Short- and long-ranged Coulomb interactions in models for ionic solutions and water John D. Weeks Institute for Physical Science and Technology and Department.

Short- and long-ranged Coulomb interactions

in models for ionic solutions and waterJohn D. Weeks

Institute for Physical Science and Technology and Department of Chemistry and Biochemistry

University of Maryland

WaterJocelyn Rodgers

Polar Molecular Liquids

Zhonghan Hu

Take home message

• LMF theory provided a general framework for understanding equilibrium   properties of realistic simulation models with strong Coulomb interactions

• LMF theory is a mapping from a full system with Coulomb interactions in   an external field to a mimic system with truncated Coulomb interactions   in an effective external field

R contains a mean-field average  over long-ranged slowly-varying   parts of Coulomb interactions

• LMF theory generalizes both reaction field and Wolf truncations of Coulomb interactions and standard Poisson-Boltzmann treatments and corrects main errors often seen in both methods

• LMF theory is derived from a controlled and physically suggestive   truncation of the exact YBG hierarchy relating forces to general   density profiles in nonuniform systems; much more accurate than   standard superposition approximation truncations

Coulomb interactions in molecular simulation modelsMolecular models: strong Coulomb interactions at short

distances compete with other strong intermolecular interactions -- LJ cores, covalent bonds …

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• Strong short-ranged Coulomb forces dominate wide class of interesting    phenomena: H-bonds in water, effective attraction between like-charged    walls, ion pairing and chain formation near ionic fluid critical point …

• Want total force on molecular charged site at r and not just Coulomb     force on infinitesimal test charge considered in classical electrostatics

Simplest idea: Truncate Coulomb interactions and hope for the best! Cf. Ion reaction field methods (Hummer), Wolf truncations,     Force-matching truncations (Voth)

Truncation captures local liquid structure in uniform LJ fluidw(r) = u0(r) +

u1(r)

Map from long to short in uniform LJ system

w(r) u0(r)

1

u0(r)

u1(r)

Attractive forces cancel

J. D. Weeks, D. Chandler, and H. C. Andersen,J. Chem. Phys. 54, 5237 (1971).

Soft-sphere u0(r) accurate everywhereHard-sphere ud(r) accurate except     in first peak near contact

LJ= 3.166 AqH  =0.424lOH=1 A

Classical water models use point charges to describe both

short-ranged H-bonds and long-ranged dipolar forcesExtended Simple Point

Charge (SPC/E) Model

H-bonds in SPC/E water result from frustrated ion pairing

Max gOO =2.75AProperly truncated Coulomb interactions can describe local H-bonds well but not long-ranged dipolar forces

Long range of Coulomb forces   causes problems

1/r = v0(r) + v1(r)

Truncation of Coulomb potential using Gaussian charge distribution

v1(r) is electrostatic potential fromGaussian charge distribution with

width

Truncated “short” models replace 1/r by v0(r)

• Screened Coulomb core potential v0(r) = 1/r - v1(r) combines with other strong core interactions.

• Force from v0(r) approaches bare Coulomb force      for r <

Choosingmin ≈ nearest neighbor spacing in short water will capture local ion pairing,

hydrogen bonding etc!

Simulations of bulk short water use v0(r) only:Assumes complete cancellation of long-ranged forces

= 4.5 A

Short water gives very good description of local H-bond network

while ignoring all effects of long-ranged dipolar interactions:

Ideal local model to test classical network picture

Very good description of dipole angle correlations

in bulk water as well!

Site-site radial distribution functions for Acetonitrile

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Truncated model describes ion pairing in uniform SAPM

Effective attraction between like-charged ions at very low density

Details of molecular cores and Coulomb potentials on scale a ≈ d2 importantMain features can be captured by mimic system of N + and N - “ions” with short     ranged properly truncated Coulomb interactions (Strong Coupling Approximation)

J. Weis & D.LevesqueChem. Phys. Lett 336,523 (2001)

Truncation captures local liquid structure in uniform LJ fluidw(r) = u0(r) +

u1(r)

Map from long to short in uniform LJ system

w(r) u0(r)

1

u0(r)

u1(r)

Attractive forces cancel

FPush from effective wall field can give same density profile

FUncanceled attractive forces pull from bulk liquidPossible drying transition

But effective field is needed in nonuniform system

Truncated models need effective Local Molecular Field (LMF)

to account for uncanceled effects of long-ranged forces

(r;[]) = R(r;[R]) In principle exactchoice for R!

Choose R so that:

w(r) = u0(r) + u1(r)

Effective field R in LMF theory is a mean-field average over slowly-varying component v1(r)

r

r´

LMF theory determines R from mean-field average over slowly-varying u1

Controlled use of mean field ideas by proper choice of u1

Theory for Coulomb interactions needs only single LMF equation for restructured electrostatic potential

involving total charge densityconvolution of full chargedensity and Gaussian-smoothed Coulomb potential

convolution of full Coulomb potential and Gaussian-smoothed charge density

LMF restructured potential satisfies Poisson’s equation

but with a Gaussian-smoothed charge density!

Integrate YBG hierarchy

Lee, McCammon, and Rossky, J. Chem. Phys. 80, 4448 (1984)

Water and short water models near hydrophobic wallsSPC/E water (with 2D Ewald) and

short water confined between hydrophobic walls; LJ 9-3 potential

Local H-bond structure  near wall (1 broken H-bond)  generates dipole layerLocal structure should be well captured by short water+− −+

Competition between local H-bond structure and long-ranged dipolar forces important for

electrostatic properties

• Short system accounts only for local     H-bonds• Neglects competing long-ranged effects    of dipole layers out to ∞ in x- and y-    directions

• This is precisely what an effective LMF    can capture!

LMF affects long-wavelength orientations of H-bond network

Gaussian-smoothing of charge density cancels out simulation noise and atomic scale fluctuations to reveal underlying long-ranged electrostatics

A self-consistent VR applies a smooth reorienting torque on water molecules mimicking the action of a dipole layer

Smooth form should permit efficient solutions of LMF equation

LMF theory and classical electrostatics: why does it work so well?

We show effective field R in LMF theory satisfies Poisson’s equation

   but with a Gaussian-smoothed (over scale ) charge density • Classical electrostatics smoothes over molecular scale fluctuations in

     deriving basic equations for polarization field P and other dielectric     properties

=

Purcell: ElectricityAnd Magnetism 1963

• LMF theory provides a general conceptual framework that shows how to carry out such smoothing in general environments and       using realistic molecular models.• may be a fundamental length scale in molecular electrostatics

Relation to standard PB treatments

LMF theory correct two main errors in PB theory:

1. Poisson part: Poisson’s equation averages over full Coulomb       interaction with nonuniform single particle density; OK for       interaction with infinitesimal test charge but not for finite       charges on molecular sites --- Main error in PB treatment

2. Boltzmann part: Density in PB theory given by Boltzmann factor      of effective field. This approximation only accurate at      very low density near ideal gas limit. LMF theory uses simulations      or DFT to determine correct density response to effective field

LMF theory reduces exactly to PB treatment of a dilute system when is set equal to zero

exact

≈0

≈0

Derivation of LMF equation

exact

Choose R so that:

exact

r

r´

Strong coupling approximation (SCA): R≈0 “complete force-cancellation” Ignore all effects of u1 on structure

LMF is theory for mapping;      not resulting structure.

Self-consistent equation Controlled use of mean field         theory

Simulations of bulk short water use SCA:Assumes complete cancellation of long-ranged forces

Short water gives very good description of local H-bond network

while ignoring all effects of long-ranged dipolar interactions

= 4.5 A

Very good description of dipole angle correlations

in bulk water as well

Acetonitrile CH3CN Model

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3.92 D

A. M. Nikitin and A P. Lyubartsev, J. Comp. Chem. Vol 28, 2020-2026, (2007)

Partial Charges:

HC 0.1904 CT -0.5503

YC 0.4917 YN -0.5126

Expt. Density: 0.777 g/cm^3

Expt. Evaporation Heat: 7.98 kcal/mol

Cal. Density:

0.782 g/cm^3 1% higher

Cal. Evaporation Heat:8.21 kcal/mol 3% higher

Work done by Zhonghan HU NSF CRC

Lee, McCammon, and Rossky, J. Chem. Phys. 80, 4448 (1984)

Water and short water models in nonuniform environmentsSPC/E water (with corrected 3D Ewald)

and short water confined between hydrophobic walls; LJ 9-3 potential

Local H-bond structure  near wall (1 broken H-bond)  generates dipole layerLocal structure should be well captured by short water+− −+

Electrostatic propertiesAway from the walls the net force from the dipole layers should be zero and the electrostatic potential should be

constant

Complete failure ofshort water!

Need effective field in nonuniform systems

Competition between local H-bond structure and long-ranged dipolar forces important for

electrostatic properties.

• Short system accounts only for local     H-bonds• Neglects competing long-ranged effects    of dipole layers out to ∞ in x- and y-    directions

• This is precisely what an effective LMF can capture!

r

r´

LMF equation determines R from density-weighted average over slowly-varying u1

Controlled use of mean field ideas by proper choice of u1

• Derived by integrating exact YBG hierarchy equation relating singlet        density gradients to forces and nonuniform pair correlations• Self-consistent equation effectively closes hierarchy• Theory very accurate when u1 is slowly varying over range of        nearest-neighbor pair correlations• LMF is theory for accurate mapping; not resulting structure• Strong-coupling (force cancellation) approximation: ignore all       effects of u1 on structure

• u1 important for thermodynamics and long wavelength structure

Cf electrostatic potential

LMF theory for Coulomb interactions alone greatly simplifies when same is used for all charges

convolution of full Coulomb potential and Gaussiansmoothed charge density

Theory reduces to single self-consistent LMF equation for restructured

electrostatic potentialVR defined with bare charge density

convolution of bare charge density and Gaussian smoothed Coulomb potential

LMF effective potential satisfies Poisson’s equation using Gaussian smoothed charge density!

Define bare charge density and exploit simple form of Coulomb interaction

Gaussian-smoothing of charge density in LMF theory cancels out simulation noise and atomic

scale fluctuations to reveal underlying long-ranged electrostatics

A self-consistent VR applies a smooth reorienting torque on water molecules, mimicking the action of a dipole layer

Smooth form should permit efficient solutions of LMF equation

LMF describes water confined by hydrophilic Pt(111) surfaces

R(x,y,z) ≈ 0(x,y,z) + R1(z)

Water in applied electric field: an even greater challenge

•Sensitive probe of effects of VR

•Wall spacing adjusted to yield B in center

•Similar to study by Yeh & Berkowitz that       lead to corrected 3D Ewald; used as        benchmark for our work here

•Can determine a dielectric constant:

Negative dielectric constant predicted for short water for using standard formula! LMF theory in excellent agreement with full Ewald.

LMF theory corrects very poor results for short water in electric field

Acetonitrile Liquid-vapor film

• Langevin dynamics simulation of a polymer (Np = 100) immersed in an

ionic solution

• 44 “hydrophilic” polymer beads carry negative point charges located

near their surface

• 56 uncharged “hydrophobic” beads interact via the attractive

Lennard-Jones potential

• Molar salt concentrations CM = 0.01-0.1M, giving 7000-10000 salt

ions in cell (GCMC)

• Salt diameter is 1/5 of the polymer bead diameter

• The box sides are aligned with the polymer axes of inertia, RG2 =

(I1 + I2 + I3)/2mNp

Charged polymer simulation model

Work done by Natasha Denesyuk

Polymer and ion density distributions

Hydrophobic Hydrophilic

Counter-ions

Co-ions

• Hydrophobic species form a dense core

• Hydrophilic species stay on the surface

• Counter-ions aggregate near the polymer surface

• Both counter- and co-ions are expelled from the

polymer interior

z

Grand Canonical System Neutrality• The mimic system does NOT have to be strictly neutral (Ewald requires

strict neutrality)

• Truncated Coulomb interactions alone results in a loss of counterions

in the simulation box

• Adding the LMF restores the missing counterions

• SPC/E water and CH3CN can be very accurately described by       short-ranged mimic system in effective external field• Effective field accounts for mean field average of special

             long-ranged slowly varying component of Coulomb interactions• Effective field satisfies Poisson’s equation with       Gaussian-smoothed charge density• Effective field corrects major errors in electrostatic properties of      nonuniform systems from simple truncations of long-ranged forces. • No Ewald sums etc. needed in mimic simulations• LMF method adapted to open-source DL-Poly MD code and in-house Langevin MD polymer simulation code• Further work on ions, water, dipolar fluids near silica surfaces,      charged polymers, etc. in progress • LMF theory provides a unified conceptual framework for wide class        of nonuniform molecular fluids: ions, polymer and water models,…

Conclusions

q

d

Y.-G. Chen and J.D. WeeksPNAS 103, 7560 (2006)

J.C. Rodgers, C. Kaur,Y.-G. Chen and J.D. WeeksPhys. Rev. Lett. 97, 097801(2006)

Ionic solutions: effective attraction between

like-charged walls

Water in applied electric field: an even greater challenge

•Sensitive probe of effects of VR

•Wall spacing adjusted to yield B in center

•Similar to study by Yeh & Berkowitz that       lead to corrected 3D Ewald; used as        benchmark for our work here

•Can determine a dielectric constant:

Short water effectively ignores Epol and has major errors

• Long-ranged forces lead to Epol

• Over-ordering of dipoles in center (bulk) region without effects of Epol

• Incorrect treatment of Epol in short water             amplified by low energy of polarized         local H-bond network

Short water oxygen density profiles incorrect in applied field

Negative dielectric constant predicted for short water for using

standard formula! LMF theory in excellent agreement with full Ewald.

LMF tames applied field systems as well

Self-consistent VR generates weak force on molecules at center

Short water feels nearly full force from bare V

everywhere