We continue our multi-part discussion of quantitative metallography. We hope this information will be usable by many in their metallurgical investigations.  

Inter-Particle Spacing
The spacing between second-phase particles such as carbides or inclusions in steels or between intermetallic particles in aluminum alloys, can affect mechanical properties and formability. A special case is the interlamellar spacing of pearlite in high-carbon steels (like rail steel) where refinement of the spacing improves both strength and toughness.

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

σ = 1/N_L       

This is not a nearest-neighbor spacing but the mean spacing between particles in the test line direction (either placed randomly or in some preferred direction, such as the through-thickness direction).  

If the amount of the second phase is determined, for example, by point counting, the mean edge-to-edge spacing, called l (or the mean free path, MFP), can be calculated by:
l = (1-P_P)/N_L        

where P_P is a fraction rather than a percentage. This is a very good structure-sensitive parameter.  

By a simple subtraction (σ - l), we can obtain the mean intercept length of the second-phase particles – without measuring any! Furthermore, if we count the number of particles within a known area to obtain N_A (including only half of the particles intersected by the field edges), we can determine the average cross sectional area of the particles, Ā, by:
Ā = P_P/N_A      

where P_P is one point fraction (as a fraction, not a %) of the second phase. Thus, the average size of particles can be determined manually without actually measuring the particles. With modern image analyzers, individual measurements of particles are fast and simple. Besides generating average particle dimensions, the distribution of particle sizes can be obtained by feature-specific image analysis.  

To determine the interlamellar spacing of pearlite (or of any eutectic or eutectoid), it is common practice to count the number of carbide interceptions with a straight test line perpendicular to the lamellae. However, because the lamellae intersect the surface at different angles, it is better to determine a mean-random spacing, σ_r, than a mean-directed spacing, σ_d. A mean-random spacing is obtained by determining N_L using randomly oriented test lines (or curved or circular lines). The mean-random spacing is easily used to calculate the mean true spacing, σ_t, by:
σ_t, = σ_r/2        

In the past, the mean-directed spacing, σ_d, was determined for the pearlite colony with the finest spacing, and this was assumed to be the mean true spacing. This is a more valid technique for isothermally formed pearlite than for pearlite formed during continuous cooling. However, the longer you search for the finest colony, the finer the measured colony size! That is, the σ_d value obtained depends upon the amount of time spent looking for the finest colony, even in isothermally formed pearlite. Any effort spent looking for a best- or worst-field condition is strongly influenced by the amount of search time, and the results are neither reproducible nor precise.