Theory of Interfacial Proton Transport

in Polymer Electrolyte Membranes

Anatoly Golovnev and Michael Eikerling

Simon Fraser University, Burnaby, BC, Canada

SSPC16. September 13, 2012

Outline

• Importance of surface proton transport

• Model of the surface (interface)

• Mechanism of surface proton transport

• Mathematical description and results

• Conclusion/Prospective

2

Motivation

Our main research focus is on polymer electrolyte membrane (PEM) fuel cells

Operation relies on the proton conducting PEM which conductivity is mainly

due to a network of water filled pores

At T>90 oC the water starts to evaporate which leads to a significant decrease

of PEM conductivity

Since the bulk water transport is out of game, one has to count on the surface

proton transport

Aim: to understand the mechanism of surface proton transport and, based on that, link surface structure with proton mobility

3

Experimental Evidences

Can the surface transport provide high enough mobility? Yes!

J. Matsui, H. Miyata, Y. Hanaoka, and T. Miyashita, ACS Appl. Mater. Interfaces, 3, 1394-1397 (2011) 4

• J. Zhang, and P. Unwin, J. Am. Chem. Soc., 124, pp 2379-2383

(2002)

• S. Serowy, S. Saparov, Yu. Antonenko, W. Kozlovsky, V. Hagen, and

P. Pohl, Biophys. J. 84,1031-1037 (2003).

Experimental Evidences

Can the surface transport provide high enough mobility? Yes!

J. Matsui, H. Miyata, Y. Hanaoka, and T. Miyashita, ACS Appl. Mater. Interfaces, 3, 1394-1397 (2011) 5

• J. Zhang, and P. Unwin, J. Am. Chem. Soc., 124, pp 2379-2383

(2002)

• S. Serowy, S. Saparov, Yu. Antonenko, W. Kozlovsky, V. Hagen, and

P. Pohl, Biophys. J. 84,1031-1037 (2003).

Model Surface

6

A. Roudgar, S.P. Narasimachary, and M. Eikerling, J.

Phys. Chem. B, 110, 20469-204777 (2006)

up-right and tilted structures (top view)

−− 33 SOCF

−3SO

+OH3

Proton Transport

Vertices: SG

Triangles: H3O+

Vertices: SG

Triangles: H3O+

Activation energy of a single hop is 0.6 eV

A. Roudgar, S.P. Narasimachary, and M. Eikerling, Chem. Phys. Lett., 457, 337-341 (2008)

Such a large value is due to uneven distribution of

hydrogen bonds

7

Solution: to keep the number of HB at each surface group constant

Concerted Motion (Soliton)

S. Vartak, A. Roudgar, A. Golovnev, and M. Eikerling, submitted to the Journal of Physical Chemistry C

Hydrogen bonds

distributed evenly

Activation energy is 0.3 eV

8

Mathematical Description

( )∑ +−+

= +

i

iiii uVuu

k

dt

dumH )(

22

2

1

2

?)( =iuV

Hydronium ions move in the potential created by surface groups

What should look like?

• Periodic

• Harmonic for small displacement

• Steep ascending and descending flanks; plateau between wells

• Asymmetric

Exact potential profile is unknown. How to model it then?

is the hydronium ion displacement

iu

1+iuu

)(uV

9

Potential Profile

How does the hydronium motion depend on the potential profile?

)(cosh)(2

1 uuV λε−−=

)(cos)( 22 ua

uVπ

ε−=

Two extreme cases of the considered potentials

Different potentials offer different behaviour of hydronium ions.

But what about soliton energy and mobility? 10

(Strong HB)

(Weak HB)

Energy and Mobility

2

0

2

2)1(

1

2

v

v

kaE

−

=ε

εµ

2

1 2)1( ka

b=

)(1 uV )(2 uV

steep wells, large plateau slanting wells, minor plateau

2

0

2

2)2(

1

22

v

v

kaE

−

=ε

π

m

kav

22

0 ≡

ε

πµ

2

1

2

2)1( ka

b=

is the maximal soliton velocity

2)2(

)1(

)1(

)2( π

µ

µ==

E

E

b is the viscous friction coefficient

introduced to derive mobility

Consider a soliton travelling with a velocity v

11

100)1(

)2(

≈N

N

Soliton size ratio

Energy and Mobility

2

0

2

2)1(

1

2

v

v

kaE

−

=ε

εµ

2

1 2)1( ka

b=

)(1 uV )(2 uV

steep wells, large plateau slanting wells, minor plateau

2

0

2

2)2(

1

22

v

v

kaE

−

=ε

π

ε

πµ

2

1

2

2)1( ka

b=

Consider a soliton travelling with a velocity v

Only two (three?) parameters define the dynamics:

ε2

ka

b

- potential well depth. Already calculated! S. Vartak, A. Roudgar, A. Golovnev, M. Eikerling, submitted to the Journal of Physical Chemistry

- hydronium ion – hydronium ion interaction.

- viscous friction coefficient12

Conclusion

• A mechanism of soliton proton transport on a surface

• The particular potential profile has a minor influence on soliton’s

energy and mobility

• A complete description of single soliton properties

Prospective

• Numerical calculation of the constants to evaluate mobility

• The step from a single soliton to soliton statistics to find conductivity

13

Acknowledgements

14

Thanks to:

•NSERC for financial support

•Michael Eikerling

•Swati Vartak

•Ata Roudgar

15

3-Step Mechanism

In 2D one can avoid

overcoming large

potential barriers

16

Constants

k

b

hydronium ion – hydronium ion interaction.

- viscous friction coefficient. Where does it come from?

( )∑

−−+−+

= +

i

iii qu

a

qq

quu

k

dt

dumH ),

)(K2(dn1

22

22

2

2

1

2ε

ε

To evaluate results one has to calculate the constants used

- potential well depth. Already calculated!S. Vartak, A. Roudgar, A. Golovnev, M. Eikerling, Collective proton dynamics at highly charged

interfaces studied by Ab Initio Metadynamics, submitted to Chemistry of Materials .

17

Quest for Conductivity

µσ qn≡

probability density for creation of a soliton

A step from one soliton consideration to a soliton statistics

?=n

dvTk

vEEdvvvW

B

d

+−=+

)(exp

2);(

π

ω

−−≡

Tk

EEE

B

T

)0()(exp)(

δδ

18

Solutions

Narrow potential wells correspond to narrow solitons with a smaller

number of participants but a higher level of collectiveness of the motion

εξ

2

)1(2

0

22

)1( v

vka

a

−

=

vtx −=ξ

19