nanostructured catalysts N. Lund 1 , X. Y. Zhang 2 , N. Gaston 1,2 , - - PowerPoint PPT Presentation

• N. Lund1, X. Y. Zhang2, N. Gaston1,2, S. C. Hendy1,2

1MacDiarmid Institute for Advanced Materials and Nanotechnology,

Victoria University of Wellington

2Industrial Research Ltd

The effective performance of nanostructured catalysts

Precious metal catalysts

• Precious metal catalysts are used for many applications

e.g. platinum and palladium in catalytic converters CO+1/2 O2 → CO2 NOx → x/2 O2+1/2 N2

• The performance of a catalyst depends on its size & shape
• How might we determine effective performance so that we

can optimise shape and size?

Nanostructured catalysts

Corners and edges tend to be more reactive

Effective catalysis problem

• E.g. oxidation of CO: CO+1/2 O2 → CO2

CO CO2 O2

• Consider an array of nanoparticles on a substrate
• How does the effective activity of the particles depend on

the arrangement of the particles on the support?

Arrays of nanoparticles

Effective catalysis problem

• E.g. oxidation of CO: CO+1/2 O2 → CO2

CO CO2 O2

• Gas A to B:

Boundary conditions

step

k

terrace

k

L

Ag A B Bg

 

g

A



i g i g i g

S B B B A S A A A S A       

4 3 2 1

k k k k

4 3 2

k k k  

absorption

• f

y probabilit 

1

k

   

i i s i s

a k dt d Q k dt d  

3 1

) 1 (    

g g

B A

Boundary conditions

L

Ag A B Bg

area unit per A gas

• f

rate collision site

• f

coverage fractional site

• f

area

g

   Q i i a

i i



In steady state:

   

i i s s

a k Q k Q k dt d dt d

3 1 1

    

g g

B A

(Langmuir eqn)

Boundary conditions

• Finally obtain a boundary condition:

where according to gas kinetic theory

• So for (i.e. absorption limited)

3 1 1

1 k Qa k Q k Dn

i A

    





A B

m T k Q   8 4 1 

 

   

A B A

m T k k Dn    8 4

1 3 1

k Qa k

i 

Boundary conditions

• Thus in terms of the length
• But again, gas kinetic theory has

so where is the mean free path of the gas molecule is a probability of absorption

 

   

A A

bn  

  m T k v D

B g

~ ~ T k m k D b

B

 2

1



1

~ k b 



1

k

• So the boundary condition under diffusion limited

conditions depends on the length

• Mean free path of gas molecule in air at 1 atmosphere

is ~ 100 nm, so b will be of O(100nm)

Boundary conditions

step

k

terrace

k

L

b

Ag A B Bg

1

~ k b 

• Now we have a diffusion problem with a heterogeneous

mixed boundary condition

• Want to replace the heterogeneous condition with a

homogeneous b.c. that gives same answer in the far field

Mathematical problem  

             

 

/ :

• n

x h z

• n

n x b z x domain some in

A A A A 2

) ( ) ( ) , (     

L

b

Direct solution approach

• Use an asymptotic expansion of boundary condition

e.g.

L b1 b2

Hendy and Lund, PRE 76, 066313 (2007)

a

b beff  

L b 

1

L b 

2

L a/  

         



y L b O u z ˆ

1 ˆ

1 ˆ

2 ˆ

        



y L b O u z 

• Use homogenization approach based on weak

formulation of Stokes equations: e.g.

Homogenization approach

~ / ~ / as   W L W h

L h

Zhang, Lund, Mahelona and SCH, submitted (2011)

1 1

2

dxdz u b h v dxdz u b v

 

 

  



Homogenization approach

1 1  

 b beff b beff  /  W L

Zhang, Lund, Mahelona and SCH, submitted (2011)

) sin( ) sin( 2

2 2 L y L x

b

 

 

Recall our results

• Small b and small roughness

L x b  ) (

) (bh O h b beff   

• Large b and small roughness

L x b  ) ( b h beff

2

1 1   

Effective rate constants

• Assume two types of rate constant with

spacing on a flat surface

• Case 1: active sites widely space w.r.t.
• Case 2: active sites closely space w.r.t.

step terrace

k k 

L

terrace

k L              



1 1

1 1 terrace eff

k k k

step

k L  

step eff

k k k   

1

terrace

k b /  

step

k b /  

• At atmospheric pressure, a nanostructured catalyst will

likely have so where a is the surface roughness

• The activity is dominated by the active sites and

enhanced by the roughness

Effective rate constants

active

k L  

active eff

k k h k ) 1 ( 1

1 2

a      

• If however as might occur for an

array of particles, then we have

• Hence, effective activity will depend on the location of the

particles

Effective rate constants

active

k L   ) (

1 1

k h O h k keff     h

• e.g. on a flat support, keeping

coverage of support by catalyst at 50%

Effective rate constants

) sin( 2

2 1 1 L x

k



 



Small particles, closely spaced Big particles, widely spaced

  L L

1 1 1  

k

1

k

Conclusions

• The macroscopic performance of

catalysts depends on the location of active sites and the surface roughness

• We have derived some simple

expressions that relate the microstructure to the macroscopic performance

• Nanostructured catalysts are a good idea

but there are diminishing returns!

• Roughness can inhibit the performance of

arrays of nanoparticles on a substrate