Dispersion and metal particle size characterization of nanocatalysts
A relationship between dispersion and surface-averaged metal particle size is provided so that either dispersion can be determined from measured surface-averaged metal particle size or surface-averaged metal particle size can be determined from measured dispersion. The method can be applied to catalysts having a single metal species as well as catalysts having multiple metal species. The size of the supported metal particles in the catalyst sample may be determined using transmission electron microscopy images of supported metal particles in the catalyst sample. The dispersion of the supported metal particles in the catalyst sample may be determined using chemisorption tests on the catalyst sample.
1. A catalyst characterization method comprising the steps of:
preparing a catalyst sample for analysis, the catalyst sample having a plurality of supported metal particles;
measuring the size of the metal particles and determining a surface-averaged diameter of the metal particles (<d> S ) therefrom;
determining an atomic volume of the metal particles (V 0 ); and
calculating a dispersion (D app ) of the metal particles based on the following relationship:
D
app
=
1
-
(
1
-
2
×
V
0
1
/
3
<
d
>
S
)
3
.
2. The method according to claim 1 , further comprising the step of obtaining transmission electron microscopy (TEM) images of the supported metal particles, wherein the size distribution of the metal particles is used to determine <d> S .
3. The method according to claim 1 , wherein the catalyst sample includes a plurality of supported metal particles containing a single metal species.
4. The method according to claim 3 , wherein the atomic volume is based on the number of atoms per unit cell of the metal species, lattice parameters of the metal species and angles associated with the lattice parameters.
5. The method according to claim 1 , wherein the catalyst sample includes a plurality of supported metal particles containing more than one metal species.
6. The method according to claim 1 , further comprising generating error ranges for the computed dispersion.
7. A catalyst characterization method comprising the steps of: preparing a catalyst sample for analysis, the catalyst sample having a plurality of supported metal particles;
determining a dispersion of the metal particles (D);
determining an atomic volume of the metal particles (V 0 ); and
calculating a surface-averaged diameter of the metal particles (<d> S app ) based on the following relationship:
<
d
>
S
app
=
2
×
V
0
1
/
3
1
-
(
1
-
D
)
1
/
3
.
8. The method according to claim 7 , further comprising the step of conducting tests on the catalyst sample to determine the dispersion of supported metal species.
9. The method according to claim 8 , further comprising the step of conducting chemisorption tests on the catalyst sample to determine the dispersion of supported metal species.
10. The method according to claim 7 , wherein the catalyst sample includes a plurality of supported metal particles containing a single metal species.
11. The method according to claim 10 , wherein the atomic volume is based on number of atoms per unit cell of the metal species, lattice parameters of the metal species and angles associated with the lattice parameters.
12. The method according to claim 7 , wherein the catalyst sample includes a plurality of supported metal particles containing more than one metal species.
13. The method according to claim 7 , further comprising generating error ranges for the computed surface-averaged diameter.
14. A computer-readable storage medium having stored thereon instructions to be executed in a computer system to determine a physical property of a catalyst sample, the instructions causing the computer system to carry out the steps of: receiving inputs specifying an input value representative of a physical property of the catalyst sample and one or more types of metal species;
computing an atomic volume of the specified metal species (V 0 ); and
computing an output value representative of a physical property of the catalyst sample based on one of the following equations:
D
app
=
1
-
(
1
-
2
×
V
0
1
/
3
<
d
>
S
)
3
.
if the surface-averaged diameter of the metal particles in the catalyst sample (<d> S ) is the input value, or
<
d
>
S
app
=
2
×
V
0
1
/
3
1
-
(
1
-
D
)
1
/
3
,
if the dispersion of the metal particles in the catalyst sample (D) is the input value.
15. The computer-readable storage medium according to claim 14 , wherein the atomic volume is computed based on crystallographic parameters of the specified metal species, including the number of atoms per unit cell of the metal species, lattice constants and associated angles.
16. The computer-readable storage medium according to claim 15 , wherein the instructions cause the computer system to further carry out the step of retrieving the crystallographic parameters of the specified metal species from a database.
17. The computer-readable storage medium according to claim 14 , wherein the instructions cause the computer system to further carry out the step of receiving an input specifying a first mode of operation or a second mode of operation.
18. The computer-readable storage medium according to claim 17 , wherein if the first mode of operation is selected, the output value representative of a physical property of the catalyst sample is computed based on the following equation:
D
app
=
1
-
(
1
-
2
×
V
0
1
/
3
<
d
>
S
)
3
,
and if the second mode of operation is selected, the output value representative of a physical property of the catalyst sample is computed based on the following equation:
<
d
>
S
app
=
2
×
V
0
1
/
3
1
-
(
1
-
D
)
1
/
3
.
19. The computer-readable storage medium according to claim 14 , wherein the instructions cause the computer system to further carry out the step of generating an error range for the computed output value.
20. The computer-readable storage medium according to claim 14 , wherein the instructions cause the computer system to further carry out the step of receiving an input specifying an estimate of the ratio of the volume-averaged diameter of the supported metal particles in the catalyst sample (<d> V ) to the surface-averaged diameter of the supported metal particles in the catalyst sample (<d> S ).
21. The computer-readable storage medium according to claim 20 , wherein the instructions cause the computer system to further carry out the step of generating a log-normal distribution of the supported metal particles based on the volume-averaged diameter of the metal particles in the catalyst sample and the surface-averaged diameter of the metal particles in the catalyst sample.
22. The computer-readable storage medium according to claim 20 , wherein the instructions cause the computer system to further carry out the step of calculating the raw moments <d>, <d 2 > and <d 3 > of the metal particles in the catalyst sample.
23. The computer-readable storage medium according to claim 20 , wherein the instructions cause the computer system to further carry out the step of calculating the dispersion using the full expression according to:
D
=
6
(
V
0
)
1
/
3
×
<
ⅆ
2
>
<
ⅆ
3
>
-
12
(
V
0
)
2
/
3
×
<
ⅆ
>
<
ⅆ
3
>
+
8
V
0
×
1
<
ⅆ
3
>
,
where <d>, <d 2 > and <d 3 > are the raw moments of the metal particles in the catalyst sample.
24. The computer-readable storage medium according to claim 20 , wherein the instructions cause the computer system to further carry out the step of calculating the dispersion using the full expression for a log-normal metal particle size distribution according to:
D
=
6
×
V
0
1
/
3
<
d
>
S
-
12
×
V
0
2
/
3
×
K
<
d
>
S
2
+
8
×
V
0
×
K
3
<
d
>
S
3
,
where <d> S is the surface-averaged size of the metal particles in the catalyst sample, K=<d> V /<d> S , and <d> V is the volume-averaged size of the metal particles in the catalyst sample.