Modeling Proton Modeling Proton Solvation in Water: Is This Solvation in Water: Is This

Easier Than Electron Easier Than Electron Solvation?Solvation?

Feng WangDepartment of Chemistry

Boston University

Greg A. VothCenter for Biophysical modeling and

Simulation

University of Utah

• Proton Chemistry (Why hydrated Proton Chemistry (Why hydrated proton?)proton?)

• Multi State-Empirical Valence Bond (MS-Multi State-Empirical Valence Bond (MS-EVB) MethodEVB) Method

• Self-consistent Iterative-MS-EVB (SCI-MS-Self-consistent Iterative-MS-EVB (SCI-MS-EVB)EVB)

• Interesting ResultsInteresting Results

Outline:Outline:

Why Hydrated Proton ?Why Hydrated Proton ?

“Proton stays on the surface in the global minima of

(H2O)21H+ !”

“Ah haa! It seems solvated

proton is easier!”

• Grotthuss mechanism.Grotthuss mechanism. (hop and turn)(hop and turn)

– Proposed by Danneel in 1905; Proposed by Danneel in 1905; 100 years after Grotthuss 100 years after Grotthuss proposed a similar mechanism proposed a similar mechanism describes water electrolysis.describes water electrolysis.

– Very difficult to describe using Very difficult to describe using conventional Molecular conventional Molecular Dynamics force fieldsDynamics force fields•Bonding topologyBonding topology changes.changes.•Atom identity changes.Atom identity changes.

Proton ChemistryProton ChemistryDifficult

• Grotthuss mechanism.Grotthuss mechanism. (hop and turn)(hop and turn)

– Proposed by Danneel in 1905; Proposed by Danneel in 1905; 100 years after Grotthuss 100 years after Grotthuss proposed a similar mechanism proposed a similar mechanism describes water electrolysis.describes water electrolysis.

– Very difficult to describe using Very difficult to describe using conventional Molecular conventional Molecular Dynamics force fieldsDynamics force fields•Bonding topologyBonding topology changes.changes.•Atom identity changes.Atom identity changes.

Proton ChemistryProton ChemistryDifficult

The one proton EVB The one proton EVB matrixmatrix

11 12 13 14

21 22

31 33

41 44

0 0

0 0

0 0

H V V V

V HH

V H

V H

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

Each state has different atomic charges and molecular topologies.

MS-EVB Description of HMS-EVB Description of H99OO44++

(H9O4)+ Eigen, (H5O2)+ Zundel

Multi-State Empirical Valence Multi-State Empirical Valence Bond Method (MS-EVB)Bond Method (MS-EVB)

• Pioneer by Pioneer by Warshel (1991)Warshel (1991)

• Schmitt and Schmitt and Voth MS-EVB 1 Voth MS-EVB 1 (1998-1999)(1998-1999)

• Day and Voth Day and Voth MS-EVB 2 (2002)MS-EVB 2 (2002)

• Wu and Voth MS-Wu and Voth MS-EVB 3 (2007)EVB 3 (2007)

Time correlation function C(t) obtained from CPMD (solid line) and EVB simulation (dashed line). The inset containsthe same correlation functions for short times. (C,Dellago, MM. Naor and G.Hummer PRL. 90, 105902,2003)

MTI MicroFuel Cells demonstrated its direct methanol micro fuel cell

system prototype for President Bush

“His cell phone battery is

pretty cool!Model this!”

Direct MethanolDirect Methanol

• NafionNafionTMTM, , developed by developed by DuPont, is DuPont, is commonly used as commonly used as the proton the proton conducting conducting membrane for direct membrane for direct methanol fuel cells.methanol fuel cells.

Structure of NafionStructure of NafionTMTM membrane membrane

• The hydrated polymer organizes into a The hydrated polymer organizes into a hydrophobic region (A,green) composed of the hydrophobic region (A,green) composed of the perfluorinated alkane backbone, a perfluorinated alkane backbone, a hydrophilic/hydrophobic interface (B,blue) hydrophilic/hydrophobic interface (B,blue) containing the perfluoroalkyl ether side chain, containing the perfluoroalkyl ether side chain, and the hydrophilic ionic region (C,red).and the hydrophilic ionic region (C,red).

H.J. Yeager, et. al., Perfluorinated Ionomer Membranes; ACS Symp. No.180, (American Chemical Society: Washington, DC, 1982)

A B

1

23

4

1

23

4A B

1

23

4

1

23

4

Big Matrix Approach to Solve Big Matrix Approach to Solve Multi-Proton EVB ProblemMulti-Proton EVB Problem

16 states required to describe

A1B3, A1B4, A2B1 ……A1B1,A1B2,

two independent (H9O4)+

Big Matrix Approach is Big Matrix Approach is Computationally Intractable for Computationally Intractable for

Any Nontrivial SystemsAny Nontrivial Systems• The matrix size scales like The matrix size scales like mm

nn, where , where mm is is the number of EVB states per proton, the number of EVB states per proton, nn is is the number of protons.the number of protons.

• In bulk, each proton typically requires 25 In bulk, each proton typically requires 25 states, thus we need states, thus we need 625x625625x625 matrices for 2 matrices for 2 protons protons 390,625x390,625390,625x390,625 matrices for 4 matrices for 4 protons.protons.

Our NafionTM simulation box contains 40 protons.

Mission Impossible !Mission Impossible !

A multi-proton problem should not A multi-proton problem should not be more difficult than a multi-be more difficult than a multi-electron problem?!electron problem?!

Linear Scaling Approach Linear Scaling Approach (SCI-MS-EVB)(SCI-MS-EVB)

― ― Iterative solution of single proton problemIterative solution of single proton problem

Each EVB complex sees all other EVB complexes as arrays of effective particles.The charges and van der Waals parameters of each effective particle is a linear combination of that of a pure hydronium or a pure water according to its local EVB vector.The MS-EVB problem is solved for each EVB complex iteratively until the coefficients of each EVB complex converge. The forces of the overall system are calculated based on a self-consistent solution of all the EVB-complexes.

A B

R

Two Proton SystemTwo Proton System

2

23

2 2

23 2

intra intra inter inter

11

H O H O H ON N,k ,kk

H OH O ,H

N,k

H O Ok kk k

H Oi H i i V iV V V

( ( , ))ijexch

ijco anns OO Oge HtVi H j A R Ri V j

AAA ARABHH HH

1( )

4s t

el

q qV r

r

12 6

( ) 4 st stvdW st

st st

V rr r

BAH 2 2

,, ,

( )s Bi sH Bj sW Bi Bj ex ijj i i j i j

q c q c q c c q A r

12 6 12 6

( ) 4 st st st stvdW st

st st st st

V rr r r r

2 2s Bi sH Bj sW

j i

c c

2 2s Bi sH Bj sW

j i

c c

1/12(4 )st st st 1/ 6(4 )st st st

1( )

4s t

el

q qV r

r

BBB BRBAHH HH

2 2 2 2

2 2, ,

, ,

2 2, ,

, ,

( )

( )

A Ai sH Aj sW Bi tH Bj tWj i j i

Ai Aj s ex ij Bi tH Bj tWi j i j j i

Bi Bj t ex i j Ai sH Aj sW Bi j i j

B

j iA

H c q c q c q c q

c c q A r c q c q

c c q A r c q c q H ABH

0 0 0 0

0 0 0 0

total AA AR BB BR

AB RR

AA BB AB AR BR RR

E a H H a b H H b

a b H a b E

E E E E E E

Hellmann-Feynman ForcesHellmann-Feynman Forces

0 0

0

0 0

0 0 0

total AA AR BB BR

AB RR

ABAA BB BRR RRA

E H H H H

H

b b

E

E

b b

E E

a

a

E EE

a

a

Hellmann-Feynman theorem holds if the total energy reaches a minimum with respect to all coefficients cAi and cBi,

simultaneously

A B

R

In Case of OverlappingIn Case of Overlapping

1

2

34

1

2

34

CA42

C B32

Double Protonation Probability

Determine basis states for each EVB-complex

Solve HEVB for each EVB-complex

Overlapping ?Yes

No

Remove conflicting EVB states

Solve HEVB for each EVB-complex

Converged ?Yes

Calculate Force

Next MD step

No

Simulation of 0.44M HCl Simulation of 0.44M HCl SolutionsSolutions

2EVBN

icec i CEC

i

r c r Feng Wang, Gregory A. Voth J. Chem. Phys., 122, 144105 (2005)

Is Two Protons Per Box Is Two Protons Per Box Enough?Enough?

• For 0.44M For 0.44M

Radial Distribution Functions of Radial Distribution Functions of 0.44M HCl Solutions0.44M HCl Solutions

Unpublished

Hydronium is Ambiphilic!Hydronium is Ambiphilic!

Proton Paring ?!Proton Paring ?!

Put These TogetherPut These Together

Unpublished

Compare with CPMD Compare with CPMD SimulationsSimulations

Unpublished

Hydrogen Bond Network Hydrogen Bond Network Inside NafionInside NafionTMTM Membranes Membranes

• The hydronium ions, The hydronium ions, sulfonate head groups, sulfonate head groups, and water form and water form extended hydrogen extended hydrogen bonded networks bonded networks surrounded by the surrounded by the polymer backbone. polymer backbone. ‘Non-vehicular’ proton ‘Non-vehicular’ proton transport occurs within transport occurs within this H-bond network.this H-bond network.

Simulation of Simulation of Poorly-hyrated Poorly-hyrated

NafionNafionTMTM MembranesMembranes

• Employing SCI-MS-Employing SCI-MS-EVB, all 40 excess EVB, all 40 excess protons are treated protons are treated with the MS-EVB with the MS-EVB method. method.

Matt Petersen, Feng Wang, Gregory A. Voth, in preparation

Sulfonic Oxygen/Hydronium Sulfonic Oxygen/Hydronium Hydrogen Radial DistributionHydrogen Radial Distribution

Excess Protons in Hydrated Nafion

Matt K. Petersen, Feng Wang, Nick P. Blake, Horia Metiu, Gregory A. Voth

J. Phys. Chem. B, 109, 3727 (2005)

AcknowledgementsAcknowledgements

•Prof. Gregory A. Prof. Gregory A. VothVoth•Prof. Kenneth D. Prof. Kenneth D. JordanJordan•Prof. H. Bernhard Prof. H. Bernhard Schlegel Schlegel •Dr. Matt PetersenDr. Matt Petersen•Dr. Sergey IzvekovDr. Sergey Izvekov