Published by Buehler Ltd. Volume 1, Issue 5

USING MICROSTRUCTURAL ANALYSIS TO SOLVE PRACTICAL PROBLEMS

Written by:George Vander VoortDirector, Research andTechnologyBuehler Ltd.,Lake Bluff, Illinois,U.S.A.

Introduction to Quantitative MetallographyQuantitative Metallography –30 Years AgoFor most of its history, metallographic obser-vations have been largely qualitative in na-ture. The structure might be described as beingrelatively coarse or fine, or layered, or uni-form. Particles might be referred to as beingglobular or spheroidal, lamellar, acicular, orblocky. Microstructures were single-phase orduplex, and so forth.

Thirty years ago when I entered industry, chartratings and visual examinations were the mainapproach toward quantification. I can well re-member the mill metallographers looking atspheroidized carbide tool steel structures andstating that it was, for example, 95%spheroidized (many raters would never say100%, just as some teachers would nevergrade an essay at 100%!) or that it was 60%spheroidized and 40% lamellar tending tospheroidize. Or, without looking at the chart (aseasoned rater never did), they would pro-nounce that the grain size was, for example,100% 6 to 8 or perhaps 70% 8 and 30% 3 to 5if it was duplex in appearance.

As a novice metallurgist, I was quite impressedby these pronouncements and tried to repeatthe practice on my own. But, I found myselfunable to repeat such ratings, if done severalweeks apart, unless I made my estimates verybroad. Later, I tried submitting the same speci-mens to different metallographers or to thesame person at different times. I quickly learnedthat their repeatability/reproducibility wasn’tthat much better than mine. (Don’t ever let amill metallographer know that you are check-ing him/her!)

The greatest mystery to me, however, was in-clusion chart ratings. I was very impressed thatthey could scan the 160 square mm area in afew minutes and, without taking any notes, jotdown worst-field ratings. Absolute black magic!I never could do this. I had to use a scale andmeasure stringer lengths or count inclusionswhen I saw a field that looked like it had a highinclusion content. And I had to keep notes. But,I was a young, “hot-shot” metallurgical engi-neer – why couldn’t I do as well? Was I overlydense? Did I go to too many parties and toofew classes?

Naturally, I tried testing their ability to repro-duce test results, and I found out that maybe Iwasn’t as slow as I thought. They didn’t do thatwell reproducing their own results. Later, whenI became active with ASTM Committee E-4, Ifound that inclusion ratings on round-robins,which had been tried numerous times, werenotoriously non-reproducible. Well, that wasgood for soothing my shattered ego, but it didnot solve my problems of describing structures.

I Discover StereologyIn the early 1970’s, I came across some ar-ticles and a book, Quantitative Stereology,written by the late Ervin E. Underwood(Addison-Wesley Publishing Co., 1970). Thiswas the beginning of a lifelong interest in quan-tifying microstructures and a long time friend-ship with a fascinating gentleman, ErvUnderwood.

Stereology is used to quantify matrix micro-structures, as opposed to standard metrologytechniques for measuring case depth, platingthickness, or particle size. Microstructural mea-surements are made on a two-dimensionalplane-of-polish through a three-dimensionalopaque metal. Stereology converts these 2-Dmeasurements into 3-D estimates of micro-structural parameters. Most procedures arevery simple to use, but there are special con-siderations for their validity.

The late Erv Underwood at his home in BigCanoe, Georgia

Metallographers commonly perform metrology type measure-ments; e.g., when measuring case depths, decarburization orplating/coating thickness. A scale is placed over the struc-ture, and the depth or thickness perpendicular to the surfaceis measured. For example, ASTM E 1077 describes the mea-surement of decarburization of steel specimens using suchmeasurements.

NomenclatureApplication of stereology has been hampered by confusiondue to the use of different mathematical symbols for the sameparameters. To minimize this problem, the International Soci-ety for Stereology has promoted a standard nomenclaturewhich is constantly evolving as new approaches are devel-oped. The most basic symbols are:

P = PointL = LineA = AreaS = SurfaceV = VolumeN = Number

These symbols can be combined in a number of ways togenerate different symbols. For example, PP represents thepoint fraction; that is, the fraction of grid points lying in a phaseof interest. While A and S seem to be the same, A is for a flatsurface while S is for a curved surface. Thus, SV representsthe grain boundary surface area per unit volume. NA is thenumber of particles per unit area while NV is the number perunit volume.

Phase ProportionsOne of the most common measurements, determination ofthe amount of phases present, can be done using three dif-ferent methods. Areal analysis, developed by Delesse in 1848,says that the area percent of a phase on a 2-D plane is equalto its volumetric percent, that is, AA = VV. However, measuringthe area of second phases is very tedious unless they arequite coarse. Lineal analysis, developed by Rosiwal in 1898,says that the lineal fraction of test lines in a phase on the 2-Dplane is equal to its volumetric percentage, that is, LL = VV. Thisis easier to determine but still rather tedious.

Starting around 1930, several workers in different fields andcountries showed that the percentage of points on a test gridlying in the phase of interest was equal to the volumetricpercentage, that is, PP = VV. Of the three methods, this is themost efficient technique; that is, it produces the best precisionfor the least effort when done manually. The point countingtechnique is described fully in ASTM E 562 (also ISO 9042).Image analyzers use essentially the same procedure; that is,the amount of a phase (usually called the area fraction orvolume fraction even if it actually is a point fraction) is deter-mined by the number of picture elements or “pixels” in thephase of interest divided by the total number of pixels; i.e., PP,expressed usually as a percentage.

Point Counting ExampleASTM E 562 describes the point counting procedure for de-termining the amount of second-phase constituents. A gridwith systematically spaced points (e.g., 10 rows of 10 equallyspaced points) is superimposed over the structure, either onan eyepiece reticle or a plastic sheet placed over or behind aground glass projection screen or on a TV monitor screen.The points are usually drawn as fine perpendicular crossinglines and the “point” is the intersection of the two lines. This isdone because actual points would be very difficult to see.

The optimum point density for manual point counting is usu-ally determined from 3/VV where the volume fraction is a frac-tion (not a percent). If the volume fraction is 0.5 (50%), thenthe optimum grid point density is 6. On the other hand, if thevolume fraction is 0.01 (1%), the optimum point density is300. The point fraction is the ratio of the points in the phase ofinterest to the number of grid points. Some people like to usea 100 point grid for all work since the division is unnecessary.Points falling on the interface are counted as ½ a hit. For bestmanual results we need to sample more fields and do as littlework as possible on each field measurement (the adage, “domore, less well”). The field-to-field variability has a greaterinfluence on measurement precision than the counting preci-sion on a given field.

The microstructure above shows the beta phase in Muntzmetal (Cu-40% Zn) preferentially colored by Klemm’s I re-agent while the alpha matrix is unaffected - ideal conditionsfor point counting. Since there is less β than α, we will countthe number of times the points fall in the colored β grains. Theamount of α is simply 100 - %β. As you can see, we havesuperimposed a 64-point test grid (8 rows of 8 points) overthe structure and we have 15 hits and 4 tangent hits. The pointfraction (volume fraction) is 17/64 = 0.266 or 26.6%.

The point counting grid would be placed randomly over thestructure a number of times so that the point fraction is deter-mined for a number of fields. The necessary number of fieldsto yield a 10% relative accuracy varies inversely with thevolume fraction (the lower the volume fraction, the greater thenumber of fields, i.e., the greater the total number of appliedgrid points).

Grain SizeGrain size is perhaps the most commonly performed micro-structural measurement, although chart ratings are more com-monly done than actual measurements (this is changing). Arecent ASTM inter-laboratory “round-robin” showed that chartratings of grain size are biased; that is, the ASTM grain sizenumber is 0.5 to 1 unit too low (See Appendix X1 of E 112 -96). No bias existed when planimetric measurements werecompared to intercept measurements by the same raters.

The ASTM grain size number, G , is defined as:

n = 2 G-1

where n is the number of grains per square inch at 100X. Toconvert n to NA (the number of grains per square mm at IX),multiply n by 15.5. The four ASTM grain size charts showgraded series of grain structures of different types.

Grain size can be measured by the planimetric method(developed by Zay Jeffries in 1916) or by the intercept method(developed by Emil Heyn in 1904). In the planimetric method,a count is made of the number of grains completely within a

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circle of known area and half of the number of grains inter-sected by the circle to obtain NA. Then, NA is related to G. Thismethod is slow when done manually because the grains mustbe marked when counted to obtain an accurate count.

In the intercept method, either straight lines, curved lines, orcircles are placed over the structure and a count is made ofeither the number of grain boundary intersections, P, or thenumber of grains intercepted, N, by the line. P or N is dividedby the true line length, LT, to determine PL or NL, the number ofintersections or interceptions per unit length (for a singlephased structure). The reciprocal of PL or NL gives the meanlineal intercept length, l,

l = 1/NL = 1/PL

a measure of grain size that can be converted to a G value.The intercept method is more efficient than the planimetricmethod yielding acceptable measurement precision (<10%relative accuracy) in much less time. ASTM E 112 contains acomplete description of these methods. A major revision of E112 was approved in 1995.

Grain Size ExampleASTM E 112 describes the manual measurement of grainsize for structures with a single grain size distribution whileASTM E 1382 covers image analysis measurements. Notethat E 112 was heavily revised in 1995 (additional minorchanges in 1996), so it is best to read the latest version. Grainsize can be measured using either the planimetric or the in-tercept methods. In the examples, we will simplify the ap-proach slightly to illustrate the methods. Additional fieldsampling should be done to obtain good statistical data.

In the planimetric method, ASTM recommends using a testcircle with a diameter of 79.8 mm (5000 sq. mm area) placedrandomly over the grain structure. To obtain an accurate countof the number of grains inside the circle and the number inter-cepted by the circle, we must mark the grains on the templateas we count which makes this method slow (although this isnot a problem by image analysis). We must know the

magnification of the image as well.

The image below shows the grain structure at 200X of a low-carbon sheet steel after color etching. A circle of known size(64.4mm diameter) has been placed over the image to illus-trate the method. There are 44 grains within the circle (ninside)and 34 grains intercepted by the circle (nintercepted). The numberof grains per sq. mm, NA, is calculated from:

NA = f {ninside + ½(nintercepted)}

The multiplier f is calculated from (M2/circle area), where M isthe linear magnification of the image. For this example,

NA = 12.28 {44 + ½(34)} = 749.1 grains /sq. mmFrom NA, we can calculate the ASTM grain size number, G,using the following formula from E 112-96:

G = {3.322 (log10 NA) – 2.954} = 6.6

The ASTM grain size can also be determined using the inter-cept method counting either the number of grains intercepted,N, or the number of grain boundaries intersected, P, with a

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test line. ASTM recommends using a grid with three concentriccircles with a 500mm total line length. To illustrate the principleof the method, we will use the same image with a single circle(shown on previous page). The count of the number of grainsintercepted by the circle is N. To calculate the number of inter-ceptions per mm, NL, we divide N by the true length (circumfer-ence) of the circle. Since the diameter of the circle is 64.4mm,its circumference is πD, that is, 202.3mm. The true length is202.3mm divided by the magnification, M, that is, 1.01mm.Hence, NL = N/LT = 34/ 1.01 = 33.6 interceptions per mm.

To calculate the grain size, we first determine the mean linealintercept length, l, which is the reciprocal of NL (or of PL, thenumber of grain boundary intersections per unit length). Thus,l = 1/ 33.6 = 0.0298mm. G is calculated from an equation from E112-96:

G = {-6.644 (log10 l) – 3.288}

where l is in mm. In this example, G = 6.85. Since the two meth-ods are sensing different geometric aspects of the three-di-mensional grain structure, they will not give exactly the samevalue, but they will be close, generally within the experimentallimitations of the measurements. In practice, we would repeatthese measurements on a number of fields in order to obtain agood estimate of the grain size.

SpacingsThe spacing between second-phase particles, such as car-bides or inclusions in steels or between intermetallic particlesin aluminum alloys, can affect mechanical properties and form-ability. A special case is the interlamellar spacing of pearlite inhigh carbon steels (like rail steel) where refinement of the spac-ing improves both strength and toughness.

Spacings are easily assessed using a simple NL (number ofparticles intercepted per unit length of test line) measurement.The mean center-to-center spacing, sometimes called σ, issimply:

σ = 1/NL

This is not a nearest-neighbor spacing, but the mean spacingbetween particles in the test line direction (either placed ran-domly or in some preferred direction, such as the through-thick-ness direction).

If the amount of the second phase is determined, for example,by point counting, the mean edge-to-edge spacing, called λ (orthe mean free path, MFP), can be calculated by:

λ = (1-PP)/NL

where PP is a fraction rather than a percentage. This is a verygood structure-sensitive parameter.

By a simple subtraction of (σ - λ), we can obtain the mean inter-cept length of the second phase particles – without measuringany particles! Furthermore, if we count the number of particleswithin a known area to obtain NA (including only half of theparticles intersected by the field edges), we can determine theaverage cross sectional area of the particles, A, by:

A = PP /NA

where PP is the point fraction (as a fraction, not a %) of thesecond phase. Thus, the average size of particles can be deter-mined manually without actually measuring the particles. Withmodern image analyzers, individual measurements of particlesare fast and simple. Besides generating average particle di-mensions, the distribution of particle sizes can be obtained byfeature-specific image analysis.

To determine the interlamellar spacing of pearlite (or of anyeutectic or eutectoid), it is common practice to count the num-

ber of carbide interceptions with a straight test line perpendicu-lar to the lamellae. However, because the lamellae intersectthe surface at different angles, it is better to determine a meanrandom spacing, σr, than a mean directed spacing, σd. A meanrandom spacing is obtained by determining NL using randomlyoriented test lines (or curved or circular lines). The mean ran-dom spacing is easily used to calculate the mean true spacing,σt, by:

σt = σr /2In the past, the mean directed spacing, σd, was determined forthe pearlite colony with the finest observed spacing, and thiswas assumed to be the true spacing. This is a better techniquefor isothermally-formed pearlite than for pearlite formed duringcontinuous cooling. However, the longer you search for thefinest colony, the finer the measured colony size! That is, the σd

value obtained depends upon the amount of time spent look-ing for the finest colony, even in isothermally-formed pearlite.Any effort spent looking for a “best” or “worst” field condition, ofany type, is strongly influenced by the amount of search time,and the results are neither reproducible nor precise.

Interlamellar Spacing ExampleTraditionally, the metallographer has searched for the finestappearing interlamellar colony and made a measurement of itsspacing using a test line perpendicular to the lamellae. Thisspacing is claimed to be the true interlamellar spacing. How-ever, this method is not reproducible as the longer you search,

the finer the measured spacing. A better approach is to mea-sure a mean random spacing and divide that by two to get themean true spacing. This method was verified and proven to becorrect (see Metallography, Vol. 17, No. 1, February 1984, pp.1-17). The micrograph shown above is that of an as-rolled car-bon steel of about 0.45% C, etched with 4% picral and photo-graphed with the SEM (specimen perpendicular to the beam)using secondary electrons at 17,800x magnification. A circlewith a diameter of 49.7mm was placed over the lamellae andthe number of carbide lamellae intercepted by the test line, N,was counted. NL was again determined as N (23) divided by thetrue line length, πD/M, where the circle diameter, D, is 49.7mmand M is 17,800x. Thus NL is 2622 interceptions per mm. Themean random spacing, σr, is given by:

σr = 1/NL = 0.381µm = 381nmThe mean true spacing, σt, is σr /2 = 190.7nm.

StatisticsOther measurements are possible, but the ones describedabove represent some of the simplest and most useful. Eachcan be repeated on a number of fields on the plane-of-polishso that a mean and standard deviation can be obtained. Thenumber of fields measured influences the precision of themeasurement.

Manual measurements are tedious and time-con-suming so sampling statistics may be less thandesired. Image analysis removes most of thebarriers to inadequate sampling.

A good measure of statistical precision is the 95%confidence interval (or confidence limit). This definesa range around the mean value where, 95 times outof 100, a subsequently determined mean will fall. Forexample, a mean volume fraction of 10% ± 2%implies that for 95 of 100 measurements, the meanvalue will be between 8 and 12%. The 95% confi-dence interval is determined by:

95% CI = ts/n½

where t is the Student’s t factor (t is a function of theconfidence level desired and the number of mea-surements, n, and can be found in standard text-books and in some ASTM standards, e.g., E 562 andE 1382) and s is the standard deviation.

The relative accuracy, RA, of a measurement isdetermined by:

%RA = 100 · (95% CI)/Xwhere X is the mean value. In general, a relativeaccuracy of 10% or less is considered to be satisfac-tory for most work.

SamplingSo far, we have discussed measurements on a singleplane-of-polish on one specimen. Because we areusually dealing with large quantities of material (suchas an entire “heat” of metal or alloy, a large heattreatment lot of bars, forgings, etc., or a large forgingor casting), a single specimen may not be represen-tative of the whole quantity. Ideally, random samplingof a large batch would be best, but practical consid-erations usually rule this out.

In most cases, sampling is done at predeterminedconvenient locations, such as the extreme ends of acoil, bar, plate, etc., or at locations which will be sub-jected to extensive machining. In some cases, ex-cess metal is added to a forging or casting to providetest material as similar as possible to that of the com-ponent. Sampling is often a compromise and is rarelyexcessive due to cost considerations. However, in-adequate sampling or sampling of nonrepresenta-tive material or locations may degrade the value ofthe measurements.

Stereological measurements are best employed onsectioning planes that sample the microstructure ran-domly. This means that any oriented plane will pro-duce the same data within the limits of statisticalprecision. However, for certain materials, the micro-structure varies with the test plane. A classic exampleis that of inclusions in wrought steels which are pref-erentially elongated in the deformation direction.Sampled perpendicularly to the deformation axis(transverse plane), the inclusions look like sphericalparticles while, when sampled on a plane parallel tothe deformation axis (longitudinal plane), they ap-pear as long, thin rods or as broken “stringers”. Ifmeasurements are made on these planes, we ob-tain different values for NA, their length (or diameter),their spacing, and even the volume fraction. Thus, if

we want to characterize their 3-D characteristics,measurements must be made on the three principalplanes and averaged (or an alternate technique suchas the trisector with vertical sectioning must be em-ployed). In practice, the true 3-D characteristics maynot be needed, and measurements are made usingone standard test plane orientation which yields datasuitable for quality control and material comparisons.This is the procedure employed in ASTM E 1245.

Specimen PreparationIf we are to obtain precise, unbiased measurements,we must prepare the specimens to reveal the truestructure. In general, it is best to prepare specimensso that only the desired aspect of the structure isvisible. In this way, the eye sees only what is needed,and other information is not present to obscure whatis of interest. To study inclusions, intermetallics, orother constituents with a reflectivity or color differentthan the matrix, etching is not required. To measurematrix phases, use a selective etch that reveals onlythe constituent of interest. For example, for a carbonsteel, etch with picral and point count the amount offerrite or pearlite (always point count the constituentwith the lower concentration). Etching with nital willbring up the ferrite grain boundaries and produceless uniform etching of the pearlite making pointcounting more difficult and less precise.

The laws of stereology for measuring microstructureare based on the assumption that the etched surfaceis planar. There are rules for non-planar (curved) sur-faces and for measurements with transmitted light orelectrons, but these are much more difficult to em-ploy. For microstructural measurements, the surfacemust be flat and etch depth should be minimal (donot over-etch!). Deep etching can dramaticallychange the apparent amount of a second phase andits size and spacing. Remember, good specimenpreparation procedures are a necessary requirementfor obtaining good data.

ConclusionsAlthough all of the most important stereological rulesfor measuring matrix microstructures were wellknown and described in Dr. Underwood’s 1970 book(and elsewhere), their application was chiefly lim-ited to research studies. Production metallographersdid not adopt these methods because of the timerequired to make measurements manually.

The incredible growth in microcomputer power andin video technology over the past fifteen years haspermitted development of very capable, reasonablypriced user-friendly image analyzers, such asBUEHLER’s OMNIMET® Advantage System. Imageanalysis permits collection of microstructural data(some of which are difficult to obtain manually) rap-idly, painlessly, and with sufficient sampling to pro-vide good statistical data. Future copies of Tech-Noteswill be devoted to the subject of image analysis andits applications.

Lord Kelvin: “When you can measure what you are speak-ing about and express it in numbers, you know some-thing about it, but when you can not, you have scarcelyadvanced to the state of science.”

T E C H - T I P SQuestion: We have been making delta ferrite mea-surements using the AMS 2301 procedures formany years, starting with the occupied squaresmethod and then with the point counting methodsadded in versions B and C of the method. Whyhave these methods been changed, which should Iuse?

Answer: When AMS 2301 was developed, it con-tained a single method for estimating the amount ofdelta ferrite in a specimen (many specifications re-quire less than 5% δ), called the “occupied squares”method. In this method, a grid consisting of 500small squares (made by intersecting horizontal andvertical lines) was placed over the microstructureand a count was made of the number of “boxes”fully occupied by delta ferrite, the number 3/4 occu-pied, 1/2 occupied, and 1/4 occupied. The resultswere tabulated and added up to determine the frac-tion of the grid area covered, or “occupied,” by thedelta ferrite. This is basically an areal analysismethod and it gives a rather precise measurementfor that particular field, but only a few fields can beevaluated because it is so tedious. Measurementprecision for the specimen is influenced much moreby the field-to-field variability of the delta ferrite thanby the precision of measuring each field. Thus, it isnecessary to measure the delta on more fields. Thismethod is too tedious for evaluating 20 or morefields, unless there is virtually no delta present. But,if that is the case, it may not be necessary to makea measurement at all. However, when the amountof delta is close to the typical 5% limit, the occupiedsquares method is too slow for practical use.

AMS 2301 next added a point counting method us-ing a 500 point grid, which was later changed to100 points. The optimum point grid density is afunction of the volume fraction and is found using

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the equation 3/VV. Thus, for 5, 3 or 1% volume frac-

tions (0.05, 0.03 and 0.01, respectively), the opti-mum point densities are 60, 100 and 300 points,respectively. A grid with 500 points takes more timeto count than a lower point grid. For example, a 500point grid takes five times as long to count as a 100point grid. To optimize the method, that is, to makeit most efficient, we want to measure as many fieldsas possible in the least amount of time (in agree-ment with the adage, “do more, less well”). To ob-tain a 10% relative accuracy with a 5% volumefraction and a 100 point grid, we do need to ana-lyze at least 50 fields, which is often viewed as pro-hibitive when done manually. Hence, the interest inimage analysis.

If you have a question that you'd like to see an-swered, or a tip that you feel would benefit ourreaders, please write, call or fax to:

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