Calculating water’s anomalous properties from first principles: Mechanisms of ion transport in the bulk and at interfaces

Mark E. Tuckerman

Dept. of Chemistry

and Courant Institute of Mathematical Sciences

New York University, 100 Washington Sq. East

New York, NY 10003

Image: news.softpedia.com

1808: “We are perhaps not far removed from the time when we shall be able to submit the bulk of chemical phenomena to calculation.”

Joseph Louis Gay-Lussac (1778-1850)

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact solution of these laws leads to equations much to complicated to be soluble.”

Paul Dirac on Quantum Mechanics (1929).

BG/L@RPI

Why study water?

Most important liquid on EarthOne of the most mysterious substances known

“Science Journal: The structure of water isn’t certain after all” -- from the Wall Street Journal March 10, 2006

Biology Atmospheric Chemistry

Image source: www.cbs.cnrs.fr

Energy Technology

From Petersen and Voth, JPCB 110 (2006)

Wernet, et al. Science (2004)

Some of water’s anomalous properties

Density maximum at 4 oC

Many stable crystalline phases

High surface tension

Anomalously high transport of protons (H+) and hydroxide (OH-) ions

PEM vs. AAEM fuel cells(AAEM=Alkali-anion exchange membrane)

3 2 2

2 2

3 2 2 2

3Cathode: O 6e 6

Anode: CH OH H O 6

3Over

H 6e C

all: CH OH O 2H

O

H 3

C2

H

O O

O2

2

3

3 2 2 2

2

2 2A

3Cathode: O 3H O 6e 6OH

3Ove

node: CH OH 6OH CO 5

rall: CH

H

OH O 2H O CO

O

2

6e

2

From Varcoe and Slade,Fuel Cells 5, 198 (2005)

1806:

Structures of the excess proton in water

H9O4+

H5O2+

H3O+

+ + +

Grotthuss Mechanism (1806)

Vehicle Mechanism

Chemistry in the “Virtual Laboratory”

On the “shelf”:

• Nuclei of the chemical elements• Unlimited supply of electrons

Instrumentation:

Fundamental laws of physics:

Nuclei: Newton’s second law mF a

Electrons: Schrödinger equation H E

ˆ ˆ ˆ ˆ ˆ ˆN e ee NN eNH T T V V V

Nuclei

Electrons

Start with nuclei Compute F

Propagate nuclei ashort time Δt with F

Add electrons

Add electrons

The Algorithm

NucleiElectrons

Ab initio molecular dynamics (AIMD)

Kohn-Sham density functional theory:

2

2

1 1 ( ) ( )[{ },{ }] [ ] [ ,{ }] ( )

2 2 '

( ) ( )

i i xc ext nni

i i j iji

n nE d d E n E n E

n

r rR r r R R

r r

r r

Nuclear evolution2

02

II

I

EdM

dt

R

R0

{ }({ }) min [{ },{ }]E E

R R

Feynman path integrals

. ... . . .......

1

2

3

P-1

P

MET, et al. JCP 99, 2796 (1993); Marx and Parrinello, JCP 104, 4077 (1996); MET, et al. JCP 104, 5579 (1996)

Feynman path integrals

lim PP

Z

Near perfect parallel scaling with increasing P

Basis Sets

Plane-waves (momentum eigenfunctions):

2

, cut

1 2 1( )

2i

i iC e ELV

g rg

g

nr g g

Discrete-variable representations [Light, et al. JCP 82, 1400 (1982)]: Begin with a set of N square-integrable orthonormal functions φi(x)

*

1

( ) ( ) ( )N

i i l i ll

u x a x x

On an appropriately chosen quadrature grid {x1,…,xN}

( ) iji j

i

u xa

(position eigenfunctions!). Expand orbitals as:

( ) ( ) ( ) ( )ii lmn l m n

lmn

C u x u y u z r

Y. Liu, D. Yarne and MET, PRB 68, 125110 (2003); H. –S. Lee and MET, JPCA 110, 5549 (2006)

Basis set size determined by # grid points. Core electrons replaced by atomic pseudopotentials

Radial distribution functions for BLYP Water

DVR

Neutron

X-ray

H. –S. Lee and MET, JPCA 110, 549 (2006)H. –S. Lee and MET JCP 125, 154507 (2006).H. –S. Lee and MET JCP 126, 164501 (2007).Neutron: Soper, et. al. JCP 106, 247 (1997)X-ray: Hura, et. al. Chem. Phys. 113, 9140 (2000)

Grid = 753, t =60 ps

Ensemble: NVT, 300 K, μ = 500 au

r(Å) 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

2 2.5 3 3.5 4 4.5 5 5.5 6

DZVPDZVP+BSSE-BLYPSCP-BLYP

gO

O(R

)

R [Å]

When basis sets are too small!from C. J. Mundy (2008)

The Grotthuss mechanism in water

MET, et al,JPC, 99, 5749 (1995); JCP 103, 150 (1995)D. Marx, MET, J. Hutter, M. Parrinello, Nature 397, 601 (1999).N. Agmon, Chem. Phys. Lett. 244, 456 (1995)T. J. F. Day, et al. J. Am. Chem. Soc. 122, 12027 (2000)

Solvent coordinate view:

P. M. Kiefer, J. T. HynesJ. Phys. Chem. A 108, 11793 (2004)

The Grotthuss mechanism in water

Second solvation shell H-bond breaking followedby formation of intermediate Zundel complex:

P

Presolvation Concept:Proton-receiving species must be“pre-solvated” like the species into which it will be transformed in theproton-transfer reaction.

MET, et al ,Nature 417, 925 (2002)

The Grotthuss mechanism in water

Computed transfer time τ = 1.5 ps

NMR: 1.3 ps

Transfer of proton resulting in “diffusion’’ ofsolvation structure:

A. Chandra, MET, D. Marx Phys. Rev. Lett. 99, 145901 (2007)

Quantum delocalization of structural defectD. Marx, MET, J. Hutter and M. Parrinello Nature 397, 601 (1999)

Ultrafast pump-probe experimentsWoutersen and Bakker, Phys. Rev. Lett. 96, 138305 (2006)

Eigen/Zundel exchange time ≈ 100 fs

1

1

2

A B

B C

k

k

k

A Chemical Master Equation Theory of PT kinetics

O*

O*

O* O*

1 1

2 1 1

[A][A] [B]

[B][B] [A]

dk k

dtd

k k kdt

A. Chandra, MET, D. Marx Phys. Rev. Lett. 99, 145901 (2007)

( ) 1h t

( ) 0h t

( ) 0h t

( ) 1h t

[A]( ) (0) ( )

[B]( ) (0)[1 ( )] ( )

t h h t

t h h t g t

Population correlation functions:

Rate equations:

( ) 1 g t in all configurations above

Chemical Master Equation Theory

( ) (0) ( )cC t h H t

Exchange time: exch 0 ( )cdt C t

1 1

slow fast

/ 2 / 21 1

/ /slow fast

2

1 1 1 2 1

1[A]( )

2

4

k K t k K t

t t

t k K e k K e

a e a e

k K k k K k k

O*

H

H

H O O*

H

H

H O

t = 0 t

fast

exch

50 fs (Bakker: 100 fs)

1.52 ps (NMR: 1.3 ps)

Liquid/vapor interface of acidic solutions

“acceptor only”

hydrogen bonded

dangling

Mucha, et al. JPCB 109, 7617 (2005) Baldelli, et al. CPL 302, 157 (1999)

Tian, et al. JACS 130, 13033 (2008)

Simulations of an HCl interface (96 waters + 1 HCl)

Petersen, et al. JPCB 108, 14804 (2004)

H. S. Lee and MET JPCA (submitted)

“Proton hole” or mirror image mechanism of hydroxide mobility

H. Daneel, Z. Elektrochem. 16, 249 (1905)E. Hückel, Z. Elektrochem. 34, 546 (1928)N. Agmon, Chem. Phys. Lett. 319, 247 (2000); Asthagiri, et al. PNAS (2004)M. L. Huggins, J. Phys. Chem. 40, 723 (1936).

OH-

H+

Spectra of 14 M KOH

IR

Raman

Librovich and Maiorov, Russian J. Phys. Chem. 56, 624 (1982)

Identified in neutron scattering of concentrated NaOH and KOH solutions: A. K. Soper and coworkers, JCP 120, 10154 (2004); JCP122, 194509 (2005).Also in other CPMD studies: B. Chen, et al. JPCB 106, 8009 (2002); JACS 124, 8534 (2002).And in X-ray absorption spectroscopy: C. D. Cappa, et al. J. Phys. Chem. A 111, 4776 (2007)

Weak H-bond donated by hydroxide also identified in neutron scattering of concentrated NaOH and KOH solutions: A. K. Soper and coworkers, JCP 120, 10154 (2004); JCP122, 194509 (2005).M. Smiechowski and J. Stangret, JPCA 111, 2889 (2007).T. Megyes, et al. JCP 128, 044501 (2008).B. Winter, et al. Nature (2008)

Hydronium:

Water:

Hydroxide:

MET, et al.Nature, 417 (2002)

Follows “presolvation” picture:

Proton-receiving species must be coordinated likethe species into which itwill be transformed beforethe proton can transfer.

Comparing IR spectra

Expt.: Bertie, et al. J. Phys. Chem. 93, 2210 (1989) (ν >700 cm-1) Zelsmann, J. Mol. Spect. 350, 95 (1995). (ν < 600 cm-1)

Z. Zhu and MET, J. Phys. Chem. B 106, 8009 (2002)

Expt.: Librovich and Maiorov, Russian J. Phys. Chem. 56, 624 (1982)

Pure water KOH solution

O

D*D

D’

O*

Expt (KOH): Librovich and Maiorov, Russian J. Phys. Chem. 56, 624 (1982)

Acknowledgments

• NSF• Alexander von Humboldt Foundation• Camille and Henry Dreyfus Foundation• ACS PRF

Postdocs

• Yi Liu (Merrill-Lynch)• Hee-Seung Lee (UNC, Wilmington)• Dawn A. Yarne (Goldman-Sachs)• Radu Iftimie (U. de Montréal)• Anatole von Lilienfeld (Sandia)• Robin L. Hayes

Funding

Students

• Yi Liu (Merrill-Lynch)• Tim Berkelbach• Zhongwei Zhu (Goldman-Sachs)• Joseph A. Morrone (Princeton)• Lula Rosso (Imperial College, London)• Peter Minary (Stanford University)• Rachel Chasin• David Krisiloff

External

• Dominik Marx (Ruhr-Universität Bochum)• Amalendu Chandra (IIT Kampur)•Alan Soper (Rutherford Appleton Lab)• Teresa Head-Gordon (UCB, LBL)• Feng Wang (BU)• Chris Mundy (PNNL)• Doug Tobias (UCI)