer. _ λ Electric field. Counterion concentration ln c λ c 2 ln E E = E 0  λ  0 + λ c = c   E0 z 0 + λ c0  z  E0 -1 -2

λ λ ln z λ λ ln z Electrostatic potential increases logarithmically e V = Edz = ln ()1+ z / λ ∫ ε lB due to balance of energy eV and entropy k ln zof counterions. Counterion Distribution near the Surface

Assumption – no correlations between counterion positions.

_ r

σ_ Average distance between counterions at concentration c

_ _ 0

_ _ ≈ -1/3 ≈ σ -2/3 -1/3 has to be larger than lB

_ r c 0 lB λ _ _ and smaller than . r > l _ λ B σ- 2 λ < 1/l B z r < σ- σ 2 3 For counterions with valence q this condition is < WC ≈ 1/(l B q )

Therefore multivalent counterions adsorb onto a charged plane σ- σ with > WC , forming strongly correlated “Wigner liquid” .

charged surface multivalent counterions

Wigner lattice Onsager-Manning Counterion Condensation r

r 0 Γ – number of charges per unit length e.g. Γ = 4 nm-1

_

_ ≈ Γ ε

_ E0 e /( r) – electric field due to charged cylinder

_

_ Electrostatic potential around a charged cylinder

_ lB

_ = − r ≈ −()()Γ ε

_ V ∫ E(r )' dr ' e / ln r / r0

r0 _

_ Entropic part of free energy of a counterion localized

_ Γ-1 _ within a cylinder of radius r is ~ kTlnr.

_

_ Electrostatic energy eV is balanced by kTlnr for all r>r _ 0

2

_ Γ ε

at OM = kT /e = 1/l B – one charge per Bjerrum length.

_

_ Γ Γ

_ If electrostatic energy eV is higher than kTlnr if ( > OM )

_ entropy cannot stabilize attraction at any r>r 0. _ Counterions condense onto polyion reducing its charge Γ density to the “universal” value OM . Spherical Regime – Poor Man’s Version Q r eff R Effective charge of the particle Qeff (or polymer as observed from distance r>R ) ε V = Qeff /( R) – potential at r ~ R (work to move a counterion from surface to distance r)

ε eQeff /( R) ≈ kT – up to logarithmic ( ln c) corrections

Qeff ≈ e R/lB –similar to Manning Qeff /(eR)≈1/lB

E.g. for R=35 nm effective charge of a particle is Qeff ≈ 50e

In order to significantly increase particle charge one needs to exponentially dilute particles (or add salt). Two-Zone Model of Dilute Polyelectrolyte Solutions Deshkovski et al ‘01 R + - r0 γ - 0 – dimensionless linear charge - + + density along the macroion - + L - = number of charges per + - + - Bjerrum length l = e 2/( εkT) - + B + - + γ Γ Γ 0 = 0 lB 0 – number of charges per unit length Cylindrically-symmetric zone near polyion Some counterions escape into the outer zone far from polyion. Cylindrical zone has net charge (not electro-neutral). γ R – dimensionless linear charge density of the cylindrical zone at R=L/2 γ Γ R = R lB Diagram of Phases 4.0 Deshkovski et al ‘01 -6 σ-3 3.5 cP = 10 -5 σ-3 Counterions leave cP = 10 = 10 -4 σ-3 3.0 cP cylindrical zone upon γ 2.5 III dilution increasing R γ R 2.0 − γ c(r) ∝ r 2 R I. Polyion charge density γ 1.5 unsaturated condensation 0 is too low to localize 1.0 counterions. ∝ −2 0.5 I II c(r) r II. Polyion charge density saturated condensation γ 0.0 0 is high. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Counterions condense γ weakly 0 − γ to reduce electrostatic ∝ 2 0 charged c(r) r energy. III. Entropy wins at very low concentrations and γ counterions leave polyions (even those with high 0). There are phase transitions between different regimes. I. Weakly Charged Polyions γ Counterions distribution in the cylindrical zone c(r) ~ r -2 0 is dominated by the outer regions ( γ0<1). γ Changes of R only change ln c γ the prefactor of c(r) ~ r -2 0 MD of 16 rod-like polymers γ γ γ σ R

) 1E-5 -3 r σ ) ( ) r

4.0 ( c 3.0 1E-6 γ 2.0 III R -1 1.0 1E-7 II 2 10 70 0.0 I 0.0 1.0 2.0 3.0 4.0 σ γ0 r ( ) Liao et al ‘03 II. Saturated Condensation Counterions are distributed self-similarly throughout the whole cylindrical zone. Onsager-Manning condensation ln c Effective charge density of a polyion together with condensed counterions is constant.

γ γ 0.01 -2 R 1 < R 2 -4 σ-3 γ cp= 10 R 1 1E-3 γ c = 10 -5 σ-3 γ 0=3 p R 2

) -2 ln r -3 1E-4 Two-zone model σ )(

-2 r predicts c(r) ~ r ( c 4.0 1E-5 3.0 γ R 2.0 III 1E-6 1.0 2 10 70 I II 0.0 r ( σ) 0.0 1.0 2.0 3.0 4.0 Liao et al ‘03 III. Unsaturated Condensation γ At R = 1 there is a transition to the unsaturated condensation regime with polyion starving for more counterions. ln c γ γ γ R 1

γ 1E-3 -4 -3 γ R1 σ -2 cp= 10 R 3 γ -5 σ-3 R2 -2 cp= 10 c = 10 -6 σ-3 ln r 1E-4 p

Two-zone model ) γ -3 -2.36 ±0.06 -2 σ R predicts c(r) ~ r 1E-5 ) ( )

4.0 r ( c 3.0 γ =1.5 γ 1E-6 0 R 2.0 III 1.0 I II 2E-7 0.0 2 10 70 0.0 1.0γ 2.0 3.0 4.0 0 r ( σ) Challenge Problem Particle with a crew-cut polyelectrolyte layer Particle R = 70 nm 150 chains 40K PSS R L L = 20 nm – chain length N = 200 monomers Estimate: (each charged)

(i) bare particle charge Qbare ;

(ii) uncondensed charge Qmanning ;

(iii) effective surface charge Qsurf of crew-cut layer of thickness L;

(iv) effective particle charge as seen from r>2R Q r eff R -6 µ -3 Assume salt concentration cs ~ 10 M and particle concentration ~ m Lecture 1 - Things to Remember

Bjerrum length – determines electrostatic interaction strength between monovalent ions 2 εεε lB =e /( kT) ≈ 7 Å Debye length – determines screening of electrostatic interactions due to salt 1/2 rD = 3 Å/[c s] Note that multivalent ions lead to stronger screening -2 π 2 2 rD = 4 lB(q + c++q - c-) Gouy-Chapman length – determines counterion distribution at charged surface λλλ= 1/( 2π2π2πlΒΒΒσσσ) Onsager-Manning condensation – linear charge density is reduced

to one charge per Bjerrum length lB by condensed counterions

General rule for charged objects – effective charge is eR/l B (if R