Soft Matter - Theoretical and Industrial Challenges Celebrating the Pioneering Work of Sir Sam Edwards

Transcription

1 Soft Matter - Theoretical and Industrial Challenges Celebrating the Pioneering Work of Sir Sam Edwards One Hundred Years of Electrified Interfaces: The Poisson-Boltzmann theory and some recent developments

2 One Hundred Years of Electrified Interfaces: Poisson-Boltzmann theory and some recent developments T. Markovich, A. Levy & D. Ben-Yaakov (Tel Aviv) H. Orland (Saclay); R. Podgornik (Ljubljana & Amherst) Y. Nakayama (Kyushu Univ); D. Harries (Hebrew Univ) The 100-year old Poisson-Boltzmann Theory Introduction and motivation Finite ion size Hydration shells & dielectric decrement Surface tension beyond Onsager-Samaras

3 Why electrostatics? Biology Cell membrane: DPPC/DPPS[], ion channels DNA/charged polymers Industry/chemistry Stabilization of colloids - DLVO Charged micelles Water based paints, aerosols, emulsions Anionic/cationic detergents lipid head micelle water colloids membrane DNA tail

4 Multi-scale interactions Collapsed vs. Swollen Chains Exploded DNA Collapsed DNA 100 nm Added multivalent counter-ions 2 m virus

5 Electrostatics in BioSoft Matter Point Ions Ionic solutions & Electrolytes d=0 Charged Polymers Polyelectrolytes Monica s talk d=1 DNA Charged Surfaces Membranes, Interfaces d=2 Charged lipids, surfactants Charged Macromolecules d=3 Na, Cl, Ca, Mg, Mn,... Proteins, Polypeptides, Colloids

6 The Grand Problem Statistical Mechanics of Coulombic Systems water 80 Entropy of ions ~ kt Dielectric solvent (water) Mobile ions around macro & charged objects Screening of Elec. Int. Coupling between conformation & ionic degrees of freedom

7 Simeon Denis Poisson Ludwig Boltzmann The Poisson-Boltzmann Theory

8 The Poisson-Boltzmann Theory electrolyte Only electrostatic interactions z water surface Point-like charges Continuum dielectric media Thermodynamical equilibrium -> electrochemistry Mean-field densities of ions & electric potentials

9 Poisson-Boltzmann Equation Boltzmann distribution for mobile ions: n n / 0e ez kt Poisson equation: 2 4 en en Poisson-Boltzmann equation (1:1 salt) 2 8 e n 0 sinh( e / kt ) Boundary conditions: Dirichlet (CP), Newman (CC), Charge regulation,

10 Adaptive Poisson-Boltzmann Solver (APBS) Software for evaluating the electrostatic properties of nanoscale biomolecular systems

11 One flat surface neutral system Gouy 1910; Chapman 1913 Negative surface charge Positive mobile ions ion densities n (0) z water 80 surface electrolyte n ( z) n ( ) 0 z

12 One flat surface neutral system Gouy 1910; Chapman kT ( z) ln( z b) e potential profle b z kt b 2 e nz ( ) 1 ~ Gouy-Chapman length kt 1 2 e ( z b) 2 2 ionic profile b Q( b) ( z) dz / 2 0 Diffusive double-layer

13 Debye-Huckel screening Electric potential 2 2 D D 8 kt ; small ( z) exp( z/ ) Debye-Huckel screening length 2 en D 1/2 0 3Å / n0[m] Salt z D : 3Å 1m l B 2 / kt 7Å Bjerrum Length 1/ 8ln d e B 0

14 Beyond Poisson-Boltzmann: Ion-specific effects finite ionic size solvent mixtures hydration shell ion-surface Int. D. Ben-Yaakov et al, COCIS (2009)

15 The Modified Poisson-Boltzmann finite-ion size Saturation at charged surfaces

16 Modified Poisson-Boltzmann Finite size of ions a S k n log n n log n (1 n n ) log(1 n n ) ion entropy solvent entropy F U TS el Saturation! Borukhov, Orland DA ; Iglic 90s, Bikermann 40s; Eigen 50s

17 Modified Poisson-Boltzmann n ne 0 1 cosh( ) Fermi-Dirac distribution ne 0 small 3 1 a large e / kt rescaled 3 2na 0 close packing MPB 2 2 sinh( ) ( ) D 1 cosh( ) saturation

18 Modified PB o o 2 1 e/ 25 A ; n 0.75M a 8 A large ions 0 - Strong effect for Large size ions

19 X-ray reflectivity at water/air interface saturation, beyond PB electron density amine surfactant 3- very large counterions tungstic acid H3PW12 O4 Rondelez & Cuvillier 98

20 Comparison with experiments MPB Exp. n(0) Area per charge Exp: Rondelez & Cuvillier 98

21 Ion hydration shell Dielectric decrement

22 Hydration shell around ions Ion surrounded by dipolar water molecules Response to the E field is reduced: dielectric decrement E water ( n) 0 Hydration shell

23 Dielectric decrement (n) Salt conc. n [M] Hasted, Ritson and Collie, 1948 ( n) n Salt 0 1 [M ] HCl 10 LiCl 7 NaCl 5.5 KCl 5 RbCl 5 KF 6.5 NaI 8

24 The Dipolar Poisson-Boltzmann (DPB) A mixture of: permanent dipoles (water) bulk conc. Ions, - bulk conc. n 0 p n d

25 The Dipolar PB The free energy of ions and dipoles F r r 8kT 3 2 dr [ E( )] 2n0 cosh [ e ( )] nd sinh[ pe] pe E ( r) Elec. field Mean-Field: the Dioplar PB equation cosh[ pe] sinh[ pe] 4 pe ( pe) io ns dipoles; e E / E 2 2n0esinh e nd p e e 2

26 Hydration layer Solve the Dipolar PB around a point ion l h profile around ion l h hydration layer thickness l h p ~ lb [1 2]Å e lb e 2 / kt 7Å Bjerrum

27 Dielectric decrement (n) Salt conc. n [M] Hasted, Ritson and Collie, 1948 ( n) n Salt 0 1 [M ] HCl 10 LiCl 7 NaCl 5.5 KCl 5 RbCl 5 KF 6.5 NaI 8

28 Dipolar PB: one-loop expansion Field-theory expansion of the free energy Fluctuations in dipole and ion densities One-loop produces a closed formula for dielectric decrement (non-linear) 3 w D D 1 D w tan ndd 3a a a Dielectric const. a D ~ 1/ n Debye length 0 minimal distance ~ few Å Linear Decrement: 1 [9 15]M for n0 1 ( n) n 0 M salt [M] Levy, DA & Orland, PRL 12, JCP 13

29 The Free Energy The dielectric constant is space-dependent n z 0 ( ) n( z) electrostatic energy ion entropy Ben-Yaakov, DA & Podgornik, 2011

30 Saturation layer at interfaces: Ion density Ions & hydration Augmented PB eqn with ( z) n( z) 0 PB ( z) ( z) APB Plateau width: l ~ ~ few Å Ben-Yaakov, DA & Podgornik, 2011

31 Sterically-Modified PB Differential Capacity C / s PB: single min K Cl smpb: steric saturation & double hump (Kornychev 07) ionic specific dielectrophoetic & steric saturation (Nakayama DA 14) KCl 10mM

32 Ion-Surface Interaction Surface tension of solutions Ion-surface

33 Surface Tension of electrolytes: Onsager-Samaras (1934) Image charge - repulsive 1 air neutral air n b nz () water - 80 water Ion depletion from neutral interface water air Onsager-Samaras Dielectric discontinuity & Debye-Huckel theory ktn ln n 0 0 increase tension with sal t conc. n ~ 2 0 D Non-specific

34 Hofmeister Series: Ionic specific effects (1888) Experiments on protein precipitation Many chemical and biological systems Ionic specific most stabilizing most destabilizing strongly hydrated anions weakly hydrated anions citrate >sulfate >phosphate >F >Cl >Br >I > NO >ClO N(CH ) NH >Cs >Rb >K > Na >H >Ca >Mg >Al weakly hydrated cations strongly hydrated cations F. Hofmeister, Arch. Exp. Pathol. Pharmakol. ) 1888(

35 Ion-Specific Surface Tension: Surface Tension Air/water interface Ion Specificity F Cl Br I Hofmeister series OS salt Exp: Matubayasi et al (2001)

36 Self-Consistent Ion-Specific Theory: Surface Tension Surface-ion interaction 1 air - : adhesivity neutral 80 water MF OS ion spec. analytical Mean-Field: Davies isotherm with One-loop expansion: OS & ion specific effects Markovich, DA, Podgornik (JCP 15)

37 Surface Tension of Electrolytes: analytical results a: distance of closest approach Analytical results for small a 1 ( ) / k T ( ) ln a const. ln a D e B e D 2 D Mean-field small Onsager-Samaras Ion-specific Effective surface charge e n0 kt ~ ( / ) small

38 Self-Consistent Ion-Specific Theory: Surface Tension air-water interface Fit with experiments F Cl Br I Strong fluctuation effect OS / kt 0.18 (NaF) 0.02 (NaI) ions at air/water ionic size r r Å

39 Markovich, DA, Podgornik (JCP 15) Ion-Specific Fit at oil/water: Surface Tension Ion specific surface-tension dodecane-water interface KCl Cl Br I KBr oil 2 KI dispersion forces induce attractive / kt 0.29 (KI?); (KBr) ; 0.08 (KCl)

40 Finite ion-size effects Conclusions Going beyond PB theory Saturation at interfaces Interaction of ions and dipoles Hydration shells & dielectric decrement 1 air 80 water Ion specific effects at interfaces: Surface tension of electrolytes More microscopic model to predict adhesively and its specificity Disturbance of water structure O H H surface O H H

41 Finite ion-size effects Conclusions Going beyond PB theory Saturation at interfaces Interaction of ions and dipoles Hydration shells & dielectric decrement Ionic profiles - NaI Na I Ion specific effects at interfaces: Surface tension of electrolytes Ion profiles Markovich, DA, Orland 16

42 100 years of the Poisson-Boltzmann theory: The future More formal understanding of long-range Coulombic liquid systems Narrow the gap between physics and biological complexity of proteins, membranes, DNA, We still do patch work : Try to extract the global picture from the specific details