New Stereology for the Recovery of Grain-Boundary Plane Distributions in the Crystal Frame

A new experimental method is given for recovering the probability-distribution function Sv (n̂A | g). The function Sv (n̂A | g) is the grain-boundary area per unit volume as a function of grain-boundary plane orientation (n̂A), given a lattice misorientation ( g) between the adjoining grains. The grain-boundary normal (n̂A) is expressed in the crystal frame in which the misorientation g originates. The proposed method recovers the three-dimensional Sv (n̂A | g) function using data taken from two-dimensional section planes. The method requires the measurement of many grain-boundary trace (in-plane) angles and lengths associated with grain boundaries of lattice misorientation. All such boundary traces may be observed from a single section plane if the crystallographic texture is sufficiently random. In heavily textured microstructures, the method requires the researcher to observe traces from multiple oblique section planes cut through the material. A method of quantitatively estimating whether the texture is sufficiently random is given. Simulations on both textured and nontextured microstructures demonstrate the validity of the method. Experimentally, the new method is used to analyze boundaries of misorientation ( 3) observed in 304 stainless steel. Calculated grain-boundary plane-probability functions are shown to be consistent with what is already known.

Using this procedure, the designation of grain A is gcncrally unicluc. This unique designation of grain A is uscft~l in avoiding error in the representation of the fi~nction S,. (& I Ag).
The given method of distinguishing Ag from Ax1' (and, ~IILIS, fi,, from fi1J does not hold whcn the axis of' disorientation for Ag falls on the pcrimctcr of the standard triangle. Although it is highly ~nilikcly tliat the calculatccl axis of disorientation will fall exactly on this pcrimctcr, given the experimental error of the ~neasurcmcnts and misorientation variations in the crystals, therc will he a zone about the perimeter in which any practical distinction hctween Ag ancl Ag7' will not exist. In these cases, peaks corrcsponding to both fijl and r"r, will likely appear in a plot of S,. ( ash~on ancl (2) that the traces sa~nplcd from each boundary type arc randolnly chosen with no orientation bias. It is clear that whcn a sample lias a completely random tcxt~~rc, both conditions may bc satisfied by traccs collected from a single section plane. In this case, the LMFs are randomly oriented with rcspcct to the laboratory framc. I-lowever, whcn the material is textured, there will bc sampling bias when a single-section plane is used for S,, (I;, I Ag) rccovcry. This bias may be overcome by obtaining traccs from scvcral oblicluc section planes, as demonstrated by simulations shown on Table 11.
In order to csti~nate whether or not a given material is sufficiently rando~ii for the stereology to be pcrfornicd, we introduce the probability-clistrib~~tion function C,, (4 I Ag). The fl~nction C,, (fi,, I Ag) is the distribution of section-plane nor-1i:rve an ;rngul:rr deviation lcss than e Sroin pcrpcndic~ilarity to the (:,: axis ( Figure 3). The value of f3(v) is given by In deriving P(v) (refer to the Appendix), it was assumed that all possible traces of' a grain houndary have an cc1~1:rl probability ol'bcing chosen (Assumption 2). Tlic Ihndamcntal relation, Ecl. 141, ilivolves a single integration over the single i~nknown parameter, v, weighted by P(v). 'l'lie factor 1~14 is the constant of proportionality between I,, and s,.I' ' I 'rlicsc I'unctions arc proportional due to representative sampling ol' all boundary types (Assumption 1).
Eqi~ation 141 rcprcscnts a classic invcrsc, or ill-posed, prob- One can create nulnerous possible ~nctliods of solving these f~~ntlamcntal equations. We have cliosen to solve the ccluations using a method proposed by Adams that makes use of surface spherical 11a1.1iionics.l'~~ 111 this method, rhc distribution i~nctions arc represented ~ising surface sphcrical harmonics and Fouricr series according to where K is the truncation order of'the aeries. Aclams obtainccl ~natrix ccluations relating the coei'ficicnts S: to the cocfficients 1: hy inserting Eqs. 161 and 171 into a fundamental equation similar to Eq. 141 and performing the proper coordinate tlansf'ormotio~is h o~n tlie S i'ramcs to tlie GMF (S, (e,, I As, ?',) t o S, (I?,\ I As)). As morc S-coortiinatc systclils arc adcled, the systems of ecluations become ovcrdctcrmined, allowing a solution of the coefficients S: ~~s i~i g singular-valuc c l e c o~n~o s i t i o n .~'~~ Tlic concepts prescntccl in tlie derivation of the f~~ndir-~nctital ecluation luay also be applied to recover the function S, (fi,, I Ag), or the gl-din-boundary area fraction. This is silnply a matter of replacing I,, ( w ( Ag, Z;), which is normalized by the total scan area, with ll ( w I B.\, Ag), wliich is normalized by the total trace length. In this case, the fundamental ey~~ation is give11 by Simulatiolis were performed to demonstrate that the technique gives viable estimates of S,, (fi,, I Ag) in both nontcxtuscd (Table 1) and textured (Table 11)  Af'ter constraining the microstructure it was necessary to select the orientations o f the section plane(s) with respect to the laboratory flame. 111 the case of rantlorn LMF distributions, a single section plane was used, but in the case of nonuniform LMF distributions, the seven section planes shown in Table 11 wcre usetl.
Having ctctermined the ~nicrostr~~cture and the sectionplane oricntations, the next step was to calculate the "observcd" traccs, expressecl in tlie GMF. Trace vectors were givcn hy the intersectiotis of all the grain-boundary planes and the section plane(s). In most si~iiulntions, thc effect of tr;~ce-~~ieas~rrement error was si~nulatecl by rotating the observed tracc in the plane of the section plane by an amount chosen r~uidomly I'rom an even distribution in the range of i 5 deg. In nontcxturecl materials, a trace from eacli of the bicrvstals was consitiercd observed. In textured microstructures, we expect the sample texture to give rise to sampling hias for bounclaries associated with the same section plane. This bias ~.esults from the fact that boundaries that are nearlv parallel to the section plane arc less likely to be observed th~un bo~rnclarics that are pcrpendiculas to the section plane. The probability that a boundary will be cut by a section plane is proportional to sin (a), where cu is the angle between the grai~i-ho~~nrlary plane and the section plane. For each section 17lanc. a fraction of observed boundaries was chosen I'rom a pool ol' "candidate" grain boundaries according to the ~xobability sin (a). A l t h o~~g h the candidate grain bounclarics were rcpresentativc of N,, (fi,, I AS), the observed boundaries wcrc not. Once all traces wcre collected ant1 expressed in the GMF accorcling to these SLIICS, N,, (fi,, I Ag) was estimated using the new n~ctiiod. The parameter e was assigned to be 3 deg in all simulations. We chose to use 500 S frames, so that the average spacing between ( ; , ; clircctions was about 6 dcg. The ortlcr K ol' series truncation (Fig. 6) was usually taken to be 34, but, in the cascs 01' trace-measurement en-or, better recoveries wcrc obtained with X = 24. The angular resolution was f'o~~nd to be about 1 .5dX.
For each recovered N,, (6, 1 Ag) value, numerical integrations wcrc pcrformecl in the regions surrounding the ten largest pc~tks in orclcr t o estimate tlie number fraction of boundaries at each peak. Known habit planes were then matched with recovcrccl peaks, anti the percent error in number I'ract ion was calc~~latecl for each peak. False peaks occurred when a recovered peak ;~ssociatcd with 110 habit plane had a higher n~~m b c r fraction than a recovered peak that was associatcd with a habit plane. The clata presented in Tables I   and 11 demonstrate that the method may be used to estimate S,,(G, I Ag) in a statistically reliable lashion. Moreover, simula-tions demonstrate that S,, (fi,, 1 Ag) may be characterized on highly textured nlaterials if several section planes are used. The silnulations also demonstrate the i~sefulness of <C,,> in quantifying the randomness of available data.

EXPERIMENTAL
The proposed rncthod was used to rccovcr the function S , (CA 1 x3) in 304 stainless steel. The sample was first treated in a vacuum filrnace at 10 10 "C for 23.5 hours and oven cooled. A subseauent heat treatment at 600 "C for 4 hours "fine tuned" the bou~idaries to lower-energy configurati~ns.~'~~ The sample was polished, and electron backscatter patterns taken from a FEI/PHILIPS'k XL 30 scanning field-emission '4:PHII,IPS is a trademark of Philips Electronics Corp., Mahwah, NJ. gun wcre processed using Orientation Imaging Microscopy (01M8') software. The step s i~e of all scans was 2 pm. The '"01M is a trademark of' TSI,, Inc., Dral~cr, UT -~p average grain diameter was approxilnately 150 pm.
A grain-boundary reconstruction algorithm was used to find all boundary lengths, trace angles, and adjacent grain orientations.lLw We estimated that in order to keep tracemeasurement error below 5 dcg, only boundaries greater than 15 times the step size of the scans could be used in the analysis. However, boundaries of this length or greater accounted for only 59 pct of the total 2 3 boundary tracc lengths. For this reason, we collected the S , (4 ( C3) function with respect to all boundaries of length greater than 30 p m . Figure 4 shows the function S,, (ri, ( x3) taken from 7 17 bicrystals observed on four scans. For these bicrystals, <C,,> was found to be 0.0003; hence, we assumed that a single section plane would provide a reasonable estimate of S, (li, 1 23). The largest peak is the coherent { 1 1 I ] / { i 1 I ) peak. An integration out to 9 deg away from this peak returned S, order. From a comparison of Figures 4 and 5 , it would seem that C3 grain boundaries tend to align on the <0 1 1 > zone. Figure 6 shows a plot of coherent trace-deviation angles of the same 717 boundaries as a function of their lengths (in ~nicrons). The coherent trace-deviation angle is the angle between the measured trace and the trace of the { 1 1 1 )/( I 1 1 ] plane.l'",'71 It is likely that most grain boundaries with coherent trace-deviation angles near 0 deg are associated with coherent boundaries. Traces with coherent trace-deviation angles 11n~1ch greater than 0 deg cannot be coherent boundaries. The striking feature of Figure 6 is that the boundaries that are known to be noncoherent ~nostly occur at lengths smaller than about 60 pnn. This result raiscs Inany f~~ndamental q~~estions about the grain-boundary planes of the small C3 boundaries that, by number and area fraction, are a major 2 3 boundary type.
In order to better understand what boundary types the noncoherent C3 types werc associated with, we decided to perform a stereological recovery only on the 146 boundaries with coherent trace-deviation angles greater than 10 deg. The result is shown in Figure 7. By far, the three largest

IV. CONC1,USIONS
Wc have tlesived the funtlamcntal ccluation ol' a new stcrcology and have shown with both sim~llations and actual data that it lnay be ~lsed 10 rccover ~~; I~I I -~o L~I~~~I plane distributions in the GMF. In the clcrivation ol'the new stcrcology, it was assunled that all bounclary types are samplccl in correct proportions and that all ho~~nclarics ;u.c sannl>letl i~n a rand0111 manner. Even in the casc of tcxturecl ~naterids, it was shown that the method is reliable if multiple scction planes are ~lscd. Wc have introclucecl thc paranneter <C,,> as an cstinna(e of the ranclomncss ol' the clata ancl have shown it to be a reliable intlicator of the accuracy of the recovery.
Trace measurements made i'rom 304 stainless steel proviclecl evidence that there is a tenclency Ibr X3 bounclaries to be tilt bouildaries on the <01 1 > /,one. It was shown that a significant proportion of x 3 grain-bo~mdary planes arc of the s~naller type, whcrc coherent bounclarics arc lcss common. Measuring these boundary traces with cno~~gln precision to determine their trace angles accurately whilc still collecting many bo~nndaries horn a large area poscs a significant expcrimental challenge. Nonethclcss, the results obtaineel show the usefulness of cluantifying the grain-boi~nclary distribution through stereological methods.
One of the authors (RJL) is gratcl'~11 to the Office of Naval Research for fellowship support. BLA acknowlctlgcs thc support of the NSF through the Matcriuls Rcscarch Science and Engineering Center, Carncgic Mcllon University (Grant No. DMR-0079996).

Derivation of P(u)
Consider a grain-boundary plane whose ~iormal lies v tlcg fro111 the ?;direction. As statcd prcvio~~sIy, P(71) is the prohability that a randonnly choscn trace angle I'rom that hountlary plane will lie within r-: tlcg oi'pcrpcndiculal.i(y to the e^: direction. To obtain P(v), it is assunnccl that all grain-bo~~nclary trace angles havc the same probability 01' being choscn.
Fro111 this i l s s l~~l l l ) t i~~i , it is clear that where 2y' is the range of trace a~iglcs on the boundary that have angular clcviations less than E from pcrpcndicularity to the ?.!, direction, as sliow~i in Figure A l . From the tlcl'inition ol'f)(u), it is o h v i o~s that f-'(u) is com-~>lctcly intlcpendcnt 01' one's placement ol' d,; and e^; . Without loss ol' gc~iuality, we arc free to place ?,: and e^; in a right-handed sense such that (; . : is perpendicular to both the houncl~try nol.liial, ti,;,,, ancl the ?; clirection, as shown in Figure A l .
In orclc~. 10 Iitbcl all ~~o s s i b l c traccs of the bou~iclary, we begin hy placing the grain-bo~~nclary nor~nal parallel to the d$ clircctio~l. Witli the bounclary normal in this position, we arc I'rcc to placc 8,: anywhcrc within the bountlary plane, since a n y choice of i ? : will he perpendicular to hot11 the houndnry nol.tnal and the ?: ' direction. Tlic angle bctwccn : I trace of tlie grain boundary ancl the ?f direction is given by y. Now, we incline the bounclal-y normal to v clcgrecs I'roln the ?.; axis by a rotation fi(v) of v deg ahout the -d: clircctio~i (wlierc 0 5 v 5 d 2 ) . R(v) is given We wish to find the two traccs (with trace cos (y') range -7d2< y S d 2 , for which the from angles of d 2 p i s with the e^: axis. Tlicse traces correspond to the daslicd vectors shown in Figure A I. Any trace such that Iyl 5 y' will have an angular deviation less than E from pcrpe~~tlicularity to the ?: direction. Using tlie dot procluct, we write when v 2 r-: When v < E , P(v) is undefined in the preceding expression.