Abstract:
Three-dimensional (3D) shapes of particles are characterized from a two-dimensional (2D) image of the particles that is obtained using TEM. The 3D shape characterization method includes the steps of obtaining a 2D image of a batch of nanoparticles, determining 2D shapes of the nanoparticles from the 2D image, and deriving six distributions, each of which corresponds to a 2D shape and a 3D shape associated with the 2D shape. The first size distribution is derived from the nanoparticles having the 2D triangle shape. The second and third size distributions are derived from the nanoparticles having the 2D tetragon shape. The fourth, fifth and sixth size distributions are derived from the nanoparticles having the 2D round shape. Based on these six size distributions, three size distributions, each of which corresponds to one of three 3D shape classes, are estimated. The size distributions corresponding to the 3D shape classes provide a better log-normal distribution than the size distributions corresponding to the 2D shapes.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention generally relates to physical characterization of particles and, more particularly, to characterization of three-dimensional (3D) shapes of nanometer-sized particles from two-dimensional (2D) images of the particles. 
     2. Description of the Related Art 
     The performance of heterogeneous catalysts is highly dependent on their physical properties, including pore size, surface area and morphology of the carrier, and size and weight of the active catalytic components. As a result, techniques for characterizing the physical properties of heterogeneous catalysts become important when assessing their performance. An article by J. Liu, entitled “Advanced Electron Microscopy Characterization of Nanostructured Heterogeneous Catalysts,” Microscopy and Microanalysis, Vol. 10, pp. 55-76 (2004), discusses various advanced electron microscopy techniques used in characterizing model and heterogeneous catalysts, including transmission electron microscopy (TEM), scanning transmission electron microscopy (STEM), and scanning electron microscopy (SEM). 
     It is understood in the art that the shape of the catalyst surface on which catalysis is carried out plays an important role in determining the performance of the heterogeneous catalyst. U.S. Pat. No. 6,746,597, for example, teaches that the crystal surface [111] of a noble metal catalyst material is selective for hydrogenation and dehydrogenation reactions. However, as the size of the catalyst materials have decreased to nanometer levels, it has become difficult to characterize the shape of the catalyst materials. 
     There have been some attempts to characterize the shapes of catalyst materials at the nanometer levels. An article by T. Ahmadi et al. entitled, “Shape-Controlled Synthesis of Colloidal Platinum Nanoparticles,” Science, Vol. 272, pp. 1924-1926 (June 1996), discloses a method in which 3D shapes of the particles were determined by tilting the samples in the TEM. An article by Y. Sun et al. entitled, “Shape-Controlled Synthesis of Gold and Silver Nanoparticles,” Science, Vol. 298, pp. 2176-2179 (December 2002), discloses another method in which 3D shapes of the particles were determined by taking an SEM image of a sample at a tilting angle of 20°. 
     The methods for characterizing the shape of catalyst materials described above have some limitations. The method employed by T. Ahmadi et al. appears to require tilting and enlargement of each of the nanoparticles being analyzed. Such a process would be too time consuming in practice, especially when a large number of nanoparticles that are less than 5 nm are present. The method employed by Y. Sun et al. addresses tilting of very large nanoparticles (˜100 nm) that resemble almost ideal metal cubes. For much smaller size nanoparticles having a number of different non-ideal possible shapes, shape characterization becomes very difficult with existing methods. In fact, the article by J. Liu explains that even for model supported nanoparticles, it is difficult, if not impossible, to obtain statistically meaningful results on the shape distributions of the metal nanoparticles. 
     SUMMARY OF THE INVENTION 
     The present invention provides a technique of characterizing 3D shapes of particles from 2D images of the particles. Using the characterized 3D shapes, a more accurate size distribution of nanoparticles can be obtained, especially when TEM images yield a somewhat small sampling set of nanoparticles. Also, the 3D shape information of the nanoparticles can be used in computer models for estimating chemical softness of the nanoparticles. 
     According to one embodiment, a 2D image of a batch of nanoparticles is obtained using a TEM and the 2D shapes of the nanoparticles are determined from the 2D image. The nanoparticles are classified into one of three 2D shape classes: triangle, tetragon and round, and one of three 3D shape classes. Based on the number of nanoparticles having the 2D triangle shape, the number of nanoparticles that are in the first of the three 3D shape classes is calculated. Based on the number of nanoparticles having the 2D triangle shape and the number of nanoparticles having the 2D tetragon shape, the number of nanoparticles that are in the second of the three 3D shape classes is calculated. Based on the number of nanoparticles having the 2D triangle shape, the number of nanoparticles having the 2D tetragon shape and the number of nanoparticles having the 2D round shape, the number of nanoparticles that are in the third of the three 3D shape classes is calculated. 
     According to another embodiment, a 2D image of a batch of nanoparticles is obtained using a TEM and the 2D shapes of the nanoparticles are determined from the 2D image. Six size distributions are determined from the nanoparticles. The first size distribution is derived from the nanoparticles having the 2D triangle shape. The second and third size distributions are derived from the nanoparticles having the 2D tetragon shape. The fourth, fifth and sixth size distributions are derived from the nanoparticles having the 2D round shape. Based on these six size distributions, three size distributions, each of which corresponds to one of three 3D shape classes, are estimated. The 3D shape classes include a first 3D shape class including a tetrahedron shape and a truncated tetrahedron shape, a second 3D shape class including a cube shape and a cub-octahedron shape, and a third 3D shape class including a truncated octahedron shape. 
    
    
     
       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 the 3D shape characterization method according to a first embodiment of the invention; 
         FIG. 2  shows the association of 2D shapes of a nanoparticle with various 3D shapes; 
         FIG. 3  is a flow diagram illustrating the 3D shape characterization method according to a second embodiment of the invention; 
         FIG. 4  is a table used in determining size distributions corresponding to 2D shapes; 
         FIG. 5  is a flow diagram illustrating the method of determining the size distributions corresponding to 2D shapes; 
         FIGS. 6A-D  shows size distributions corresponding to 2D shapes; 
         FIGS. 7A-D  shows size distributions corresponding to 3D shapes; and 
         FIG. 8  graphically illustrates the goodness of log-normal fits for size distributions corresponding to 3D shapes and size distributions corresponding to 2D shapes. 
     
    
    
     DETAILED DESCRIPTION 
     A shape characterization method according to a first embodiment of the invention is illustrated in the flow diagram of  FIG. 1 . In Step  110 , a TEM sample of a batch of nanoparticles is prepared. For this step, the TEM sample preparation method disclosed in U.S. patent application Ser. No. 11/016,578, entitled “Method of Preparing Nanoparticle Samples,” filed Dec. 17, 2004, incorporated by reference herein in its entirety, may be used. A TEM image of the sample is then obtained (Step  111 ). In Step  112 , the area and the perimeter of each nanoparticle appearing in the TEM image is measured. Then, in Step  113 , the 2D shape of each nanoparticle appearing in the TEM image is determined. The 2D shape is determined to be one of the following major types: tetragon, round, and triangle. The 2D shape determination of a nanoparticle may be performed visually from the TEM image or based on the form factor of the nanoparticle. The form factor of a nanoparticle is derived from the measured area (A) and the measured perimeter (P) of the nanoparticle. The form factor is defined as 4π*A/P^2, which can also be expressed in terms of the harmonic parameter, h, as 2π*h/P, where h=2A/P. The form factor by its definition represents the similarity between 2D shapes and circles, which have a form factor of exactly 1. Nanoparticles having form factors less than or equal to 0.75 are classified as triangles. Nanoparticles having form factors greater than or equal to 0.85 are classified as round. Nanoparticles having form factors between 0.75 and 0.85 are classified as tetragons. 
     In Step  114 , each nanoparticle appearing in the TEM image is associated with one or more 3D shapes. The association of a nanoparticle having a particular 2D shape with one or more of the 3D shapes is shown in  FIG. 2 . The matrix shown in  FIG. 2  is referred to as a 3D-to-2D projection matrix. The association is made based on expected 2D projections of nanoparticles having various 3D shapes. When there is more than one possible 2D projection, weight factors are assigned to each of the possible 2D projections, such that the sum of the weight factors for any one 3D shape is one. The weight factors represent the probability of having a particular 2D projection among all possible projections of the 3D shapes. For example, the probability of having a 2D tetragon shape projected from a cube shape and a cub-octahedron shape is 34%, and that of a 2D round shape projected from a cube shape and a cub-octahedron shape is 66%. 
     The 3D shapes include a 3D tt shape, which is a tetrahedron shape or a truncated tetrahedron shape, a 3D cc shape, which is a cube shape or a cub-octahedron shape, and a 3D to shape, which is a truncated octahedron shape. Each nanoparticle having a 2D triangle shape is associated with a 3D tt shape. Each nanoparticle having a 2D square shape is associated with a 3D cc shape and a 3D tt shape. Each nanoparticle having a 2D round shape is associated with a 3D cc shape, a 3D tt shape and a 3D to shape. 
     In Step  115 , 3D shapes of the nanoparticles in the batch are derived from their 2D shapes based on the relationships between 3D shapes and 2D shapes set forth in the projection matrix. The equations for deriving the 3D shapes based on the 2D shape data are shown below: 
               M   cc     =       1   0.34     ⁢     (       M   Tet     -       0.09   0.82     ⁢     M   Tri         )                     M   tt     =       1   0.82     ⁢     M   Tri                     M   to     =       M   R     -       0.66   0.34     ⁢     M   Tet       +       (       0.32   0.34     ×     0.09   0.82       )     ⁢     M   Tri               
where M cc , M tt  and M to  represent the number of nanoparticles having 3D cc, tt and to shapes, respectively; and M Tet , M R  and M Tri  are measured values that represent the number of nanoparticles having the 2D tetragon, round and triangle shapes, respectively. Since M cc , M tt  and M to  cannot be less than zero, the above equations are valid so long as the measured values of M Tet , M R  and M Tri  meet the following inequalities:
 
     
       
         
           
             
               M 
               Tet 
             
             &gt; 
             
               
                 0.09 
                 0.82 
               
               ⁢ 
               
                 M 
                 Tri 
               
             
           
         
       
       
         
           
             
               M 
               R 
             
             &gt; 
             
               
                 
                   0.66 
                   0.34 
                 
                 ⁢ 
                 
                   M 
                   Tet 
                 
               
               - 
               
                 
                   ( 
                   
                     
                       0.32 
                       0.34 
                     
                     × 
                     
                       0.09 
                       0.82 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   M 
                   Tri 
                 
               
             
           
         
       
     
       FIG. 3  illustrates a shape characterization method according a second embodiment of the invention. In this method, three distributions, one each for the 3D cc shape (G cc ), 3D tt shape (G tt ), and 3D to shape (G to ), that are defined with respect to the number of atoms (N), are determined from six distributions (g 1 , g 2 , g 3 , g 4 , g 5  and g 6 ), that are defined with respect to the number of atoms (N), based on the following matrix equation:
 { G}=[CP′]×{g}   
where:
 
                 {   G   }     =     {             G   cc     ⁡     (   N   )                   G   tt     ⁡     (   N   )                   G   to     ⁡     (   N   )             }       ;                         {   g   }     =     {             g   1     ⁡     (   N   )                   g   2     ⁡     (   N   )                   g   3     ⁡     (   N   )                   g   4     ⁡     (   N   )                   g   5     ⁡     (   N   )                   g   6     ⁡     (   N   )             }       ;     ⁢     
     [           ⁢     CP   ′     ]     =     [           ⁢           CP   ⁡     (     1   ,   1     )           0         CP   ⁡     (     1   ,   3     )           0       0       0           0         CP   ⁡     (     2   ,   2     )           0         CP   ⁡     (     2   ,   4     )           0       1           0       0       0       0         CP   ⁡     (     3   ,   5     )           0         ⁢           ]       ⁢           ;                   CP   ⁡     (     1   ,   1     )       =     1   -       0.09   0.82     ⁢       M   Tri       M   Tet             ;                   CP   ⁡     (     1   ,   3     )       =       0.66   0.34     ⁢     (         M   Tet       M   R       -       0.09   0.82     ⁢       M   Tri       M   R           )         ;                   CP   ⁡     (     2   ,   2     )       =       0.09   0.82     ⁢       M   Tri       M   Tet           ;                   CP   ⁡     (     2   ,   4     )       =       0.09   0.82     ⁢       M   Tri       M   R           ;   and                   CP   ⁡     (     3   ,   5     )       =     1   -       0.66   0.34     ⁢       M   Tet       M   R         +       0.09   0.82     ⁢     0.32   0.34     ⁢       M   Tri       M   R             ,         
and where M Tet , M R  and M Tri  are measured values that represent the total number of nanoparticles having the 2D tetragon, round and triangle shapes, respectively. In order for the matrix equation, {G}=[CP′]×{g}, to hold, the contributions to {G} by {g} must be greater than zero. It then follows that the measured values of M Tet , M R  and M Tri  must meet the same inequalities as above:
 
     
       
         
           
             
               M 
               Tet 
             
             &gt; 
             
               
                 0.09 
                 0.82 
               
               ⁢ 
               
                 M 
                 Tri 
               
             
           
         
       
       
         
           
             
               M 
               R 
             
             &gt; 
             
               
                 
                   0.66 
                   0.34 
                 
                 ⁢ 
                 
                   M 
                   Tet 
                 
               
               - 
               
                 
                   ( 
                   
                     
                       0.32 
                       0.34 
                     
                     × 
                     
                       0.09 
                       0.82 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   M 
                   Tri 
                 
               
             
           
         
       
     
     In Step  310 , a TEM sample of a batch of nanoparticles is prepared. For this step, the TEM sample preparation method disclosed in U.S. patent application Ser. No. 11/016,578 may be used. A TEM image of the sample is then obtained (Step  311 ). In Step  312 , the six distributions (g 1 (N), g 2 (N), g 3 (N), g 4 (N), g 5 (N) and g 6 (N)) are determined in discrete form.  FIG. 4  is a table used in deriving the six distributions in discrete form. 
     The g 1 (N) distribution is derived from the nanoparticles having the 2D triangle shape, and based on the knowledge that the 2D triangle shape is associated with a 3D tt shape. The value corresponding to g 1 (N L →N U ) represents the number of nanoparticles having the 2D triangle shape that have a number of atoms, as calculated from the 2D area of the nanoparticle and the associated 3D tt shape of the nanoparticle, that fall within the range defined by N L  and N U . 
     The g 2 (N) and g 3 (N) distributions are derived from the nanoparticles having the 2D tetragon shape, and based on the knowledge that the 2D tetragon shape is associated with either a 3D cc shape or a 3D tt shape. The value corresponding to g 2 (N L →N U ) represents the number of nanoparticles having the 2D tetragon shape that have a number of atoms, as calculated from the 2D area of the nanoparticle and the associated 3D cc shape of the nanoparticle, that fall within the range defined by N L  and N U . The value corresponding to g 3 (N L →N U ) represents the number of nanoparticles having the 2D tetragon shape that have a number of atoms, as calculated from the 2D area of the nanoparticle and the associated 3D tt shape of the nanoparticle, that fall within the range defined by N L  and N U . 
     The g 4 (N), g 5 (N) and g 6 (N) distributions are derived from the nanoparticles having the 2D round shape, and based on the knowledge that the 2D round shape is associated with a 3D cc shape or a 3D tt shape or a 3D to shape. The value corresponding to g 4 (N L →N U ) represents the number of nanoparticles having the 2D tetragon shape that have a number of atoms, as calculated from the 2D area of the nanoparticle and the associated 3D cc shape of the nanoparticle, that fall within the range defined by N L  and N U . The value corresponding to g 5 (N L →N U ) represents the number of nanoparticles having the 2D tetragon shape that have a number of atoms, as calculated from the 2D area of the nanoparticle and the associated 3D tt shape of the nanoparticle, that fall within the range defined by N L  and N U . The value corresponding to g 6 (N L →N U ) represents the number of nanoparticles having the 2D tetragon shape that have a number of atoms, as calculated from the 2D area of the nanoparticle and the associated 3D to shape of the nanoparticle, that fall within the range defined by N L  and N U . 
       FIG. 5  illustrates Step  312  in additional detail. In Steps  512 - 526 , the nanoparticles appearing in the TEM image are processed one at a time. In Step  526 , a check is made to see if all nanoparticles have been processed. If all nanoparticles have been processed, the process ends. If not, the process returns to Step  512 , where the next nanoparticle to be processed is selected. In Step  513 , the area (A) and the perimeter (P) of the nanoparticle selected in Step  512  are measured, and in Step  514 , its 2D shape is determined. The 2D shape is determined to be one of the following major types: tetragon, round, and triangle. The 2D shape determination of a nanoparticle may be performed visually from the TEM image or based on the form factor of the nanoparticle, in the same manner as in Step  113  of the first embodiment. 
     If the 2D shape is determined to be a triangle in Step  515 , Steps  516 - 517  are carried out. According to the projection matrix of  FIG. 2 , the 2D triangle shape is associated with the 3D tt shape, so in Step  516 , the number of atoms in the nanoparticle determined to have the 2D triangle shape in Step  515  is calculated based on this association. The number of atoms, N 1 , is calculated based on the crystal structure of the element constituting the nanoparticle, its area (A), and the associated 3D shape. For a platinum nanoparticle having the associated 3D tt shape, N 1 =0.040*A 3/2 . In Step  517 , the g 1 (N L →N U ) value corresponding to N 1  is incremented. Step  526  is then executed to see if all nanoparticles have been processed. If all nanoparticles have been processed, the process ends. If not, the process returns to Step  512 , where the next nanoparticle to be processed is selected. 
     If the 2D shape is determined to be a tetragon in Step  518 , Steps  519 - 521  are carried out. According to the projection matrix of  FIG. 2 , the 2D tetragon shape is associated with the 3D cc shape or the 3D tt shape, so in Step  519 , the number of atoms in the nanoparticle determined to have the 2D tetragon shape in Step  518  is calculated twice, once for the association with the 3D cc shape (N 2 ) and once for the association with the 3D tt shape (N 3 ). The number of atoms is calculated based on the crystal structure of the element constituting the nanoparticle, its area (A), and the associated 3D shape. For a platinum nanoparticle having the associated 3D cc shape, N 2 =0.050*A 312 . For a platinum nanoparticle having the associated 3D tt shape, N 3 =0.023*A 3/2 . In Step  520 , the g 2 (N L →N U ) value corresponding to N 2  is incremented, and in Step  521 , the g 3 (N L →N U ) value corresponding to N 3  is incremented. Step  526  is then executed to see if all nanoparticles have been processed. If all nanoparticles have been processed, the process ends. If not, the process returns to Step  512 , where the next nanoparticle to be processed is selected. 
     If the 2D shape is determined to be neither a triangle nor a tetragon, it is determined that the 2D shape is round and Steps  522 - 525  are carried out. According to the projection matrix of  FIG. 2 , the 2D round shape is associated with the 3D cc shape or the 3D tt shape or the 3D to shape, so in Step  522 , the number of atoms in the nanoparticle determined to have the 2D tetragon shape in Step  518  is calculated three times, once for the association with the 3D cc shape (N 4 ) and once for the association with the 3D tt shape (N 5 ) and once for association with the 3D to shape. The number of atoms is calculated based on the crystal structure of the element constituting the nanoparticle, its area (A), and the associated 3D shape. For a platinum nanoparticle having the associated 3D cc shape, N 4 =0.045*A 3/2 . For a platinum nanoparticle having the associated 3D ft shape, N 5 =0.028*A 3/2 . For a platinum nanoparticle having the associated 3D to shape, N 6 =0.036*A 3/2 . In Step  523 , the g 4 (N L →N U ) value corresponding to N 4  is incremented. In Step  524 , the g 5 (N L →N U ) value corresponding to N 5  is incremented. In Step  525 , the g 6 (N L →N U ) value corresponding to N 6  is incremented. Step  526  is then executed to see if all nanoparticles have been processed. If all nanoparticles have been processed, the process ends. If not, the process returns to Step  512 , where the next nanoparticle to be processed is selected. 
     After the six distributions, g 1 (N), g 2 (N), g 3 (N), g 4 (N), g 5 (N) and g 6 (N), have been determined in discrete form in accordance with Steps  512 - 526 , the solution to the equation [CP′]×{g} is computed for each N L →N U  range to obtain G cc , G tt  and G to , values for each N L →N U  range (Step  313 ).  FIGS. 6A-6C  show size distributions corresponding to 2D shapes for a batch of platinum nanoparticles, and  FIGS. 7A-7C  show size distributions corresponding to 3D shapes that were computed in the above manner.  FIG. 6D  shows the combined distribution of the size distributions corresponding to 2D shapes, and  FIG. 7D  shows the combined distribution of the size distributions corresponding to 3D shapes. 
     The distribution shown in  FIG. 7D  provides a better log-normal distribution than the distribution shown in  FIG. 6D , and this is an indication a more accurate model because, according to published literature, the size distribution of particles is expected to have a log-normal distribution. See, e.g., Kiss, L. B., et al., “New Approach to the Origin of Lognormal Size Distributions of Nanoparticles,” Nanotechnology 10 (1999), pp. 25-28; and Granqvist, C. G., et al., “Ultrafine Metal Particles,” Journal of Applied Physics, Vol. 47, No. 5 (May 1976), pp. 2200-2219. With the second embodiment of the present invention, the improvement in the log-normal distribution becomes more pronounced for smaller samples.  FIG. 8  provides a comparison of the log-normal fit between size distributions determined using 2D shapes and size distributions determined using 3D shapes. It is shown that the improvement in the log-normal fit for small datasets (˜100) is greater than for large datasets (˜1000). This is noteworthy because nanoscale modeling that relies on log-normal size distributions, e.g., Monte Carlo simulations and atomistic simulations, becomes much easier and more practicable when smaller datasets are used. 
     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.