Research Article Fractal Dimension Calculation of a ...downloads.hindawi.com/journals/jnm/2015/384835.pdfResearch Article Fractal Dimension Calculation of a Manganese-Chromium Bimetallic

Research ArticleFractal Dimension Calculation of a Manganese-ChromiumBimetallic Nanocomposite Using Image Processing

Amir Lashgari1 Shahriar Ghamami1 Saeedeh Shahbazkhany1

Guillermo Salgado-Moraacuten2 and Daniel Glossman-Mitnik3

1Department of Chemistry Faculty of Science Imam Khomeini International University Qazvin 34148-96818 Iran2Departamento de Ciencias Quımicas Facultad de Ciencias Exactas Universidad Andres Bello Sede Concepcion4070000 Concepcion Chile3Laboratorio Virtual NANOCOSMOS Departamento de Medio Ambiente y EnergıaCentro de Investigacion en Materiales Avanzados (CIMAV) 31136 Chihuahua CHIH Mexico

Correspondence should be addressed to Daniel Glossman-Mitnik danielglossmancimavedumx

Received 11 February 2015 Revised 9 April 2015 Accepted 16 April 2015

Academic Editor Ungyu Paik

Copyright copy 2015 Amir Lashgari et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Bimetallic materials which have the ability to convert heat change into mechanical movement normally consist of two bondedstrips of dissimilar metals that expand at different rates We describe how we made a manganese-chromium (Mn-Cr) bimetallicnanocomposite using the centrifuge method and a low-to-high approach We conducted scanning electron microscope (SEM)imaging energy-dispersive X-ray spectroscopy (EDX) analysis and X-ray diffraction spectra of the nanocomposite to prove itsidentityWe examined how centrifuge speed process time and the use of an ldquointruder agentrdquo affected the properties of thematerialThe fractal dimension is a significant factor that can be used to approximate the surface roughness the texture segmentation and animage of the studied compounds We calculated the technique of fractal dimensions using image-processing values on a computerand histogramplot with the SEM image of theMn-Cr bimetallic nanocomposite usingMATLAB softwareWe applied the StatisticalPackage for the Social Sciences software for statistics data extracted from the SEM image of the nanocomposite and obtained thefollowing results mean = 1778 median = 1770 max = 198 min = 160 skewness = 0177 range = 038 and harmonic mean = 1771for fractal dimension of the SEM image

1 Introduction

Metalmatrix composites (MMCs) are an important engineer-ing material due to their various applications LightweightMMCs due to their significant hardness and strength havefound wide applications in the aerospace and automotiveindustries However these applications have remained lim-ited due to the lack of stretch of the composites Compositeconversion nanocomposites increase the yield strength andremove these limitations [1 2] MMCs have steadily becomemore favored in recent years for their high strength and stressresistance properties [3ndash11]

Chromium (Cr) and chromium compounds are widelyused in many industrial procedures such as textile dyeingprinting inks electroplating leather tanning chromiumplating and pigments as serious manufacturing resources

[12 13] Manganese (Mn) is a mineral created in largeamounts in both plant and animal tissues Manganese lowersthe temperature at which austenite converts into ferritetherefore avoiding cementite precipitation at ferrite grainboundaries and purifies the resulting pearlitic structuresAlloying elements such as manganese also enable solution-hardening of ferrite but this effect is limited compared withthat of carbon nitrogen phosphorus and even silicon [14 15]

Image processing of microstructures involves severalpreparation steps division and postprocessing that finallylead to the extraction of quantitative data Recently com-puter-based image-processing methods have advancedquickly making it possible to quantitatively describe com-plex colors shapes texture properties and sizes Image-processing methods are regularly used in tandem withmechanical and instrumental devices to change human

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 384835 9 pageshttpdxdoiorg1011552015384835

2 Journal of Nanomaterials

manipulative work in the show of an assumed process[16 17]

The term ldquofractal dimensionrdquo has been used to describeseveral aspects of particle morphologyThe fractal dimensionof an individual particle which can be a main particle or acomposite particle such as a floc or aggregate denotes theindiscretion and self-similarity of the particle The fractaldimension parameter is determined by computing the lengthof the predictable perimeter by taking steps of a known lengthalong the border and including the number of ladders Theborder length increases with decreasing step size because asmaller step size allows for the presence of more detail inthe particle outline [18ndash20] The word ldquofractalrdquo is derivedfrom the Latin ldquofractiousrdquo meaning fragmented a fractal isdefined as a mathematical object with a fractional (nonin-teger) dimension It seems that wherever there is structuraldisorder and chaos fractal geometry can be an efficient wayof describing and analyzing it

Fractal dimension is a very good mathematical methodfor the study of unequal geometric substances Fractal dimen-sion can reflect characteristics appearing in nature Theapplication of fractal system allows us to explain numerousstates of fragmenting and branching occurring in ecologi-cal biological and other systems [21ndash24] The output is abinarized image with a one-pixel-wide boundary appropriateto be used for box-counting calculations to approximate thefractal dimension of the boundaryWe developed aMATLAB(MathWorksNatickMassachusetts USA) programbased ontheory to implement the box-counting technique for fractaldimension calculations

Several specific theoretical and numerical models havebeen proposed in order to assess the fractal dimensionRecent studies have examineddimensions but themajority ofthem were only one- or two-dimensional These dimensionsare a clear limitation to the study of complex and moregeneral strengthening geometries [25 26]

Manganese and chromium have been used in this nano-composite potassium permanganate and potassium dichro-mate were used as the starting materials Scanning electronmicroscope (SEM) images of the nanocomposites may beregular or irregular and their fractal dimension can be cal-culated It is interesting to compare the fractal images of thenanocomposites For example in this paper we investigatedthe size of the nanoparticles and calculated their fractaldimension We also measured the average minimum andmaximum of fractal dimensions and imagersquos histogram andnormalized imagersquos histogramplot of the SEM image [27ndash29]

2 Experimental

21 Materials and Instruments The starting materials wereobtained from Merck (Berlin Germany) and were usedwithout further purification The surface morphology of theproduct was characterized by using a LEO-1430VP scanningelectron microscope (Zeiss Germany) with an acceleratingvoltage of 15 kV X-ray powder diffraction (XRD) measure-ments were performed using a Philips Xrsquopert diffractometer(PANalytical Almelo Netherlands) with monochromatized

CuK120572 radiation The sizes of the selected samples wereestimated using the Scherer method For identification theSEM samples were coated with gold

22 Synthesis of Nanocomposite Manganese and ChromiumPotassium permanganate (KMnO

4) and potassium dichro-

mate (K2Cr2O7) were used as starting materials to prepare

this nanocomposite Initially 02 g of potassium perman-ganate was transferred to a test tube and 001 g of claywas added as an intruder agent Ethanoic acid CH

3COOH

(4mL) was then added to the mixture and the tube wasquickly transferred to a centrifuge and run at 3000 rpm for45 minutes The sample was then taken out of the centrifugeand put asideThen 05 g of potassium dichromate and 001 gof clay were added to another test tube NH

3(5mL) solution

was added and the tube was placed in a centrifuge and run at3000 rpm for 45 minutes

23 Characterization of the Nanocomposite The XRD tech-nique was used to determine the composition of the milledpowder The morphology and weight and atomic percentageof the nanocomposite were observed using a SEM image andEDX analysis respectivelyThe contents of the first tube werethoroughly stirred and transferred to a second tubeThis sec-ond tube was returned to the centrifuge and run at 3000 rpmfor 45 minutes The test tube was then removed fromthe centrifuge and the precipitate was separated out usingfilter paper After drying the precipitate was imaged withSEM instrument and the atomic percentages were achievedby EDX analysis So the XRD spectrum was obtained toshow that a manganese-chromium nanocomposite had beenproduced

24 Calculation of Fractal Dimension and Image AnalysisDimension is one of the most significant ideas in fractalgeometryNow there are numerous descriptions of this quan-tity including packing dimension box-counting dimensionHausdorff dimension and modified box-counting dimen-sion Since the mathematic design and approximation ofthe box-counting dimension are straightforward this quan-tity has become one of the most commonly used generaldimensions [30 31] The calculations and analysis were doneby programming with MATLAB and Statistical Package forthe Social Sciences (SPSS IBM Armonk New York USA)software [32] Moreover by declaring that the calculationresults were comparable all of the images were processedusing a similar crop size and threshold value Using thismethod we calculated the fractals dimension theoretically byapplying the following formula

119863 = lim119903rarr0

log119873119903

log (1119903) (1)

where 119863 is the fractal dimension and 119873119903is the number of

squares including a part of the considered fractal dimensionsConsidering the resulting numbers we drew a loga-

rithmic diagram in which the vertical and horizontal axeswere log(1119903) and log119873

119903 respectively and the coefficient of

Journal of Nanomaterials 3

2120579

4 10 20 30 40 50 60 700

10

20

30

Lin

(cou

nts)

(a)

2120579

4 10 20 30 40 50 60 700

10

20

30

Lin

(cou

nts)

(b)

Figure 1 XRD spectrum for the manganese-chromium bimetallic nanocomposite produced via centrifuge (a) poor crystalline structure(amorphous) and (b) good crystalline structure

the box size was 13 We used a program written in MATLABto calculate the fractal dimension There are several methodsfor calculating the fractal definition including (1) the self-similarity definition [33] (2) themass fractal dimension [34](3) the spare-sphere fractal dimension [35] (4) the Eucliddimension [36] (5) the grid style [37] (6) the analyticalapproach for fractal diminishing [38] and (7) the box-counting fractal dimension [39]

We used the self-similar method for determining thefractal dimension from the SEM image which employed thebox-counting method for the fractal dimension

For the box-counting fractal dimensionwe suppose that119865is a nonempty and bounded subset of 119877119899 and we suppose that119873119903(119865) is the fewest number of collections with a maximum

dimension of (119903) that can cover (119865) The box-countingdimensions under and above (119865) if they are equal we saythey are equal to the box-counting dimension or the 119865 boxdimension then we define

dim119861

119865 = lim119903rarr0

log119873119903(119865)

minus log 119903 (2)

The circle of cubes in 119903 coordinates from 119877119899 implies cubes inthe following form

If [1198981119903 (1198981+ 1)119903] times sdot sdot sdot times [119898

119899119903 (119898119899+ 1)119903]

then119898119899 119898

1are integers

Finally119873119903radic119899(119865) le 119873

1015840

119903(119865)

If 119903radic119899 lt 1then log119873

119903radic119899(119865) log(119903radic119899) le log1198731015840

119903(119865)(minus logradic119899minus

log 119903)If we limit dim

119861119865 = lim

119903rarr0(log1198731015840

119903(119865) minus log 119903)

then dim119861119865 le lim

119903rarr0(log1198731015840

119903(119865) minus log 119903)

Finally1198731015840119903(119865) le 3

119899119873119903(119865)

3 Results and Discussion

31 Analysis of the Effective Factors for Creating the Nano-material It is important to note that we used centrifugesto make the nanomaterials with the low-to-high method

After we mixed the solutions they were immediately put intothe centrifuge and an intruder agent was used to ensurethat the particles were not connected This intruder agentmust be inert and must not participate in the main reactionCentrifugal force ensures that the particles produced are notconnected and remain on the nanoscale The advantages ofthis method are its low cost rapidity and ease of application

Here we examine the preparation of a manganese-chromium bimetallic nanocomposite The XRD spectrumand SEM image and EDX analysis prepared for the sampleboth confirmed the creation of a nanocomposite In the XRDspectrum (Figure 1) no distinguishing peaks are visible andthese data did not identify the substance This outcome isnormal because composites are amorphous and their spectraare wide and uniform unlike the spectra of crystallinematerials that have clear and sharp peaks

The SEM image (Figure 2) shows that the particles aresmaller than 100 nm which also confirms that this materialis a nanomaterial We were therefore able to confirm that wemade amanganese-chromiumbimetallic nanocompositeWethen obtained 30 random pictures from the SEM image andcalculated their fractal dimensions using MATLAB software[40] For these 30 fractal dimensions we obtained imagersquoshistogram and normalized imagersquos histogram mean medianmax and min range skewness and harmonic mean usingSPSS software [41]

The EDX analysis (IDFix report) for the manganese-chromium bimetallic nanocomposite is shown in Table 1 andFigure 3 As shown in the IDFix report the gold present(1189) was used for SEM imaging and was not part ofthe nanocomposite The primary elements of the nanocom-posite were manganese (2433) chromium (384) carbon(677) nitrogen (261) potassium (873) and oxygen(3634) other elements (silicon (224) sulfur (025)copper (207) and calcium (092)) were related to the clayintruder compound (Figure 3) These results confirmed thecorrect synthesis of nanocomposite

32 Analysis Using MATLAB and SPSS Data Thirty pictureswere selected at random from the SEM image We firstcalculated the fractal dimension for these 30 images usingMATLAB software (Table 2) Mean median max and min


Figure 2 SEM images of manganese-chromium bimetallic nanocomposite produced via centrifuge

Table 1 IDFix report for the manganese-chromium bimetallic nanocomposite

Elta Intb 119882c119860d Elta Intb 119882c

119860d

C 802 677 1422 Ca 798 092 058N 170 261 469 Cr 1863 384 186O 9324 3634 5729 Mn 9739 2433 1117Si 3121 224 201 Cu 329 207 082S 307 025 020 Au 121 1189 152K 8276 873 563aElement bintensity cweight percentage datomic percentage

15000

10000

5000

0

Cou

nts (

eds)

0 5 10

(keV)

AuL120572AuLI

CuL120572C K120572

K K120572

AuM120572AuM120573

S K120573 K K120573CaK120572 CrK120572

MnK120572

MnK120573CrK120573 CuK120572 CuK120573CaK120573

S K120572SiK120572

O K120572

CrL120572

MnL120572CaL120572N K120572

Figure 3 IDFix plot for the manganese-chromium bimetallicnanocomposite

range skewness and harmonic mean were also obtained forthese data (Table 2) using SPSS software (Table 3) In Table 3themean of the fractal dimension is 1778 themedian is 1770the minimum is 160 the maximum is 198 the range is 038the skewness is 0177 and the harmonic mean is 1771

In statistics and probability theory the skewness reflectsthe degree of asymmetry of the probability distribution

Skewness is a measure of the presence or absence of dis-tribution function symmetry For a perfectly symmetricaldistribution the skewness is zero for an asymmetric distri-butionwith stretching toward higher quantities the skewnessis positive Conversely for an asymmetric distribution withstretching toward smaller quantities the skewness is negativeGiven that the amount of skewness for particles in thisnanocomposite was positive (+0177) this result reveals anasymmetrical distribution of fractal data with stretchingtoward higher quantities (Table 3) The harmonic mean is ameasure of central tendency and is typically calculated whenthe average of the data is importantThe harmonic mean wasfound to be 1771 for this nanocomposite (Table 3)

A grayscale digital image consists of separate points ofgray tones or brightness before incessantly variable tones Anormal image is separated into a number of specific pointsof brightness and each of those points is defined via adigital data value A pixel is the greatest fundamental elementof some digital image Each brightness point is a pixel ofthe digital image Essentially the image histogram showsthe distribution of the pixel intensities in the image and iscontinuously used as a reference Interactive thresholdingcan be very effective and afford fast precise information[42 43] To establish the frequency distribution model for


Table 2 Fractal dimension for 30 random images from the SEM image of the manganese-chromium bimetallic nanocomposite

Pic number Pic Fractal dimension Pic number Pic Fractal dimension

1 184 16 189

2 173 17 166

3 169 18 184

4 175 19 161

5 16 20 182

6 176 21 19

7 184 22 196

8 177 23 166

9 182 24 167

10 194 25 17

11 186 26 188

12 169 27 177

13 195 28 163

14 165 29 17

15 178 30 198

fractal data we plotted imagersquos histogram and normalizedimagersquos histogram (Figure 4) by MATLAB software for thefractal data distribution

The descriptive statistics of the fractal dimension waslisted in Table 4 This table shows fractal dimensions of 30random separated pieces of the SEM image with frequencyand valid and cumulative percent of each pieceWhen dealingwith statistical information we come through numbersSometimes the numbers are repeated After a value repeated

more than once it is recognized as the frequency of anumber The total frequency of all modules less than thehigher class boundary of a quantified class is named thecumulative frequency of that class Cumulative frequency isfound by addition of the frequency of a class interval and thefrequencies of the former intervals up to that class intervalThis is another way of expressing frequency distributionCumulative frequency is used to determine the number ofexplanations that lie below or above a specific value The best


Table 3 Analysis of fractal data for the manganese-chromium bimetallic nanocomposite

Mean Median Minimum Maximum Range Skewness Harmonic mean17780 17700 160 198 038 0177 17713

Image Image in grayscale

Imagersquos histogram

6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250

Normalized imagersquos histogram1

08

06

04

02

00 50 100 150 200 250

Figure 4 Imagersquos histogram and normalized imagersquos histogram of the fractal dimension for the SEM image for the manganese-chromiumbimetallic nanocomposite

advantage of using cumulative percentage is that it offers aneasier method to compare different sets of data

Figure 5 shows a diagram in which there are 30 particleson the horizontal axis and the vertical axis shows the fractaldimension of these particles which ranges from 160 to 198The differences in the fractal dimensions of the particles canbe clearly seen which reveals the nonhomogeneity of theproduced particles

Normal probability plots were generated to check forerrors in the normal distribution If the points of these plotswere in a straight line then normally distributed errors areassumed According to Figure 6 therefore which shows anearly straight line we conclude that the fractal dimensiondistribution of the nanocomposite particles is a normallydistributed error Plotting residuals is a useful way of eluci-dating the goodness of fit of the data Plotting the residualsversus the corresponding fitted values is a useful modelfor understanding the common types of disproportion If

198

17

17

163 1

671

661

961

91

821

611

841

661

891

781

651

951

691

861

941

821

77 184

176

16

175

169

184

173

Frac

tal d

imen

sion

Accidental 30 particles

177

188

123456789101112131415161718192021222324252627282930

Figure 5 Diagram of 30 particles on the horizontal axis and theamount of fractal dimension on the vertical axis

the fitted model is appropriate the curve should be symmet-rical and the points should be uniformly distributed Basedon Figure 7 one can see that the points are not uniformly


Table 4 Descriptive statistics of the fractal dimension of themanganese-chromium bimetallic nanocomposite

Descriptive statisticsFractaldimension Frequency Valid percent Cumulative

percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000

distributed revealing an asymmetric curve that conforms toa skewness of 0177

4 Conclusion

Our calculations of the fractal dimensions of 30 randomlyselected particles from the SEM image of the nanomaterialand the max min and range of the data can be used foranalyzing the nanoparticlesWe considered homogeneity anduniformity of the SEM image by selecting the 30 randomparticles from it In fact the fractal dimension can beobtained from the size and morphology of the particles Ifthe max and min of the data are close to each other or therange of data is small then we can conclude that the sizeand dispersion of the particles in the SEM image are similarand that these created nanoparticles show good homogeneityand uniformity In the SEM image of this nanocomposite therange was large (about 038) For this reason we concludedthat the sizes andmorphologies of the particles were differentThis conclusion was consistent with the SEM image of thenanocomposite Initially it seemed that the picture wasuniform However after doing the calculations it becameclear that the produced particles were not homogenous

Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20

21Normal Q-Q plot of nanocomposite

Figure 6 Normal probability curve of the fractal data distributionfor the manganese-chromium bimetallic nanocomposite

minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value

Detrended normal Q-Q plot of nanocomposite

Figure 7 Residuals curve versus the values of fractal variables forthe manganese-chromium bimetallic nanocomposite

despite the uniform dispersion of the particles their sizes andmorphologies were different

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

The authors gratefully acknowledge the financial supportfrom the Research Council of Imam Khomeini Interna-tional University andMollasadra student home researchTheauthors also warmly acknowledge the support of ShahrbanoMaleki Islamic Azad University Ardabil Branch Facultyof Science chemistry group for his MS degree thesis oninorganic chemistryDanielGlossman-Mitnik is aResearcherat CONACYT andCIMAV and acknowledges partial supportfrom both institutions

References

[1] M Surender B Basu and R Balasubramaniam ldquoWear char-acterization of electrodeposited Ni-WC composite coatingsrdquoTribology International vol 37 no 9 pp 743ndash749 2004

[2] N Sombatsompop K Sukeemith T Markpin and N TareelapldquoA new experimental apparatus of electro-codeposited systemfor Ni-WC composite coatingsrdquoMaterials Science and Engineer-ing A vol 381 no 1-2 pp 175ndash188 2004

[3] G L Povirk ANeedleman and S RNutt ldquoAn analysis of resid-ual stress formation in whisker-reinforced Al-SiC compositesrdquoMaterials Science and Engineering A vol 125 no 2 pp 129ndash1401990

[4] G L Povirk A Needleman and S R Nutt ldquoAn analysis of theeffect of residual stresses on deformation and damage mecha-nisms in AlSiC compositesrdquo Materials Science and EngineeringA vol 132 pp 31ndash38 1991

[5] Y-L Shen A Needleman and S Suresh ldquoCoefficients ofthermal expansion of metal-matrix composites for electronicpackagingrdquoMetallurgical and Materials Transactions A vol 25no 4 pp 839ndash850 1994

[6] L L Shaw and D B Miracle ldquoEffects of an interfacial regionon the transverse behavior ofmetal-matrix compositesmdasha finiteelement analysisrdquoActaMaterialia vol 44 no 5 pp 2043ndash20551996

[7] P J Ward H V Atkinson P R G Anderson et al ldquoSemi-solidprocessing of novel MMCs based on hypereutectic aluminium-silicon alloysrdquoActaMaterialia vol 44 no 5 pp 1717ndash1727 1996

[8] J C Michel and P Suquet ldquoAn analytical and numerical studyof the overall behaviour of metal-matrix compositesrdquoModellingand Simulation in Materials Science and Engineering vol 2 pp637ndash658 1994

[9] N Ramakrishnan ldquoAn analytical study on strengthening of par-ticulate reinforced metal matrix compositesrdquo Acta Materialiavol 44 no 1 pp 69ndash77 1996

[10] D B Zahl S Schmauder and R M McMeeking ldquoMechanicalbehavior of residually stressed composites with ductile andbrittle constituentsrdquo Modelling and Simulation in MaterialsScience and Engineering vol 2 pp 267ndash276 1994

[11] M Suery C Teodosiu and L F Menezes ldquoThermal residualstresses in particle-reinforced viscoplastic metal matrix com-positesrdquo Materials Science and Engineering A vol 167 no 1-2pp 97ndash105 1993

[12] L Khezami and R Capart ldquoRemoval of chromium(VI) fromaqueous solution by activated carbons kinetic and equilibriumstudiesrdquo Journal of Hazardous Materials vol 123 no 1-3 pp223ndash231 2005

[13] X Guo G T Fei H Su and L De Zhang ldquoHigh-performanceand reproducible polyaniline nanowiretubes for removal of

Cr(VI) in aqueous solutionrdquo The Journal of Physical ChemistryC vol 115 no 5 pp 1608ndash1613 2011

[14] S R Yeduru A Backen S Fahler L Schultz and M KohlldquoTransformation behaviour of freestanding epitaxial Ni-Mn-Gafilmsrdquo Journal of Alloys andCompounds vol 577 no 1 pp S353ndashS357 2013

[15] K Hayashi K Miyata F Katsuki T Ishimoto and T NakanoldquoIndividual mechanical properties of ferrite and martensite inFe-016 mass C-10 mass Si-15 mass Mn steelrdquo Journal ofAlloys and Compounds vol 577 no 1 pp S593ndashS596 2013

[16] Q Li M Wang and W Gu ldquoComputer vision based systemfor apple surface defect detectionrdquo Computers and Electronicsin Agriculture vol 36 no 2-3 pp 215ndash223 2002

[17] A Hosseini S M Jafari H Mirzaei A Asghari and SAkhavan ldquoApplication of image processing to assess emulsionstability and emulsification properties of Arabic gumrdquo Carbo-hydrate Polymers vol 126 pp 1ndash8 2015

[18] D H De Boer and M Stone ldquoFractal dimensions of suspendedsolids in streams comparison of sampling and analysis tech-niquesrdquoHydrological Processes vol 13 no 2 pp 239ndash254 1999

[19] D H de Boer ldquoAn evaluation of fractal dimensions to quantifychanges in the morphology of fluvial suspended sedimentparticles during baseflow conditionsrdquo Hydrological Processesvol 11 no 4 pp 415ndash426 1997

[20] B H Kaye ldquoImage analysis techniques for characterizingfractal structuresrdquo in The Fractal Approach to HeterogeneousChemistry Surfaces Colloids Polymers D Avnir Ed WileyChichester UK 1989

[21] P A Burrough ldquoFractal dimensions of landscapes and otherenvironmental datardquo Nature vol 294 no 5838 pp 240ndash2421981

[22] B T Milne ldquoMeasuring the fractal geometry of landscapesrdquoApplied Mathematics and Computation vol 27 no 1 pp 67ndash791988

[23] B J West and A L Goldberger ldquoPhysiology in fractal dimen-sionsrdquo The Journal of American Science vol 75 pp 354ndash3651987

[24] C J Du and D W Sun ldquoRecent developments in the applica-tions of image processing techniques for food quality evalua-tionrdquo Trends in Food Science amp Technology vol 15 no 5 pp230ndash249 2004

[25] R J Arsenault and M Taya ldquoThermal residual stress in metalmatrix compositerdquoActaMetallurgica vol 35 no 3 pp 651ndash6591987

[26] P JWithersWM Stobbs andO B Pedersen ldquoThe applicationof the eshelby method of internal stress determination to shortfibre metal matrix compositesrdquoActaMetallurgica vol 37 no 11pp 3061ndash3084 1989

[27] B B Mandelbrot The Fractal Geometry of Nature W HFreeman and Company New York NY USA 1982

[28] J-F Gouyet and B B Mandelbrot Physics and Fractal Struc-tures Springer New York NY USA 1996

[29] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons New York NY USA 2003

[30] X C Jin S H Ong and Jayasooriah ldquoA practical method forestimating fractal dimensionrdquo Pattern Recognition Letters vol16 no 5 pp 457ndash464 1995

[31] A K Bisoi and J Mishra ldquoOn calculation of fractal dimensionof imagesrdquo Pattern Recognition Letters vol 22 no 6-7 pp 631ndash637 2001


[32] J Zhang R Zhang B Hu and S Bai ldquoFractal image processingand analysis by programming in MATLABrdquo in Proceedings ofthe 8th WSEAS International Conference on Mathematics andComputers in Biology and Chemistry Vancouver Canada June2007

[33] BMandelbrot ldquoHow long is the coast of Britain Statistical self-similarity and fractional dimensionrdquo ScienceMagazine vol 156no 3775 pp 636ndash638 1967

[34] E Perfect ldquoEstimating soil mass fractal dimensions from waterretention curvesrdquo Geoderma vol 88 no 3-4 pp 221ndash231 1999

[35] M BorkovecW de Paris andR Peikert ldquoThe fractal dimensionof theApollonian sphere packingrdquo Fractals vol 2 no 4 pp 521ndash526 1994

[36] Z Khanbabaei A Karam and G Rostamizad ldquoStudyingrelationships between the fractal dimension of the drainagebasins and some of their geomorphological characteristicsrdquoInternational Journal of Geosciences vol 4 pp 636ndash642 2013

[37] P Parsons-Wingerter B Lwai M C Yang et al ldquoA novel assayof angiogenesis in the quail chorioallantoicmembrane stimula-tion by bFGF and inhibition by angiostatin according to fractaldimension and grid intersectionrdquo Microvascular Research vol55 no 3 pp 201ndash214 1998

[38] E Guerin and E Tosan ldquoFractal inverse problem approx-imation formulation and differential methodsrdquo Fractals inEngineering New Trends in Theory and Applications pp 271ndash285 2005

[39] M K Biswas T Ghose S Guha and P K Biswas ldquoFractaldimension estimation for texture images a parallel approachrdquoPattern Recognition Letters vol 19 no 3-4 pp 309ndash313 1998

[40] The MathWorks MATLABmdashStatistics Toolbox Release 2012The MathWorks Natick Mass USA 2012

[41] SPSS SPSS Statistics for Windows Version 170 SPSS 2008[42] G F Voort ASM Handbook Volume 9 Metallography and

Microstructures ASM InternationalMaterials Park Ohio USA2004

[43] L Wojnar Image Analysis Applications in Materials Engineer-ing CRC Press Boca Raton Fla USA 1999

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Journal ofNanomaterials


manipulative work in the show of an assumed process[16 17]

The term ldquofractal dimensionrdquo has been used to describeseveral aspects of particle morphologyThe fractal dimensionof an individual particle which can be a main particle or acomposite particle such as a floc or aggregate denotes theindiscretion and self-similarity of the particle The fractaldimension parameter is determined by computing the lengthof the predictable perimeter by taking steps of a known lengthalong the border and including the number of ladders Theborder length increases with decreasing step size because asmaller step size allows for the presence of more detail inthe particle outline [18ndash20] The word ldquofractalrdquo is derivedfrom the Latin ldquofractiousrdquo meaning fragmented a fractal isdefined as a mathematical object with a fractional (nonin-teger) dimension It seems that wherever there is structuraldisorder and chaos fractal geometry can be an efficient wayof describing and analyzing it

Fractal dimension is a very good mathematical methodfor the study of unequal geometric substances Fractal dimen-sion can reflect characteristics appearing in nature Theapplication of fractal system allows us to explain numerousstates of fragmenting and branching occurring in ecologi-cal biological and other systems [21ndash24] The output is abinarized image with a one-pixel-wide boundary appropriateto be used for box-counting calculations to approximate thefractal dimension of the boundaryWe developed aMATLAB(MathWorksNatickMassachusetts USA) programbased ontheory to implement the box-counting technique for fractaldimension calculations

Several specific theoretical and numerical models havebeen proposed in order to assess the fractal dimensionRecent studies have examineddimensions but themajority ofthem were only one- or two-dimensional These dimensionsare a clear limitation to the study of complex and moregeneral strengthening geometries [25 26]

Manganese and chromium have been used in this nano-composite potassium permanganate and potassium dichro-mate were used as the starting materials Scanning electronmicroscope (SEM) images of the nanocomposites may beregular or irregular and their fractal dimension can be cal-culated It is interesting to compare the fractal images of thenanocomposites For example in this paper we investigatedthe size of the nanoparticles and calculated their fractaldimension We also measured the average minimum andmaximum of fractal dimensions and imagersquos histogram andnormalized imagersquos histogramplot of the SEM image [27ndash29]

2 Experimental

21 Materials and Instruments The starting materials wereobtained from Merck (Berlin Germany) and were usedwithout further purification The surface morphology of theproduct was characterized by using a LEO-1430VP scanningelectron microscope (Zeiss Germany) with an acceleratingvoltage of 15 kV X-ray powder diffraction (XRD) measure-ments were performed using a Philips Xrsquopert diffractometer(PANalytical Almelo Netherlands) with monochromatized

CuK120572 radiation The sizes of the selected samples wereestimated using the Scherer method For identification theSEM samples were coated with gold

22 Synthesis of Nanocomposite Manganese and ChromiumPotassium permanganate (KMnO

4) and potassium dichro-

mate (K2Cr2O7) were used as starting materials to prepare

this nanocomposite Initially 02 g of potassium perman-ganate was transferred to a test tube and 001 g of claywas added as an intruder agent Ethanoic acid CH

3COOH

(4mL) was then added to the mixture and the tube wasquickly transferred to a centrifuge and run at 3000 rpm for45 minutes The sample was then taken out of the centrifugeand put asideThen 05 g of potassium dichromate and 001 gof clay were added to another test tube NH

3(5mL) solution

was added and the tube was placed in a centrifuge and run at3000 rpm for 45 minutes

23 Characterization of the Nanocomposite The XRD tech-nique was used to determine the composition of the milledpowder The morphology and weight and atomic percentageof the nanocomposite were observed using a SEM image andEDX analysis respectivelyThe contents of the first tube werethoroughly stirred and transferred to a second tubeThis sec-ond tube was returned to the centrifuge and run at 3000 rpmfor 45 minutes The test tube was then removed fromthe centrifuge and the precipitate was separated out usingfilter paper After drying the precipitate was imaged withSEM instrument and the atomic percentages were achievedby EDX analysis So the XRD spectrum was obtained toshow that a manganese-chromium nanocomposite had beenproduced

24 Calculation of Fractal Dimension and Image AnalysisDimension is one of the most significant ideas in fractalgeometryNow there are numerous descriptions of this quan-tity including packing dimension box-counting dimensionHausdorff dimension and modified box-counting dimen-sion Since the mathematic design and approximation ofthe box-counting dimension are straightforward this quan-tity has become one of the most commonly used generaldimensions [30 31] The calculations and analysis were doneby programming with MATLAB and Statistical Package forthe Social Sciences (SPSS IBM Armonk New York USA)software [32] Moreover by declaring that the calculationresults were comparable all of the images were processedusing a similar crop size and threshold value Using thismethod we calculated the fractals dimension theoretically byapplying the following formula

119863 = lim119903rarr0

log119873119903

log (1119903) (1)

where 119863 is the fractal dimension and 119873119903is the number of

squares including a part of the considered fractal dimensionsConsidering the resulting numbers we drew a loga-

rithmic diagram in which the vertical and horizontal axeswere log(1119903) and log119873

119903 respectively and the coefficient of


2120579

4 10 20 30 40 50 60 700

10

20

30

Lin

(cou

nts)

(a)

2120579

4 10 20 30 40 50 60 700

10

20

30

Lin

(cou

nts)

(b)






dim119861

119865 = lim119903rarr0

log119873119903(119865)




119899119903 (119898119899+ 1)119903]

then119898119899 119898

1are integers

Finally119873119903radic119899(119865) le 119873

1015840

119903(119865)





119861119865 = lim

119903rarr0(log1198731015840

119903(119865) minus log 119903)


119903rarr0(log1198731015840

119903(119865) minus log 119903)

Finally1198731015840119903(119865) le 3

119899119873119903(119865)












119860d


15000

10000

5000

0

Cou

nts (

eds)

0 5 10

(keV)

AuL120572AuLI

CuL120572C K120572

K K120572

AuM120572AuM120573

S K120573 K K120573CaK120572 CrK120572

MnK120572


S K120572SiK120572

O K120572

CrL120572

MnL120572CaL120572N K120572









1 184 16 189

2 173 17 166

3 169 18 184

4 175 19 161

5 16 20 182

6 176 21 19

7 184 22 196

8 177 23 166

9 182 24 167

10 194 25 17

11 186 26 188

12 169 27 177

13 195 28 163

14 165 29 17

15 178 30 198









6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250


08

06

04

02

00 50 100 150 200 250





198

17

17

163 1

671

661

961

91

821

611

841

661

891

781

651

951

691

861

941

821

77 184

176

16

175

169

184

173

Frac

tal d

imen

sion


177

188

123456789101112131415161718192021222324252627282930






percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000


4 Conclusion


Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20



minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value







Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




2120579

4 10 20 30 40 50 60 700

10

20

30

Lin

(cou

nts)

(a)

2120579

4 10 20 30 40 50 60 700

10

20

30

Lin

(cou

nts)

(b)






dim119861

119865 = lim119903rarr0

log119873119903(119865)




119899119903 (119898119899+ 1)119903]

then119898119899 119898

1are integers

Finally119873119903radic119899(119865) le 119873

1015840

119903(119865)





119861119865 = lim

119903rarr0(log1198731015840

119903(119865) minus log 119903)


119903rarr0(log1198731015840

119903(119865) minus log 119903)

Finally1198731015840119903(119865) le 3

119899119873119903(119865)












119860d


15000

10000

5000

0

Cou

nts (

eds)

0 5 10

(keV)

AuL120572AuLI

CuL120572C K120572

K K120572

AuM120572AuM120573

S K120573 K K120573CaK120572 CrK120572

MnK120572


S K120572SiK120572

O K120572

CrL120572

MnL120572CaL120572N K120572









1 184 16 189

2 173 17 166

3 169 18 184

4 175 19 161

5 16 20 182

6 176 21 19

7 184 22 196

8 177 23 166

9 182 24 167

10 194 25 17

11 186 26 188

12 169 27 177

13 195 28 163

14 165 29 17

15 178 30 198









6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250


08

06

04

02

00 50 100 150 200 250





198

17

17

163 1

671

661

961

91

821

611

841

661

891

781

651

951

691

861

941

821

77 184

176

16

175

169

184

173

Frac

tal d

imen

sion


177

188

123456789101112131415161718192021222324252627282930






percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000


4 Conclusion


Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20



minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value







Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







119860d


15000

10000

5000

0

Cou

nts (

eds)

0 5 10

(keV)

AuL120572AuLI

CuL120572C K120572

K K120572

AuM120572AuM120573

S K120573 K K120573CaK120572 CrK120572

MnK120572


S K120572SiK120572

O K120572

CrL120572

MnL120572CaL120572N K120572









1 184 16 189

2 173 17 166

3 169 18 184

4 175 19 161

5 16 20 182

6 176 21 19

7 184 22 196

8 177 23 166

9 182 24 167

10 194 25 17

11 186 26 188

12 169 27 177

13 195 28 163

14 165 29 17

15 178 30 198









6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250


08

06

04

02

00 50 100 150 200 250





198

17

17

163 1

671

661

961

91

821

611

841

661

891

781

651

951

691

861

941

821

77 184

176

16

175

169

184

173

Frac

tal d

imen

sion


177

188

123456789101112131415161718192021222324252627282930






percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000


4 Conclusion


Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20



minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value







Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






1 184 16 189

2 173 17 166

3 169 18 184

4 175 19 161

5 16 20 182

6 176 21 19

7 184 22 196

8 177 23 166

9 182 24 167

10 194 25 17

11 186 26 188

12 169 27 177

13 195 28 163

14 165 29 17

15 178 30 198









6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250


08

06

04

02

00 50 100 150 200 250





198

17

17

163 1

671

661

961

91

821

611

841

661

891

781

651

951

691

861

941

821

77 184

176

16

175

169

184

173

Frac

tal d

imen

sion


177

188

123456789101112131415161718192021222324252627282930






percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000


4 Conclusion


Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20



minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value







Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials








6000

5000

4000

3000

2000

1000

0

0 50 100 150 200 250


08

06

04

02

00 50 100 150 200 250





198

17

17

163 1

671

661

961

91

821

611

841

661

891

781

651

951

691

861

941

821

77 184

176

16

175

169

184

173

Frac

tal d

imen

sion


177

188

123456789101112131415161718192021222324252627282930






percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000


4 Conclusion


Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20



minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value







Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






percent160 1 33 33161 1 33 67163 1 33 100165 1 33 133166 2 67 200167 1 33 233169 2 67 300170 2 67 367173 1 33 400175 1 33 433176 1 33 467177 2 67 533178 1 33 567182 2 67 633184 3 100 733186 1 33 767188 1 33 800189 1 33 833190 1 33 867194 1 33 900195 1 33 933196 1 33 967198 1 33 1000


4 Conclusion


Expe

cted

nor

mal

val

ueObserved value

15 16 17 18 19 20 2115

16

17

18

19

20



minus004

minus002

000

002

004

006

16 17 18 19 20

Dev

iatio

n fro

m n

orm

al

Observed value







Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Acknowledgments


References





















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article Fractal Dimension Calculation of a ...downloads.hindawi.com/journals/jnm/2015/384835.pdfResearch Article Fractal Dimension Calculation of a Manganese-Chromium Bimetallic