What a section shows

A metallographer never sees a microstructure. What reaches the eye is a flat, polished slice through it, a few square millimetres of a part that may weigh many kilograms, viewed through a microscope one small field at a time. The grains, nodules, pores and inclusions in that slice are cuts through solid objects whose real sizes, shapes and connections lie above and below the plane. A good measurement starts from knowing exactly what such a slice can and cannot tell.

This chapter sets out the geometry of sectioning, the quantities that a section shows faithfully and those it distorts, and the scale of what the microscope actually looks at. Its error is the most common one in any laboratory: a report built on the single field that looked worst.

A slice through three dimensions

Cutting a solid with a plane removes one dimension from everything in it. A grain or a pore, which occupies a volume, shows up as an area. A grain boundary, which is a surface, shows up as a line. A dislocation line or the edge where three grains meet shows up as a point. Anything that is itself a point in space, such as the centre of a particle, almost never lies exactly in the plane and is not seen at all.

Figure 1What a plane section does to each kind of feature

Feature in the solid	Seen in the section as	Example	What can be measured directly
Volume, such as a phase, pore or particle	an area	ferrite, graphite, a shrinkage pore	area fraction, section sizes
Surface, such as a grain or phase boundary	a line	grain boundaries in steel	boundary length per unit area
Line, such as a triple junction or a crack front	a point	where three grains meet	points per unit area
Point, such as a particle centre	nothing	the number of nodules per volume	only by inference from sections

That last row is the source of much confusion. The number of graphite nodules in each cubic millimetre of a casting is a real and useful property, but no section shows it. A section shows the number of nodules cut per square millimetre, which depends on both how many there are and how big they are, because a large nodule is more likely to be cut by a given plane than a small one. Chapter 6 deals with that relation in detail.

The quantities a section shows faithfully

The great simplification of quantitative metallography is that some quantities pass through the section without bias. The best known is the volume fraction of a phase. If a structure is sampled by sections placed at random through it, the average fraction of section area occupied by a phase equals its volume fraction. The same value is obtained from the fraction of a test line that falls in the phase, and from the fraction of a grid of test points that land in it. The principle was stated for rocks by the geologist Delesse in the nineteenth century and extended to lines and points later; it is the foundation of every phase fraction in this book.

Figure 2The fraction relations of stereology

V_V = A_A = L_L = P_P (each as an expected value over randomly placed sections, lines or points)

V_V

volume fraction of the phase in the solid

A_A

area fraction of the phase in the section

L_L

fraction of the length of test lines lying in the phase

P_P

fraction of test points falling in the phase

1. Casing matrix: 160 of 500 grid points fell in pearlite, so P_P = 0.32 and the estimated pearlite volume fraction of the matrix is 0.32

Two other quantities also pass through without needing any shape assumption. The surface area of boundaries per unit volume equals twice the number of boundary crossings per unit length of a randomly oriented test line, and the length of lines per unit volume equals twice the number of line crossings per unit area of section. Grain size, in Chapter 4, rests on the first of these: counting where a test line crosses grain boundaries measures how much boundary there is, and so how fine the grains are.

The catch: random sampling

All of these hold only for sections that sample the structure without bias. In practice that means two things. The position of the section and of the fields within it must not be chosen by what they show, and, where the structure has a direction, the orientation of the section or test lines must be chosen deliberately. Rolled steel has inclusions drawn out along the rolling direction and grains that may be elongated; a casting has columnar grains growing in from the mould walls and coarser structure at its thermal centre. A section in one orientation, or at one place, gives a true answer for that orientation or place and not necessarily for the part. Chapter 2 turns these requirements into a sampling plan.

What a section distorts

Other properties do not survive the section intact. Each needs either an assumption about shape or a different measurement.

• Size. A plane through a sphere cuts it in a circle no larger than the sphere’s diameter and usually smaller. Only planes through the centre show the full size. Sections of identical spheres therefore show a spread of circle sizes, most of them smaller than the spheres.
• Number. Large particles are hit more often than small ones, so the sections over-represent the large particles and the number per area reflects size as well as number.
• Connectivity. Two pores that look separate in the section may join above or below it, and a network that looks broken may be continuous. Whether porosity forms a leak path through a wall cannot be seen in one section.
• Shape and orientation. An elongated inclusion looks like a long streak when the section lies along it and like a small dot when the section crosses it.

Figure 3Identical spheres cut by one plane

Sections, projections and fracture faces

Not every image of a microstructure is a section. The distinction matters because the relations in this book apply only to true plane sections.

A polished section is a plane, and features in it have zero thickness. Everything in this book assumes this kind of image. A thin foil or a thick slice viewed in transmitted light or in a transmission electron microscope is a projection: everything within its thickness is superimposed. In a projection, particles overlap and hide each other, and small particles appear in greater numbers than in a section of the same area because the slice has depth. Area fractions measured on projections overstate volume fractions, by an amount that depends on the slice thickness and the particle size. A fracture face or an etched-out deep relief is not a plane at all. A fracture runs preferentially through weak features, so a fracture face shows far more inclusions or pores than a random plane would, which makes it excellent for finding the cause of a failure and useless for estimating how common that cause is.

The same distinction applies to non-destructive images. A radiograph is a projection of the whole wall thickness, and the porosity it shows cannot be compared directly with porosity measured in a section. Computed tomography reconstructs the volume itself and avoids the sectioning problems altogether, but at a resolution limited by the size of the part, so in a casting the size of Plant P’s it resolves pores of a fraction of a millimetre and misses the micrometre-scale structure that the microscope shows.

Which numbers need a model

It helps to sort every quantity in a metallographic report by what it assumes. Some follow from the section alone, given unbiased sampling. Others need a model of particle shape or a statement about orientation, and their accuracy is only as good as that model.

Figure 4What each common measurement assumes beyond unbiased sampling

Measurement	Extra assumption	Chapter
Phase or porosity volume fraction	none	3 and 8
Grain boundary area per volume, and hence the intercept grain size	none if test lines are randomly oriented; otherwise holds for the orientation used	4
Number of grains per area in a section	none, as a section property; converting it to grains per volume needs a shape model	4
Particle or pore size distribution in the solid	particles of known shape, usually spheres	6 and 8
Number of particles per volume	particle shape and size distribution	6
Largest inclusion in a volume	a distribution for the largest values, usually Gumbel	9

The practical rule is simple. Report section quantities as section quantities, with their sampling, and present anything three-dimensional as an estimate under a stated model. A report that says “nodule count 200 per mm²” makes a correct statement about the section. A report that says “200 nodules per mm³” has made a units error, and one that says “about 6,700 nodules per mm³, assuming spheres of one size” has made a model-based estimate and said so.

The scale of a field

It is easy to forget how little of a part the microscope sees. Plant P’s camera captures a field 1.60 mm wide and 1.20 mm high at 100× magnification, an area of 1.92 mm². At 200× the field is a quarter of that, 0.48 mm², and at 500× it is 0.0768 mm². A polished specimen cut from the casing flange measures 28 by 24 mm, or 672 mm², so a single field at 200× covers well under a thousandth of the face, and the face itself is one plane through a flange that holds several hundred cubic centimetres of iron.

Figure 5Area of one camera field at each magnification

50×7.68 mm²

100×1.92 mm²

200×0.48 mm²

500×0.0768 mm²

The choice of magnification is a compromise. It must be high enough that the features of interest are many pixels across, so that their boundaries are located well, and low enough that each field contains enough of them to count. A useful rule is to choose the lowest magnification at which the smallest feature that matters is still clearly resolved, and then to measure enough fields to reach the precision needed, which Chapter 3 shows how to calculate.

Calibrating the scale

Every length in a metallographic report depends on the scale of the image, which is set by the objective, any intermediate optics and the camera. Nominal magnifications are not accurate enough for measurement. Plant P’s technician calibrates each objective by photographing a stage micrometer, a glass slide ruled with a 1 mm scale in 10 µm divisions, and entering the measured number of pixels per micrometre into the image analysis software. The calibration is repeated after any change to the optics and checked monthly. An error of 3% in the scale becomes a 3% error in every length and a 6% error in every area.

The error: the worst field

When the first leaking casing reached the laboratory, the technician sectioned its flange across the leak path, polished the section, and searched it under the microscope until finding the region of greatest porosity. That field, photographed at 100×, showed a cluster of irregular cavities covering 5.6% of its area. The draft report stated that the flange contained 5.6% porosity.

The engineer who read the report asked how the field had been chosen. The 5.6% was a true measurement of one field, but the field had been chosen because it was the worst, so it said nothing about the flange as a whole. A picture of the worst field is useful: it shows what the defect looks like and where to look. A number from it is not an estimate of anything.

The technician repeated the measurement on 30 fields placed on a regular grid, 4 mm apart, starting from a randomly chosen point, across the whole flange section. The average porosity was 0.8%, with the porous fields concentrated near the junction between the flange and a thick boss, and almost none elsewhere. Chapter 8 follows that result to its cause.

The corrected report kept the photograph of the worst field, labelled as the worst field found in a search, alongside the systematic result and a sketch of the section showing where the porous fields lay. That combination answered three different questions at once: what the defect looked like, how much of it there was on average, and where it was concentrated. The third answer turned out to matter most, because porosity concentrated at one junction points to a feeding problem in the casting process, while porosity spread evenly through a section points to gas. A single number, however carefully measured, would have hidden that pattern.

Figure 6Two answers to "how porous is the flange?"

5.6%porosity in the single field chosen because it looked worst

0.8%average porosity over 30 systematically placed fields

30fields on a 4 mm grid from a random start

1junction where the porous fields were concentrated

How this applies to a small Australian business

Whether a business runs its own microscope or sends parts to a laboratory, it should ask how every number in a report was obtained: which section, which orientation, how many fields, and how they were placed. A report that gives a single field, or fields described as representative, gives a picture rather than a measurement. Ask also whether the image scale was calibrated with a stage micrometer and when. These questions cost nothing and catch most of the errors that matter, and accredited laboratories will answer them as a matter of course.

Exercise: size bias in a nodule count

A casting contains equal numbers per cubic millimetre of two kinds of graphite particle: small spheres 10 µm across and large spheres 40 µm across. (a) On a random section, in what ratio will the large and small particles appear? (b) For the large spheres, what fraction of their sections will look smaller than 20 µm across, given that the fraction of sections of a sphere of diameter D smaller than x is 1 − √(1 − (x ÷ D)²)?

Worked answer. (a) A sphere is cut by a plane if its centre lies within half a diameter of the plane on either side, so the chance of being cut is proportional to the diameter. With equal numbers per volume, the large spheres appear 4 times as often as the small ones, 40 ÷ 10, even though there are equally many of each. (b) x ÷ D = 20 ÷ 40 = 0.5, so the fraction is 1 − √(1 − 0.25) = 1 − 0.866 = 0.134. About one section in seven of a 40 µm sphere looks smaller than 20 µm, and would be mistaken for a much smaller particle by anyone reading the section at face value.