Polyelectrolyte Chains At Finite Concentrations Counterion Condensation

Polyelectrolyte Chains at Finite Concentrations Counterion Condensation 1 Polyelectrolyte Chains at Finite Polyelectrolyte Chains at Finite ConcentrationsConcentrations Counterion CondensationCounteri[.]

Polyelectrolyte Chains at Finite Concentrations Counterion Condensation c3 = 1.5 10-4 c3 = 1.5 10-2 N=187, f=1/3, LJ =1.5, u=3 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 counterions Polymer chain 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) At equilibrium the sum of all forces Force balance on element with area xy acting on the element xy is equal to p(x,y+y)x y zero p(x,y)y p(x+x,y)y Projection on x-axis: E(x) p ( x, y ) y  p ( x  x, y ) y  E ( x)eqc( x )xy 0 E(x)eqc(x)xy  p(x,y)x p ( x, y )  E ( x)eqc( x ) 0 x By introducing electrostatic potential  ( x) p(x,y) is a (osmotic) pressure at point (x,y) x  p ( x, y )  ( x)  eqc( x ) 0 x x c( x) p ( x) k BTc ( x)  k BT  eqc ( x) ( x) 0 x we can rewrite the last equation as In ideal solution x E ( x ) Electrochemical potential   ( x) k BT ln cx  eq ( x) Counterion Condensation Two-State Model (Oosawa-Manning ) r0 Counterion in solution are divided into: State 1: Counterion localized inside potential valleys of radius State r0 along polymer backbones; R L 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 Npv (Np number of chains in a system) – State localization volume and another outer region (state 2) with volume V- Npv Partitioning of counterions between these two volumes is determined by equality of electrochemical potentials  el  n1   n2    1 ln   2 ln  N v V  N v k BT p   p   Let us introduce fraction of condensed counterions =n1/(n1+n2) and polymer volume fraction =Npv/V       For cylindrical symmetry  ln    ln 1   1   l fN  r0   21    B ln  1   0 ln   ,where L  R 0 4 l B fN L Counterion Condensation Two-State Model Relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is given by the following equation     1  1   0  ln  ln  1   Dependence of the inverse reduced effective linear charge density on the Oosawa-Manning condensation parameter 0 (1-)0 00  10  10  10  10 2 3 4 5 State     (1-) L State 0 -1 -1 100 101 0 102 Counterion Condensation Two-State Model for Flexible Polyelectrolyte Chains Size of a polyelectrolyte chain depends on fraction of free counterions (1-)f L bN (1   ) / (uf )1/ Thus, the counterion condensation parameter is equal to l B fN u / f 1/ 0   L (1   ) / In this case the relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is written as   ln  1     1/    1    u / f 1/ ln   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