1. 31 Polyelectrolyte Chains at Finite Concentrations Counterion Condensation N=187, f=1/3,  LJ =1.5, u=3 c  3 = 1.5 10 -4 c  3 = 1.5 10 -2

2. 32 Counterion Condensation The electrostatic attraction between polyelectrolyte chain and counterions in solutions Can results in condensation of counterions on polyelectrolyte chain. The counterion condensation appears to be due to a fine interplay between the electrostatic attraction and the loss of the translational entropy by counterions due to their localization in the vicinity of polymer chain. Polymer chain counterions

3. 33 Tutorial: Electrochemical Potential Equilibrium distribution of charge density in external electric fields Consider distribution of a charged particles with charge eq in nonhomogeneous external electric field E(x) E(x)eqc(x)  x  y Force balance on element with area  x  y At equilibrium the sum of all forces acting on the element  x  y is equal to zero. Projection on x-axis: By introducing electrostatic potential we can rewrite the last equation as In ideal solution Electrochemical potential y x p(x,y)  y p(x+  x,y)  y E(x) p(x,y) is a (osmotic) pressure at point (x,y). p(x,y)  x p(x,y+  y)  x

4. 34 Counterion Condensation Two-State Model ( Oosawa-Manning ) Counterion in solution are divided into: State 1: Counterion localized inside potential valleys of radius r 0 along polymer backbones; State 2: Counterions freely moving outside the region occupied by polyelectrolyte chains. The total solution volume V is divided into two regions. One with volume N p v (N p number of chains in a system) – localization volume and another outer region (state 2) with volume V- N p v. Partitioning of counterions between these two volumes is determined by equality of electrochemical potentials Let us introduce fraction of condensed counterions  =n 1 /(n 1 +n 2 ) and polymer volume fraction  =N p v/V For cylindrical symmetry,where

5. 35 Counterion Condensation Two-State Model Relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is given by the following equation Dependence of the inverse reduced effective linear charge density on the Oosawa-Manning condensation parameter  0 L State 1 State 2 L State 1 State 2 (1-  )  0 00

6. 36 Counterion Condensation Two-State Model for Flexible Polyelectrolyte Chains Comment: The counterion condensation phenomena obtained in the framework of the Two-State model is a specific feature of cylindrical symmetry of the system (even at infinite dilution) for which the electrostatic potential varies with distance r logarithmically. The original Oosawa-Manning treatment of counterion condensation corresponds to the case of semidilute polyelectrolyte solutions when the distance between chains R is smaller than their length L (R<<L). This relation between R and L is assumed even at infinite dilution,. Size of a polyelectrolyte chain depends on fraction of free counterions (1-  )f Thus, the counterion condensation parameter is equal to In this case the relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is written as

7. 37 Counterion Condensation Tutorial: Poisson-Boltzmann Equation Two-zone Model Cylindrical zone Spherical zone At equilibrium the electrochemical potential of counterions is constant Distribution of the electrostatic potential is described by Poisson equation Poisson-Boltzmann Equation

8. 38 Two-Zone Model of Dilute Polyelectrolyte Solutions - L R - r0r0 + + + + + + + + - - - - - - - - + Poisson-Boltzmann equation for the cylindrical cell Boundary condition at the surface of the rod Boundary condition at the surface of the outer cylinder, where Two-parameter solution for counterion density, where

9. 39 Diagram of Phases Counterions leave cylindrical zone upon dilution increasing  R I. Polyion charge density  0 is too low to localize  counterions. II. Polyion charge density  0 is high. Counterions condense to reduce electrostatic energy. III. Entropy wins at very low concentrations and counterions leave polyions (even those with high  0 ). There should be phase transitions between different regimes. weakly charged 00 RR Deshkovski et al ‘01

10. 40 I. Weakly Charged Polyions Counterions distribution in the cylindrical zone c(r) ~ r -2   is dominated by the outer regions (    ). Changes of  R only change the prefactor of c(r) ~ r -2   ln c ln r -2   RR 1 RR 2 RR 3 RR 3 RR 2 RR 1 <<  0 =0.5 MD of 16 rod-like polymers with N=97 and l B =3  r

11. 41 II. Saturated Condensation -2 Counterions are distributed self-similarly throughout the whole cylindrical zone. Effective charge density of a polyion together with condensed counterions is constant.  0 =3 ln c ln r -2 RR 1 RR 2 RR 2 RR 1 < Two-zone model predicts c(r) ~ r -2

12. 42 III. Unsaturated Condensation -2 -2.36±0.06 At  R = 1 there is a transition to the unsaturated condensation regime with polyion starving for more counterions. ln c ln r -2 RR 1 RR 2 -2  R 3 RR 3 RR 2 RR 1 <<1<  0 =1.5 Two-zone model predicts c(r) ~ r -2  R

13. 43 Osmotic Coefficient Osmotic coefficient (ratio of osmotic pressure to that of an ideal gas of counterions).  =  /(k B Tc) Osmotic coefficient increases towards unity with decreasing concentration of dilute solutions. Prediction of 2-zone model

14. 44 Diagram of Phases for Flexible Chains unsaturated condensation saturated condensation weakly charged polyions Two-zone model for flexible chains R=R e /2 ReRe RR For flexible chains both parameters  0 and  R are functions of polymer concentration. 00

15. 45 Osmotic Coefficient of Flexible Chains Osmotic coefficient of flexible chains Prediction of two-zone model for flexible chains

16. 46 Ion-binding and Ion Localization Models of Counterion Condensation These models also separate counterions into two different classes: bound and free counterions free counterions bound counterions This leads to nonlinear equation for the fraction of bound counterions At equilibrium the chemical potentials of counterions in two states are equal to each other  bound =  free  -solvent poor  solvent Continuous change with  Avalanche-like condensation with change of 

17. 47 Dependence of a Chain Size on Polymer Concentration Counterion condensation leads to reduction of a chain size with increasing polymer concentration. Poor solvent  -solvent Dilute Solution Regime Dilute Solution Regime

18. 48 Electrostatic Persistence Length In salt solutions the electrostatic interactions are screened at distances larger than the Debye screening length . How do electrostatic interactions influence chain conformations at length scales larger than the Debye screening length r D ? + + + + + + + bnbn b1b1 Directional memory between vectors b 1 and b n Polyelectrolyte chain in salt solutions behaves as semiflexible polymer with salt dependent persistence length.

19. 49 Electrostatic Persistence Length Odijk-Skolnick-Fixman approach Distance between two monomers separated by n-bonds Excess of the electrostatic energy in the bended conformation In the OSF derivation of the electrostatic persistence length it is assumed that the energy change is k B T per persistence length (n p  ~1) This leads to electrostatic persistence length to be equal to

20. 50 Electrostatic Persistence Length Flory-like Calculations Correlations between bond vectors in freely -rotating chain model The distance between monomers separated by n-bonds Excess of the electrostatic energy in deformed conformation For large n the chain is Gaussian with

21. 51 Electrostatic Persistence Length Flory-like Calculations Entropy change due to chain alignment Free energy of a bond with angular (bending) constraints Electrostatic persistence length b x y z   This free energy has a minimum at Unrestricted rotationRestricted rotation number of states 