quantifying the dispersion of mixture microstructures · journal of microscopy, vol. 225, pt 2...

Journal of Microscopy, Vol. 225, Pt 2 February 2007, pp. 118–125

Received 29 October 2005; accepted 30 August 2006

Quantifying the dispersion of mixture microstructures

Z . P. L U O ∗ & J. H . KO O†∗Microscopy and Imaging Center, Biological Sciences Building West, Texas A&M University,College Station, TX 77843-2257, USA

†Department of Mechanical Engineering, The University of Texas at Austin, Austin,TX 78712-0292, USA

Key words. Composite, dispersion, fibre, inclusion, microstructure, mixture,multiphase, particle, quantification, section.

Summary

A general method to quantify the inclusion dispersion ofmixture microstructures has been developed. The dispersionquantity, D, is defined as the probability of inclusion particlefree-path spacing falling into a certain range of the meanspacing µ, according to the particle spacing data frequencydistribution. Two quantities, D0.1 and D0.2, are proposed,which are the probabilities of the particle free-path spacingfalling into the ranges of µ± 0.1 µ and µ± 0.2 µ, respectively.Both normal and lognormal distributions are discussed, andin both cases, the quantities D0.1 and D0.2 are specifiedas monotonous increasing functions of µ/σ , where µ andsigma; are the mean particle free-path spacing and standarddeviation, respectively. Examples of composite are presented toquantify the dispersion of foreign reinforcements based on theproposed method.

Introduction

Quantitative measurements of material microstructures arebeneficial for a better understanding of correlations betweenmaterial processing-microstructure-property characteristics(Higginson & Sellars, 2003). Mixture microstructures, such asmicrostructures with multiphases, precipitation in the matrix,chemical reactants and products and foreign reinforcementsin composites are commonly studied (Higginson & Sellars,2003; Chung, 2005). Developments involving image-processing techniques have made it possible to quantifythe particle dimensions (diameter, perimeter and areas),mass centre locations (x and y positions) as well as thefractional concentrations based on these measurements.These processing techniques are basically determined by thethreshold of a certain image grey level to select the particles(Russ, 1991).

Correspondence to: Dr. Z. P. Luo. Tel: +1 979 845 1129; Fax: +1 979 847 8933;

e-mail: [email protected]

To study the dispersion of the mixture microstructure, the‘dispersion value’ is an important characteristic. Generally,a good dispersion of the inclusions is favourable to thematerials properties, whereas large aggregation with poordispersion reduces or compromises the properties. Moore(1970, 1972a, b) pointed out that the dispersion of theinclusion particles could be characterized using the followingthree characteristics:

1. Variability index (VI): The variability index is defined as

VI = σobs

σp,

where σ obs is the standard deviation of the material, and σ p

is the Poisson standard deviation from a uniformly randommixture containing the same number of particles.

2. Anisotropy ratios: The anisotropy is specified by the ratios ofmean intercept size (l1/l2 ) (on the particles or the matrix)as measured along these two different principal directions.

3. Degree of patterness (slice index): This characteristic of sliceindex (SI) is used to indicate the severity of laminated orfibrous nature of the microstructure. Lower SI correspondsto a better dispersion. It approaches zero for a uniformdistribution of the particles.

The above three characteristics only reflect the dispersiongrade in different ways, but they do not provide quantitativedefinitions of the dispersion. Recently, Tscheschel et al. (2005)studied the dispersion of high-level silica-filled rubber using avariance σ 2 and a median range h0.5. The former one, σ 2, is ameasure of the grade of the micro-dispersion of filler particles(smaller values of σ 2 imply higher grade of dispersion); thelater one, h0.5, is related to the size of the object (smaller valuesof h0.5 correspond to smaller objects and thus a higher gradeof dispersion).

In this publication, a quantitative definition of the inclusiondispersion is given. A fundamental principle of stereology

C© 2007 The AuthorsJournal compilation C© 2007 The Royal Microscopical Society

D I S P E R S I O N O F M I X T U R E M I C RO S T RU C T U R E S 1 1 9

says that the volume fraction VV of a certain phase ismeasured from the point counting- (P P ), areal analysis- (AA)and lineal analysis (L L )-related measurements (Moore, 1970;Underwood, 1972; Weibel, 1979; Stoyan et al., 1995; Howard& Reed, 1998; Higginson & Sellars, 2003), that is,

VV = P P = AA = L L , (1)

where P P is the fraction of points falling in the interestedphase in a random array of a total number of points, andAA and L L are the corresponding area and length fractions,respectively (for details, see Higginson & Sellars, 2003).This equation serves as the fundamental relationship inquantitative measurements.

Quantification method

The dispersion quantity is related to the free-path spacingbetween the inclusion particle surfaces (rather than theparticle centre spacing), regardless of their shape or size. Themore uniform the spacing between these inclusion surfaces,the higher the dispersion grade will be. If all of the inclusionparticles are distributed at an equal free-path distance, thedispersion is defined as 100%.

Figure 1(a) depicts schematically a solid sample with variousirregular inclusions (in dark colour) in a matrix. A section ismade,asshowninFig.1(b),wherethese inclusionsareseenina2-dimensional (2D) plane view. Because the inclusion particleshave very irregular shape and are in reality three-dimensional(3D), it is impossible to calculate the spacing between them.However, one may use a random line to intercept these particlesto obtain the free-path spacing measurements between theseinclusions, x1, x2, . . ., xi , . . ., xN , as shown in Fig. 1(b). If twoparticles attach together on this line, their spacing is countedas x = 0. These measurements are then divided into groupswith an interval of �x, and the frequency density f i of eachgroup is thus

fi = ni

N �x, (2)

where ni is the number of data falling into this group, and N isthe total number of the measurements. Therefore, a histogramcan be constructed to exhibit the spacing data distribution, asshown in Fig. 1(c). If the interval �x is small enough (usuallydivided into 10–100 classes) and the number of measurementsN is sufficiently large (usually over 100), then one can obtain asmooth curve of the frequency function, that is, the probabilitydensity function, f (x), as shown in Fig. 1(d). It has a mean valueof µ and a standard deviation of σ . It is well known that thestandard deviation σ is a measurement of the data dispersion,that is, smaller σ indicates more data closer to the mean µ.However, σ does not quantify the dispersion ‘value’. It canbe noted that the area below the curve is the probability of acertain range of the spacing data. For example, the probabilityof the spacing data falling into the range of a–b is

∫ ba f (x)d x.

Certainly,∫ +∞

0 f (x)d x = 1. The probability of a certain rangeof spacing around the mean µ can be defined as the dispersionD. In the range of µ ± 0.1 µ, the dispersion D0.1 is

D0.1 =∫ 1.1µ

0.9µ

f (x)d x. (3)

Similarly, in the range of µ ± 0.2 µ, D0.2 is

D0.2 =∫ 1.2µ

0.8µ

f (x)d x. (4)

The dispersion quantities of D0.1 and D0.2 are the shadowedareas below the f (x) curve as shown in Fig. 1(d), correspondingto the possibilities of the spacing data falling into the rangeof µ ± 0.1 µ and µ ± 0.2 µ, respectively. Higher D valuesindicate more spacing data closing to the mean µ, and thusmore uniform distribution of the inclusion particles. In thisway, one may define the dispersion in other data ranges aroundthe mean spacing µ, such as D0.3, D0.4, D0.5, etc. It is evidentthat for smaller σ , the probability f (x) around µ is higher,and thus, D0.1 and D0.2 are also higher for the smaller σ ,although the Eqs (3) and (4) do not contain σ directly.

The quantity of the dispersion D is related to the accuratemeasurement of the spacing x. If the inclusions have regularspherical shape and their locations (x, y, z) are known, thespacing between particles 1 at centre position (x1, y1, z1) withradius r1, and particle 2 at (x2, y2, z2) with radius r2 is

x =√

(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 − r2 − r1.

(5)

For some soft samples, such as some biological and polymericspecimens, the dissector method can be applied to assist thespacing measurements (Gundersen, 1986; Howard & Reed,1998), for example, a physical dissector using two physicallyseparated sections with known thickness as prepared bymicrotome (two parallel sections in Fig. 1(a) with knownthickness), or an optical dissector using two images obtainedat two different focal planes by a confocal microscope(Howard & Reed, 1998; Gundersen, 2001). However, theseapproaches may not be applicable to most hard materials,such as metal- and ceramic-matrix composites (Luo et al.,2001; Chung, 2005) and some polymer-matrix composites,depending on their hardness and transparency (Koo et al.,2003). Sectioning these hard materials, such as by the ionmilling method as described later, looses the thickness controlduring the preparation process. In this case, isotropic uniformrandom sampling should be used to make the microstructurerepresentative (Howard & Reed, 1998). In fact, under the mildcondition that the inclusions are isotropic in the matrix, themean free path in 2D is an unbiased estimator of the mean freepath in 3D (Underwood, 1970, pp. 82–83).

It should be pointed out that in this publication, the spacingis measured directly. If one calculates the fraction of xi/lj,

C© 2007 The AuthorsJournal compilation C© 2007 The Royal Microscopical Society, Journal of Microscopy, 225, 118–125

1 2 0 Z . P. L U O & J. H . KO O

(c)

Freq

uenc

y, f

Spacing, xx

(d)

D0.2

D0.1

x

Freq

uenc

y, f

µ0.1µ

0.2µ 0.2µ

0.1µ

(a) (b)

x1 x2 xi

ljxN

x=0

(c)

Freq

uenc

y, f

Spacing, xx

(c)

Freq

uenc

y, f

Spacing, x∆x

(d)

D0.2

D0.1

x

Freq

uenc

y, f

µ0.1µ

0.2µ 0.2µ

0.1µ

(d)

D0.2

D0.1

x

Freq

uenc

y, f

σ σ

µ0.1µ

0.2µ 0.2µ

0.1µ

(a) (b)

x1 x2 xi

ljxN

x=0

x1 x2 xi

ljxN

x=0

Fig. 1. (a) A solid specimen contains various irregular inclusions (dark colour) in the matrix; (b) a section from the solid specimen, where free-path spacingof the inclusions are measured; (c) construction of histogram; (d) definition of dispersion D0.1 and D0.2 quantities from the frequency function f (x).

where lj is the length of the random jth line in Fig. 1(b), thefraction of the matrix space can be obtained according toEq. (1). In fact, Moore (1970, 1972a, b) used the ratio ofthe mean spacing along two different directions to specify theanisotropy character, whereas here we define the dispersionfrom the data scattering, which can be measured along anydirection, that is, the dispersion can be quantified in alldirections. Therefore, one may use the dispersion D alongdifferent directions to specify the material anisotropy.

Consider the situations of particle distributions as discussedby Russ (1991). If the particles are regularly distributed, asshown in Fig. 2(a), the frequency curve only shows one peak,as schematically shown in Fig. 2(b), with a mean µ and astandard deviation σ . A normal and lognormal distributioncurves are superimposed, with the same mean µ and standarddeviation σ , as shown in thin line in Fig. 2(b). The normaldistribution is defined as

f (x) = 1

σ√

2πe− 1

2 ( x−µ

σ)2, (6)

and the lognormal distribution is defined as

f (x) =

1

xn√

2πe− 1

2 ( ln x−mn )2 (x > 0)

0 (x ≤ 0)(7)

where

m = lnµ2√

µ2 + σ 2(8)

and

n =√

lnµ2 + σ 2

µ2. (9)

One may choose the normal or lognormal distribution to fitthe real frequency distribution depending on which one givesthe best fit. However, when clusters are present, as shown inFig. 2(c), the frequency distribution curve exhibits two peaks ofbimodal distribution (or more peaks depending on the cluster



1

(b)(a)

(d)(c)

µ

µFr

eque

ncy,

fFr

eque

ncy,

f

Spacing, x

Spacing, x

normallognormal

norm

al logn

orm

al

µ1

1 2

µ2

2σ1

(b)(a)

(d)(c)

µσ σ

µ

σ σ

Freq

uenc

y, f

Freq

uenc

y, f

Spacing, x

Spacing, x

normallognormal

norm

al logn

orm

al

µ1

σ1 σ2

µ2

σ2

Fig. 2. (a) Regularly distributed particles; (b)schematic spacing distribution of particles in(a); (c) clustered particles; (d) schematic spacingdistribution of particles in (c). Normal and lognormaldistribution curves are superimposed on (b) and (d),with the same mean µ and standard deviation σ .

distance distribution), as shown in Fig. 2(d). The first onewith (µ1, σ 1) reflects the distribution inside the clusters withshorter spacing, and the second one with (µ2, σ 2), thedistribution of these clusters with larger spacing. If one usesthese two peaks to quantify the dispersions D1 and D2 andthen take the average of them for the sample dispersion D,this dispersion maybe overestimated, because the D1 onlyrepresents the dispersion within the clusters and D2 thedispersion associated with the clusters (both of them maybe good ones). However, this problem may be solvedby superimposing the normal distribution curve and thelognormal distribution curves with the same mean µ andstandard deviation σ of the entire data (mixture of the twopeaks), as shown in Fig. 2(d). Such distribution curves reflectthe scattering of the entire data, and thus one may use them toquantify the dispersion in this case. It is seen that when clustersare present, the dispersion grade is very decreased as expected.

The treatments by normal and lognormal distributionsare presented in the appendix separately to deduce generaldispersion formulas, which show that the dispersions D0.1

and D0.2 are only monotonous increasing functions of µ/σ ,as shown in Fig. 3. Once the spacing data x1, x2, . . . , xi, . . . ,xN are measured, one may use the data sample mean x =

∑i xi

N

and data sample standard deviation s =√∑

i (xi −x)2

N−1 to estimatethe bulk material population mean µ and population standarddeviation σ , respectively, if the number of measurementsN is very large, that is, µ̂ = x and σ̂ = s, where µ̂ andσ̂ are the estimators of µ and σ , respectively. Therefore,

D can be obtained accurately from (12) and (13) for thenormal, and (17) and (18) for the lognormal distributions,respectively, or estimated fairly rapidly from (14) and (15) forthe normal, and (19) and (20) for the lognormal distributions,respectively.

Examples

Two examples of the microstructure of carbon nanofibre(CNF)-reinforced polymer matrix composite (Koo et al., 2003)obtained by transmission electron microscope (TEM) arepresented in Figs. 4 and 5. The samples were prepared byion milling (for hard materials), or ultra-microtome (for softmaterials) using a diamond knife at 5–6◦ cutting angle. Theselection of the sample preparation method depends on thesample hardness. The ion mill sample preparation processwere as follows: the bulk sample was first cut into slices about1-mm thick, then these slices were mechanically groundedand polished until 50–100 µm and then punched into smalldisks with 3-mm diameter. These disks were further thinnedby dimple grinding to achieve the centre thickness of 10–30 µm, and finally were thinned by ion beam until perforation.The image processing was performed using an IMAGEJprogram.

The sample prepared by ion milling has non-uniformthickness, as shown in Fig. 4(a). The sample by ultra-microtome has uniform thickness but cutting artefacts areintroduced,as indicatedbyarrowheadsinFig.5(a).After image


1 2 2 Z . P. L U O & J. H . KO O

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8 9 10

Normal Lognormal

D0.1

D0.2

Dis

pers

ion

µ/σ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8 9 10

Normal Lognormal

D0.1

D0.2

Dis

pers

ion

µ/σ

Fig. 3. Calculated dispersion D0.1 and D0.2 as functions of µ/σ for normal and lognormal distributions. The lognormal distribution shows slight highervalues over the normal distribution.

0

0.0005

0.001

0.0015

0.002

0.00250

20

0

40

0

600

800

100

0

120

0

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00

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Spacing (nm )

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cy

(a) (b) (c)500 nm

0

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Fre

qu

en

cy

(a) (b) (c)500 nm

Fig. 4. (a) TEM image of CNF composite (sample prepared by ion milling); (b) processed image by removing the background; (c) spacing frequencydistribution with lognormal distribution fitting (smooth curve). The arrow indicates the mean position.

(a) (b) (c)

0

0.0005

0.001

0.0015

0.002

0.0025

0

20

0

40

0

60

0

80

0

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00

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00

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00

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300

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500 nm(a) (b) (c)

0

0.0005

0.001

0.0015

0.002

0.0025

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20

0

40

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Spacing (nm)

Fre

qu

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cy

(a) (b) (c)

0

0.0005

0.001

0.0015

0.002

0.0025

0

20

0

40

0

60

0

80

0

10

00

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Spacing (nm)

Fre

qu

en

cy

500 nm

Fig. 5. (a) TEM image of CNF composite (sample prepared by ultra-microtome); (b) processed image by removing the background; (c) spacing frequencydistribution with lognormal distribution fitting (smooth curve). The arrow indicates the mean position.



Table 1. Statistics of figures 4 and 5.

Figure x(nm) s (nm) Minimum x (nm) Maximum x (nm) N

Figure 4 455.3 374.1 28.2 2745.4 338Figure 5 508.1 493.5 46.3 2990.8 313

processing to show the reinforcement edges, these samplepreparation artefacts by both methods are removed, as shownin Figs 4(b) and 5(b), respectively. From such images, thespacing data between these CNF are measured from 20 × 20equal distance horizontal and vertical grid lines, respectively.The histograms are shown in Figs 4(c) and 5(c), respectively,and the statistical data of the measurements are given inTable 1.

From the histogram in Figs 4(c) and 5(c), it is found that thelognormal distribution gives the best fit, as shown by smoothcurves that are superimposed. In Fig. 4, x/s = 1.2171, andin Fig. 5, x/s = 1.0296. Using x and s to estimate µ andσ , respectively, therefore, D0.1 and D0.2 for the lognormaldistribution are calculated out by the Eqs (17) and (18) foraccurate results, and by (19) and (20) for approximations, asshown in Table 2. Basically, the results by these two differentmethods are reasonably close to each other. Visual inspectionshows that Fig. 4 has better dispersion over Fig. 5, becauseFig. 5 contains some unoccupied areas in the centre. Thequantified results of D, as listed in Table 2, reveal that thedispersion of Fig. 4 is higher than Fig. 5.

Concluding remarks

1. An universal method to quantify the dispersion D of mixturemicrostructures was developed. Two quantities, D0.1 andD0.2, are proposed, which are the probabilities of the particlespacing falling into the ranges of µ ± 0.1 µ and µ ± 0.2 µ,respectively, according to the spacing data frequencydistribution.

2. Two cases of normal and lognormal distributions ofthe spacing data were considered, and the dispersionquantities were formularized in Eqs. (12)–(15) and (17)–(20), respectively. In both cases, it was demonstrated thatthe dispersion D is only a monotonous increasing functionof µ/σ .

Table 2. Quantified dispersions of Figs 4 and 5.

Lognormal distribution Lognormal distribution,by (17) and (18) approximation by (19) and (20)

Figure x/s

D0.1 D0.2 D0.1 D0.2

Figure 4 1.2171 10.45% 21.08% 10.45% 21.14%Figure 5 1.0296 9.04% 18.28% 9.04% 18.27%

3. The challenge to quantify the dispersion D is the accuratemeasurement of the free path distance between theinclusions in the matrix. Once the spacing data aremeasured and the mean x and standard deviation s areobtained, depending on data distribution, the dispersionquantity D can be simply calculated from these equationsby using x and s to estimate the material populationmean µ and population standard deviation σ , respectively,if the number of measurements N is sufficiently large.The selection of normal or lognormal distribution dependson which one fits the spacing data distribution, buteach of them gives consistent quantifications. Usuallythe measurements obey the lognormal distribution, asdemonstrated in the examples. Even in the case of thepresence of clusters, as depicted in Fig. 2(c), one may usethe lognormal (or normal) distribution to compare thedispersions quantitatively.

4. The method proposed in this publication can be applied foreither irregular inclusion particles, as shown in Fig. 1(a)and (b), or regular shape particles, as shown in Figs 2(a)and (c). In the later case, the particle spacing can becalculated from their centre coordinates using Eq. (5), andthus the dispersion can be obtained from their spacing datascattering.

Acknowledgements

The authors are grateful to a reviewer for critical comments,and to Dr. Stan Vitha, Microscopy and Imaging Center, TexasA&M University, for some valuable discussions. This workwas supported by the Air Force Office of Scientific Research(AFOSR), Arlington, Virginia.

References

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Dougherty, E.R. (1990) Probability and Statistics for the Engineering,Computing, and Physical Sciences. Prentice-Hall, Inc., Englewood Cliffs,New Jersey.

Gundersen, H.J.G. (1986) Stereology of arbitrary particles: a review ofunbiased number and size estimators and the presentation of some


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new ones, in memory of William R. Thompson. J. Microsc. 143,3–45.

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Higginson, R.L. & Sellars, C.M. (2003) Worked Examples in QuantitativeMetallography. Maney Publishing, London.

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Appendix

Dispersion for the normal distribution case

Let us start with the simple case of a normal distribution,with a mean of µ and standard deviation of σ . This normaldistribution can be converted to a standard normal distribution(µ = 0, σ = 1) using a substitution of

z = x − µ

σ, (10)

and thus we have dx = σdz (Dougherty, 1990). Hence,the probability (dispersion) between a–b is calculated

as

D =∫ b

a

1√2πσ

e− 12 ( x−µ

σ)2d x

=∫ b−�

σ

a−µ

σ

1√2π

e− z22 d z

= �

(b − µ

σ

)− �

(a − µ

σ

).

(11)

Here �(z) is the probability from –∞ to z, �(z) =∫ z−∞ e−y2/2d y, whose value can be found in statistical

references or a number of online calculators. Therefore, forD0.1, a = 0.9µ and b = 1.1µ, hence

D0.1norm = �(0.1µ/σ ) − �(−0.1µ/σ ). (12)

Similarly we have

D0.2norm = �(0.2µ/σ ) − �(−0.2µ/σ ). (13)

From the above (Eqs (8) and (9)), it is clear that D0.1 and D0.2

are only functions of (µ/σ ). Therefore, D0.1norm and D0.2norm

are plotted as functions of (µ/σ ) from 0 to 10, as shown in Fig. 3.It can be seen that the dispersion D is a monotonous increasingfunction of (µ/σ ), that is, D increases with increasing µ,which is reasonable because for larger µ the sample looksmore uniform. This case is similar to the study by Tscheschelet al. (2005) who showed that smaller values of h0.5 indicatesmaller objects and thus higher dispersion. D also increaseswith decreasing σ , because smaller σ indicates less spacingdata scattering.

For convenience, we give here the regression results ofD0.1norm and D0.2norm by curve fitting as follows:

D0.1norm = 6.8843 × 10−5 + 7.964 × 10−2(µ/σ )

+ 1.043 × 10−4(µ/σ )2 − 1.6286 × 10−4(µ/σ )3

+ 3.8639 × 10−6(µ/σ )4 (14)

andD0.2norm = −4.0117 × 10−4 + 0.16056(µ/σ )

− 2.8118 ×10−4(µ/σ )2 −1.1826 × 10−3(µ/σ )3

+ 5.6084 × 10−5(µ/σ )4 (15)

Dispersion for the lognormal distribution case

Lognormal distribution should also be considered, becausemost of the measurements are found to obey the lognormaldistribution instead of the normal distribution (Carpenter et al.,1998; Velecky & Pospisil, 2000), as will be demonstrated in thefollowing examples. In the case of lognormal distribution, asdefined in (7), let z = lnx, then we have dz = dx/x. Hence, theprobability (dispersion) between a–b is calculated as

D =∫ b

a

1

xn√

2πe− 1

2 ( ln x−mn )2d x



=∫ ln b

ln a

1

n√

2πe− 1

2 ( z−mn )2

d z

= �

(ln b − m

n

)− �

(ln a − m

n

). (16)

Therefore for D0.1, taking a = 0.9 µ and b = 1.1 µ andsubstituting m and n using (8) and (9) yield

D0.1 log = �

ln 1.1µ − lnµ2√

µ2 + σ 2√ln

µ2 + σ 2

µ2

− �


µ2 + σ 2√ln

µ2 + σ 2

µ2

= �

ln 1.1

√1 + (µ/σ )2

(µ/σ )√ln

1 + (µ/σ )2

(µ/σ )2

−�

ln 0.9

√1 + (µ/σ )2

(µ/σ )√ln

1 + (µ/σ )2

(µ/σ )2

.

Similarly,

D0.2 log = �


µ2 + σ 2√ln

µ2 + σ 2

µ2

(17)

− �


µ2 + σ 2√ln

µ2 + σ 2

µ2

= �

ln 1.2

√1 + (µ/σ )2

(µ/σ )√ln

1 + (µ/σ )2

(µ/σ )2

−�

ln 0.8

√1 + (µ/σ )2

(µ/σ )√ln

1 + (µ/σ )2

(µ/σ )2

. (18)

As for the case of normal distribution, D0.1log and D0.2log

are only functions of µ/σ . For a given value of µ/σ , D0.1log

and D0.2log can be calculated according to Eqs (17) and (18).These relations are also plotted in Fig. 3. It is seen that D0.1log

and D0.2log are slightly higher than D0.1norm and D0.2norm,respectively, in the range computed. D0.1log and D0.2log areregressed as

D0.1log = 1.1539 × 10−2 + 7.5933 × 10−2(µ/σ )

+ 6.6838 × 10−4(µ/σ )2 − 1.9169 × 10−4(µ/σ )3

+ 3.9201 × 10−6(µ/σ )4 (19)

and

D0.2log = 2.266 × 10−2 + 0.15629(µ/σ )

+ 4.442 × 10−4(µ/σ )2 − 1.2738 × 10−3(µ/σ )3

+ 5.9978 × 10−5(µ/σ )4. (20)


quantifying the dispersion of mixture microstructures · journal of microscopy, vol. 225, pt 2...

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