# Patent

## Title
Characterization of metal particle distributions in a supported catalyst

## Abstract

A catalyst sample may contain both small and large metal particle distributions simultaneously. Characterizing the properties of the metal particles contained in each distribution is important to help describe catalytic performance and optimize catalysts. Monte Carlo simulations and dispersion measurements are employed to determine the relationship between dispersion parameters of each metal particle distribution. Various properties, such as the atom fraction and the surface atom fraction of each distribution can be determined.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention generally relates to catalyst characterization and, more particularly, to determining the large metal particle fraction in a heterogeneous catalyst.
2. Description of the Related Art
The performance of catalysts is highly dependent on their physical and chemical properties. However, it is often difficult to directly measure physical and chemical properties of supported metal particles in catalysts, especially catalysts whose average metal particle sizes are less than 100 nm, hereafter called “nanocatalysts.” Catalyst developers therefore rely on sophisticated characterization techniques to determine the physical and chemical properties and performance characteristics of new catalyst designs.
Dispersion and metal particle size distribution influence the performance of catalysts. Dispersion D is defined according to D=NTotS/NTot, where NTotS is total amount of metal surface atoms and NTot is all metal atoms in the catalyst. Dispersion is an important property in catalysis because only atoms that are exposed to the surface are able to play a role in catalytic surface reactions. Catalyst metal particle size is also an important property because larger metal particles have less percentage of their atoms at the surface. As a consequence, a catalyst with smaller metal particles will usually outperform an equivalent amount of catalyst with identical metal concentration, but having larger metal particles.
When a catalyst contains a combination of a large metal particle fraction and a small metal particle fraction, conventional characterization methods for determining catalyst dispersion and metal particle size may not work very well to adequately describe the properties of all metal particles in the sample. A catalyst can contain two distinct metal particle fractions for a variety of reasons. They may result from a particular synthesis method that was applied and/or how the initial distribution of metal particles on the support material is changed upon aging (e.g., hydro-thermal treatment, exposure to poisons, etc.). It would be possible to quantify the large metal particle fraction by imaging all of the metal particles in a catalyst sample and measuring the size of each one, but this process would be too slow and too expensive to be applied in practice.
SUMMARY OF THE INVENTION
Embodiments of the present invention provide an improved method for determining the nature of large metal particles in a heterogeneous catalyst. The improved method is more practical than the brute-force microscopic techniques described above because it relies on measurements that can be made more easily. With this ability to determine the nature of the large metal particle fraction in a catalyst, one can better quantify metal use efficiency in a catalyst sample and optimize the synthesis techniques to maximize performance.
A catalyst characterization method according to an embodiment of the present invention includes the steps of providing a catalyst having small and large metal particle size distributions (P1 and P2, respectively), estimating a dispersion of the P1 metal particles (D1); D1=N1S/N1, where N1S is the number of surface atoms in P1 and N1 is the total number of atoms in P1, estimating a total dispersion of the catalyst (DTot), and determining the atomic fractions of the P2 metal particles, as well as the corresponding dispersion (D2); D2=N2S/N2, where N2S is the number of surface atoms in P2 and N2 is the total number of atoms in P2. This determination is based on D1 and DTot, where D1 may be estimated from transmission electron microscopy (TEM) or scanning transmission electron microscopy (STEM) measured sizes of the P1 metal particles and DTot may be estimated from chemisorption measurements. If D1 is not significantly different from DTot, within the accuracy of the experimental techniques used to determine them, the fraction of P2 metal atoms in the catalyst sample is below the detection limit of the method or there is no large particle fraction.
A method for evaluating the effects of aging on a catalyst according to an embodiment of the present invention includes the steps of preparing a fresh catalyst sample and performing appropriate characterization methods. Thereafter, the nature of P2 (i.e., properties or existence of) is determined based upon size measurements of the metal particles in the fresh catalyst sample and chemisorption measurements performed on the fresh catalyst sample. Subsequently, the fresh catalyst is subjected to aging and characterized by determining the nature of P2 (i.e., properties or existence of) in the aged catalyst based on TEM or STEM size measurements of the metal particles in the aged catalyst sample and chemisorption measurements performed on the aged catalyst. The properties of P2 in the fresh and aged catalyst samples are then compared to determine the aging effects.
Another embodiment of the present invention provides a computer-readable storage medium having stored thereon instructions to be executed in a computer system to determine a physical property of a catalyst. According to this embodiment, the instructions stored on the computer-readable storage medium cause the computer system to carry out the steps of calculating D1 and DTot of the catalyst, using these numbers to determine if there is a large metal particle fraction in the catalyst, and if so determining the properties of such a fraction.



BRIEF DESCRIPTION OF THE DRAWINGS
So that the manner in which the above recited features of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments.
FIG. 1 is a flow diagram illustrating a design process for a catalyst in which embodiments of the present invention can be practiced.
FIG. 2 is a TEM image of a catalyst sample that illustrates metal particles of various sizes.
FIG. 3 graphically illustrates the size distribution of the catalyst sample shown in FIG. 2.
FIG. 4 is a flow diagram illustrating the steps carried out to determine properties of large metal particles in a catalyst.
FIG. 5 illustrates the variation of a correction factor as a function of an estimated upper bound for the size range of large metal particles in a catalyst.
FIG. 6 is a flow diagram illustrating the steps carried out to evaluate the effects of aging on a catalyst using the process according to FIG. 4.
FIG. 7 is a flow diagram illustrating the steps carried out by a computer system in accordance with a computer program module for characterizing a physical property of a catalyst.



DETAILED DESCRIPTION
FIG. 1 is a flow diagram illustrating the design process of catalysts in which embodiments of the present invention can be practiced. In step 110, the composition and the properties of a target catalyst are specified. The composition indicates the metal species and the support material. It also specifies the weight ratio of the metal species if it includes more than one metal species. The properties may include the metal particle size distribution. A catalyst with the target composition is then synthesized in step 120. In addition, a sample of the synthesized catalyst is prepared for subsequent characterization in step 130, where the properties of the catalyst sample are determined. Then, the properties of the target catalyst are compared against the properties of the synthesized catalyst in step 140. If there is a good match, the synthesized catalyst is accepted, and the process ends. If a match is not obtained within a given predetermined range of acceptable values, the process returns to step 120, where a catalyst with the target composition is synthesized according to a different synthesis method.
FIG. 2 is a TEM image of a catalyst sample that illustrates metal particles of various sizes. According to embodiments of the present invention, this system is modeled as two distinctly separated particle size distributions P1 and P2, where P1 contains particles that are smaller than those in P2. FIG. 3 is a graphical illustration of the catalyst sample of FIG. 2 having small metal particles belonging to P1 and a few large metal particles belonging to P2. P1 can often be modeled with a log-normal distribution while P2 is modeled as a stochastic distribution. The P2 metal particles contain a disproportionately large number of bulk atoms and therefore may not contribute significantly to catalytic activity. The dispersion values corresponding to P1 and P2 are D1 and D2, respectively, and the dispersion value corresponding to all supported metal particles (i.e., P1+P2) is defined as DTot. Other properties of the catalyst sample that can be determined include: N2/NTot, N1/NTot, N2S/NTotS, N1S/NTotS, where N1 represents the number of atoms in the P1 metal particles and N2 represents the number of atoms in the P2 metal particles. NTot is the sum of N1 and N2 and the superscript ‘S’ signify surface atoms. The derivation for N1/NTot and N1S/NTotS follow directly when N2/NTot and N2S/NTotS have been determined because NTot=N1+N2 and NTotS=N1S+N2S.
If NTot is known (easily calculated using the total amount of metal in the catalyst sample) the absolute numbers of N2, N1, N2S, N1S and NTotS can be determined. For a given catalytic reaction rate RTot, the turnover frequency (TOFTot) can be calculated according to TOFTot=RTot/NTotS. We can also define one turnover frequency for P1: TOF1=R1/N1S and one turnover frequency for P2: TOF2=R2/N2S.
FIG. 4 is a flow diagram illustrating the steps carried out to determine the properties of the large metal particle fraction in a catalyst (i.e., P2). In step 410, a catalyst, e.g., the synthesized catalyst, is provided. A statistically valid sample size for TEM measurements is required for accurate characterization of P1. It has been observed that about 1000 particles should be analyzed to obtain statistical validity. Then, the dispersion D1 is estimated in step 420, and DTot is estimated in step 430. In step 440, the atomic fraction of P2 metal particles in the catalyst may be determined using the relationship:




  
    
      
        N
        2
      
      
        N
        Tot
      
    
    =
    
      
        
          D
          1
        
        -
        
          D
          Tot
        
      
      
        D
        1
      
    
  
  ,




where the dispersion D2 of the P2 metal particles in the catalyst is assumed to be zero or in practice negligible compared to D1. If D2 is not assumed to be negligible compared to D1, the following relationship is used:





  
    
      
        N
        2
      
      
        N
        Tot
      
    
    =
    
      (
      
        
          
            D
            1
          
          -
          
            D
            Tot
          
        
        
          D
          1
        
      
      )
    
  
  ,




where y2 is a correction factor based on a size range for P2. This correction factor is determined through Monte Carlo simulations, as further described below. Hence, N2 and N1 are simply calculated by





  
    N
    2
  
  =
  
    
      N
      Tot
    
    ×
    
      y
      2
    
    ×
    
      (
      
        
          
            D
            1
          
          -
          
            D
            Tot
          
        
        
          D
          1
        
      
      )
    
  




and N1=NTot−N2. When y2 is determined, the dispersion D2 for P2 metal particles is determined by:





  
    D
    2
  
  =
  
    
      (
      
        1
        -
        
          1
          
            y
            2
          
        
      
      )
    
    ×
    
      
        D
        1
      
      .
    
  




The ratio of the number of surface atoms of the P2 metal particles, to the total number of surface metal atoms in the catalyst is determined by applying:




  
    
      N
      2
      S
    
    
      N
      Tot
      S
    
  
  =
  
    
      (
      
        
          y
          2
        
        -
        1
      
      )
    
    ×
    
      
        (
        
          
            
              D
              1
            
            
              D
              Tot
            
          
          -
          1
        
        )
      
      .
    
  




Thereafter, it is straightforward to calculate NTotS=DTot×NTot,





  
    N
    2
    S
  
  =
  
    
      N
      2
      S
    
    ×
    
      (
      
        
          y
          2
        
        -
        1
      
      )
    
    ×
    
      (
      
        
          
            D
            1
          
          
            D
            Tot
          
        
        -
        1
      
      )
    
  




and N1S=NTotS−N2S. Having determined all dependent variables, the numbers of core atoms: NTotS=NTotS−NTotS, N1C=N1−N1S and N2C=N2−N2S, can be determined as well. Thus, for given catalytic reaction rates (RTot, R1, R2), the corresponding turnover frequencies are well defined and simple to determine using formulae defined above.

The dispersion D1 may be estimated using TEM, STEM, and other methods known to those skilled in the art. D1 may also be estimated using the method described in U.S. patent application Ser. No. 11/695,943, entitled “Dispersion and Metal Particle Size Characterization of Nanocatalysts,” filed Apr. 3, 2007, the entire contents of which are incorporated by reference herein. In this method, the estimate of D1 is primarily based on the statistical raw moments of a length variable that describes the particles under investigation (e.g., diameter for spheres). Both the surface-averaged size <d>S and volume averaged size <d>V are defined in terms of the raw moments according to <d>S=<d3>/<d2> and <d>V=<d4>/<d3>, respectively, where < > denotes the arithmetic mean.
The total dispersion of the metal particles in the catalyst (DTot) may be estimated from chemisorption measurements. First, the adsorption (α) of the catalyst is measured by chemisorption using techniques well known in the art. Second, the saturation coverage (θ) of the catalyst for a given probe molecule or adsorbate is determined either experimentally, e.g., using infrared spectroscopy or time-resolved X-ray photo-electron spectroscopy, as described in A. Baraldi, et al., “Real-time X-ray Photoelectron Spectroscopy of Surface Reactions,” Surface Science Report 49, 169 (2003), or by computational modeling, e.g., using density functional theory combined with kinetic Monte Carlo simulations as described in E. Hansen and M. Neurock, “First-Principles Based Kinetic Simulations of Acetic Acid Temperature Programmed Reaction on Pd(111),” J. Phys. Chem. B, 105, 9218 (2001). The total dispersion of the catalyst (DTot) is determined from the relationship:




  
    D
    Tot
  
  =
  
    
      α
      θ
    
    .
  




The correction factor (y2) is determined by carrying out Monte Carlo simulations using the following equation that describes the influence of a known distribution P1 and a variable distribution P2 on the total dispersion (DTot):




  
    
      D
      Tot
    
    =
    
      
        <
        
          n
          1
          S
        
        >
        
          +
          
            
              m
              2
            
            
              m
              1
            
          
        
        <
        
          n
          2
          s
        
        >
      
      
        <
        
          n
          1
        
        >
        
          +
          
            
              m
              2
            
            
              m
              1
            
          
        
        <
        
          n
          2
        
        >
      
    
  
  ,




where <n1S> and <n2S> are the mean numbers of surface atoms, <n1> and <n2> are the mean total number of atoms and m1 and m2 are the number of metal particles in P1 and P2, respectively. The lower bound for the size of particles in P2 (dlow) can be determined as the value that satisfies the following equality:





  
    
      
        ∫
        0
        
          d
          low
        
      
      ⁢
      
        
          
            P
            1
          
          ⁡
          
            (
            x
            )
          
        
        ⁢
        
          
        
        ⁢
        
          d
          x
        
      
    
    =
    ξ
  
  ,




where ξ is typically 0.95 and P1(x) is the log-normal probability density function corresponding to P1. Alternatively, the lower bound may be defined in terms of the surface-averaged diameter of P1 (i.e., <d1>S=<d13>/<d12>), e.g., dlow=ν<d1>S, where ν is a number greater than 1, typically around 3. The upper bound dhigh must be greater than the lower bound.

When the above-described Monte Carlo simulations are carried out using increasing values of m1 and m2, the influence of distributions P1 and P2 on the total dispersion (DTot) remains linear after a certain value of m1 according to the following equation:




  
    
      N
      2
    
    
      N
      Tot
    
  
  =
  
    
      y
      2
    
    ×
    
      
        (
        
          
            
              D
              1
            
            -
            
              D
              Tot
            
          
          
            D
            1
          
        
        )
      
      .
    
  




At this point, the correction factor (y2) can be calculated using the formula for the total dispersion (DTot) shown above.

FIG. 5 illustrates the variation of the correction factor (y2) as a function of the upper bound (dhigh) in the case of an actual supported platinum catalyst, where dlow=3<d1>S, <d1>S=23 nm, and <d1>V=26 nm. <d1>V (i.e., <d1>V=<d14>/<d13>) represents the volume-averaged diameter of the P1 metal particles. FIG. 5 confirms that, for very large values of dhigh, the number of atoms in the P2 metal particles may be approximated with y2=1 or according to the formula:




  
    
      N
      2
    
    
      N
      Tot
    
  
  =
  
    
      
        
          D
          1
        
        -
        
          D
          Tot
        
      
      
        D
        1
      
    
    .
  




FIG. 6 is a flow diagram illustrating the steps carried out to evaluate the effects of aging on a catalyst using the process according to FIG. 4. In step 610, a fresh catalyst is prepared. In step 620, the fraction of P2 metal particles in the fresh catalyst is determined from the size measurements carried out using the process according to FIG. 4. In step 630, the fresh catalyst is subjected to aging conditions, e.g., exposed to high temperatures under a given atmosphere of gas for an extended period of time. In step 640, the fraction of metal particles in P2 in the aged catalyst is determined from the size measurements carried out using the process according to FIG. 4. A disproportionately high degree of sintering will give rise to an increased fraction of metal atoms in P2 as determined in step 640 relative to the value determined in step 620. Different catalysts may be evaluated using this method to determine which develop the least amount of wasted metal under given aging conditions.
FIG. 7 is a flow diagram illustrating the steps carried out by a computer system in accordance with a computer program module for characterizing a physical property of a catalyst. In step 710, the computer system receives inputs from a user. The inputs identify the metal species (one or more) of the catalyst. In step 720, the computer system receives further inputs, such as those representative of measured catalyst metal particle sizes, e.g., the surface-averaged diameter <d1>S of the P1 metal particles and measured α and θ. Then, the computer system calculates D1 from <d1>S in step 730, and a second dispersion value (DTot) from α and the θ in step 740. In step 750, the atomic fraction of metal atoms in P2 is determined using the relationship:




  
    
      
        N
        2
      
      
        N
        Tot
      
    
    =
    
      
        
          D
          1
        
        -
        
          D
          Tot
        
      
      
        D
        1
      
    
  
  ,




where the dispersion D2 of the metal particles in P2 is assumed to be zero or negligible compared to D1. The atomic fraction of metal atoms in P1 follows directly from





  
    
      N
      1
    
    
      N
      Tot
    
  
  =
  
    1
    -
    
      
        
          N
          2
        
        
          N
          Tot
        
      
      .
    
  




In an alternative embodiment where D2 is not assumed to be negligible compared to D1, the correction factor y2 is input and, in step 750, the atomic fraction of metal atoms in P2 is determined using the relationship:




  
    
      N
      2
    
    
      N
      Tot
    
  
  =
  
    
      y
      2
    
    ×
    
      
        (
        
          
            
              D
              1
            
            -
            
              D
              Tot
            
          
          
            D
            1
          
        
        )
      
      .
    
  




Again, the atomic fraction of metal atoms in P1 follows directly from





  
    
      N
      1
    
    
      N
      Tot
    
  
  =
  
    1
    -
    
      
        
          N
          2
        
        
          N
          Tot
        
      
      .
    
  




When y2 has been determined, the dispersion for P2 is calculated by





  
    D
    2
  
  =
  
    
      (
      
        1
        -
        
          1
          
            y
            2
          
        
      
      )
    
    ×
    
      
        D
        1
      
      .
    
  




The ratio of the number of surface atoms of the P2 metal particles, to the total number of surface metal atoms in the catalyst is determined by:





  
    
      N
      2
      S
    
    
      N
      Tot
      S
    
  
  =
  
    
      (
      
        
          y
          2
        
        -
        1
      
      )
    
    ×
    
      
        (
        
          
            
              D
              1
            
            
              D
              Tot
            
          
          -
          1
        
        )
      
      .
    
  




The corresponding fraction of metal surface atoms in P1 is





  
    
      N
      1
      S
    
    
      N
      Tot
      S
    
  
  =
  
    1
    -
    
      
        
          N
          2
          S
        
        
          N
          Tot
          S
        
      
      .
    
  




Thereafter, it is straightforward to calculate





  
    
      N
      Tot
      S
    
    =
    
      
        D
        Tot
      
      ×
      
        N
        Tot
      
    
  
  ,
  
    
  
  ⁢
  
    
      N
      2
      S
    
    =
    
      
        N
        Tot
        S
      
      ×
      
        (
        
          
            y
            2
          
          -
          1
        
        )
      
      ×
      
        (
        
          
            
              D
              1
            
            
              D
              Tot
            
          
          -
          1
        
        )
      
    
  




and N1S=NTotS−N2S. Similarly, N2 and N1 are simply calculated by





  
    N
    2
  
  =
  
    
      N
      Tot
    
    ×
    
      y
      2
    
    ×
    
      (
      
        
          
            D
            1
          
          -
          
            D
            Tot
          
        
        
          D
          1
        
      
      )
    
  




and N1=NTot−N2. Having determined all dependent variables, the number of core atoms: NTotC=NTot−NTotS, N1C=N1−N1S and N2C=N2−N2S, can be determined. Thus, for given catalytic reaction rates (RTot, R1, R2), the corresponding turnover frequencies are well defined and simple to determine according to: TOFTot=RTot/NTotS, TOF1=R1/N1S and TOF2=R2/N2S, respectively.

While particular embodiments according to the invention have been illustrated and described above, those skilled in the art understand that the invention can take a variety of forms and embodiments within the scope of the appended claims.

## Claims

1. A catalyst characterization method comprising the steps of:
providing a catalyst having a small metal particle distribution P1 and a large metal particle distribution P2;
estimating dispersion D1 of the metal atoms in P1;
estimating a total dispersion DTot of the catalyst; and
determining the amount of metal atoms in P2 based on D1 and DTot, wherein the ratio of metal atoms in P2 (N2) to the total number of atoms in the catalyst (NTot) is determined from the relationship:





  
    
      N
      2
    
    
      N
      Tot
    
  
  =
  
    
      
        
          D
          1
        
        -
        
          D
          Tot
        
      
      
        D
        1
      
    
    .

2. The method according to claim 1, wherein the step of estimating the dispersion D1 of the metal particles in P1 comprises the steps of:
obtaining TEM images of a representative sample of the catalyst;
determining the statistical raw moments of a length variable that describes the metal particles in P1 from the TEM images; and
estimating D1 based on the statistical raw moments of the length variable.

3. The method according to claim 2, wherein a dispersion D2 of the metal particles in P2 is determined from the relationship:




  
    D
    2
  
  =
  
    
      (
      
        1
        -
        
          1
          
            y
            2
          
        
      
      )
    
    ×
    
      
        D
        1
      
      .

4. The method according to claim 1, further comprising the steps of:
determining the total number of metal atoms in the catalyst (NTot); and
determining N2 and N1 using:





  
    N
    2
  
  =
  
    
      N
      Tot
    
    ×
    
      y
      2
    
    ×
    
      (
      
        
          
            D
            1
          
          -
          
            D
            Tot
          
        
        
          D
          1
        
      
      )
    
  




and N1=NTot−N2.

5. The method according to claim 4, further comprising the step of determining NTotS, N2S, and N1S using: NTotS=DTot×NTot,




  
    
      N
      2
      S
    
    =
    
      
        N
        Tot
        S
      
      ×
      
        (
        
          
            y
            2
          
          -
          1
        
        )
      
      ×
      
        (
        
          
            
              D
              1
            
            
              D
              Tot
            
          
          -
          1
        
        )
      
    
  
  ,




and N1S=NTotS−N2S.

6. The method according to claim 5, further comprising the step of determining turnover frequencies for given catalytic reaction rates RTot, R1, and R2 using: TOFTot=RTot/NTotS, TOF1=R1/N1S and TOF2=R2/N2S, respectively.

7. The method according to claim 5, further comprising the step of determining the numbers of core atoms NTotC, N1C, and N2C using: NTotC=NTot−NTotS, N1C=N1−N1S and N2C=N2−N2S.

8. The method according to claim 1, wherein the total dispersion of the catalyst sample (DTot) is estimated through chemisorption measurements carried out on the catalyst sample.

9. The method according to claim 8, wherein the total dispersion of the catalyst sample (DTot) is estimated using the following relationship:




  
    
      D
      Tot
    
    =
    
      α
      θ
    
  
  ,




where α is adsorption of the catalyst and θ is saturation coverage of the catalyst sample.

10. A method for evaluating effects of aging on a catalyst, comprising the steps of:
preparing a fresh catalyst, the fresh catalyst having a small metal particle distribution P1 and a large metal particle distribution P2;
determining an amount of metal atoms in P2 based on size measurements of metal particles in the fresh catalyst and chemisorption measurements performed on the fresh catalyst;
subjecting the fresh catalyst to aging; and
determining an amount of metal atoms in P2 after aging, based on size measurements of metal particles in the aged catalyst and chemisorption measurements performed on the aged catalyst,
wherein each step of determining includes the steps of:
estimating a dispersion D1 of metal particles in P1 based on the size measurements of the metal particles in P1;
estimating a total dispersion DTot of the catalyst based on the chemisorption measurements performed on the catalyst; and
determining the number of metal atoms in P2 of the catalyst based on D1 and DTot, wherein the ratio of metal atoms in P2 (N2) to the total number of atoms in the catalyst (NTot) is determined from the relationship:





  
    
      N
      2
    
    
      N
      Tot
    
  
  =
  
    
      
        
          D
          1
        
        -
        
          D
          Tot
        
      
      
        D
        1
      
    
    .

11. A computer-readable medium having stored thereon instructions to be executed in a computer system to determine a physical property of a catalyst having a small metal particle distribution P1 and a large metal particle distribution P2, the instructions causing the computer system to carry out the steps of:
determining a first dispersion value (D1) representative of metal particles in P1;
determining a second dispersion value (DTot) representative of all metal particles in the catalyst; and
determining an amount of metal atoms in P2 based on D1 and DTot, wherein the ratio of metal atoms in P2 (N2) to the total number of atoms in the catalyst (NTot) is determined from the relationship:





  
    
      N
      2
    
    
      N
      Tot
    
  
  =
  
    
      
        
          D
          1
        
        -
        
          D
          Tot
        
      
      
        D
        1
      
    
    .

12. The computer-readable medium according to claim 11, wherein the dispersion value D1 is determined from measured sizes of the metal particles in P1.

13. The computer-readable medium according to claim 11, further comprising the step of receiving inputs specifying a first input value representative of a total adsorption (α) of the catalyst and a second input value representative of a saturation coverage (θ) of the catalyst, wherein the second dispersion value (DTot) is determined from the relationship:




  
    D
    Tot
  
  =
  
    
      α
      θ
    
    .