Particle shape characterization from 2D images
The size distributions corresponding to the three-dimensional (3D) shapes of particles are estimated from a 3D-to-2D projection matrix and a two-dimensional (2D) image of the particles that is obtained using TEM. Two different methods to generate a 3D-to-2D projection matrix are described. The first method determines the matrix coefficients assuming equal probability for all high-symmetry projections. The second method employs a large set of 3D-to-2D projection matrices with randomly generated coefficients satisfying the high-symmetry projection constraint. The second method is a general method to determine 3D-to-2D projection matrices based on the assumption that certain high-symmetry projections are present for the system under investigation.
1. 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.
2. 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)*[M 1 −(β1/β3)*M 3 ]>0, and M 2 −(α2/α1)*M 1 +(α2/α1)*(β1/β3)−(β2/β3)*M 3 >0.
3. 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.
4. 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.
5. 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.
6. The method according to claim 5 , further comprising the step of determining the total number of nanoparticles, M, wherein the selected sets do not include any set for which the distributions of the 3D shapes are such that: the total number of nanoparticles having the first 3D shape is less than 10%*M or greater than 90%*M; or the total number of nanoparticles having the second 3D shape is less than 10%*M or greater than 90%*M; or the total number of nanoparticles having the third 3D shape is less than 10%*M or greater than 90%*M.