NANO EXPRESS Open Access

A Two-Step Methodology to Study theInfluence of Aggregation/Agglomeration ofNanoparticles on Young’s Modulus ofPolymer NanocompositesXinyue Ma1, Yasser Zare2 and Kyong Yop Rhee3,4*

Abstract

A two-step technique based on micromechanical models is suggested to determine the influence of aggregated/agglomerated nanoparticles on Young’s modulus of polymer nanocomposites. The nanocomposite is assumed toinclude nanoparticle aggregation/agglomeration and effective matrix phases. This method is examined for differentsamples, and the effects of important parameters on the modulus are investigated. Moreover, the highest and thelowest levels of predicted modulus are calculated based on the current methodology. The suggested techniquecan correctly predict Young’s modulus for the samples assuming the aggregation/agglomeration of nanoparticles.Additionally, the aggregation/agglomeration of nanoparticles decreases Young’s modulus of polymer nanocomposites. Itis demonstrated that the high modulus of nanoparticles is not sufficient to obtain a high modulus in nanocomposites,and the surface chemistry of components should be adjusted to prevent aggregation/agglomeration and to dispersenano-sized particles in the polymer matrix.

Keywords: Polymer nanocomposites, Aggregation/agglomeration, Multi-step method, Young’s modulus

BackgroundMany researchers have focused on polymer nanocom-posites in recent years in order to determine theeffective parameters in processing-structure-propertiesrelationships and to optimize the overall performance asmeasured by mechanical, thermal, physical, and barrierproperties [1–4]. A low content of nanoparticles inpolymer nanocomposites produces large interfacial area,high modulus, low weight, and inexpensive productsthat are extremely attractive in the composite industry.Accordingly, the application of nanoparticles is an easy,efficient, and economical way to improve the perform-ance of polymer matrices. The effects of many materialand processing parameters on the properties of polymernanocomposites containing silicate layers (nanoclay),carbon nanotubes (CNT), and inorganic fillers such as

silica (SiO2), and calcium carbonate (CaCO3) have beeninvestigated [5–8].The size and dispersion/distribution quality of nano-

particles in polymer matrix change the general proper-ties of polymer nanocomposites. The nanoparticles tendto aggregate and agglomerate, due to the attraction be-tween nanoparticles such as van der Waals forces andchemical bonds [9] or the strong reduction in surfaceseparation as filler size decreases [10]. Therefore, it isdifficult to disperse the nanoparticles in polymermatrices at nanoscale. Both aggregation and agglomer-ation are assemblies of nanoparticles, where aggregationincludes strong and dense colonies of particles, butagglomeration comprises loosely combined particles thatcan be disrupted by mechanical forces. Agglomeration/aggregation is evident at high filler contents, whichdeteriorates the nanoscale of filler and produces manydefects and stress concentrations in nanocomposites[11–13]. Agglomeration/aggregation also reduces theinterfacial area between polymer matrix and nanoparti-cles, which decreases the mechanical involvement of

* Correspondence: rheeky@khu.ac.kr3Department of Mechanical Engineering, College of Engineering, Kyung HeeUniversity, Yongin 446-701, Republic of Korea4Yongin, Republic of KoreaFull list of author information is available at the end of the article

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Ma et al. Nanoscale Research Letters (2017) 12:621 DOI 10.1186/s11671-017-2386-0

polymer chains in nanoparticles and eliminates the stiffen-ing effect. Our recent findings [14, 15] and the study of Jiet al. [16] on mechanical properties have indicated thatany aggregation/agglomeration severely damages the stiff-ening effect of nanoparticles in polymer nanocomposites.In addition to the experimental characterization of

nanocomposites, the theoretical investigations thatquantify the dependence of mechanical behavior on theproperties of constituent phases and the geometricmorphology of nanoparticles have introduced attractivechallenges in recent research. Theoretical studies mayhelp to elucidate the experimental results and facilitatethe optimal synthesis of highly promising nanocompos-ites. The nanoparticles in nanocomposites introducedisorder into the adjacent matrix, leading to formationof interphase zones surrounding the filler, which showdifferent properties from bulk matrix and nanoparticles[17–19]. Theoretical studies on the interphase propertieshave shown attractive results, justifying the use of nano-particles in polymer nanocomposites [20–22].The effects of aggregation/agglomeration on the

mechanical performance of nanocomposites were inves-tigated in previous works [11, 14, 23, 24]. These studiesgenerally considered the aggregation/agglomeration bylarge particles. Recently, multiscale modeling methodshave been used to study the properties of nanocomposite[25–27]. In the current paper, a two-step method issuggested to examine the role of nanoparticleaggregation/agglomeration in Young’s modulus of poly-mer nanocomposites assuming the fraction of aggrega-tion/agglomeration phase in nanocomposite and theportion of nanoparticles in aggregates/agglomerates.In this regard, two micromechanical models of Pauland Maxwell are applied to express Young’s modulusof nanocomposites. Numerous experimental data arepresented to evaluate the predictions. Moreover, theeffects of aggregation/agglomeration parameters onYoung’s modulus of nanocomposites are studied.

MethodsWhen a fraction of nano-sized particles aggregates/ag-glomerates, a non-uniform distribution of nanoparticles isshown in nanocomposite. As a result, some nanoparticlescan be assumed in spherical regions in the matrix asaggregation/agglomeration phase and others are uniformlydispersed in the polymer matrix, as illustrated in Fig. 1.Accordingly, the nanofiller shows two parts with differentreinforcement which can be considered two differentphases in calculation as aggregation/agglomeration andeffective matrix phases which demonstrate the regionsinside and outside the spheres, respectively (Figure 1).The following two parameters are suggested for the

aggregation/agglomeration level of nanoparticles in poly-mer nanocomposite:

z ¼ V agg

Vð1Þ

y ¼ V aggf

V fð2Þ

where “Vagg” and “V” denote the total volumes ofaggregation/agglomeration phase and nanocomposite,respectively. Also, “Vf

agg” and “Vf” show the volumes ofnanoparticles in the aggregation/agglomeration phaseand whole nanocomposite, respectively. The volumefraction of nanoparticles incorporated in the aggrega-tion/agglomeration phase is presented by:

ϕaggf ¼ V agg

f

V agg¼ yϕf

zð3Þ

where “ϕf” is the volume fraction of nanofiller innanocomposites. Also, the volume fraction of well-dispersed nanoparticles incorporated in the effectivematrix phase (out of aggregation/agglomeration phase)phase is calculated by:

ϕmatf ¼ V f −V

aggf

V−V agg¼ 1−yð Þϕf

1−zð4Þ

In this study, a two-step methodology based on themicromechanical models is used to determine the aggre-gation/agglomeration parameters (z and y) in polymernanocomposites by Young’s modulus. Firstly, the modu-lus of aggregation/agglomeration and effective matrixphases is calculated by Paul’s model. Secondly, theaggregation/agglomeration phase is assumed as sphericalinclusions in the effective matrix, and Young’s modulusof nanocomposite is calculated by Maxwell’s model for acomposite containing dispersed particles.

Fig. 1 Schematic illustration of aggregation/agglomeration andeffective matrix phases in polymer nanocomposites containinglayered and spherical nanoparticles

Ma et al. Nanoscale Research Letters (2017) 12:621 Page 2 of 7

Paul [28] suggested a model which assumes themacroscopically homogeneous stress in two componentsof composite as:

E ¼ Em

1þ a−1ð Þϕ2=3f

1þ a−1ð Þ ϕ2=3f −ϕf

� � ð5Þ

a ¼ Ef

Emð6Þ

where “Em” and “Ef” are Young’s moduli of polymermatrix and filler phases, respectively. At the first step,the modulus of aggregation/agglomeration (Eagg) andeffective matrix (Emat) phases are calculated by Paul’smodel through replacing “ϕf” with “ϕagg

f ” and “ϕmatf ” as:

Eagg ¼ Em

1þ a−1ð Þϕaggf 2=3

1þ a−1ð Þ ϕaggf 2=3−ϕagg

f

� � ¼ Em

1þ a−1ð Þ yϕ f

z

� �2=3

1þ a−1ð Þ yϕ f

z

� �2=3−

yϕf

z

� �

ð7Þ

Emat ¼ Em

1þ a−1ð Þϕmatf 2=3

1þ a−1ð Þ ϕmatf 2=3−ϕmat

f

� �

¼ Em

1þ a−1ð Þ 1−yð Þϕf

1−z

h i2=3

1þ a−1ð Þ 1−yð Þϕf

1−z

� �2=3−

1−yð Þϕf

1−z

� �2=3� �

ð8ÞAlso, the Maxwell model [29] for composites containing

dispersed filler is given by:

E ¼ Em1þ 2ϕf a−1ð Þ= aþ 2ð Þ1−ϕf a−1ð Þ= aþ 2ð Þ ð9Þ

At the second step, the Maxwell model is applied forcalculation of modulus in a composite containing an ef-fective matrix (matrix and well-dispersed nanoparticles)and aggregation/agglomeration phases by replacing “ϕf”with “z” (see Eq. 1), “Ef” with the modulus of aggrega-tion/agglomeration phase (Eagg) and “Em” with themodulus of effective matrix (Emat) as:

E ¼ Emat1þ 2z k−1ð Þ= k þ 2ð Þ1−z k−1ð Þ= k þ 2ð Þ ð10Þ

k ¼ Eagg=Emat ð11Þwhich correlates Young’s modulus of nanocomposites tothe moduli of aggregates/agglomerates and the effectivematrix as well as the “z” parameter. When “Eagg” and“Emat” from Eqs. 7 and 8 are input into the latter equa-tions, the modulus of nanocomposites is expressed usingfiller concentration, filler modulus, matrix modulus, and“z” and “y” parameters. The dependency of modulus onthese parameters is reasonable, because the properties of

polymer and nanoparticles as well as the extent of filleraggregation/agglomeration control the modulus ofnanocomposites. In the present methodology, y > z ismeaningful, because VV agg

f > V f V agg.

Results and DiscussionThe proposed method is applied to evaluate nanoparticleaggregation/agglomeration in several samples fromprevious studies including PVC/CaCO3 [30], PCL/nano-clay [31], ABS/nanoclay [32], PLA/nanoclay [33], PET/MWCNT [34], and polyimide/MWCNT [35]. Figure 2shows the experimental results of Young’s modulus aswell as the predictions of the two-step method. The cal-culations properly follow the experimental data at differ-ent nanofiller concentrations, illustrating the correctnessof the suggested method. However, the highest agree-ment between the experimental and theoretical data isobtained when the aggregation/agglomeration of nano-particles are assumed by proper levels of “z” and “y”parameters. The highest predictions of “z” and “y” pa-rameters are calculated as z = 0.2 and y = 0.95 for PVC/CaCO3 nanocomposite. Also, (z, y) values of (0.3, 0.75),(0.1, 0.99), and (0.35, 0.7) are obtained for PCL/nanoclay,PLA/nanoclay, and PET/MWCNT samples, respectively.Moreover, (z, y) levels of (0.2, 0.93) and (0.15, 0.9) arecalculated for PET/MWCNT and polyimide/MWCNTnanocomposites, respectively. These levels of “z” and “y”parameters demonstrate the formation of aggregated/ag-glomerated nanoparticles in the mentioned nanocom-posites. The small improvement of modulus in thesesamples confirms the weak dispersion and high level ofnanoparticle accumulation in polymer matrices. Forexample, the addition of 7.5 wt% CaCO3 to PVC onlyincreases the modulus of neat PVC (1.13 GPa) to1.3 GPa. Also, the incorporation of 10 wt% of nanoclayin PCL only improves the modulus of neat PCL from0.22 to 0.37 GPa. However, the nanoparticles show ahigh modulus compared to polymer matrices. Young’smodulus of CaCO3, nanoclay, and MWCNT werereported as 26, 180, and 1000 GPa [36], respectively,while Young’s modulus of the present polymer matriceshardly reaches 2.5 GPa. As a result, the aggregated/ag-glomerated nanoparticles significantly decrease themodulus in nanocomposites, and the present method-ology suggests acceptable data for aggregation/agglomer-ation of nanoparticles in polymer nanocomposites.The highest and smallest moduli predicted by the

current methodology are calculated and illustrated inFig. 3 at an average Em = 2 GPa and Ef = 200 GPa. Themaximum modulus is obtained by the smallest values of“z” and “y” parameters; for example, z = 0.00001 and y =0.00001 (they cannot be 0). On the other hand, the “y”level of 0.99 results in the aggregation/agglomeration ofall nanoparticles, which significantly reduces the

Ma et al. Nanoscale Research Letters (2017) 12:621 Page 3 of 7

modulus. Also, the highest level of “z” (maximum ex-tent of agglomeration) causes the minimum modulus.“z” as the volume fraction of agglomerated filler inthe nanocomposite is smaller than the volume frac-tion of all nanoparticles (ϕf ). So, z = ϕf can suggestthe slightest level of modulus. The significant differ-ence between the upper and lower values of modulusshows the important role of aggregation/agglomer-ation of nanoparticles in the stiffness of nanocompos-ites. The aggregation/agglomeration of nanoparticlesin the nanocomposites greatly decreases Young’smodulus at different filler concentrations, while a finedispersion of nanoparticles without aggregation/ag-glomeration produces a good modulus. Also, the highaggregation/agglomeration at great nanofiller contentsdecreases the rate of modulus growth upon increasingin “ϕf”. Therefore, it is important to adjust the mater-ial and processing parameters to prevent the

Fig. 2 The difference between experimental and theoretical results assuming the aggregation/agglomeration of nanoparticles for a PVC/CaCO3

[30], b PCL/nanoclay [31], c ABS/nanoclay [32], d PLA/nanoclay [33], e PET/MWCNT [34], and f polyimide/MWCNT [35] samples

Fig. 3 The maximum and minimum levels of modulus predicted bythe present methodology at average Em = 2 GPa and Ef = 200 GPa

Ma et al. Nanoscale Research Letters (2017) 12:621 Page 4 of 7

aggregation/agglomeration of nanoparticles that pro-mote stress concentration and defects or debondingin polymer nanocomposites [37, 38].Figure 4 illustrates the effects of “z” and “y” parameters

on the modulus at Em = 3 GPa, Ef = 150 GPa, and ϕf=0.02. The highest modulus is obtained at the smallestlevels of “z” and “y” parameters, confirming the positiverole of good dispersion/distribution of nanoparticles onthe modulus of nanocomposites. However, the modulusdecreases greatly as “y” parameter increases. Accordingto Eq. 2, “y” shows the concentration of nanoparticles inthe agglomeration/aggregation phase. A low modulus isobserved at high “y” level, which shows that a large frac-tion of nanoparticles in the agglomeration/aggregationphase weakens a nanocomposite. Accordingly, agglomer-ated/aggregated nanoparticles cause a negative effect onthe modulus of nanocomposites. Therefore, much effortshould be made to facilitate nanoparticle dispersion/dis-tribution in the polymer matrix, which depends on theinterfacial interaction/adhesion between polymer andnanoparticles and the processing parameters. Previousstudies have reported valuable results in this area andhave suggested various techniques to improve thisdispersion [39–41].

Figure 5 demonstrates the dependence of predictedmodulus on “Em” and “Ef” parameters at average ϕf =0.02, z = 0.3, and y = 0.5 with the current technique. It isobserved that the modulus depends on both “Em” and“Ef” factors at low Ef < 150 GPa. However, a highermodulus of nanoparticles does not change the modulusof the nanocomposite. As a result, the modulus of nano-composites only depends on “Em” when “Ef” is higherthan 150 GPa. This suggests that high nanoparticle stiff-ness does not play a main role in the nanocompositemodulus, and much attention should be paid to the dis-persion/aggregation/agglomeration of nanoparticles.

ConclusionsA two-step technique was suggested to determine theinfluences of aggregated/agglomerated nanoparticles onYoung’s modulus of polymer nanocomposites. The Pauland Maxwell models were applied to calculate the moduliof aggregation/agglomeration and effective matrix phases.The predictions of the suggested methodology showedgood agreement with the experimental data of differentsamples, assuming correct aggregation/agglomeration pa-rameters. Accordingly, the present methodology can giveacceptable results for aggregation/agglomeration of

Fig. 4 a, b The calculations of modulus by Eqs. 10–11 as a function of “z” and “y” at Em = 3 GPa, Ef = 150 GPa, and ϕf = 0.02

Fig. 5 a, b The effects of “Em” and “Ef” on the predicted modulus by Eqs. 10–11 at average ϕf = 0.02, z = 0.3, and y = 0.5

Ma et al. Nanoscale Research Letters (2017) 12:621 Page 5 of 7

nanoparticles in polymer nanocomposites. The aggrega-tion/agglomeration of nanoparticles significantly de-creased Young’s modulus, whereas a fine dispersion ofnanoparticles produced a high modulus. The highestmodulus was obtained at the smallest “z” and “y” parame-ters, which confirmed the positive role of good dispersion/distribution of nanoparticles in the modulus of nanocom-posites. However, the modulus decreases as the “y” param-eter increased. Moreover, it was found that the excellentcharacteristics of nanoparticles such as high modulus arenot sufficient to achieve the optimal properties in polymernanocomposites. Accordingly, much attention should befocused on the dispersion/distribution of nanoparticles inthe polymer matrix depending on the interfacial inter-action/adhesion between polymer and nanoparticles andthe processing parameters.

FundingNo funding.

Authors’ ContributionsAll authors contributed to the calculations and discussion. All authors readand approved the final manuscript.

Competing InterestsThe authors declare that they have no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Author details1Jilin Agricultural Science and Technology University, Jilin 132101, China.2Young Researchers and Elites Club, Science and Research Branch, IslamicAzad University, Tehran, Iran. 3Department of Mechanical Engineering,College of Engineering, Kyung Hee University, Yongin 446-701, Republic ofKorea. 4Yongin, Republic of Korea.

Received: 18 July 2017 Accepted: 27 November 2017

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