Particle Size Distribution (PSD)

P_10: Initial Particle Size Distribution of Fe-Cr-C

On the Precipitation Calculator you can set the pre-exisiting particle size distribution (PSD) parameters in the Show Details section. The size distribution can be generated from a probability distribution function, by importing a file, calibrating a probability distribution function to experimental data, or by manually entering information into a table. Uni-modal distributions can be created within the Graphical Mode interface. Pre-existing multimodal distributions can be loaded from a file.

Accessing the 'Preexisting Size Distribution' Settings

To the right of the Precipitate Phase settings fields, click Show Details.
At the bottom of the Precipitate Phase section, click to select the Preexisting size distribution checkbox. Then click Edit Particle Size Distribution.
The Preexisting Particle Size Distribution window opens as shown for example P_10 CEMENTITE phase.

Defining the Preexisting Size Distribution

Compositions

Choose a Dependent component from the list and enter the composition for the other component.

Length Unit

Choose a Length unit: Meter, Micrometer, Nanometer, or Ångström.

Amount of Phase

For all options, choose an Amount of phase: Volume percent or Volume fraction and then enter a number in the field.

Input Type

Choose an Input type: Generate 3D PSD from distribution function, From File, or Approximate the 3D PSD from experimental data. See the following sections for the details about each setting.

Generate 3D PSD from Distribution Function

This option allows the generation of the size distribution from a list of probability distribution functions. The Distribution types include LSW, Normal, Log normal, and Weibull.

Mean radius The mean particle radius.
Sigma The standard deviation for the chosen length unit.
Alpha A dimensionless shape factor that should vary between 1.5 and 10.
Fraction offset The fraction offset applied to the mean radius to shift the start of the distribution. This must be greater than or equal to zero, and less than 0.95.

Once the Distribution is defined, click Generate to create the size distribution.

You can also click in the table to manually enter numbers into the cells rather than use one of the provided probability distribution functions. The radius values must monotonically increase in size.

From File

This option lets you load the initial size distribution from a file. Click Import and navigate to the file containing the required information and click Open. This file can be in .xls, .xlsx, .csv, or .txt formats. The file should consist of two columns with values where the first column contains radius data and the second contains number density data.

The example P_10 (available from the Example files → Precipitation Module - TC-PRISMA folder) shows this for the precipitate phase M23C6. There is also a file in this folder called P_10_Precipitation_Initial_PSD_FeCrC_psd_3D_M23C6.csv that is an example of a correctly formatted text file.

The imported size distribution is normalized into a probability distribution function by ensuring that the zeroth moment is unity. If you wish to import a histogram that has un-even bin spacing, it is necessary to convert the Y-axis counts of the histogram to a count density first to avoid changes to the shape of the distribution. The count density is obtained simply by dividing the count of a bin by the bin's width.

Approximate the 3D PSD from Experimental Data

The Approximate the 3D PSD from experimental dataInput type allows you to calibrate a continuous probability distribution function to an experimentally measured size distribution, and account for stereological considerations. There are additional settings related to this.

Dimensionality of Exp. Data and Import Exp. Data

The Dimensionality of exp. data setting must be defined.

For precipitate radius measurements that are a good approximation of the 3D size of the particles, select 3D. Measurements from quantitative image analysis of Transmission Electron Micrographs (TEM) provide 3D size data, however the precipitates can still be truncated within the TEM sample foil thickness.
2D data is obtained from a cross section taken through the microstructure. Examples include micrographs taken from Scanning Electron Microscopy (SEM), and Optical Microscopy (OM). The radius measurements are obtained from calculating the equivalent radius of a circle from the areas of identified objects from quantitative image analysis of the micrographs. It is possible that the samples of interest are over-etched, revealing the full size of the precipitates. In this case the data may be more descriptive of the 3D size, however smaller particles are removed by the etching, distorting the obtained size distribution.
1D data is obtained from applying the linear intercept method to micrographs descriptive of a cross section through the dispersion. A half-linear intercept is used in this calculation, which is analogous to the radius of the particle, opposed to the diameter.

Click Import exp. data to import the experimental data. Navigate to the file containing the required information and click Open. This file can be in .xls, .xlsx, .csv, or .txt formats. The file should consist of two columns with values where the first column contains radius data and the second contains number density data.

The example P_10 (available from the Example files → Precipitation Module - TC-PRISMA folder) shows this for the precipitate phase M7C3. There is also a file in this folder called P_10_Precipitation_Initial_PSD_FeCrC_psd_1D_M7C3.csv that is an example of a correctly formatted text file.

Distribution

Choose a Distribution: LSW (Lifshitz-Slyozov-Wagner), Normal, Log normal, or Weibull.

Click Generate to calibrate the chosen Distribution to capture the Import exp. data accounting for the chosen stereological Dimensionality of exp. data.

For 3D data, there are two datasets shown in the preview; the experimental data and the 3D model data. For 2D and 1D data, the sectioned model data is also shown and compared with the experimental data

The Optimization error is output into the text-field highlighted below. This can be used to determine the best distribution type to use for a given dataset.