Grain size

Looking At A Grain!

Sia Afshari, Global Marketing Manager, EDAX

November seems to be the month when the industry tries to squeeze in as many events as possible before the winter arrives. I have had the opportunity to attend a few events and missed others, however, I want to share with you how much I enjoyed ICOTOM18*!

ICOTOM (International Conference on Texture of Materials) is an international conference held every three years and this year it took place in St. George, Utah, the gateway to Zion National Park.

This was the first time I have ever attended ICOTOM which is, for the most part, a highly technical conference, which deals with the material properties that can be detected and analyzed by Electron Backscatter Diffraction (EBSD) and other diffraction techniques. What stood out to me this year were the depth and degree of technical presentations made at this conference, especially from industry contributors. The presentations were up to date, data driven, and as scientifically sound as any I have ever seen in the past 25 years of attending more than my share of technical conferences.


The industrial adaptation of technology is not new since X-ray diffraction has been utilized for over half a century to evaluate texture properties of crystalline materials. At ICOTOM I was most impressed by the current ‘out of the laboratory’ role of microanalysis, and especially EBSD, for the evaluation of anisotropic materials for quality enhancement.

The embracing of the microanalysis as a tool for product enhancement means that we equipment producers need to develop new and improved systems and software for EBSD applications that will address these industrial requirements. It is essential that all technology providers recognize the evolving market requirements as they develop, so that they can stay relevant and supply current needs. If they can’t do this, then manufacturing entities will find their own solutions!

*In the interests of full disclosure, I should say that EDAX was a sponsor of ICOTOM18 and that my colleagues were part of the organizing committee.

Aimless Wanderin’ – Need a Map?

Dr. Stuart Wright, Senior Scientist, EDAX

In interacting with Rudy Wenk of the University of California Berkeley to get his take on the word “texture” as it pertains to preferred orientation reminds me of some other terminologies with orientation maps that Rudy helped me with several years ago.

Map reconstructed form EBSD data showing the crystal orientation parallel to the sample surface normal

Joe Michael of Sandia National Lab has commented to me a couple of times his objection to the term “IPF map”. As you may know, the term is commonly used to describe a color map reconstructed from OIM data where the color denotes the crystallographic axis aligned with the sample normal as shown below. Joe points out that the term “orientation map” or “crystal direction map” or something similar would be much more appropriate and he is absolutely right.

The reason behind the name “IPF map”, is that I hi-jacked some of my code for drawing inverse pole figures (IPFs) as a basis to start writing the code to create the color-coded maps. Thus, we started using the term internally (it was TSL at the time – prior to EDAX purchasing TSL) and then it leaked out publicly and the name stuck – my apologies to Joe. We later added the ability to color the microstructure based on the crystal direction aligned with any specified sample direction as shown below.

Orientation maps showing the crystal directions aligned with the normal, rolling and transverse directions at the surface of a rolled aluminum sheet.

The idea for this map was germinated from a paper I saw presented by David Dingley where a continuous color coding schemed was devised by assigning red, green and blue to the three axes of Rodrigues-Frank space: D. J. Dingley, A. Day, and A. Bewick (1991) “Application of Microtexture Determination using EBSD to Non Cubic Crystals”, Textures and Microstructures, 14-18, 91-96. In this case, the microstructure had been digitized and a single orientation measured for each grain using EBSD. Unfortunately, I only have gray scale images of these results.

SEM micrograph of nickel, grain orientations in Rodrigues-Frank space and orientation map based on color Rodrigues vector coloring scheme. Source: Link labeled “Full-Text PDF” at www.hindawi.com/archive/1991/631843/abs/

IPF map of recrystallized grains in grain oriented silicon steel from Y. Inokuti, C. Maeda and Y. Ito (1987) “Computer color mapping of configuration of goss grains after an intermediate annealing in grain oriented silicon steel.” Transactions of the Iron and Steel Institute of Japan 27, 139-144.
Source: Link labeled “Full Text PDF button’ at www.jstage.jst.go.jp/article/isijinternational1966/27/4/27_4_302/_article

We didn’t realize it at the time; but, an approach based on the crystallographic direction had already been done in Japan. In this work, the stereographic unit triangle (i.e. an inverse pole figure) was used in a continues color coding scheme were red is assigned to the <110> direction, blue to <111> and yellow to <100> and then points lying between these three corners of the stereographic triangle are combinations of these three colors. This color coding was used to shade grains in digitized maps of the microstructure according to their orientation. Y. Inokuti, C. Maeda and Y. Ito (1986) “Observation of Generation of Secondary Nuclei in a Grain Oriented Silicon Steel Sheet Illustrated by Computer Color Mapping”, Journal of the Japan Institute of Metals, 50, 874-8. The images published in this paper received awards in 1986 by the Japanese Institute of Metals and TMS.

AVA map and pole figure from a quartz sample from “Gries am Brenner” in the Austrian alps south of Innsbruck. The pole figure is for the c-axis. (B. Sander (1950) Einführung in die Gefügekunde der Geologischen Körper: Zweiter Teil Die Korngefüge. Springer-Vienna)
Source: In the last chapter (Back Matter) in the Table of Contents there is a link labeled “>> Download PDF” at link.springer.com/book/10.1007%2F978-3-7091-7759-4

I thought these were the first colored orientation maps constructed until Rudy later corrected me (not the first, nor certainly the last time). He sent me some examples of mappings of orientation onto a microstructure by “hatching” or coloring a pole figure and then using those patterns or colors to shade the microstructure as traced from micrographs. H.-R. Wenk (1965) “Gefügestudie an Quarzknauern und -lagen der Tessiner Kulmination”, Schweiz. Mineralogische und Petrographische Mitteilungen, 45, 467-515 and even earlier in B. Sander (1950) Einführung in die Gefügekunde Springer Verlag. 402-409 . Sanders entitled this type of mapping and analysis as AVA (Achsenvertilungsanalyse auf Deutsch or Axis Distribution Analysis in English).

Such maps were forerunners to the “IPF maps” of today (you could actually call them “PF maps”) to which we are so familiar with. It turns out our wanderin’s in A Search for Structure (Cyril Stanley Smith, 1991, MIT Press) have actually not been “aimless” at all but have helped us gain real insight into that etymologically challenged world of microstructure.

Aimless Wanderin’ in 3D (Part 3)

Dr. Stuart Wright, Senior Scientist, EDAX

In my research on the origins of the term texture to describe preferred lattice orientation I spent some time looking at one of the classic texts on the subject: Bunge’s “red bible” as we called it in our research group in grad school – Texture Analysis in Materials Science Mathematical Methods (1969). As I was reading I found an interesting passage as it relates to where we are with EBSD today:

“In a polycrystalline material crystallites of different shape, size and orientation are generally present. It can thus also occur that regions of different orientation are not separated from one another by unequivocally defined grain boundaries, but that, on the contrary, the orientation changes continuously from one point to another. If one desires to completely describe the crystal orientation of a polycrystalline material, one must specify the relevant orientation g for each point with coordinates x, y, z within the sample:

g=g(x,y,z)           (3.1)

If one writes g in EULER’s angles, this mean explicitly

φ_1=φ_1 (x,y,z);  Φ=Φ(x,y,z);  φ_2=φ_2 (x,y,z);           (3.2)

One thus requires three functions, each of these variables, which are also discontinuous at grain boundaries. Such a representation of the crystal orientation is very complicated. Where therefore observe that it has as yet been experimentally determined in only a very few cases (see, for example, references 139-141, 200-203), and that its mathematical treatment is so difficult that it is not practically applicable.”

I don’t quote these lines to detract in any way from the legacy of Professor Bunge in the field of texture analysis. I did not know Professor Bunge well but in all my interactions with him he was always very patient with my questions and generous with his time. Professor Bunge readily embraced new technology as it advanced texture analysis forward including automated EBSD. I quote this passage to show that the ideas behind what we might today call 3D texture analysis were germinated very early on. The work on Orientation Coherence by Brent Adams I quoted in Part 2 of this series was one of the first to mathematically build on these ideas. Now with serial sectioning via the FIB or other means coupled with EBSD as well as high-energy x-ray diffraction it is possible to realize the experimental side of these ideas in a, perhaps not routine but certainly, tractable manner.

A schematic of the evolution from pole figure-based ODF analysis to EBSD-based orientation maps to 3D texture data.

Others have anticipated these advancements as well. In chapter 2 of Rudy Wenk’s 1985 book entitled Preferred Orientation in Deformed Metal and Rocks: An introduction to Modern Texture Analysis it states:

“Pole figures and fabric diagrams provide information only about the orientation of crystals. It may be desirable to know the relation between the spatial distribution of grains and grain shape with respect to crystallographic orientation. Orientation relations between neighboring grains further defined the fabric and help to elucidate its significance.”

But let us return to the theme of aimless wanderin’s in texture terminology. The title for Chapter 4 of Bunge’s book is “Expansion of Orientation Distribution Functions in Series of Generalized Spherical Harmonics”. This chapter describes a solution the determination of the three-dimensional ODF (orientation distribution function) from two-dimensional pole figures. The chapter has a sub-title “Three-Dimensional Textures”. The three dimensions in this chapter of Bunge’s book are in orientation space (the three Euler Angles). What we call today a 3D texture is actually a 6D description with three dimensions in orientation space and three spatial dimensions (e.g. x, y and z). And those working with High-Energy x-rays have also characterized spatially resolved orientation distributions for in-situ experiments thus adding a seventh dimension of time, temperature, strain, …

It is nice to know in the nearly 50 years since Bunge’s book was published that what can sometimes appear to be aimless wanderin’s with mixed up terminology has actually lead us to higher dimensions of understanding. But, before we take too much credit for these advances in the “metallurgical arts”, as it says on the Google Scholar home page we “stand on the shoulders of giants” who envisioned and laid the groundwork for these advances.

Fine-Tuning the Microstructure

Dr. Stuart Wright, Senior Scientist, EDAX

Since my blog about piano wires back in November 2014, I’ve continued to think about what it all means in terms of music. As I mentioned in my posting, my friend Keith Kopp provided the wires for me to look at. Keith is very generous with his time. I’ve seen him many times help out with music at various church and neighborhood socials. I’m always impressed that someone like Keith can clearly hear when an instrument is even slightly out of tune and also recognize how to fix the problem. I certainly don’t have the ear for that kind of thing. Keith mentioned to me that he can hear the difference between the two wires he supplied to me. I realized quickly I had no hope of picking that up with my insensitive ears. I then realized that Keith was saying he could hear the difference even when the two wires are tuned. I wondered what it was that he was hearing. I turned to Wikipedia for some insight and stumbled across an entry on “inharmonicity” which is “the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency”.  I realized that the sound waves are travelling through the wires at slightly different rates due to elastic anisotropy coupled with grain-to-grain differences in orientation. Thus, while the average pitch of the wire will be in tune there will actually be a spread about that pitch. I might be able to estimate that spread using the principles of elastic anisotropy.

For a single crystal the elastic behavior is anisotropic as is illustrated in a plot for the elastic modulus for an iron single crystal below (courtesy of Megan Frary at Boise State University).

Figure 1

The elastic properties of a single crystal can be expressed in terms of a tensor. This is handy, because rotating the property tensor to reflect the grain orientation with respect to a set of samples axes is fairly straight forward (C is a fourth order elasticity tensor and g is the orientation matrix):

equation

The next step was to simply go to a reference volume and find the single crystal elastic constants for fcc and bcc iron, plug them into OIM Analysis and then “Bob’s your uncle”. However, I learned it wasn’t nearly as straightforward as I thought. Once again, some searching on the internet led me to some papers on first principle calculations of elastic constants and I quickly discovered that estimating elastic constants at room temperature is not as simple as I would have thought. I found several papers by Levente Vitos and co-workers. Professor Vitos was kind enough to teach me a little about this field and after some correspondence with him I decided to use the elastic constants for Fe-Mn found in Zhang, H., Punkkinen, M. P., Johansson, B., & Vitos, L. (2012). Elastic parameters of paramagnetic iron-based alloys from first-principles calculations. Physical Review B, 85:  054107-1. My thinking was that while the absolute values of the constants were probably not accurate, the ratios between the components would be constant enough to at least give me a rough idea of the distribution. (I tried some of the other constants in this paper and the results were all pretty similar.)

I then calculated the distribution of elastic moduli parallel to the longitudinal direction of the wire thinking this might give me an idea of the differences in the distribution of pitches one might hear when a piano wire is struck. The results are shown below – on the left for actual elastic moduli and then on the right for how this might translate to a distribution of pitches. However, this second plot is only a schematic to illustrate my thinking – I have no idea how the elastic moduli variations would translate to pitch variations, the horizontal scale could be much wider, i.e. the flats and sharps could be much farther away from the center pitch and it may also not be a linear relationship. Also the choice of C is completely arbitrary – the actual pitch will depend on the diameter and tension on the wire.

Figure 2
The bad wire (in terms of breakage), which according to Keith’s ear has a clearer sound than the wire less prone to breaking, has a narrower distribution of elastic moduli. Of course, I may be completely off-base as a “fuller” sound may correspond to the broader distribution as well. Perhaps what Keith can hear is the fine balance between clarity and fullness. So if my large set of assumptions is correct then, while I may not be able to hear the difference, I can at least see the difference in the texture data.

 

When a picture is worth only a single word….

Matt Nowell – Product Manager EBSD, EDAX

I’ve been at EDAX, and formerly TSL, for 20 years now, and given that OIM makes such beautiful images, one of the more ironic facts about my career is that I am color blind.  That can sometimes make interpreting colored microstructural images a bit more challenging, and I’m very grateful for the flexibility in coloring within OIM Analysis that the software guys have put in for me (although I think they keep the default first 2 colors in phase maps red and green just because I won the last golf Burrito Open).

Occasionally, however, it’s very easy to read the microstructure.  Take this image for example:

Inverse Pole Figure showing crystallographic orientation.

This image is an Inverse Pole Figure (IPF) map showing the crystallographic orientation.  While I’m sure if one were properly motivated, you might find the right vector in sample space to turn this IPF map into a test for colorblindness, even I can see that it spells out DOE.  This very cool example was created by researchers at Oak Ridge National Laboratory, where they used an additive manufacturing process called Electron Beam Melting (EBM) to spatially modify the solidification texture development in a nickel-based superalloy.  One can easily imagine that if you can control the local microstructure, you can then design and engineer the microstructure to optimize properties spatially for specific loads and applications.  You can learn more about the work at Oak Ridge at: http://3dprint.com/19477/ebm-printing-3d-ornl/ or http://web.ornl.gov/sci/manufacturing/research/additive/.

Other approaches have also been used to write into the microstructure, which I guess is the equivalent to changing the font and font size.

Dimes

In this example from the Else Kooi Laboratory, formerly known as the Dimes Technology Center, at the Delft University of Technology (http://www.dimes.tudelft.nl/EKL/Home.php) a laser beam was used to locally induce recrystallization in polycrystalline silicon.  This approach has been used to develop thin film transistors used in things like liquid crystal displays.  The writing is visible in both the OIM image quality (IQ) map(top) and the grain map (bottom), where adjacent measurement pixels of similar orientations are grouped together as grains, and then these resolved grains are randomly colored to show size and morphology.  That approach gives each letter a different color.

OIM has even been used to read the deformation in metals to recover destroyed serial numbers in metal objects like firearms.  In the images below, an “X” has been stamped into a piece of stainless steel (a), and then polished to visually remove the marker (b).

Figure2

Researchers at NIST have then used OIM to map over the area, with the corresponding IQ map shown here:

ImageJ=1.47v unit=um

The residual plastic deformation present in the microstructure causes a lower EBSD IQ value which is used to image the stamped X.  Years ago EDAX was featured on the TV show CSI for our Orbis µXRF product.  With this forensic application, we are finally ready for a sequel.  More information about this application can be found in a paper by Ryan White and Bob Keller in Forensic Science International (R.M. White and R.R Keller, Restoration of firearm serial numbers with electron backscatter diffraction (EBSD) Forensic Science International 249 (2014) pp 266-270) and at http://www.nist.gov/mml/acmd/ebsd-021115.cfm.

While all of these examples have used OIM to visualize the text within the microstructure, my first introduction to this literary metallurgical engineering was observable by eye:

RexGG

This sample was created for the International Conference on Grain Growth (ICGG), held back in 1995.  In keeping with theme of this conference, the characters were placed by locally inhibiting the grain growth while the bulk material was recrystallized.

So, while these pictures many not be worth a thousand words, they do contain at least a thousand grains.  The fact that a few words have been engineered into the microstructure by various means is pretty incredible.

Many thanks to Ryan Dehoff at Oak Ridge National Lab, Ryan White and Bob Keller at NIST, and David Field at Washington State University for allowing the use of their images for this blog.

Resolving Matters

Dr. René de Kloe, Applications Specialist, EDAX

Every now and then a new critical parameter in EBSD analysis comes up. This parameter is often related to a new trend in materials science research or industrial development. One of the latest buzz-words is “nano”. Things are getting smaller and smaller and EBSD technology has to keep up to be able to provide the microstructural answers. This drive to investigate the tiniest details led, for example, to the emergence of transmission EBSD. But it seems that not everyone means the same thing when they talk about nano; Where does nano begin and where does it end? For example, is “nano” anything smaller than 1 micron, or anything below 100 nm, or perhaps only things smaller than 10 nm? The answer is of course: all of them. The feature scale really depends on your application and field of science. For example in natural geological materials, grain sizes smaller than 1 micron are not very common and such rocks could be described as nano-structures. But in semiconductor or photovoltaic applications nano may range from single atomic layers to perhaps 100 nm.

In the EBSD community the emergence of nano-analysis has sprouted the use of another buzz-word: ‘resolution’. But exactly as with nano, ‘resolution’ can have several different meanings depending on who you talk to. So let’s start with the basics, what is resolution?

resolution
Pronunciation: /rɛzəˈluːʃ(ə)n/
The smallest interval measurable by a telescope or other scientific instrument; the resolving power.
The degree of detail visible in a photographic or television image
A firm decision to do or not to do something
The action of solving a problem or contentious matter
(from http://www.oxforddictionaries.com/definition/english/resolution)

Of these definitions the first two have obvious relevance to EBSD analysis, but at the same time define something completely different. The first one deals with the feature size on the sample, whose limits are defined by the combination of SEM beam settings and physical sample properties. The second one describes the amount of detail observable in a diffraction pattern collected from an area of a sample, which is affected by the number of pixels available on the imaging sensor. These two definitions are easily confused, but are not directly related.

And of course there is a third definition of resolving something which is a well-known strength of EBSD. It is the power to resolve the difference between different phases. For example here is a geological material, a granitic rock with several minerals with low-symmetry crystal structures. In order to resolve the phase differences, the analysis of a rock like this does not require any high resolution settings. A detector resolution of only 120 pixels and step size of 0.5 micron was sufficient.

Figure 1
But things are not always what they seem at first glance as is illustrated by the sequence of images below. All these maps are collected with the same low detector resolution of only 80 pixels, but with very different spatial resolution, or step size, on the sample. The sample is a piece of the Gibeon meteorite, an iron meteorite that was discovered in Namibia in 1836. The structure is very coarse grained and the individual grains can easily be seen with the naked eye.

Field of View_2

But looks can be deceiving. My first attempt to characterize the structure resulted in a speckled map that looked pretty bad and made me doubt my polishing skills. Subsequent maps with higher and higher spatial resolution (i.e. smaller step size) started to resolve a fine-grained structure with many grains being smaller than 100 nm. So for this example, high spatial resolution maps were obtained using low resolution camera settings.

Figure 3

For phase identification and characterization of deformation microstructures the pixel resolution of the detector becomes more important, but within reason. For phase identification you want to be able to identify small details in a pattern and this is also helpful when you are looking at small orientation changes due to dislocation structures in a material. This biotite pattern displays a pseudosymmetry where the difference between similar orientations is defined by the position of some relatively weak bands in the pattern. The correct indexing result is outlined in red.

Figure 4

Left: Biotite patterns with pseudosymmetric indexing results – red pattern is correct
Right: Kernel average misorientation map in deformed iron alloy illustrating precise locations of subgrain boundaries

In such cases, what you need to identify the difference between these orientations is having enough pixels in the band pattern across the weaker and thinner bands to be able to detect them automatically. Therefore the band detection capabilties dictate the resolution that you need to resolve the difference between these two orientations and not the number of pixels that you have available on your EBSD detector. Typically a pattern of 200 pixels is sufficient on any material. And that resolution is also enough to be able to measure orientation changes down to 0.05 degrees, which allows accurate identification of subgrain structures as shown in the kernel misorientation map.

If even smaller orientation changes are the target of your analysis or if you want to measure minute shape changes of the crystal lattice due to elastic strains, the standard band detect routines are insufficient. For such analyses you would need to use a pattern with more pixels and a dedicated technique based on cross correlation of sections of the diffraction patterns. For this tool, using many more pixels would appear to be better. But keep in mind that other variables in the system geometry such as the exact detector positioning, signal-to-noise level, lens artifacts, or pattern center calibration errors can introduce uncertainties that may easily exceed the improvements gained by using more pixels. In practice, patterns with 480 pixels or more have been used successfully. What you see is not always what you get.

The above examples have highlighted a number of different uses of the word resolution and it appears that you have to be pretty careful in describing what you really want to accomplish. Going for a high resolution camera will not give you better spatial resolution in your EBSD maps. Similarly low resolution maps may be measured using a high resolution camera which shows that the number of pixels on your detector is totally independent of the spatial resolution that may be obtained on your samples.

Therefore to conclude I would like to appeal to the last definition of resolution mentioned above: The action of solving a problem or contentious matter.
With the different meanings of resolution clouding the waters and creating confusion I would like to propose the use of “high resolution EBSD ” to describe the application of mapping with small step-size to resolve the fine details in a material. “High precision EBSD” would then describe the application to map out small orientation changes in a material in order to investigate and understand the deformation mechanisms and finally “high definition EBSD” to describe the technique of investigating minute changes in pattern geometry to characterize (elastic) strains in the crystal lattice by applying cross-correlation methods.

My $0,02

Time For A Change – New Perspectives in Grain Size Analysis

EBSD geometry

Dr. Stuart Wright, Senior Scientist EBSD, EDAX

For better or worse, I’ve been long involved in trying to set forth some guidelines for the measurement of grain size using EBSD. This involvement has included serving on some of the standard committees, advising customers and reading, reviewing and publishing papers [1]. However, I’ve always felt unsettled about the outcome of those efforts. Part of that discomfort comes from the fear that an inexperienced EBSD user could be misled to an incorrect conclusion based on using a canned procedure. However, some recent experiments have given me more confidence in obtaining reliable grain size statistics using EBSD.

There are a couple of challenges associated with the measurement of grain size using EBSD. First, there are the usual factors associated with collecting good EBSD data: determining a polishing procedure that will produce good patterns, finding good SEM and EBSD camera settings, ensuring the sample is in the expected geometry [2]. However, I will assume these factors are well under control so that good quality EBSD data is obtained, that is data with a high indexing rate so that very little clean-up is required. Now, the challenge is making sure the choices for the parameters associated with grain size are appropriate. The first of these is the grain tolerance angle. I will focus on recrystallized materials at this point so the choice of grain tolerance angle is not as critical as for a deformed material. In the case of deformed materials, I’ve found you need to experiment with different values to get a feel for the best choice. However, for recrystallized material using the default value of 5° generally works well. The next critical parameter is the choice of the minimum grain size in terms of the number of grid points. Once this value is selected the analysis software will exclude grains with less than this value from the grain size distribution.

There are several approaches to selecting a good value for the minimum grain size – I will call it Nmin for brevity. It is entirely possible that a single point in an OIM scan could be a grain. Imagine the grain structure in three dimensions; it is easy to imagine a single scan point being associated with the top or bottom tip of a grain just intersecting the sampling plane. Thus a single point in an OIM scan could represent a grain especially if that point has a high confidence index. Of course, we have also seen many cases where an individual point is located at a grain boundary with a low confidence index. This arises because the resulting pattern is a combination of two competing patterns from the two grains on either side of the grain boundary (or three at a triple point) and thus the Hough transform will find bands from all the competing patterns resulting in an incorrect indexing [3]. Another consideration that has gone in to the choice of Nmin is is how well a shape can be reconstructed from a given number of grid points. For example, consider a circle. The following figure shows how well a circle can be approximated by a given number of grid points. To approximate the area of circle with less than 5% error requires about 30 points. However, if very many grains are measured the errors in approximating a given grain will be averaged out so I personally don’t think this argument should carry much weight in the choice of Nmin. In fact, I hope by the end of these ramblings that I will have convinced you that if using the right approach, the choice of Nmin is not as important as one might presume.

Grid points in a circle

Figure 1: Approximation of the area of a circle using a square grid and a hexagonal grid.

I’ve done some experiments with a very nice set of data provided by my colleague Matt Nowell. These measurements were performed as part of a Round-Robin test conducted for the development of ISO Guidelines for the measurement of grain size using EBSD [4]. The figure below shows five data sets measured on one sample merged together. I have excluded the grains touching the edges from my analysis. On average, each field contains 750 grains. The step size is 2 microns and each scan contains almost 250,000 data points. The total number of grains analyzed was 3748 grains. Prior to any analysis a grain dilation cleanup was applied to each individual dataset using a 5° tolerance angle, two pixels for the minimum grain size and the grains were required to span multiple grid rows. The percentage of points changed in the clean-up process was only 0.7% confirming the high fidelity of the data.

Figure 2: Grain map of the merged data.

Figure 2: Grain map of the merged data.

The grain size distributions for the individual data sets and the merged data set are shown in Figure 3 using both a linear axis and a log axis for the grain diameter (the horizontal axis). The vertical axis is the usual area fraction.

Stu June3a Stu June3b
Figure 3: Log and linear grain size distributions of the individual data sets (colors) and the merged data set (black) 

The next step in such analyses is to calculate the average grain size (I will use the diameter). This is simply done by adding up the diameters of all the grains and then dividing by the total number of grains. Figure 4 shows the average grain diameter overlaid on the distribution for the merged data.

Figure 4: Area fraction grain diameter distribution for the merged data overlaid with the average diameter.

Figure 4: Area fraction grain diameter distribution for the merged data overlaid with the average diameter.

As usual, the location of the average diameter does not correlate well with the center of the distribution. I always find this disconcerting. A common response is to simply pass this off to more data being needed especially since the distribution curve is still a bit jagged even with 3748 grains. You can experiment with the number of bins used to create the distribution as well as the grain definition parameters but the average grain size is always a bit left of where you think it should be. The reason for this mismatch is that the distribution is given in area fraction but the average is calculated as a number average. If you plot the distribution as a number fraction then the number average appears to fit the data better. However, what originally approximated a Gaussian curve in the area fraction plot now becomes skewed to the left in the number fraction plot.

Figure 5: Number fraction grain diameter distribution for the merged data overlaid with the average diameter.

Figure 5: Number fraction grain diameter distribution for the merged data overlaid with the average diameter.

Another solution to overcoming this mismatch between the average and the peak in the distribution is, instead of using a number averaging approach, using an area weighted averaging approach. The area weighted average is a relatively simple calculation given by:
Equation
Where n is the number of grains, Ai is the area of grain i and di is the diameter.
The area weighted averaging leads to an average value that matches that approximated “by-eye”.

Figure 6: Area fraction grain diameter distribution for the merged data overlaid with the average diameter calculated using number averaging and area weighted averaging.

Figure 6: Area fraction grain diameter distribution for the merged data overlaid with the average diameter calculated using number averaging and area weighted averaging.

In fact, I’ve found that the weighted area average provides a very good seed to any automatic fitting of the distribution data. The advantage of the area average relative to a curve-fit determination, is that the area average is calculated from the raw grain size data independent of the binning used to create the distribution plot.

Figure 7: Gaussian distributions for the merged data grain diameter distributions.

Figure 7: Gaussian distributions for the merged data grain diameter distributions.

So why do I bring this whole area weighted averaging approach up when the accepted approach is the number averaging approach? What does it have to do with selecting an appropriate Nmin ? The reason lies in the following plot of the area and number averages versus Nmin. In this plot the area average appears less sensitive to the choice of Nmin than the number average. This observation has held up on across many different samples on which I’ve performed grains size analysis.

Plot of average grain diameter as a function of the choice of Nmin

Figure 8: Plot of average grain diameter as a function of the choice of Nmin.

The sensitivity of the area average to Nmin seems to increase in the plot at about a value of 50. If we look at this same data plotted on a log scale the increase in sensitivity at higher Nmin become more apparent. These curves start at a value of Nmin = 5 due to the fact that this was the minimum grain size I selected for the grain dilation clean-up.

Figure 9: Log plot of average grain diameter as a function of the choice of Nmin.

Figure 9: Log plot of average grain diameter as a function of the choice of Nmin

The argument among the grain size community is to select a much larger Nmin value than that typically used in the EBSD community (2 to 5 points). The ASTM standard states that the minimum grain that should be included should contain 100 points and that the average grain should contain at least 500 points [4]. This standard evolved to this methodology primarily for historical consistency with optical measurements. However, these results show that the choice of Nmin is not so critical in the determination of the average value provided it is not too large. Another way to look at this is to plot the data differently. If we change the horizontal axis to Nmin divided by the number of points corresponding to the average grain size and change the vertical axis to be the average diameter divided by the average diameter at Nmin equal to five then we get the following plot.

Figure 10: The normalized average grain size as a function of the ration of the number of points in the minimum sized grain to the number of points in the average sized grain.

Figure 10: The normalized average grain size as a function of the ration of the number of points in the minimum sized grain to the number of points in the average sized grain.

From this plot it appears that using a small number for Nmin is fine. The key is not to pick too large of a number – it should not exceed 10% of the number of points in the averaged sized grain. However, this is with the proviso that area weighted averaging is used in the analysis. The contribution of the small grains to the average is very much less for area weighted averaging as opposed to number averaging. Thus using a small number, even just 1, will lead to nearly the same result as using a larger value such as the value of 5 I used in this example. Because we (the EBSD community) can very much more definitively calculate grains of any size relative to traditional optical measurements, I believe this approach is the correct one despite its inconsistency with the historical approach.

References
[1] Wright, S. I. (2010). “A Parametric Study of Electron Backscatter Diffraction based Grain Size Measurements.” Practical Metallography 47: 16-33.
[2] Nolze, G. (2007). “Image distortions in SEM and their influences on EBSD measurements.” Ultramicroscopy 107: 172-183.
[3] Wright, S. I., M. M. Nowell, R. de Kloe and L. Chan (2014). “Orientation Precision of Electron Backscatter Diffraction Measurements Near Grain Boundaries.” Microscopy and Microanalysis: (in press but available on First View).
[4] ISO 13067: Microbeam analysis – Electron backscatter diffraction – Measurement of average grain size.
[5] ASTM E2627-10: Standard practice for determining average grain size using electron backscatter diffraction (EBSD in) fully recrystallized polycrystalline materials.