On the Characterization of Directionally Solidified Dendritic Microstructures

On the Characterization of Directionally Solidified DendriticMicrostructures

NILS WARNKEN and ROGER C. REED

A novel method for the stereological assessment of arrays of directionally solidified dendrites ispresented. It is suitable particularly for the evaluation of the distribution of primary dendritearm spacings and the dendrite coordination number. The method involves (1) determination ofthe center of gravity of each dendrite in the array under consideration, (2) calculation of thedistances to all neighboring dendrites in the data set (3) sorting of the distances determined intoascending order (4) under the assumption of a minimum number of nearest neighbors—typicallythree—calculation of the average distance to these and associated standard deviation, (5) anassessment of whether the remaining neighbors are within the range of proposed nearest-neighbor distances and (6) repetition of the previous to determine a set of values for the entiredata set and, thus, the distributions of dendrite arm spacings and coordination numbers. Byapplication of the method to experimental microstructures, it is demonstrated that a detailedstatistical assessment of directionally solidified dendritic arrays can be accomplished.

DOI: 10.1007/s11661-010-0544-4� The Minerals, Metals & Materials Society and ASM International 2010

I. INTRODUCTION

THE microstructure of a material is rarely homoge-neous or entirely regular. Instead, spatial distributionsof microstructural features are usually present, e.g.,bimodal grain structures, precipitates of varying dimen-sions, and defects such as pores or microcracks. Thissituation should be acknowledged more widely in thefield of materials. Why? Because although the mean sizeof any microstructural feature is of considerable impor-tance, the material’s behavior is often controlled bystatistical deviations from the mean values observed.For example, fracture behavior is controlled usually bythe largest rather than the mean defect size. A variationin fatigue performance is linked intimately to thestandard deviation in the grain size. And for bimodaldistributions, cavitation damage arising from creepdeformation is found often to be greater at the grainboundaries of the smaller grains.

This article is written with the previous description inmind. A method is presented for the stereological assess-ment of arrays of dendrites of the type arising duringdirectional solidification. In particular, it allows spatialdistributions of dendrite arm spacings to be estimated. Itshould be noted that this article deals strictly withdirectionally solidified structuress, the primary lengthscale of equiaxed structures is defined by the grain size.Characterization of the microstructure of directionallysolidified dendritic arrays is important because (1) its scaledetermines the extent of microsegregation and interden-

dritic porosity, and the details of the subsequent process-ing needed to remove these[1,2]; (2) the propensity forprocessing-related defects such as freckling—caused bythermal-solutal convection currents—are influenced byit[3–5], and (3) the mechanical performance of manysystems are influenced strongly by the scale and distribu-tionof the dendritic structure inherited fromprocessing.[6]

In practice, the characterization of dendritic microstruc-tures presents an additional difficulty as a result of theirinherent anisotropy, because their morphologies arestrongly inherited from the heat flow arising duringfreezing. Thus, the characterization of directionallysolidified dendritic arraysmakes an interesting and highlytopical test case for new stereological assessment proce-dures of the type presented in this article.As will be discussed, the method presented has wider

applicability, but in what follows we choose to apply it tothe dendritic microstructures arising from the directionalsolidification of the nickel-based superalloys.[7] These findwidespread application in the hottest sections of the gasturbines, e.g., those used for jet engines and electricitygeneration.[8,9] When solidified in this way, arrays ofdendrites are formed that are strongly aligned with afibrous h100i texture, because of the competitive growthprocesses that occur. The characterization of such den-dritic arrays on transverse sections (whose normal is theheat flow direction) is of considerable practical impor-tance. Unfortunately, no existing consensus exists for thebest methods for quantifying stereological informationfrom this kind of microstructure. This situation hasmotivated the work reported in this article.

II. CLASSIC APPROACHES

The characterization of a dendritic array by deter-mining the primary dendrite arm spacing (PDAS) k1 has

NILS WARNKEN, Research Fellow, and ROGER C. REED,Professor, are with the Department of Metallurgy and Materials,University of Birmingham, Edgaston B15 2TT, UK. Contact e-mail:[email protected].

Manuscript submitted May 11, 2010.Article published online December 16, 2010

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 42A, JUNE 2011—1675

been the subject of several works. The most commonmethod used is counting the number of dendrites n in animage plane and evaluating the PDAS according to

k1 ¼ c

ffiffiffiffi

A

n

r

½1�

where A is the area analyzed. The first use this methodfor dendritic arrays is probably from Jacobi andSchwerdtfeger[10], but it was adopted quickly by otherresearchers.[11,12] Previous works on directional den-dritic solidification determine the primary dendritespacing using line intersect methods or manual mea-surements of distance between adjacent dendrites.[13,14]

However, equations similar to Eq. [1] have be used inquantitative microscopy for the characterization ofmicrostructures (compare Reference 15). In most worksconcerned with dendrites, the factor c is taken as unity(compare Reference 3). It is obvious that this relationdescribes a tesselation in which the the area of theaverage tile is equal k1

2. A mosaic consisting only of

average tiles would fill an area. If the variation of the tilesize follows a normal distribution, then the average ofthat distribution is still obtained by applying Eq. [1].According to McCartney and Hunt,[12] this applies tostructures possessing a cubic structure, whereasc = 1.075 applies to hexagonal structures.[10] For arandom distribution of particles in a matrix, it has beenshown that the right average spacing is obtained forc = 0.5.[16] This was obtained by assuming a normalsize distribution, from which the appearance of thestructure in a metallographical section has been workedout. As the underlying structure is known, the correla-tion with the metallographic section can be worked out.It is obvious that the average tile for c = 0.5 would notbe space (respectively area) filling. Voids in the tessela-tion have to be filled by larger tiles. A common featureof random structures is the presence of clusters, whichresult in a large number of closely spaced lattice pointsand a small number of widely spaced ones. This explainsthe form of Eq. [1] for random structures. From thesethree cases—cubic, hexagonal, and random—it becomes

(a) (b)

(c) (d)

Fig. 1—Simple cubic arrays with different noise parameters applied: (a) 0 pct, no noise; (b) noise fraction 50 pct, noise level 0.33 (50 pct, of thelattice sites displaced by up to 1/3 of the initial lattice spacing); (c) noise fraction 50 pct, noise level 0.66 (50 pct, displaced by 2/3), and (d) noisefraction 90 pct, noise level 0.33 (90 pct, displaced by 1/3).

1676—VOLUME 42A, JUNE 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A

clear that c accounts for differences in the packingdensity of a structure.

Different approaches were presented in the past,which were based on area filling tesselations. In caseswhere the area covered by each dendrite can bedetected easily, the diameter of an equivalent circlewith the same area gives the PDAS of each dendrite.Instead of the area, the perimeter can also be used.[17]

From this data, spacing distributions can be obtained.The drawback is that this method is close to impossibleto apply to multiphase microstructures, with eitherlarge amounts of interdendritic phases or quenchedliquid, and therefore it is mainly appropriate to analyzecellular structures. Another approach is based onvoronoi tesselations,[18] which are based on construct-ing a mesh of lines with equal distances to at least twoadjacent dendrite cores. This requires the determina-tion of the location of all dendrite cores in thestructure. For each tile, the diameter of an circle withan equivalent area is then calculated, which thendefines the local PDAS.

III. THE METHOD

The approach presented here aims at determining thePDAS distribution of a dendritic array, together withother characteristics such as the standard deviation andaverage number of nearest neighbors.To put our new approach into context, consider the

analysis of dendritic arrays that are perfectly arrangedon a two dimensional (2-D) transverse section whosenormal is the direction of solidification. A simple 2-Dcubic lattice is assumed for the sake of simplicity, butothers might also be considered, e.g., a 2-D hexagonallattice. Thus, each lattice point has four nearest neigh-bors, and the lattice parameter denoted Dx is simply thedistance between any given dendrite and its nearestneighbors. The second nearest neighbors, again four innumber, are found at a distance of

ffiffiffi

2p

Dx, which isconsiderably larger than the lattice parameter (ca.41 pct). In this ideal case, the coordination number isequal to the number of nearest neighbors. In a moregeneral case, to determine the lattice parameter andcoordination number within an array of points—with-out any prior assumptions concerning periodicity orstructure—the task is to determine for any given pointthe nearest neighbors and the average nearest-neighbordistance. Note that if the points are not at their idealpositions so that considerable statistic noise is present,the transition from nearest to second-nearest neighborswill not be sharp; however, in principle, it is detectable.Based on these considerations, our new approach

proceeds as follows. One determines first the distancesbetween each point in the array and those surroundingit. In principle, the average distance �k1 to the nearestneighbors can then be calculated, provided that anassumption is made for the number of nearest neigh-bors; a minimum number might be three, but in practice,the number (to be determined) will be greater than this.Subsequent nearest neighbors have to be within closeproximity to the nearest-neighbor distance assumed;thus, they must be within �k1 þ Dk1; where Dk1 is the

(a)

(b)

Fig. 2—Results obtained from analyzing the test structures showingthe mean spacing (a) and its standard deviation (b) as functions ofnoise level and noise fraction ða ¼ 1:6Þ.

Fig. 3—The coordination number obtained from analysing the teststructures as in Fig. 2 as functions of noise level and noise fractionða ¼ 1:6Þ.


search radius around �k1: The chief difficulty lies infinding a rule to determine Dk1: The approach used hereis to use the standard deviation of �k1 scaled by a factor awhere a > 1 is a factor used to ‘‘widen’’ the standarddeviation artificially . This is necessary for two reasons.First, the distance to the next nearest neighbor is greaterthan to the ones taken into account in the standarddeviation; thus, the search radius has to be widened.Second, analyzing the distance to the neighbors takeninto account for the calculation of Dk1; it is found thatthe one furthest away already exceeds �k1 þ Dk1: If thedistance to the next neighbours is within �k1 þ aDk1; it isidentified as another nearest neighbor.

The proposed algorithm can be summarized asfollows:

(a) Determine the location of each dendrite in thearray to be analyzed. Image analysis software canbe used for this purpose; however, in our experi-ence, some manual intervention is advisable tolocate the centers of gravity in order to avoid miss-ing dendrites in the data set. This yields the x andy coordinates of each dendrite in the array.

(b) Calculate the distance between each dendritei and every other dendrite jðj 6¼ iÞ according to

dij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðxj � xiÞ2 þ ðyj � yiÞ2q

(c) For each dendrite i, sort the distances dij intoascending order.

(d) For a given dendrite i and its ki nearest neighbors,calculate the local average nearest neighbor dis-tance �ki1 and its standard deviation Dki1; assume aminimum number—typically three—of nearestneighbors, i.e., ki ¼ 3

(e) Assess whether the distance to the next nearestdendrite (ki+1) falls within �ki1 þ aDki1; if so, countit also as a nearest neighbour, increase ki andupdate �ki1=Dki1 accordingly.

(f) Continue step (e) until the distance exceeds�ki1 þ aDki1: At this point, the actual number ofnearest neighbors ki is the coordination number forthe dendrite i and �ki1 is its primary dendrite armspacing.

(g) Repeat steps (d), (e), and (f) for all other dendritesin the array.

(h) When finished, calculate the average PDAS �k1; itsstandard deviation Dk1; and the average coordina-tion number �k of the dendritic array from the�ki1 þ aDki1 and ki.

IV. APPLICATION OF METHOD: TEST CASES

To demonstrate the application of the algorithm, it isnow applied to several test cases. In the first instance,the primitive cubic lattice is considered. Additional non-ideal lattices were generated by applying an offset ofrandom direction and magnitude to a fraction f of thelattice points. Thus, two random variables (fraction, andmagnitude of offset) influence the test cases. Theoffset—referred to as noise level—was chosen between0 Dx and 1 Dx in both the x and y directions; thus, the

total offset range varied from 0 toffiffiffi

2p

Dx: For noiselevels greater than 0.3, the maximum offset of latticepoints is such that the distance to one of the second-order neighbors becomes less than one. This meansthat an increasing number of lattice points can havemore than four nearest neighbors. Figure 1 illustratessome parts of the test structures considered. The actualtests were performed on grids of size 50 by 50 latticepoints. The structures were then analyzed using thealgorithm described previously, using different valuesfor the factor a.The mean spacing, its standard deviation, and the

corresponding average coordination numbers, which areestimated for various test structures as function of thenoise level, are shown in Figure 2. Different curvesrepresent different noise fractions. As a general trend themean spacing decreases with increasing noise levels andnoise fractions (Figure 2(a)). At the same time, thestandard deviation increases (Figure 2(b)), consistentwith a widening of the spacing distribution. The meanspacing obtained using Eq. [1] is within the standard

(a)

(b)

Fig. 4—Results obtained from analyzing the test structures showingthe mean spacing (a) and the coordination number (b) as functionsof noise level and a factor (noise fraction 50 pct).


distribution. This observation is consistent with theanalysis presented in Reference 16, in which the meanspacing was found to be lower for disordered structures.However, the effect is less pronounced in the structuresanalyzed here, which can be attributed to the 2-D natureof the structure, as opposed to the three-dimensional(3-D) structure considered in Reference 16. The 2-Dstructure resembles rather fibrous structures, whereas the3-D one is appropriate for dispersed particles in a matrix.

With just a few points in the grid being displaced(noise fraction around 10 pct) by a small amount (noiselevel less than 0.2, Figure 3), the coordination numberdecreases. This is because the displacement is too smallto increase the coordination number of the adjacentpoint but large enough to decrease the one of the pointunder consideration. This effect is more pronouncedwhen large fractions of points are affected. Structureslike this do not show clusters of points, but rather theyresemble well-defined, grid-like structures. For the same

reasons, the mean spacing decreases and the standarddeviation increases. However, the initial grid spacing of1.0 is still within the standard deviation.Figure 4 illustrates the effect of the a factor on the

mean spacing and coordination number (noise fractionof 50 pct). Increasing the a factor widens the spacingdistribution. For noise levels up to 10 pct the a factor hasno effect on the mean spacing (Figure 4(a)), and only amild effect on the coordination number (Figure 4(b)).However, for larger noise levels, higher values for aincrease the mean spacing and the coordination number.This effect is more pronounced for a > 1.4.With increasing noise levels and higher noise frac-

tions, the structure changes from well-defined and grid-like to one consisting of an agglomeration of clusters.The concept of nearest neighbors as described previ-ously becomes less applicable. Naturally, the standarddeviation gets larger. As a result, the coordinationnumber increases for sufficiently large a factors.

Fig. 5—Histograms of the spacing distributions obtained from analyzing a cubic structure using different a values. The initial structure was ran-domised using noise fraction of 50 pct and noise level of 0.5.


To investigate the effect of the a factor, a structurewith a noise fraction of 50 pct and a noise level of0.5 was analyzed using different values for the a factor.The histograms of the spacing distribution shown inFigure 5 summarize the results. In all cases, most of thelattice points have spacings at approximately 1.0 and fora < 1.6, normal distributions around unity are observed.However, at a = 1.6 and above, a second mode isobserved at

ffiffiffi

2p

; and at a = 2.0, another one at isobserved 2.0. Second and third modes are not stronglyvisible. These results show clearly that for larger valuesof a, secondary and tertiary neighbors are taken intoaccount in the analysis and indicate that a should chosento be 1.6 or less. It should, however, be pointed out thatin structures with less noise, the results will depend lesssensitively on the value of a. The mean spacing,standard deviation, and coordination number increasesonly slightly with the a value, except for large values(a > 1.8), as can be observed in Figure 6.

It is also interesting to observe how the mean spacingdepends on the number of grid points that are taken intoaccount in the analysis shown in Figure 6. Structureswith a noise fraction of 50 pct and noise levels between

0.2 and 0.6 were used for this analysis. Starting from thecenter of the array, subsets were selected and the meanspacing was calculated for each subset. Figure 7 showsthe resulting mean spacings as functions of the cumula-tive number of grid points. The mean spacings convergequickly—after�40 grid points—toward steady values. Itshould be noted that the mean spacings are alreadywithin the standard deviation of the final spacing longbefore convergence is reached. Structures with greaterdisorder (higher noise level) show stronger variation.

V. APPLICATION OF METHOD: DENDRITICARRAYS

In this section, the algorithm is applied directly toexperimentally observed dendritic microstructures of thetype obtained from nickel-based superalloys cast bydirectional solidification. The microstructure shown inFigure 8(a) was obtained from directionally solidifiedCMSX-4 with the thermal gradient being ca. 4 K/mmand the solidification rate 3.8 mm/min. By using Eq. [1],the PDAS is determined k1 = 302 lm. It should bementioned that measurements based on Eq. [1] areinherently prone to counting errors and the incorrectdetermination of area sizes. The number of dendritesobserved in Figure 8(a) is 148; this value includes alsohalf dendrites on the edges of the figure. If, for example,the number of dendrites differs by ±10, the estimatedPDAS differs from 293 lm to 313 lm. A rigorousestimate would have to discard all dendrites not fullyviewed in the micrograph and then determine the truearea covered by the remaining dendrites, which is ofcourse irregular in shape. If dendrites on the edges of theimage are removed—by discarding all dendrites lessthan 240 lm away from the edges—the number ofdendrites reduces from 148 to 105. On Figure 8, theseare marked with a circle. If it is assumed that the edgedendrite then contribute on average as half a dendrite,the number of dendrites in the image is 126 and theresulting PDAS is 328 lm.

(a)

(b)

Fig. 6—Mean spacing (a) and coordination number (b) correspond-ing to the histograms shown in Fig. 5 (noise fraction 50 pct noiselevel 0.5).

Fig. 7—Mean spacing as function of cumulative number of gridpoints taken into account, for test structures with noise fraction50 pct and noise levels 0.2, 0.4, and 0.6.


The advantage of the new method presented in thisarticle is that it is less prone to these kind of errors; evenif individual dendrites are ‘‘missing’’ in the analysis, itwill only affect the number of neighbors taken intoaccount in the analysis. The positions of the centers ofthe dendritic cores are shown in Figure 8(b). To avoidboundary effects, only dendrites that were more that240 lm away from the edges were taken into account(compare Figure 8(b)), but dendrites on the boundarieswere taken into account as neighbors of other dendrites.It can be observed that the positions are arranged gridlike with straight lines parallel to the y axis butconsiderable shifts parallel to the x axis. In some areas,more random arrangements are observed. From thestructure, the mean PDAS was determined using differ-ent a factors, which ranged from 1.0 to 2.0 (Figure 8(c)).The mean PDAS is found at ~300 lm, which is in goodagreement with the value obtain by using Eq. [1]. Amoderate dependency of the PDAS on a is observed,with a change in slope of the curve a~1.45. This iseven more pronounced in the coordination numberFigure 8(d). This behavior becomes understandablewhen comparing the PDAS histograms for a ¼ 1:4and a ¼ 1:6; which reveal the influence of second-orderneighbors on the PDAS for a = 1.6 (Figure 9(a)). Thelatter is revealed by the appearance of a second mode at

420 lm, which isffiffiffi

2p� 300 lm: However, the influence

of the a factor on the PDAS is rather weak and largelysuperseded by the standard deviation (compareFigure 8(c)), although the standard deviation alsoincreases with a. The importance of taking into accountall first nearest neighbors becomes readily apparentwhen comparing the PDAS histograms in Figure 9(a)with the one obtained by taking into account onlyone—the closest—neighbor Figure 9(b). The meanvalue in Figure 9(b) is approximately 50 lm lower thanin Figure 9(a). These results show that a values between1.4 and 1.6 yield reasonable results when analyzingdirectionally solidified dendritic arrays. The analysisalso tends to yield slightly lower PDAS values than theones obtained by using Eq. [1]. This indicates a lowerdegree of ordering in the directionally solidifieddendritic array.In a second example, the algorithm is applied to

dendritic arrays consisting of large numbers of den-drites. These were obtained from an experimentalnickel-based superalloy cast by directional solidifica-tion[19] using two different sets of solidification param-eters. Figures 10 and 11 show the microstructures andthe results of the analysis; the minimum number ofneighbors was assumed to be 3, and an a factor of 1.4was used. In both cases, a normal distribution of the

(a)

(b)

(d)(c)

Fig. 8—The directionally solidified dendritic structure (a), with the center of dendrites show in (b), was analyzed using different values for a.(c) and (d) show the PDAS and coordination number respectively; only the circled dendrites in (a) were analyzed.


PDAS is found. As expected, a higher thermal gradientG and greater solidification rate v result in a lowerPDAS, and to a more narrow PDAS distribution, i.e., a

smaller standard deviation. The coordination number inboth cases is found to be 4.8, which is greater than the 4expected for a simple primitive cubic lattice but signif-icantly smaller than the value of 6 expected for ahexagonal lattice. The PDAS values determined for thelow and high G cases are 380 � 67:7 lm and147 � 21:1 lm, respectively. The mean spacings agreewith the values obtained by the square-root method ofEq. [1], which are 385 lm and 152 lm, respectively. Thelatter values are influenced not only by errors during thecounting of the dendrites but also by uncertainties inestimating the area covered by the dendrites.

VI. DISCUSSION

The analysis of the test structures demonstrated thatthe proposed method can detect the average spacing andvarious degrees of deviation from the ideal structure.The decrease of the mean spacing in comparison withthe spacing obtained from Eq. [1], the standard devia-tion, and the coordination number could serve as ameasure for the degree of disorder. Application ofexperimental dendritic structures show that these struc-tures are not ideal and possess a certain degree ofdisorder. This is consistent with the observations ofother researchers—McCartney and Hunt,[12] for exam-ple, found good agreement with Eq. [1] for c = 0.8.With the proposed method, such assumptions are notnecessary; what is necessary, however, is a reasonablechoice of the a factor. This can be achieved, as indicatedby the results shown, by choosing a in a way that thehigher order modes at

ffiffiffi

2p

�k1 and 2�k1 do not appear inthe spacing distribution histogram. As pointed outpreviously, these particular higher order modes repre-sent second-order and higher order neighbors. It can beexpected that a is a characteristic of each class of alloysand can be taken as constant within each class. Thismeans that a values exist for superalloys, aluminumalloys, etc. More work is required to establish this. Toachieve a representative PDAS analysis, a sufficientlylarge number of dendrites is required, but this is also thecase for the classics methods. For a low number of

(a)

(b)

Fig. 9—Histogram of the PDAS distribution obtained from analyz-ing the structure shown in Fig. 8, for two different a factors (a) andtaking into account only the first nearest neighbor (b).

0 100 200 300 400 500 600 700

Primary Dendrite Spacing λ1 [μm]

0

10

20

30

40

50

Cou

nts

G = 4 K/mmv = 3 mm/min

Fig. 10—Measured primary dendrite spacing distribution for �300observations in a cross section of a directionally solidified superalloysolidified under G ¼ 4K=mm and v ¼ 3mm=min: The average coor-dination number is 4.8. An image of the experimental microstructureis also shown.

0 50 100 150 200 250 300

Primary Dendrite Spacing λ1 [μm]

0

100

200

300

400

500

Cou

nts

G = 20 K/mmv = 5 mm/min

Fig. 11—Measured primary dendrite spacing distribution for �2000observations on a cross section of a directionally solidified superal-loy, solidified under G ¼ 20K=mm and v ¼ 5mm=min: The averagecoordination number is 4.8. An image of the experimental micro-structure is also shown.


dendrites, the method still yields insight into thestructure of a dendritic array, but the mean PDAS thenhas less significance.

Practical applications for the method can be observedin research and industrial environments to gain addi-tional insight into the mechanism of spacing selectionand for quality control of directionally solidified cast-ings. Although several steps must be performed duringthe analysis, the actual time required for the calculationsis not high on a modern computer system. The analysisof medium size arrays as shown in Figure 8 takes lessthan a minute to run—although the analysis timedepends strongly on the size of the array being analyzed;the 50 by 50 test structures required a few minutes each,which is still acceptable. To analyze the structure withthe proposed method, the centers of the dendrites needto be known; this is the most laborious and potentiallytime-consuming part of the procedure and can be doneeither by manual ‘‘clicking’’ or using an image-analysissoftware. The latter is more error prone as it issometimes hard to distinguish dendrites automatically.Identifying the center of a dendrite is not necessarilystraightforward. The need to determine the centers ofdendrites is, however, not a specific characteristic of theproposed method, as this is also a requirement by all theclassic approaches described earlier. More relevant isthe ability of the method to handle a data set with‘‘missing’’ dendrites. This can happen when large arraysare handled. Single missing dendrites should not have abig effect on the results; more critical are islands ofmissing points, as these can result locally in largespacings and high coordination numbers. Future workwould, therefore, include a mechanism to handle thisproblem. Besides this, the dependency on the initialnumber of dendrites requires that an assumption ismade on the structure that is analyzed.

Future work could comprise the extension of themethod to analyze nondirectionally solidified structures,to quantify and analyze grain size distributions. Thesestructures, however, are a result of the formationmechanism that is more random in nature.

VII. SUMMARY AND CONCLUSIONS

The conclusions drawn from this work can besummarised as follows:

1. A novel algorithm for the estimation of the primarydendrite arm spacing distribution in dendritic arraysis presented. The method is based on the determina-tion of the number of nearest neighbors of eachdendrite in a dendritic array, as well as the determi-nation of the average distance to these. The applica-tion to experimentally determined directionallysolidified arrays has been demonstrated.

2. The method not only gives the average primarydendrite arm spacing and additional characteris-tics of dendritic arrays, for example, a frequency

distribution of the spacing and a coordination num-ber distribution.

3. With the current method, a more detailed analysisof dendritic arrays and their characteristics becomespossible, which will give better indication of thelikelihood of the appearance of other phenomenacommonly associated with the occurrence of PDASabove given thresholds, e.g., initiation of freckling.The response of dendritic arrays to changes ingrowth conditions can be described more realisti-cally, as more parameters, i.e., PDAS distributionwidth and coordination number, to describe den-dritic arrays, are introduced.

4. Subsequent enhancements of the method wouldimprove the ability to handle ‘‘missing position’’data in dendritic arrays. In our experience, experi-mental data sets can be flawed in this way, espe-cially when automated image analysis methods areapplied to large data sets; single dendrites can bemissed on the local scale. One might also proposechanges to remove the dependence of the resultson the initial minimum number of nearest neigh-bors assumed.

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2009, vol. 25 (2), pp. 179–185.3. T.M. Pollock and W.H. Murphy: Metall. Mater. Trans. A, 1996,

vol. 27A, pp. 1081–1094.4. P. Auburtin, T. Wang, S. Cockcroft, and A. Mitchell: Metall.

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14. M.C. Flemings: Solidification Processing, McGraw-Hill, NewYork, 1974.

15. R.T. DeHoff and F.N. Rhines, eds.: Quantitative Microscopy,McGraw-Hill, New York, NY, 1968.

16. J. Gurland: in Quantitative Microscopy, R.T. DeHoff andF.N. Rhines, eds., McGraw-Hill, New York, NY, 1968,pp. 279–90.

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On the Characterization of Directionally Solidified Dendritic Microstructures