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

The effective performance of nanostructured catalysts N. Lund 1 , X. Y. Zhang 2 , N. Gaston 1,2 , S. C. Hendy 1,2 1 MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington 2 Industrial Research Ltd

Precious metal catalysts • Precious metal catalysts are used for many applications e.g. platinum and palladium in catalytic converters CO+1/2 O 2 → CO 2 NO x → x/2 O 2 +1/2 N 2

Nanostructured catalysts • The performance of a catalyst depends on its size & shape Corners and edges tend to be more reactive • How might we determine effective performance so that we can optimise shape and size?

Effective catalysis problem • E.g. oxidation of CO: CO+1/2 O 2 → CO 2 CO CO 2 O 2

Arrays of nanoparticles • 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?

Effective catalysis problem • E.g. oxidation of CO: CO+1/2 O 2 → CO 2 CO CO 2 O 2

Boundary conditions    • Gas A to B: 0 A g B g A g B A k terrace L k step   A S A  probabilit y of absorption k k g i 1 1   A A S  k k 0 g i 2 2   A B k k k 3 3 4   B B S k g i 4

Boundary conditions B g  area of site a i A g i   fractional coverage of site i B A i  collision rate of gas A per unit area Q g L In steady state:       A d A B d d g       g g s k ( 1 ) Q s s 1 i dt dt dt   B d  k Q g    s 1 k i 3 a i (Langmuir eqn) k dt  i k Q 3 1 a i

Boundary conditions • Finally obtain a boundary condition: k Q      1 Dn  A k Qa  1 1 i k 3 1 8 k T   B where according to gas kinetic theory Q  A 4 m i  • So for (i.e. absorption limited) k Qa k 1 3 k 8 k T       1 B Dn    A A 4 m

Boundary conditions  D 2 m  • Thus in terms of the length b k k T 1 B       bn   A A k T   • But again, gas kinetic theory has B D ~ v ~ g m  so b ~ k 1  where is the mean free path of the gas molecule k is a probability of absorption 1

Boundary conditions • So the boundary condition under diffusion limited  conditions depends on the length b ~ k 1 A g B g B A b k k L step terrace • Mean free path of gas molecule in air at 1 atmosphere is ~ 100 nm, so b will be of O(100nm)

Mathematical problem • Now we have a diffusion problem with a heterogeneous mixed boundary condition        2 0 in some domain ( x , z ) A         : b ( x ) n on z h ( x )   A A       / on A 0 b L • Want to replace the heterogeneous condition with a homogeneous b.c. that gives same answer in the far field

Direct solution approach • Use an asymptotic expansion of boundary condition b  b  b 1 b 2 L L e.g. 2 1   a / L a L   b       ˆ 1 u z O 0 y  ˆ   0 L  b eff  b   b       ˆ 2 u z O y 1  ˆ   0 L Hendy and Lund, PRE 76 , 066313 (2007)

Homogenization approach • Use homogenization approach based on weak formulation of Stokes equations: e.g.   2 1 h 1    0 v u dxdz v u dxdz b b    0   as h / W ~ L / W ~ 0 h L Zhang, Lund, Mahelona and SCH, submitted (2011)

Homogenization approach     2 y b 2 sin( ) sin( ) 2 x L L b eff  b 1  1  b  b eff  L / W 0 Zhang, Lund, Mahelona and SCH, submitted (2011)

Recall our results  • b ( x ) L Small b and small roughness    b eff b h O ( bh )  b ( x ) L • Large b and small roughness 2   1 h 1  b eff b

Effective rate constants  • Assume two types of rate constant with k k terrace step spacing on a flat surface L   • Case 1: active sites widely space w.r.t. b / k terrace  1    1 k      L terrace k     eff k k 1   terrace 1   • Case 2: active sites closely space w.r.t. b / k step      L k k k eff 1 step k step

Effective rate constants • At atmospheric pressure, a nanostructured catalyst will  likely have  L k active 2       a so k 1 h k ( 1 ) k eff 1 active where a is the surface roughness • The activity is dominated by the active sites and enhanced by the roughness

Effective rate constants   • If however as might occur for an L k active array of particles, then we have  h    k eff h O ( ) k k 1 1 • Hence, effective activity will depend on the location of the particles h

Effective rate constants    1  • e.g. on a flat support, keeping k 2 sin( 2 x ) L 1 coverage of support by catalyst at 50% Small particles, closely spaced k 1 Big particles, widely spaced  L  1   1 k L 1

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