The method according to claim 4, further comprising the step of generating a distribution corresponding to a third 3D shape based on the matrix and the numbers in the selected sets.Ħ. The method according to claim 3, further comprising the step of generating a distribution corresponding to a second 3D shape based on the matrix and the numbers in the selected sets.ĥ.
A method of estimating a distribution of nanoparticles based on a matrix for mapping at least three 2D shapes onto at least three 3D shapes, comprising the steps of: establishing matrix variables that define the relationships between the 2D shapes and the 3D shapes generating multiple sets of numbers, wherein the numbers in each of the multiple sets are assigned to the matrix variables to fix the relationships between the 2D shapes and the 3D shapes, wherein each set defines five numbers, α1, α2, β1, β2, β3 computing the mean and the standard deviation for each of the numbers, α1, α2, β1, β2, β3 selecting those sets whose numbers all lie within their computed mean ± computed standard deviation and generating a distribution corresponding to a first 3D shape based on the matrix and the numbers in the selected sets.Ĥ. The method according to claim 1, further comprising the step of determining the number of 2D tetragon shapes, M 1, the number of 2D round shapes, M 2, and number of 2D triangle or unique shapes, M 3 in the particles, wherein the numbers in each of the selected sets satisfy the following inequalities: (1/α1)*>0, and M 2−(α2/α1)*M 1+(α2/α1)*(β1/β3)−(β2/β3)*M 3>0.ģ. A method of estimating 3D shape distributions of particles, comprising the steps of: defining variable relationships between 2D shapes and 3D shapes generating multiple sets of numbers, wherein the numbers in each of the multiple sets fix the variable relationships between the 2D shapes and the 3D shapes, wherein each of the multiple sets includes five numbers, and three of the five numbers, α1, β1, β2, in each of the multiple sets are randomly generated, and the remaining two of the five numbers are derived using the following equations: α2=1−α1, and β3=1−β1−β2 selecting a number of sets from the multiple sets and estimating the 3D shape distributions of the particles based on the selected sets.Ģ. Granqvist et al., “Ultrafine Metal Particles,” Journal of Applied Physics, May 1976, vol. 2176-2179.Īmanda Crowell, “Shaping Nanoparticles,” Research Horizons: Georgia Institute of Technology, 1996, vol. Yugang Sun et al., “Shape-Controlled Synthesis of Gold and Silver Nanoparticles,” Science, Dec. Ahmadi et al., “Shape-Controlled Synthesis of Colloidal Platinum Nanoparticles,” Science, Jun. Kiss et al., “New Approach to the Origin of Lognormal Size Distributions of Nanoparticles,” Nanotechnology, 1999, vol. Jingyue Liu, “Advanced Electron Microscopy Characterization of Nanostructured Heterogeneous Catalysts,” Microsc.
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