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Polymer 53 (2012) 1571e1580

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Polymer

journal homepage: www.elsevier .com/locate/polymer

Automatic quantification of filler dispersion in polymer composites

Zhuo Li a,1, Yi Gao b,1, Kyoung-Sik Moon a, Yagang Yao a, Allen Tannenbaumb, C.P. Wong a,c,*

a School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive NW, Atlanta, GA 30332, United Statesb School of Electrical and Computer Engineering, Georgia Institute of Technology, 313 Ferst Drive NW, Atlanta, GA 30318, United StatescDepartment of Electronic Engineering, Faculty of Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong, PR China

a r t i c l e i n f o

Article history:Received 16 December 2011Received in revised form23 January 2012Accepted 25 January 2012Available online 1 February 2012

Keywords:Polymer compositesFiller dispersionImage analysis

* Corresponding author. School of Materials ScienInstitute of Technology, 771 Ferst Drive NW, AtlanTel.: þ1 404 894 8391; fax: þ1 404 894 9140.

E-mail address: cp.wong@mse.gatech.edu (C.P. Wo1 Contributed equally to this work.

0032-3861/$ e see front matter Crown Copyright � 2doi:10.1016/j.polymer.2012.01.048

a b s t r a c t

A facile and objective method is introduced to automatically quantify the filler dispersion in polymercomposites through image analysis. This method consists of automatic identification of the fillers in theimage and a rigorous measurement of the filler dispersion within the space of functions. Compared withprevious methods, this method has the advantages of 1) automatically recognizing the fillers, 2) mini-mizing the subjectivity induced by the inhomogeneity and noise of the background in the images, 3)a mathematically more rigorous definition of the deviation of the filler dispersion from uniformity, 4)a single performance metric reflecting both the distribution and the size of fillers. Both synthetic and realimages from model compounds are used to demonstrate the sensitivity of the proposed method to thedispersion and aggregation of fillers. The computed dispersion index shows good agreement with visualobservation of synthetic images and mechanical properties of the model compounds.

Crown Copyright � 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Filler dispersion plays a critical role in determining variousproperties of polymer composites, including electrical conductivity[1], dielectric constant [2], heat transfer [3], mechanical strength[4,5], optical transparency [6], wear resistance [7], etc. Uniformfiller dispersion has become one of the ultimate goals in thepreparation of composites. Although methods to improve fillerdispersion have been intensively explored [8e10], it is hard tocompare their dispersion quality and thus select the best methoddue to a deficiency of objective and universal metrics for fillerdispersion.

In most prior studies, the dispersion status is characterized bythe evaluation of the physical properties of composite materials,including the electrical conductivity, rheology, mechanical prop-erties, etc [8,11e13]. In the meantime, some other dispersionevaluation techniques were also proposed for specific categories ofnanocomposites, such as Raman mapping and microscale dynamicmechanical analyzer (DMA) mapping of carbon nanotube (CNT)based nanocomposites [14,15]. However, these indirect approachescannot provide adequate dispersion information because not onlyfiller dispersion but also many other factors could affect the

ce and Engineering, Georgiata, GA 30332, United States.

ng).

012 Published by Elsevier Ltd. All

measured properties. For example, in the case of a filled compositewith a loading level near the percolation threshold, even if the fillerdispersion keeps constant, slight difference in filler loading couldlead to orders of magnitude change in the electrical conductivityand viscosity. Therefore, direct assessment of filler dispersionthrough visualization will be favored in terms of reliable datainterpretation.

Direct assessment usually includes three steps. Firstly, imagesrepresenting the microstructure of composites are captured bymicroscopes, such as optical microscope (OM), scanning electronmicroscope (SEM), transmission electron microscope (TEM), etc.Secondly, the fillers are identified from the obtained photo images.Thirdly, certain statistical analyses are made to give a numericalvalue that indicates the level of dispersion. While most of theprevious researchers took such a procedure, they simply focused ondeveloping a more objective statistic method in the last part of thepipeline and neglected the second step of filler identification,namely it was often assumed that the positions of the fillers hadbeen obtained, in many cases, manually [16e22]. Unfortunately,given the large number of fillers that may exist in the image,manually picking them out is an extremely tedious work. Thus it isimpossible for such an analysis to be extended to a high throughputscale, which is necessary for more data sets and statistically morerobust results. In addition to the time consumption, an even morefundamental problem lies in the subjectivity induced by sucha manual tracing process. Therefore, in more recent papers, iden-tification of the fillers is givenmoreweight. For instance, Pegel et al.set a threshold for the input image so that pixels in the image with

rights reserved.

Fig. 1. (a) The original synthetic image. (b) the binary image of extracted fillers. (c) identified fillers are shown as crosses in the image.

Z. Li et al. / Polymer 53 (2012) 1571e15801572

higher intensity values than the threshold are identified as fillers.Based on that, the spatial statistics is computed as an indication ofdispersion [23]. Using a similar way to identify the fillers, Sul et al.set a global threshold and extract the particles from the originalimage. After that, the Delaunay triangulation is computed based onwhich the Lennard-Jones potential is calculated as the dispersionmeasurement [24]. In the work of Glaskova et al, the brightnessthreshold is optimized by the so-called isodata algorithm beforesubsequent analysis [25].

All the above methods take advantage of the strong contrastbetween fillers and the polymer matrix and set a threshold toseparate the fillers from the background. However, due to the noisein the imaging devices and spatial difference in electrical conduc-tivity/topography, the images in many cases may display inhomo-geneous brightness. Under such scenarios, a single threshold is notsufficient to correctly extract the fillers in the entire image anda more accurate method is needed.

Therefore, in this work, we present an entire pipeline for theanalysis of the filler dispersion with more accurate filler identifi-cation and dispersion evaluation processes. In the filler identifica-tion step, the images are divided into finite elements and a localcomparison in each element is used to separate the fillers frommatrix. By doing so, we promote the tolerance of inhomogeneityand noise in the image. With the filler positions obtained, in thedispersion step, the well-developed measurement in probabilitytheory is used to evaluate the (dis)-similarity between two sets ofsamples. Specifically, the fillers in the image are considered asrealizations of a random variable distributed in the image field ofthe view. Then, the probability density function (PDF) is estimatedand the mathematically rigorously defined “distance” between twoPDFs is employed to measure the (dis-) similarity between the fillerdispersion under investigation and a uniform distribution. It not

Fig. 2. An example demonstrating using a local threshold versus a global threshold for extrusing the local threshold. (c) the binary image for extracted fillers using a single threshold

only represents a rigorous definition for the dispersion index, butalso can reflect the filler size and the aggregation of fillers in thecomposites, which have rarely been achieved in previous disper-sion metrics involving comparison with uniform distribution [18].With the developed technique, we are able to compare thedispersion efficiency of three commonly used processing methodsfor polymer composites and identify the best processing methodfor the model compound.

2. Experimental

2.1. Composite preparation and characterization

Carbon black (CB)/silicone composites were used as the modelcompound. CB (Vulcan. XC72, Cabot) has an aggregate density of1.8 g/cm3 and a surface area of 254 m2/g (tested under N2). Siliconeresin (SR) is a peroxide-curable methyl phenyl silicone gumprovided by Rockbestos-Surprenant Cable Co.

Generally there are two strategies to promote dispersion ofcarbon fillers in a polymer matrix, namely the surface modificationof fillers or the optimization of processing conditions [26]. Formanyindustrial applications, chemical modification may be undesirabledue to the possible change in the properties or performance of thematerial. As a result, enhancing dispersion through the processingoptimization is preferred. The fabrication processes of carbon/polymer composites can be generally divided into four categories,namely 1) no fluid mixing by shear force, 2) melt mixing throughextrusion, 3) dispersion-dissolution-precipitation, where polymersand fillers are dissolved/dispersed in solvents for mixing and thenevaporate the solvents, and 4) dispersion reaction, which is appliedto liquid monomers with a low viscosity [8].

acting filler. (a) the original (synthetic) image. (b) the binary image for extracted fillersat 150. (d) the binary image for extracted fillers using a single threshold at 200.

Fig. 3. (a) The original uniform dispersion is composed of 1024 particles in a 1000 � 1000 grid. The effect of adding (c) 10%, (d) 20% or missing (e) 10%, (f) 20%, (g) 40%, (h) 60%random particles to the original image. (b) the corresponding dispersion index of each image.

Z. Li et al. / Polymer 53 (2012) 1571e1580 1573

Z. Li et al. / Polymer 53 (2012) 1571e15801574

In the case of the SR used here, three of the above dispersionprocesses were tested.

1) No-fluid mixing by three-roll mill. Dry CB powders weredispersed in SR by shear force provided by the roll mill. Thedispersion level was set by the number of rolling cycles. Theinitial ten cycles of mixing was used to incorporate all the CBfillers into SR.

2) Dispersion-dissolution-precipitation via ultrasonication. TheCB fillers were dispersed in toluene for 1 h under ultra-sonication (Misonix 3000) while the SR was dissolved inanother beaker with toluene. Then the two solutions werecombined and further mixed by ultrasonication for variousdurations (0e4 h). The mixture was then dried under vacuumfor 12 h before curing.

3) Dispersion-dissolution-precipitation via high shear blending;This method is basically the samewith the previous one, exceptthat all the mixing was performed by high shear blending(Talboys laboratory stirrer 104A). Also different time intervals(0e4 h) of high shear blending were examined to compare theeffect of mixing intensity.

All the samples had a constant filler loading of 15 wt%. In theend, themixturewas pressed into a Teflonmold and cured for 1 h at180 �C under 1.6 MPa pressure.

Fractured surfaces of the cured specimens were obtained inliquid nitrogen. Three fractured surfaces per specimenwere imagedand analyzed to give the averaged dispersion index. SEM observa-tion was conducted in LEO 1530 FE-SEM with an accelerationvoltage of 15 kV. Dynamic mechanical measurements were per-formed on a DMA (TA instruments 2980) tension mode. Frequencysweep was performed at 50 �C with a strain of 0.5%.

2.2. Filler identification

Usually, the image is contaminated by the noise from theimaging process. Hence, it is pre-processed to remove the noise bypassing through the median filter with a neighborhood size of3 � 3. Then, denote the (noise removed) image as I(x; y) : R2 / Rand discretely, it can be written as a matrix: I(i,j), where i ¼ 1,., Nx

and j ¼ 1, ., Ny. Subsequently, the task of filler identification is toobtain a set of points c1;.; cM3R2, where each Ci represents thecoordinates of a single filler element.

Conductive fillers such as CB appear to be bright spots in SEM athigh accelerating voltages while the insulative polymer matrixremains dark [27,28]. Indeed, this is also the insight behind thethresholding technique described previously [23e25]. However,due to the qualities of the sample material, imaging device, or both,often times the brightness of the field is not as homogeneous asexpected. In such cases, using a single threshold to determine the

Fig. 4. Filler identification for CB. (a) original SEM image. (b) the outpu

filler locationwould fail. Actually, the brighter intensity of one filleris only relative to its neighboring region. Hence, an adaptivethreshold is necessary as the filter marches through the image. Thatis, the filter takes the input grayscale image I and outputs anotherimage H in which the particle regions are highlighted. Mathemat-ically it is defined as:

Hði; jÞ ¼�1; if Iði; jÞ ¼ max

ðu;vÞ˛Bði;jÞIðu; vÞ

0 otherwise(1)

where B(i, j) is a neighboring region around the position (i, j):B(i,j) ¼ {x, y; |x-i|<5, |y-j|<5}.

In the image H, each particle is represented as an island of value1. But it can be seen in Fig.1 that the islands do not always consist ofa single point. Indeed, there are some very bright regions in theoriginal image where the image intensity is all saturated. In thesegmentation process, each of those regions is considered asa single particle. In order to identify different islands, a distinctlabel needs to be assigned to each of the islands. To this end, theconnected components (CC) are computed in the H image and wedenote the total number of CCs as M. Then, the center-of-mass foreach CC is computed. The M center points c1;.; cM3R2 are theoutput of the filler identification process, which are shown inFig. 1(c).

2.3. Dispersion computation

After the centers of the fillers having been identified, the fillersin the image are viewed as a random variable in the image field ofthe view. Specifically, all the center points c1;.; cM3R2 are treatedasM samples drawn from a randomvariable S. The more evenly thefillers are distributed, the more such a random variable S is similarto the uniform random variable, denoted as U. In order to estimatethe PDF of S, the kernel density estimation method is used. Thekernel density estimation is a widely used method to infer the PDFfrom samples in cases where no prior information of the PDF isavailable [29]. The PDF is computed as the sum of a series ofkernels–usually Gaussian functions. More explicitly, the PDF for S isestimated as:

f ðxÞ ¼ 1M

XMi¼1

1ffiffiffiffiffiffiffiffiffi2ps

p e�ðx�ciÞ2

2s2 (2)

where the standard deviation of the kernel s is computed with themethod proposed in [30].

Then the discrepancy between the PDF obtained and that of theuniform randomvariable is computed bymeasuring the L2 distancebetween them (In the area of functional analysis, the L2 distance isone of the most widely used distance for measuring the discrep-ancy between two functions [31].):

t image H. (c) final extraction result for the centers of fillers (CB).

Fig. 5. The first set of synthetic images, illustrating the effect of local dispersion. The left column (a,d,g,j) shows four synthetic images. In the middle column (b,e,h,k), the fillers areidentified and are represented as red markers. The right column (c,f,i,l) shows the estimated PDFs obtained from the identified fillers, which is to be compared with uniform densityfunction to assess the dispersion. From the top row to the bottom row, the dispersion is getting worse locally. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Z. Li et al. / Polymer 53 (2012) 1571e1580 1575

dðf ; gÞ ¼�Z �

f ðxÞ � gðxÞ2dx��1

2

(3)

where we have g(x) ¼ 1/(area of the image), to represent a randomvariable uniformly distributed on the image area. While theEuclidean distance defines the distance between two points apartin space, the above defined L2 distance gives such a “distance”between two functions. Many previous analyses investigatedcertain aspects of the distribution of the fillers [18,23,24,32],however, either the analysis stops at providing the distributionhistograms which need further human assessment, or the final

dispersion index is computed through some heuristics. Contrast-ingly, with the L2 distance, we give a mathematically more rigorousframework that outputs a single value to measure the discrepancybetween the fillers in the current image with the ideal uniformdistribution. Its capability and benefit will be shown in the nextsection for both the synthetic and real images. Intuitively, whencenters of the fillers, c1;.; cM are distributed evenly, its corre-sponding random variable S is more likely to be close to uniformdistribution and thus the L2 distance d is close to 0. On the contrary,the more non-uniform the fillers spread, the larger d will be ob-tained. The MATLAB� code to perform the above computation can

Z. Li et al. / Polymer 53 (2012) 1571e15801576

be downloaded from the following website: http://tinyurl.com/autodispersion.

3. Results and discussion

3.1. Filler identification at inhomogeneous background

Fig. 2(a) is a synthetic image with inhomogeneous backgroundand Fig. 2(b) shows the fillers extracted by using the local infor-mation based adaptive method. As a comparison, the singlethreshold method is also tested (Fig. 2(c) and (d)). As can be

Fig. 6. The second set of synthetic images, illustrating the effect of global dispersion. The leftare identified and are represented as red markers. The right column (c,f,i,l) shows the estimdispersion is getting worse globally. (For interpretation of the references to colour in this fi

observed, it is very difficult, if not impossible, to find a suitableglobal threshold to characterize all the spots. In Fig. 2(c), a globalthreshold at 150 is able to identify about two thirds of the spots;however a large region around the center region is erroneouslyincluded because of the brighter background there. Similarly,a higher threshold at 200 helps differentiate the spots near thecenter, but misses too many relatively darker spots around theimage boundary, as shown in Fig. 2(d).

Since filler identification using a global threshold will probablymiss or add some fillers in the image, these artifacts in the iden-tification process will further lead to inaccurate assessment in the

column (a,d,g,j) shows four synthetic images. In the middle column (b,e,h,k), the fillersated PDFs obtained from the identified fillers. From the top row to the bottom row, thegure legend, the reader is referred to the web version of this article.)

Z. Li et al. / Polymer 53 (2012) 1571e1580 1577

dispersion analysis. In Fig. 3 (a), 1024 particles are uniformlydispersed in the 1000 � 1000 grid; the corresponding dispersionindex is 0.33. When randomly taking away 10%, 20%, 40%e60% ofthe total number of particles, the dispersion index increasesalmost linearly and the corresponding dispersion index is 0.49,0.65, 1.05 and 1.38, respectively. Similarly, introducing extraparticles will also increase the dispersion index, indicating a non-uniform distribution. Although the effect of missing or addingparticles on the ultimate dispersion assessment shown in thisexample seems trivial when comparing with the effect of realaggregations shown later, this effect can become significant in realcases. This is because in the synthetic images, the removal and

Fig. 7. The third set of testing images, illustrating the effects of the combination of local andcolumn (b,e,h,k), the fillers are identified and are represented as red markers. The right copretation of the references to colour in this figure legend, the reader is referred to the web

addition of particles is still random, whereas missing or addingparticles caused by manual counting or by using universalthreshold is caused by brightness difference and is probably notrandom any more.

Having illustrated the filler identification for synthetic images,we show an example on a SEM image of CB/SR in Fig. 4.

3.2. Dispersion analysis e synthetic samples

To test the developed dispersion assessment method, a seriesof images were generated to simulate the situation of real imagesof the polymer composites. Since an ideal dispersion of fillers

global dispersion. The left column (a,d,g,j) shows four synthetic images. In the middlelumn (c,f,i,l) shows the estimated PDFs obtained from the identified fillers. (For inter-version of this article.)

Fig. 8. Dispersion measurements for all the 12 cases in Figs. 5-7.

Z. Li et al. / Polymer 53 (2012) 1571e15801578

usually includes both the randomness at both global and locallevels, we synthesized three sets of images to represent the effectof dispersion locally (Fig. 5), globally (Fig. 6), and a combination ofboth (Fig. 7), respectively. In each set, four cases with differentlevels of filler dispersion are shown. In particular, in order to morefaithfully represent the real photos, we made the background ofthese images with highly inhomogeneous brightness.

In Fig. 5, the fillers are distributed in a rectangular grid. Thedifferences among Fig. 5(a), (d), (g) and (j) lie in the degree of thelocal uniformity. In the middle column, the identified fillers arerepresented by red markers. Although the background of theimages is very inhomogeneous, the synthetic fillers are correctlyidentified. After that, the right column shows the estimated PDFsobtained from the identified fillers. Such density functions are thencompared with the uniform distribution to determine the level ofdispersion. Similar experiments are conducted to show the effect ofdispersion globally in Fig. 6 and a more general case in Fig. 7. Fig. 8

Fig. 9. Effect of particle size on the dispersion index. The two synthetic images (a) and (d) haThe middle column (b, e) shows the identified fillers in each image and the right column (

shows the quantified results of all the 12 cases that the dispersionindices increase monotonically as the dispersion gradually getsworse. The dispersion index can provide sensitive response to thedispersion status of fillers at both local and global levels, while thefailure of random dispersion in both levels can give rise to an evenlarger dispersion index. The results are consistent with visualassessment, confirming its applicability to composites withdifferent dispersion states.

3.3. Sensitivity to particle size

One of the drawbacks of many previously developed fillerdispersion analyses is that they are unable to identify the effectof particle size, because these methods are established on thecomparison of sample images against a “uniform distribution”that is independent of any measurement of the density ofdispersion [18]. This limits their application in predicting themechanical properties of composites, which is very sensitive tothe particle size [4]; Moreover, these methods would also beincompetent to provide the information of the agglomeration ofthe particles. To solve this problem, Burris et al. utilized thedistance between neighboring particles as the computationmetric [18]. In a constant filler loading, the smaller particle sizemeans larger numbers of particles (i.e. larger density of parti-cles) and thus a smaller interparticle distance. In our case, sincewe are using the PDF, the smaller particle size will directly bereflected in the density of particles. That is, our method is able torecognize the difference of particle size. To illustrate this, twoimages (Fig. 9 (a) and (d)) with an identical filler loading butdifferent particle sizes are compared. Although both the casesdisplay a perfectly uniform distribution, larger particles formindividual density peaks in the PDF while smaller particlesconstruct a more flat and continuous PDF. Therefore, the ob-tained dispersion index is 56.74 of larger particles and 16.27 forsmaller particles, indicating a better “dispersion” in the case of

ve the same filler loading but different filler size. The filler size in (d) is 1/3 of that in (a).c, f) shows the PDF of fillers in each image, respectively.

Fig. 10. The images before and after filler identification of the samples processed by three-roll milling. The inset textboxes show the number of cycles that the composites passthrough the roll mill.

Z. Li et al. / Polymer 53 (2012) 1571e1580 1579

smaller particle sizes. This will correspond to the real case thatbreaking up agglomerates during processing renders a smallerparticle size and better dispersion.

3.4. Application to real composite systems

Besides the synthetic images, CB/SR is used as a modelcompound to validate the developed dispersion assessmentmethod. With the help of the dispersion index, we aim at selectingthe best processing methods among the three and also elucidatethe dispersion as a function of mixing intensity. In the meanwhile,

Fig. 11. The computed dispersion index of samples processed by different methods andat different mixing intensity.

the mechanical properties of all samples are measured to correlatewith the dispersion index.

Fig. 10 shows the typical cross-sections of the samples pro-cessed by three-roll mill at different mixing intensities. Comparedto the simplified synthetic images, it is even more difficult tomanually pick out all the CB fillers from these real images. Thus itis necessary to utilize the automatic filler identification anddispersion analysis for the real cases. The dispersion indices ob-tained from the developed method are shown in Fig. 11. For allthree processing methods, there is a general trend that thedispersion index decreases with a longer mixing time. This is

Fig. 12. The storage modulus of samples processed by different methods and atdifferent mixing intensity.

Z. Li et al. / Polymer 53 (2012) 1571e15801580

intuitively understandable since a longer mixing time wouldresult in better filler dispersion. A comparison among the threemixing methods clearly shows that the sequence of dispersionefficiency is ultrasonication gt; three-roll mill gt; high shearblending. The dispersion index of ultrasonication for 0 h is lowerthan any roll mill or high shear blending samples, suggesting thatevenwithout any further mixing after combining CB and dissolvedSR together, the pre-dispersion of CB in solvent is already efficientenough to give a good dispersion in the final composite. Incontrast, high shear blending is not able to break down CBaggregate in solvent in the premixing stage, therefore fails toobtain a good dispersion of the composite. In the case of three-rollmill, the breakdown process is facilitated by the large shear forceduring roll milling as well as the gel formation in the interfacebetween SR and CB [33]. Hence, the dispersion efficiency of three-roll mill is better than that of high shear blending but inferior tothat of ultrasonication.

The storage modulus at a frequency of 0.01 Hz and 50 �C wellcorresponds to the dispersion index results (Fig. 12). Becauseultrasonication is able to break down the aggregation of CB andleads to a good dispersion, the storage modulus of ultrasonicatedsamples is almost 50% higher than those made by the other twomethods. For each individual processing method, a slight increasecan be observed of high shear blending samples and three-roll millsamples in the first 400 cycles, while this trend is not very obviousfor the ultrasonication samples. The sudden drop of storagemodulus after 400 rolling cycles in the three-roll mill processingprobably results from the degradation of polymer chains of thefierce mechanical mastication [34].

Overall, the ultrasonication shows the best dispersion and thusthe best mechanical properties among the three processingmethods. The dispersion indices can well represent the efficienciesin dispersion and can also be used to predict the mechanicalproperties of nanocomposites when no interfering factors such aspolymer degradation exist.

3.5. Limitation of the analysis

Although the current analysis is universal for all filler reinforcedcomposites, it may lose its accuracy in some cases.

First, this method could be the best for analyzing particulatefillers.When the composites contain high aspect ratio fillers such asrods and fibers, these fillers will be simplified as points in the filleridentification step, which may decrease the accuracy in thedispersion analysis and lose the information of orientation of thesefillers. The study on quantification of the dispersion and orientationof high aspect ratio fillers in composites is currently undergoing.

Second, when the image has lots of noise, the proposed methodmay not perform a good automatic extraction of the fillers. Indeed,the noise in the images is random perturbation of the imagegrayscale intensities. Therefore, if the random perturbation raisesthe intensities higher than the neighbors at certain non-fillerpixels, its location could be mistakenly regarded as a filler.Evidently, the subsequent dispersion analysis would also beaffected.

4. Conclusion

A quantitative and automatic analysis has been developed in thecurrent study to estimate the dispersion of fillers in polymer

composites. The importance of automatic filler identification isstressed and a local threshold is used to separate the fillers from thebackground in order to eliminate the interference from inhomo-geneous background. The dispersion metric is based on thedistance between the PDF function of the sample image and thePDF of uniformity. This metric can not only represent the distri-bution status of the fillers but also reflect the filler size, asdemonstrated by a series of synthetic images. Application of themethod to the model composite also shows that it can accuratelypredict the dispersion and mechanical properties of modelcomposites. This method is applicable to different scales of fillerreinforced composites, but is best applied for particulate fillers.Study on dispersion and orientation of composites with high aspectratio fillers is under way.

Acknowledgment

This work was supported by the US Department of Energythrough project number DE-SC0001967.

References

[1] Sumita M, Sakata K, Asai S, Miyasaka K, Nakagawa H. Polym Bull 1991;25(2):265e71.

[2] Xu J, Wong CP. Composites Part A 2007;38(1):13e9.[3] Le H, Kolesov I, Ali Z, Uthardt M, Osazuwa O, Ilisch S, et al. J Mater Sci 2010;

45(21):5851e9.[4] Coleman JN, Khan U, Blau WJ, Gun’ko YK. Carbon 2006;44(9):1624e52.[5] Montes H, Chaussée T, Papon A, Lequeux F, Guy L. Eur Phys J E 2010;31(3):

263e8.[6] Pu Z, Mark JE, Jethmalani JM, Ford WT. Chem Mater 1997;9(11):2442e7.[7] Dasari A, Yu Z-Z, Mai Y-W, Hu G-H, Varlet J. Compos Sci Technol 2005;

65(15e16):2314e28.[8] Grady BP. Macromol Rapid Commun 2010;31(3):247e57.[9] Supova M, Martynkova GS, Barabaszova K. Sci Adv Mater 2011;3(1):1e25.

[10] Moniruzzaman M, Winey KI. Macromolecules 2006;39(16):5194e205.[11] Lim SW, Park EY, Cha KS, Lee J, Shim SE. Macromol Res 2010;18(8):766e71.[12] Ha H, Kim SC, Ha K. Macromol Res 2010;18(5):512e8.[13] Schroder A, Briquel L, Sawe M. KGK-Kautsch Gummi Kunstst 2011;64(1e2):

42e7.[14] Linton D, Driva P, Sumpter B, Ivanov I, Geohegan D, Feigerle C, et al. Soft

Matter 2010;6(12):2801e14.[15] Gershon AL, Cole DP, Kota AK, Bruck HA. J Mater Sci 2010;45(23):6353e64.[16] Ciecierska E, Boczkowska A, Kurzydlowski K. J Mater Sci 2010;45(9):2305e10.[17] Kashiwagi T, Fagan J, Douglas JF, Yamamoto K, Heckert AN, Leigh SD, et al.

Polymer 2007;48(16):4855e66.[18] Khare HS, Burris DL. Polymer 2010;51(3):719e29.[19] Liang JZ. Composites Part A 2007;38(6):1502e6.[20] Liang JZ, Li RKY. J Reinf Plast Compos 2001;20(8):630e8.[21] Mills SL, Lees GC, Liauw CM, Lynch S. Macromol Mater Eng 2004;289(10):

864e71.[22] Mills SL, Lees GC, Liauw CM, Rothon RN, Lynch S. J Macromol Sci Part B Phys

2005;44(6):1137e51.[23] Pegel S, Potschke P, Villmow T, Stoyan D, Heinrich G. Polymer 2009;50(9):

2123e32.[24] Sul IH, Youn JR, Song YS. Carbon 2011;49(4):1473e8.[25] Glaskova T, Zarrelli M, Borisova A, Timchenko K, Aniskevich A, Giordano M.

Compos Sci Technol 2011;71(13):1543e9.[26] Grossiord N, Loos J, Regev O, Koning CE. Chem Mater 2006;18(5):1089e99.[27] Kovacs JZ, Andresen K, Pauls JR, Garcia CP, Schossig M, Schulte K, et al. Carbon

2007;45(6):1279e88.[28] Peter TL, Jae-Woo K, Luke JG, Cheol P. Nanotechnology 2009;20(32):

325708.[29] Duda RO, Hart PE, Stork DG. Pattern classification. Wiley; 2001.[30] Botev ZI, Grotowski JF, Kroese DP. Ann Stat 2010;38(5):2916e57.[31] Kreyszig E. Introductory functional analysis with applications. Wiley; 1989.[32] Zhu Y, Allen GC, Adams JM, Gittins D, Heard PJ, Skuse DR. Compos Struct 2010;

92(9):2203e7.[33] Karásek L, Sumita M. J Mater Sci 1996;31(2):281e9.[34] Watson WF. Ind Eng Chem 1955;47(6):1281e6.