Counterions at charged polymers [Elektronische Ressource] / vorgelegt von Ali Naji

Counterions at Charged Polymers
Ali Naji
Munich, 2005Counterions at Charged Polymers
Ali Naji
Dissertation
der Fakult¨at fu¨r Physik
der Ludwig–Maximilians–Universit¨at
Mu¨nchen
vorgelegt von
Ali Naji
geboren am 11. September 1976 in Karaj (Tehran)
Mu¨nchen, August 2005Erstgutachter: Prof. Dr. Roland R. Netz
Zweitgutachter: Prof. Dr. Joachim O. R¨adler
Tag der mu¨ndlichen Pru¨fung: 10. November 2005To my wife–Hoda–,
my parents–Masoumeh and Sohrab–,
my brothers–Majid and Hamid–,
and the memory of
my grandfather, RazzaghContents
Abstract xi
Zusammenfassung xiii
1 Introduction 1
2 Counterion at Charged Objects: General aspects 7
2.1 Length scales in a classical charged system . . . . . . . . . . . . . . . . . . . . 7
2.1.1 From mean-field to strong-coupling regime . . . . . . . . . . . . . . . . 8
2.2 Counterion distribution at a charged surface . . . . . . . . . . . . . . . . . . . 10
2.2.1 Weak-coupling or mean-field regime . . . . . . . . . . . . . . . . . . . 10
2.2.2 Strong-coupling (SC) regime . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 The role of curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Binding-unbinding transition of counterions . . . . . . . . . . . . . . . 13
2.4 Interactions between like-charged surfaces . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Mean-field regime: Repulsion . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Strong-coupling regime: Attraction . . . . . . . . . . . . . . . . . . . . 16
2.4.3 Rouzina-Bloomfield criterion . . . . . . . . . . . . . . . . . . . . . . . 17
3 Counterion-Condensation Transition (CCT) at Charged Cylinders 19
3.1 Cell model for charged rod-like polymers . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Dimensionless description . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 CCT as a generic binding-unbinding process . . . . . . . . . . . . . . . . . . . 23
3.2.1 Onsager instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.2 Beyond the Onsager instability . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Mean-field theory for the CCT . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Non-linear Poisson-Boltzmann (PB) equation . . . . . . . . . . . . . . 25
3.3.2 Onset of the CCT within mean-field theory . . . . . . . . . . . . . . . 27
3.3.3 Critical scaling-invariance: Mean-field exponents . . . . . . . . . . . . 28
3.4 Strong-coupling theory for the CCT . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Monte-Carlo study of the CCT in 3D . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1 The centrifugal sampling method . . . . . . . . . . . . . . . . . . . . . 33
3.5.2 Simulation model and parameters . . . . . . . . . . . . . . . . . . . . 35
3.6 Simulation results in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6.1 Overall behavior in the infinite-system-size limit . . . . . . . . . . . . 35
3.6.2 Critical Manning parameter ξ . . . . . . . . . . . . . . . . . . . . . . 41c
3.6.3 Scale-invariance near the critical point . . . . . . . . . . . . . . . . . . 44viii CONTENTS
3.6.4 Critical exponents: the CCT universality class . . . . . . . . . . . . . 47
3.7 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Counterion-Condensation Transition in Two Dimensions 53
4.1 The 2D model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.1 Rescaled representation . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Simulation results in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 The order parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.2 Energy and heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.3 Condensation singularities in 2D: an analytical approach . . . . . . . . 57
4.2.4 Critical point and the continuum limit . . . . . . . . . . . . . . . . . . 60
4.2.5 The condensed fraction . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 Strong-Coupling Interactions 63
5.1 Strong-coupling theory: General formalism . . . . . . . . . . . . . . . . . . . 64
5.1.1 The strong-coupling free energy . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Two like-charged rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2.1 Threshold of attraction . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2.2 Equilibrium axial distance . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.3 Comparison with numerical simulations . . . . . . . . . . . . . . . . . 72
5.3 Two like-charged spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.1 Attraction threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3.2 Comparison with numerical simulations . . . . . . . . . . . . . . . . . 77
5.4 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 Polyelectrolyte Brushes: Non-linear osmotic regime 83
6.1 Non-linear scaling theory for the osmotic brush . . . . . . . . . . . . . . . . . 85
6.1.1 Comparison with Molecular Dynamics simulations . . . . . . . . . . . 88
6.1.2 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Non-linear mean-field theory for the osmotic brush . . . . . . . . . . . . . . . 91
6.2.1 The cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.2 The electrostatic free energy . . . . . . . . . . . . . . . . . . . . . . . 92
6.2.3 The elastic free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.4 Optimal brush height and its limiting behavior . . . . . . . . . . . . . 95
6.2.5 Comparing mean-field results with simulations . . . . . . . . . . . . . 99
6.3 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7 Charged Polymers in Electric Field 105
7.1 Model and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.1.1 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.2 Rescaled parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.3 Langevin Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 109
7.1.4 Models for charge pattern . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2 Simulations results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2.1 Counterion distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2.2 Hydrodynamic “evaporation”mechanism . . . . . . . . . . . . . . . . . 114
7.2.3 Electrophoresis: Mobility in an external field . . . . . . . . . . . . . . 116Table of Contents ix
7.2.4 Counterion condensation and electrophoretic mobility . . . . . . . . . 120
7.2.5 Self-diffusion at a charge array: An analytical approach . . . . . . . . 122
7.3 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A Field Theory for Macroions in an Ionic Solution 127
A.1 Weak-coupling limit: Mean-field theory . . . . . . . . . . . . . . . . . . . . . 129
A.2 Strong-coupling limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B Notes on the Onsager instability 133
C Mean-Field Theory for Charged Cylinders: Asymptotic results 135
C.1 Limiting behavior of β for large Δ . . . . . . . . . . . . . . . . . . . . . . . . 135
PBC.2 Finite-size scaling for β close to ξ : . . . . . . . . . . . . . . . . . . . . . . . 136c
C.3 The PB cumulative density profile . . . . . . . . . . . . . . . . . . . . . . . . 136
C.4 Asymptotic behavior of S within PB theory . . . . . . . . . . . . . . . . . . 137n
C.5 PB potential, counterion density and free energy . . . . . . . . . . . . . . . . 138
C.6 PB solution in an unbounded system (Δ =∞) . . . . . . . . . . . . . . . . . 139
D Periodic Cell Model: Summation techniques for simulations in 3D 141
D.1 Counterions at charged cylinders: MC simulations . . . . . . . . . . . . . . . 141
E Strong-Coupling Interactions: Asymptotic analysis 145
E.1 Two like-charged rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
E.2 Two like-charged spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
F Free Energy of a Charged Brush 149
F.1 Freely-jointed-chain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
F.2 PB free energy: Asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . 150
Bibliography 153
List of Publications 165
Acknowledgment 167
Curriculum Vitae 169x Table of Contents